Download Lecture Notes in Statistical Mechanics and Mesoscopics

Lecture Notes in Statistical Mechanics and Mesoscopics
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Contents
Core Topics
2
1 The statistical picture of classical mechanics
2
2 Calculating Z (β) and N (E)
10
3 The canonical formalism
19
4 The chemical potential
24
5 The grand canonical formalism
28
6 The kinetic description of a gas
37
7 Systems with interactions
39
8 Phase Transitions
47
Special Topics
63
9 Ergodicity, entropy, irreversibility and dissipation
63
10 Work and Heat
69
11 Fluctuations and Dissipation
76
12 Quantum fluctuations (I)
80
13 Quantum fluctuations (II)
86
14 Linear response theory
89
15 The Kubo formula for non-interacting Fermions
93
16 Scattering approach to mesoscopic transport
98
17 The modeling of the environment
104
18 The Feynman Vernon formalism
107
19 The dissipative dynamics of a two level system
112
2
Core Topics
[1] The statistical picture of classical mechanics
Distribution function as a basic function
Gases, µ definition
The grand canonical ensemble
Ideal gases (Boltzmann, Boze, Ferme)
The Viral expansion
The information is in the ”energy spectrum”
Definition of ρ(x, p)
Wigner function. Consequences of trace(ρ2 ) = 1.
Evolution in case of free particle and oscillator.
Stationary states, microcanonical states.
Quantization, plank cell, Weyl law.
Chaotic systems, ergodicity (illustrations)
Canonical ensemble justification (pr ∝ e−βEr
Free spreading, mixing of two gases
Ergodization and entropy growth in sudden process H1 → H2
The case H(Q, P ; X) with Ẋ = V
P
Classical point of view on S = − pr ln pr
The arbitrariness of the coarse graining
ergodization → entropy growth → diffusion → dissipation
====== [1.1] Random variables
Random variable/observation
x̂
3
Distribution function
for discrete spectrum
ρ(x)
ρ (x) ≡ Prob (x̂ = x)
for continuous spectrum
ρ (x) dx ≡ Prob (x < x̂ < x + dx)
State definition in statistical / quantum mechanics
P
Expectation values hx̂i ≡ x ρ (x) x
Changing variables ŷ = f (x̂)
(1)
ρ̃ (y) dy = ρ (x) dx
Joint Distribution function of two variables
ρ (x, y)
Ŝ = x̂ + ŷ
(2)
hŜi = hx̂i + hŷi
(3)
2
2
2
h(S − hSi) i = h(x − hxi) i + h(y − hyi) i + h(x − hxi) (y − hyi)i
(4)
Central limit theorem.
====== [1.2] Dynamics of a free particle
Free particle
H=
p2
2m
(5)
at t = 0 the particle is at (x0 , p0 )
ẋ =
p
∂H
=
∂p
m
(6)
∂H
=0
∂x
(7)
ṗ = −
x(t) = x0 +
t
p0
m
p(t) = p0
Evolution in the phases space
Liouville theorem
(8)
(9)
4
Generalization: systems described by the Hamiltonian
H=
p2
+ V (x)
2m
(10)
potential well
harmonic oscillator
Bonded systems in 1D.
Integrability versus chaos for d > 1.
====== [1.3] Statistical description of a classical particle
ρ(x, p)
dxdp
≡ PROB (x < x̂ < x + dx, p < p̂ < p + dp)
2π
(11)
Normalization:
Z
dxdp
ρ (x, p) = 1
2π
(12)
Expectation values:
Z Z
hAi =
dxdp
ρ (x, p) A(x, p)
2π
(13)
Probability function of x
Z
dp
ρ (x, p)
2π
ρ (x) =
(14)
Expectation value of x and of x2
Z Z
hx̂i =
2
hx̂ i =
Z
dxdp
ρ (x, p) x =
2π
Z
dxρ (x) x2
dxρ (x) x
(15)
(16)
Variance:
2
σx2 = h(x̂ − hx̂i) i = hx̂2 i − hx̂i2
(17)
The ”energy” of the system
E = hH (x̂, p̂)i
(18)
5
====== [1.4] From classical to statistical mechanics
Description of classical dynamics in the framework of statistical mechanics.
A free particle has been prepared in a “deterministic preparation” in point (x0 , p0 ). Find its state after time t.
ρt=0 (x, p) = 2πδ (p − p0 ) δ (x − x0 )
(19)
After time t
p0 ρt (x, p) = 2πδ (p − p0 ) δ x − x0 + t
m
(20)
Above ρt (x, p) is the probability function of the random variables
x̂t = x̂0 +
t
pˆ0
m
(21)
p̂t = pˆ0
(22)
For not deterministic preparation, we observe the spreading phenomenon
s
σx (t) =
σx2
(0) +
σp (0)
m
σp (0)
t
m
t2 ≈
p
(23)
====== [1.5] The definition of N (E) and Z (E)
These functions characterize the energy spectrum of a bounded systems. Consider for example
H=
p2
+ V (x)
2m
(24)
where the energy Vmin < E < ∞ is a constant of motion.
V(x)
Vmin
x
6
p
The number of states up to energy E
Z Z
Z Z
dxdp
Θ (E − H (x, p)) =
2π
N (E) =
H(x,p)<E
dxdp
2π
(25)
The partition function
Z Z
Z (β) =
dxdp −βH(x,p)
e
=
2π
Z
g (E) dE e−βE
(26)
Note about semiclassical quantization.
====== [1.6] The route to ergodicity
∂ρ
= [H, ρ]PB
∂t
(27)
Stationary state means
∂p
=0
∂t
(28)
Anharmonic oscillator in 1D: ergodic-like state due to spreading (not for mono-energetic distribution!)
Global mixing leads to ergodicity in the case of a chaotic system. e.g. Sinai billiard and Lorentz gas.
To give example of systems with mixed phase space.
In what sense ρ (t) becomes microcanonical?
The concept of ”coarse graining” (cells).
Implications of quantum mechanics?
Spreading
p
free particle
x
ergodic like state due to spreading but not for mono−energetic distribution
7
ergodic like state due to spreading but not for mono−energetic distribution
Mixing
0
consider mono−energetic preparation
,
sm θ
+1
ψ
−1
energy surface
x,y
poincase section
====== [1.7] The derivation of the canonical state
Let us assume the following total Hamiltonian:
Htotal = H (Q) + Henv (Qα ) + Hint (Q, Qα )
(29)
For sake of presentation we do not write the conjugate momenta. In what follows Q stands for (Q, P ) and dQ should
be understood as dQdP .
Q
Qa
Assume that the universe is in a microcanonical state. Then it follows that
Z Z
ρ (Q, P ) =
dQα
1
gtot (Etot )
Z
=
1
gtot (Etot )
Z
=
1
δ (Etot − Htot )
gtot (Etot )
(30)
dQα δ (Etot − − Henv ) δ ( − H)
(31)
Z
d
d genv (Etot − ) δ ( − H) =
1
genv (Etot − H)
gtot (Etot )
(32)
Using
genv (Etot − ) ≈ genv (Etot ) e−β
(33)
We get:
ρ (Q, P ) ∝ e−βH(Q,P )
(34)
8
Another way of writing this result is
pr ∝ e−βEr
(35)
Where r is phase space cell index. The normalization factor is
Z (β) ≡
X
e−βEr
Z Z
≡
r
dQdP
d
(2π)
e−βH(Q,P )
(36)
====== [1.8] Mathematical digression
f (x) = xN
(37)
f (x + δx) = xN + N xN −1 δx + N 2 xN −2 δx2
(38)
δx x/N
(39)
S (x) ≡ ln f (x) = N ln (x)
(40)
S (x + δx) = N ln (x) +
δx x
N
N
δx + 2 δx2
x
x
(41)
(42)
9
Thus we have the recipe:
f (x + δx) ≈ f (x) eβδx
(43)
where
β ≡
∂ ln f (x)
∂x
(44)
In complete analogy we have:
g (E0 + δE) ≈ g (E0 ) eβδE
(45)
where
g (E) ≡
dN (E)
dE
(46)
10
[2] Calculating Z (β) and N (E)
classical:
one dimensional potential well
d dimensional box, general dispersion relation
The effect of A(x), V (x) potential
oscillator
N particles (Boltzmann)
Two particles with interaction
The Gibbs factor
quantum:
spin
oscillator
particle in a box
Two identical particles - without interaction (Fermi/Bose)
Two identical particles - with interaction (via phase shifts)
Molecules
====== [2.1] The classical calculation
In what follows the Gaussian integral is useful:
Z
1
2
e− 2 ax dx =
2π
a
12
(47)
Potential well:
Assume 1D box 0 < x < L.
H=
p2
+ Vbox (x)
2m
N (E) = rectangle area =
Z
Z (β) =
Z
dx
(48)
1
1
L
L 2 (2mE) 2 = 2
2π
λE
dp −β p2
e 2m = L
2π
m
2πβ
12
=
L
λT
(49)
(50)
11
where
λE = √
2π
= DeBroglie wavelength
2mE
r
λT =
En ≈
(51)
2π
= Thermal wavelength
mT
π2 2
n
2mL2
(52)
for n 1
(53)
oscillator:
H=
p2
1
− mω 2 x2
2m 2
(54)
N (E) = ellipse area =
Z
Z (β) =
En =
dx e
−β 12 mx2
Z
1
π
2π
2E
mω 2
dp −β p2
e 2m
2π
12
1
(2mE) 2
=
2π
βmω 2
1
+n ω
2
=
21 E
ω
m
2πβ
(55)
12
=
1
βω
(56)
(57)
arbitrary scalar potential:
H=
p2
+ V (x)
2m
Z
Z(β) =
dx dp −βH
e
=
2π
(58)
m
2πβ
1/2 Z
dx e−βV (x)
(59)
In three dimensions:
H=
p2
+ V (~x)
2m
(60)
12
Z
Z(β) =
d3 x d3 p −βH
e
=
2πβ
m
2πβ
3/2 Z
d3 x e−βV (~x)
(61)
For N non-interacting particles:
H=
N 2
X
p~
α
α=1
2m
"
ZN (β) =
+ V (~xα )
m
2πβ
32 Z
(62)
#N
3
d xe
−βV (x)
(63)
For N interacting particles:
H=
N
X
p~2α
+ U (~x1 ...~xN )
2m
α=1
ZN (β) =
m
2πβ
3N
Z
2
(64)
d3 x1 ...d3 xN e−βU (~x1 ...~xN )
(65)
Particles in magnetic field:
H=
1
2
(p − eA (x)) + V (x)
2m
Z
Z
=
dx dp
2
(2π)
e
1
−β [ 2m
(p−eA(x))2 +V (x)]
(66)
Z
=
dx dp0
2
(2π)
e
−β
h
1
2m
2
(p0 )
+V (x)
i
(67)
The result does not depend on A (x). The energy spectrum is not affected from the existence of A (x).
E=
1 2
mv + V (x)
2
(68)
irrespective of A (x).
To explain that the implicit assumption is having relaxation processes that makes the dynamics irrelevant. The
classical diamagnetic effect disappears in the presence of relaxation. To discuss the effect of quantization.
13
Φ (t)
====== [2.2] The quantum mechanical calculation
Assume a time independent bounded system which is described by a Hamiltonian H whose eigenvalues are Er .
N (E) ≡
X
Θ (E − Er ) =
r
Z (β) ≡
X
X
1
(69)
Er <E
e−βEr
(70)
r
If we have a large system we can smooth N (E), and then we can define the density of states as
g (E) ≡
dN (E)
dE
=
X
δ (E − Er )
(71)
r
Note that
Z
Z (β) =
g (E) dE e−βE
(72)
The definition given above is consistent with the definition given for a classical particle, provided one interprets the
sum over states as follows:
X
Z Z
7−→
r
dxdp
2π
(73)
Each ”cell” in phase space represents a state.
Two level system:
H =
1
σ3
2
(74)
14
The eigenstates are |+i and |−i with eigenvalues E± = ±/2. Accordingly
Z (β) = e
−β (− E
2 )
+e
−β ( E
2 )
= 2 cosh
1
β
2
(75)
Note that if we write the energies as Er = n with n = 0, 1 then
Z(β) = 1 + e−β
(76)
Harmonic oscillator:
H=
p2
1
+ mω 2 x2
2m 2
(77)
The eigenstates are |ni with eigenvalues En =
∞
X
Z (β) =
e−β ( 2 +n)ω
1
n=0
=
1
2
+ n ω. Accordingly
1
2 sinh
1
2 ωβ
(78)
Note that if we write the energies as Er = ωn with n = 0, 1, 3, 4, ... then
Z(β) = 1 + e−βω
−1
(79)
A particle in 1D space: The simplest is to assume periodic boundary conditions
H=
p2
2m
(80)
The eigenstates are the momentum states |pi with
p=
2π
π
L
where n = 0, ±1, ±2...
(81)
Hence the eigenvalues are
En =
1
2m
2π
n
L
2
(82)
The number of states up to energy E is
N (E) = 2
1
L
(2mE) 2
2π
(83)
and the density of states is
L
g (E) =
2π
2m
E
1/2
(84)
15
This is a pathological case. In general for N particles in d dimensional space and dispersion relation E = C|p|ν we
get a rapidly increasing density of states g(E) ∝ e(N d/ν)−1 .
∞
X
Z (β) =
2π 2
e−β mL2 n
Z
2
∞
2π 2
dn e−β mL2 n
≈
2
= L
−∞
n=−∞
m
2πβ
21
(85)
A particle in 3D space:
H=
3
X
p2i
2m
i=1
p~ =
(86)
2π
(n1 , n2 , n3 )
L
E n1 n2 n3
=
1
2m
2π
L
(87)
2
n21 + n22 + n23
(88)
The summation over the states factorizes:
!3
Z (β) =
X
e
−βEn1 n2 n3
=
n0 n1 n2
X
n
e
−βEn
=V
m
2πβ
32
(89)
For N distinct particles with the same mass the calculation is similar:
N X
3
X
p2α
2m
α=1 t=1
(90)
N
ZN (β) = Z1 (β)
(91)
H=
where Z1 is the one particle result.
====== [2.3] two spins system in interaction
H = εσ a · σ b = 2S 2 − 3 ε
(92)
where S = 12 σ a + 12 σ b
Z (β) = e3βε + 3e−βε
(93)
16
+ε
Sz
−3ε
====== [2.4] Z (β) for identical particles
Eαβ = Eα + Eβ
(94)
X
1 X −β(Eα +Eβ )
1
Zα =
e
+
e−β(2Eα )
0
2
α
(95)
α6=β

=
=
1
2

X
e−β(Eα +Eβ ) ±
X
e−2βEα 
(96)
α
αβ
1
2
Z1 (β) ± Z1 (2β)
2
(97)
More generally we have the Gibbs approximation
ZN ≈
1 N
Z
N! 1
(98)
Note that for d = 3 we get
1 2
1
λ3T
Z2 = Z1 × 1 ± 3/2
2
volume
2
(99)
The Fermi case is similar to hard sphere:
1 2
sphere volume
Z2 = Z1 × 1 −
2
box volume
(100)
====== [2.5] particle in a box


 2  L d
Ω
d
d/2
π
Ld
N (E) =
(2mE)
=
d
4π/3 λE
d
(2π)
1
(101)
17
1
g (E) = Ωd
2E
d
L
λE
∝ E (d/2)−1
(102)
for d = 1 we have
Z1 (β) = L
m
2πβ
1/2
=
L
λT
(103)
and in general
Z1 (β) =
L
λT
d
(104)
1 2
Z1 ± 2−d/2 Z1
2
Z2 =
ZN ≈
(105)
1 N
Z
N! 1
(106)
====== [2.6] Two quantum particles in a box with interaction
H=
p2
p2
+
+ V (r)
4m m
|knlm >⇒ eik+R
(107)
U ns (r)
Y cm (Θ, ϕ)
r
(108)
2
Eknlm =
Z2 =
(~m)
+ Enc
2m
V
2 2
λ
3
2
·
1
X
(109)
!
e
−βEnc
(110)
nc
V(x)
R
18
Where the sum is for even or for odd values of l, therefore, if the particles are fermions or bozons. We’ll define
ζ≡
1
X
e−βEnc
(111)
nc
We have to calculate ζ, therefore we’ll write it as the sum
ζ=
X
e−βEB
(112)
n
∞
Z
g (k) dke−β
+
(~k)2
2m
(113)
0
in the first constant sum on the relevant states, and in the second on the sequence states. We’ll calculate g (k),
sequence states density.
1
U nc (π) → sin nR − cn + δc
2
(114)
U nc (π) = 0 boundary conditions
(115)
1
K R̄ − lπ + δc = nπ
2
(116)
2l + 1
∂δc (k)
gc (k) =
R̄ +
π
∂k
(117)
g (k) − g 0 (k) =
1
1X
∂δc
(2l + 1)
π %
∂k
(118)
from that we get the result
ζ − ζ0 =
X
e−βEB +
X
e−βEB +
B
∞
(g (k) − g (k)) dke−p
(~k)2
2m
(119)
0
B
=
Z
Z ∞
(~k)2
λ2 X
(2l
+
1)
δ (k) kdke−β 2m
2
π
0
(120)
when zeta0 is in the limited distribution function for two particles without interaction
ζ0 =
1
S
22
V
λ3
1 λ3
1± V
2
V
(121)
19
[3] The canonical formalism
====== [3.1] Stationary states
We consider systems that in the absence of driving are assumed to be prepared in a stationary state. This means that
ρ 7→ diag{pn } is diagonal in the energy representation. In particular the microcanonical state is
ρ =
1
δ(H − E)
g(E)
(122)
and the microcanonical state is
ρ =
1 −βH
e
Z(β)
(123)
where the density of states and the partition function are defined as
g(E) = trace(δ(E − H)) =
X
δ(E − En )
(124)
n
Z(β) = trace(e
−βH
) =
X
e
−βEn
Z
=
g(E)dE e−βE
(125)
n
It is convenient to write the probability density to find the system at some particular energy E as
ρ(E) = g(E) f (E)
(126)
where
f (E) =
f (E) =
1
g(E) δ(E − E)
1
−βE
Z(β) e
[microcanonical]
(127)
[canonical]
(128)
If we have a many body system of non-interacting participles we can re-interpret f (E) as the occupation function,
and accordingly ρ(E) becomes the energy distribution of the particles (with normalization N ).
Z (β; X) =
X
e−βEr
(129)
r
pr =
1 −βEr
e
Z
E = hHi = −
y =
(130)
∂ ln Z
∂β
∂H
−
∂X
=
(131)
1 ∂ ln Z
β ∂X
(132)
20
====== [3.2] ideal gas in a box
H=
N X
3
X
p2
ai
a=1 i=1
Z (β, V ) = V
E=−
(133)
2m
N
m
2πβ
3N
2
3
∂ ln Z
= N β −1
∂β
2
(134)
(135)
P =
1 ∂ ln Z
N
= β −1
β ∂V
V
(136)
E=
3
PV
2
(137)
Note:
This system can be used as a thermometer. The empirical temperature is defined as follows:
θ =
PV
N
=
1
β
(138)
21
====== [3.3] Generalized forces, temperature, entropy
dE =
X
dpr Er +
X
pr dEr
(139)
r
X
pr dEr = −ydX
(140)
pr Er = T dS
(141)
r
X
Absolute temperature
T = integration factor =
1
β
(142)
Entropy
S=−
X
pr ln pr
(143)
note additivity. We have the identity
dE = T dS − ydX
(144)
====== [3.4] The thermodynamic formalism
We can define the basic function
F (T, V ) ≡ −
1
ln Z
β
(145)
from which we can get
∂F
∂T
∂F
y = −
∂X
S = −
(146)
(147)
and
E = F + TS
(148)
State functions that describe physical systems can be derived from basic functions. The basic functions should be
expressed as a functions of their canonical variables. The basic functions do not always have a physical meaning.
F (T, X) :
dF = −SdT − ydX
(149)
(150)
22
G (T, y) ≡ F + yX
dG = −SdT + Xdy
(151)
(152)
E (S, X) ≡ F + T S
dE = T dS − ydX
(153)
(154)
S (E, X) :
1
y
dS = dE + dX
T
T
(155)
(156)
====== [3.5] Heat capacity of harmonic Bath
Z
C = const
0
ωc
ω ω 2
eω/T
c
α−1
α
ω
dω
=
const
T
f
T
(eω/T − 1)2 T
(157)
Debay function (e.g. α = d = 3)
Z
f (ν) ≡
0
ν
ex
x1+α dx
(ex − 1)2
(158)
f( ν)
να
ν
C
Tα
23
====== [3.6] Heat capacity in different systems
spin
exp
ε
1
OSC
exp
w
N
Tα
debay
(α=d=3)
wc
N
Tα
Tc
N
fF
fermi
24
[4] The chemical potential
µ (T, V, N ) ≡
∂F
∂N
(159)
This can be calculate exactly if the particles occupy a set of sites (or modes) that have the same binding energy.
====== [4.1] Thermal occupation of a site system
Using the above definition we can get results for the thermal occupation of an M site system. Since we assume that
the biding energy is the same for all sites, it follows that estimating Z1 is essentially a combinatorial problem. We
assume n 1 so we can approximate the derivative of ln(n!) as ln(n). We also write the result for the most probable
n which is obtained given µ. Note that ”n̄” is meaningless for small M
Fermi/Langmuir site: each site can have at most one particle
M!
e−βεn
n!(M − n)!
n
µ = ε + T ln
M −n
Zn =
n̄ = M (eβ(ε−µ) + 1)−1
(160)
(161)
(162)
Bose site: each site can have any number of particles. The combinatorial problem is solved by asking how many ways
to divide n particles in a row with M − 1 partitions. If the particles were distinct the result would be (n + (M − 1))!.
Taking into account that the particles are indistinguishable we get
(n + M − 1)! −βεn
e
n!(M − 1)!
n
µ = ε + T ln
(M − 1) + n
Zn =
n̄ = (M − 1)(eβ(ε−µ) − 1)−1
(163)
(164)
(165)
Boltzmann/Gibbs approximation: assuming dilute occupation (n M ) we get a common approximation for both
Fermi and Bose case:
Zn =
M n −βεn
e
n!
n
µ = ε + T ln
M
n̄ = M e−β(ε−µ)
(166)
(167)
(168)
Also photons can be regarded as particles. Each mode of the electromagnetic field is like a site. Since n is not
constrained it follows that the formally we have
µ=0
====== [4.2] The Gibbs approximation
(169)
25
How do we calculate F
If the particles are distinct...
If the particles are identical...
Gibbs paradox?
ZN
1 N
Z
N! 1
=
µ = T ln
N
Z1
(170)
(171)
we note that the inverse relation is
N
= Z1 eβµ
(172)
where
V X −βεbound
e
λ3T
Z1 =
(173)
The summation is over the non-translational freedoms of the particle.
====== [4.3] The notion of chemical equilibrium
(a)
A[a] A[b]
(b)_
(174)
Given N particles we characterize the macroscopic occupation of the two phases by a coordinate n, such that N − n
particles are in phase [a] and n particles are in phase [b]. In what follows the second equality assumes Boltzmann/Gibbs
approximation.
ab
ZN
=
X
n
p(n) =
a
b
ZN
−n Zn =
N
1
Z1a + Z1b
N!
N −n
a
b
ZN
(Z1a )
Z1b
1
−n Zn
=
ab
ab
(N − n)!n!
ZN
ZN
(175)
n
(176)
26
hni =
X
p(n) n = N
n
Z1a
Z1b
+ Z1b
(177)
In general it is too difficult to calculate hni exactly, so we use the following strategy:
p(n) = const e−β (F
a
(N −n)+F b (n))
(178)
Looking for the most probable n we get the equation
− µa (N − n) + µb (n) = 0
(179)
Within the framework of the Boltzmann/Gibbs approximation it leads to the equation
n
Zb
= 1a
N −n
Z1
(180)
with the solution
n̄ = N
Z1b
Z1a + Z1b
(181)
Thus the typical (most probable) value of n coincides with its expectation value. In the more general case of chemical
equilibrium as discussed below this is an approximation that becomes valid for N 1 in accordance with the central
limit theorem.
====== [4.4] The law of mass action
This procedure is easily generalized. Consider for example
2C 5A + 3B
(182)
Given that initially there are NA particles of type A, NB particles of type B, and NC particles of type C we define a
macroscopic reaction coordinate n such that NC −2n is the number of particles of type C, and NA +5n is the number
of particles of type A, and NB +3n is the number of particles of type B. Accordingly
Z abc =
X
c
ZN
Za
Zb
C −2n NA +5n NB +3n
(183)
n
and
p(n) = const e−β (F
c
(NC −2n)+F a (NA +5n)+F b (NB +3n))
(184)
leading to the equation
− 2µc (NC −2n) + µa (NA +5n) + 3µb (NB +3n) = 0
(185)
which with Boltzmann/Gibbs approximation becomes
5
(Z1a ) Z1b
(NA +5n)5 (NB +3n)3
=
(NC −2n)2
(Z1c )2
3
(186)
27
or
NA +5n 5 NB +3n 3
V
V
NC −2n 2
V
= K(T )
(187)
====== [4.5] Equilibrium in many body systems
Consider the reaction
γ + γ e+ + e−
(188)
The electromagnetic field is like a ”bath”, so if we regard it as part of the environment we can also write
vacuum e+ + e−
(189)
In any case we get at equilibrium
+
−
µe (n1 ) + µe (n2 ) = 0
(190)
where in the Boltzmann/Gibbs approximation
2
µ(n) ≈ mc + T ln
nλ3T
V
(191)
leading to
n1 n2 =
V
λ3p
2
2
e−2mc
/T
(192)
10>
28
[5] The grand canonical formalism
|Ri = a many body eigenstate
N̂ |Ri = NR |Ri
(193)
Ĥ|Ri = ER |Ri
(194)
N ≡ hN̂ i =
X
pR N R =
R
E − µN = −
y≡
−
∂H
∂X
∂ ln Z
∂β
=
FG (T, V, µ) ≡ −
N =−
1 −β(ER −µNR )
e
Z
pR =
∂FG
∂µ
1 ∂ ln Z
β ∂µ
(195)
(196)
(197)
1 ∂ ln Z
β ∂X
(198)
1
ln Z
β
(199)
(200)
29
y=−
∂FG
∂X
(201)
E = FG + T S + µN
(202)
where
S=−
∂FG
∂T
(203)
In the thermodynamic limit FG is extensive, also in the case of non ideal gas. Consequently
FG (β, µ, V ) = −V P (β, µ)
(204)
====== [5.1] Justification
pR =
env
e−βER · ZN̄
−N
(205)
Z sys+env
env
ZN̄ −N ≈ e−β (FN̄
−µN )
(206)
Occupying orbital in close system (E262):
ε
ο
if the particles are different
hni
N −hni
= e−βε
hni = eβεN+1 = N ∗ (fermi function with µ = 0)
if the particles are identical
Z=
∞
X
n=0
hni = −
N +1
1 − e−βε
=
1 − (e−βE )
(207)
1 ∂
1
N +1
ln Z = βε
−
ε ∂β
e − 1 (eβε )N +1 − 1
(208)
e
−βε n
30
in every N → limit M = 0 with bose function. it is formally identical to ”photon” gas.
fermi:
(n)
1−(n)
= eβε
probability ratio
bose:
1+(n)
(n)
= eβε
detailed balance
====== [5.2] Fermi occupation
A site or mode can occupy n = 0, 1 particles. The binding energy is . the site is in thermochemical equilibrium with
a gas in temperature β and chemical potential µ.
(209)
(210)
Nn = n
En = n
pn =
1 −β(−µ)n
e
Z
(211)
Z(β, µ) = 1 + e−β(−µ)
N (β, µ) = hn̂i =
X
pn n =
n
(212)
1
≡ f ( − µ)
eβ(−µ) + 1
E(β, µ) = hn̂i = f ( − µ)
(213)
(214)
We have defined the Fermi occupation function 0 ≤ f ( − µ) ≤ 1
====== [5.3] Bose occupation
A site or mode can occupy n = 0, 1, 2, 3... particles. The binding energy is . the site is in thermochemical equilibrium
with a gas in temperature β and chemical potential µ.
Nn = n
En = n
(215)
(216)
31
pn =
1 −β(−µ)n
e
Z
(217)
−1
Z(β, µ) = 1 − e−β(−µ)
N (β, µ) = hn̂i =
X
pn n =
n
1
≡ f ( − µ)
eβ(−µ) − 1
E(β, µ) = hn̂i = f ( − µ)
(218)
(219)
(220)
We have defined the Bose occupation function 0 ≤ f ( − µ) ≤ ∞. If < µ then hni → ∞. If = µ then the site may
have any occupation. If < µ then hni is finite.
====== [5.4] Quantum ideal gases
We assume one particle states |ri that have the density
g() = V c α−1
(221)
For a particle in d dimensional box α = d/ν where ν is the exponent of the dispersion relation ∝ |p|ν , and c is a
constant which is related to the mass m. For example, in the case of spin 1/2 particle in 3D space we have
g() = 2 × V
(2m)3/2 1
2
(2π)2
(222)
In what follows, unless written otherwise = 0 is the ground state and
X
∞
Z
→
r
g()d
(223)
0
The stationary states of the multi particle system are occupation states
|ni = |n1 , n2 , n3 , ..., nr , ...i
(224)
where nr = 0, 1 for Fermi occupation and nr = 0, 1, 2, 3, 4, ... for Bose occupation. For these states we have
Nn =
X
nr
(225)
nr Er
(226)
r
En =
X
r
In thermo-chemical equilibrium we have
pn ∝ e−β
P
r (r −µ)nr
which can be factorized. This means that each site or mode can be treated as an independent system.
(227)
32
We use E and N without index for the expectation values in an equilibrium state:
N =
X
hn̂r i =
r
E =
X
r hn̂r i =
g()d f ( − µ)
(228)
0
r
X
∞
Z
f (r − µ) =
r
∞
Z
X
f (r − µ)r =
g()d f ( − µ)
(229)
0
r
where f (r − µ) is the appropriate occupation function. Using integration by parts we also have
ln Z =
X
ln(1 ± e−β(−µ) )±1 = β
Z
∞
N ()d f ( − µ)
(230)
0
r
where N () is the integrated g(). It follows that for the pressure we have
P
=
1 ln Z
β V
Z
∞
N ()d f ( − µ) =
=
0
1
α
E
V
(231)
The last equality assumes a density of states that is characterized by a well defined α.
====== [5.5] The state equations of a quantum ideal gas
The following integral is useful:
Z
0
∞
1 x
e ±1
z
−1
xα−1 dx ≡ Γ(α)Fα (z)
(232)
where
Fα (z) =
∞
X
±(±1)`
`α
`=1
z`
(233)
For small values of z we have
Fα (z1) ≈ z
(234)
For larger values of z this function grows faster in the case of a Bose occupation. In the latter case it either diverges
or attains a finite value as z → 1. Namely,
∞
X
1
Fα (z=1) = Γ(α)ζ(α) = Γ(α)
`α
(235)
`=1
√
The latter expression gives a finite result for α > 1. In particular we have Γ(3/2) = π/2 and ζ(3/2) ≈ 2.612. In the
Fermi case the integral is always finite. Using the step-like behavior of the Fermi occupation function we get for z 1
Fα (z1) ≈
1
α
(ln z)
α
(236)
We can express the state equations using this integral. For this purpose we define the fugacity
z = eβµ
(237)
33
Hence we get
N = cΓ(α) V T α Fα (z)
E = cΓ(α+1) V T α+1 Fα+1 (z)
(238)
(239)
We cite below the specific results in case of a spin 0 non-relativistic particle:
1
F3/2 (z)
λ3T
3 T
E = V 3 F5/2 (z)
2 λT
T
P = 3 F5/2 (z)
λT
N = V
(240)
(241)
(242)
For more details see Huang p.231-232;242.
Fα(z)
α>1
Bose
Fermi
1
z
====== [5.6] Ideal gases in the Boltzmann approximation
basic state energy is E0 = 0
The Boltzmann approximation is
f ( − µ) ≈ e−β(−µ)
(243)
It holds whenever the occupation is f () 1. It is valid for the ground state = 0 and hence globally for all levels if
z = eβµ 1
(244)
In the case of standard 3D gas the Boltzmann approximation gives N = (V /λ3T )z in consistency with Gibbs expression.
Hence an equivalent condition for the validity of Boltzmann is
N λ3T V
(245)
Within the framework of the Boltzmann approximation we can re-derive the classical equation of an idea gas. In
particular we find that
V µ/T
e
λ3T
3
E = NT
2
N
P =
T
V
N =
Note that within this approximation E and P do not depend on the mass of the particles.
(246)
(247)
(248)
34
E
E
E
fermi
E 0 =0
0
boltzman
bose
µ
g(E)
1\2
1
f(E− µ )
====== [5.7] The Bose Einstein condensation
N
(β, µ) = c
V
Z
α−1 d
1
(249)
eβ(−µ) − 1
we get
N
(β, µ → 0− ) = ∞ for α ≤ 1
V
(250)
while for α > 1 only a finite fraction can be accommodated:
N
(β, µ → 0− ) = cΓ(α)ζ(α)T α for α > 1
V
(251)
In the latter case for µ = 0 we have
N
(β, µ=0) =
V
hni0
+ cΓ(α)ζ(α)T α
V
(252)
and
E = cV Γ (α+1) ζ (α+1) T α+1
1 E
P =
α V
(253)
(254)
In particular the standard results for condensation in 3D are
3
3
m 2 3
N = n0 + V ζ
T2
2
2π
3
5
m 2 5
P = ζ
T2
2
2π
(255)
(256)
The pressure P is independent of the total number of particles, because the condensate does not have any contribution.
Hence the compressibility κ ∝ (∂P /∂V )−1 = ∞. If we change the volume the extra/missing particles just come from
the ground state, which is like a reservoir of µ = 0 particles.
N
Τ 1 <Τ 2 <Τ 3
for α <1
N
V
T1 <T2 <T3
µ
V
µ
for 1< α
35
µ
µ
(α < 1)
(1< α )
T
T
Τc
Now we would like to eliminate µ as a function of T given N . Best is to do it graphically. It leads to the identification
of the condensation temperature:
Tc =
1
N
cΓ (α) ζ (α) V
1/α
(257)
and the observation that
µ(T <Tc ) = 0
(258)
Consequently in such circumstances the occupation of the ground state is
hni0
V
=
N
N
−
β, µ → 0− =
V
V
1−
T
Tc
α N
V
(259)
It is easily verified that V /lambdaT is like (T /Tc )α and therefore the limit T Tc coincides with Boltzmann.
====== [5.8] Fermi gas at low temperatures
At low temperatures the Fermi function is step-like:
f ( − µ) = Θ (µ − E) −
π2 2 0
T δ ( − µ) + O T 4
6
(260)
This means that if G () is a smooth function then at low temperatures
Z
Z
G()df ( − µ) ≈
µ
G () d +
0
π2 2 0
T G (µ) + O(T 4 )
6
(261)
In particular if G() is the density of states g() then the first integral will give N (µ). Solving for zero temperature we
find µ = F . For the leading order correction due to the temperature one finds
µ ≈ F −
π 2 g0 (F ) 2
T + ...
6 g (F )
(262)
E
E
E
x
g(E)
0
f(E− µ )
E 0 =0
Ν
Βοse
case
1
fermi
T
fermi
zero T
36
Specific results are:
π2
1 + α (α − 1)
6
1
N = cV µα
α
!
2
T
+ ...
µ
(263)
leading to
F =
αN
cV
α1
(264)
and in leading order
µ=
π2
1 − (α−1)
6
T
F
!
2
+ ... F
(265)
In particular in 3D we have
E = V
!
2
T
+ ...
µ
!
2
5π 2 T
1+
+ ...
8
µ
π2
1+
8
3
3
1
N = V 2 (2m) 2 µ 2
6π
3
5
3 1
(2m) 2 µ 2
2
5 6π
(266)
(267)
leading to
1
F =
2m
6π
2N
23
(268)
V
and
µ=
π2
1−
12
E=
5π 2
1+
12
T
F
!
2
+ ... F
(269)
and
T
F
!
2
+ ...
3
N F
5
(270)
Note that in zero temperature, the pressure given by the equation
2 1
1
6π 2 3
P =
5
m
N
V
53
(271)
37
[6] The kinetic description of a gas
The number of one particle states within a phase space volume is dN = d3 rd3 p/(2π~)3 . The occupation of this phase
space volume is:
dN
≡ f (r, p)
drdp
(2π~)3
(272)
where f (r, p) is called Boltzmann distribution function. In equilibrium we have
f (r, p)
= f (p − µ)
(273)
eq
where f ( − µ) is either the Bose or the Fermi occupation function, or possibly their Boltzmann approximation. If we
use (r, v) instead of (r, p) we have
F (r, v) =
m 3
2π
f (r, p)
(274)
By integrating over r and over all directions we get the velocity distribution
F (v) = L3 × 4πv 2
m 3 1
f
mv 2 − µ
2π
2
(275)
If we use Boltzmann approximation for the occupation function and express µ using N and T we get
F (v) = N
m 3/2
2
1
4πv 2 e− 2 mv /T
2πT
(276)
We note that
Z Z
N=
drdp
f (r, p) =
(2π~)3
Z
Z
dg()f ( − µ) =
F (v)dv
(277)
Given N gas particles that all have velocity v we can calculate the number of particles that hit a wall element per
unit time (=flux), and also we can calculate the momentum transfer per unit time (=force). One obtains (per unit
area):
1 N
v
J =
4 V
1 N
P =
mv 2
3 V
(278)
(279)
If we have a distribution of velocities N should be replaced by F (v) and the result should be integrated over v. In
the case of flux calculation:
Z
Jincident =
0
∞
1
4
F (v)dv
V
v
In the case of the pressure P one obviously recovers the Grand canonical result.
(280)
38
====== [6.1] Photon gas and blackbody radiation
The modes of the electromagnetic field are labeled by the wavenumber k and the polarization α. For historical reasons
we use k instead of p for the momentum and ω instead of for the energy. The dispersion relation is linear ω = |k|.
The density of modes is
V
g(ω)dω = 2 ×
3 4πω
2
(2π)
dω
(281)
The canonical equilibrium state can be formally regarded as the grand canonical thermochemical equilibrium of µ = 0
Bose particles:
pn ∝ e−β
P
kα
ωkα nkα
(282)
For the occupation we have
1
hnkα i =
eβωkα −1
≡ f (ωkα )
(283)
For the energy we have
Z
E
∞
=
Z
∞
ωdω g(ω) f (ω) = V
dω
0
0
1
π2
ω3
βω
e −1
(284)
The energy flow of photons that hit the wall per unit area per unit time is
∞
Z
Jincident =
dω
0
ω3
1 1
c
4 π 2 eβω − 1
(285)
where c is the speed of light. Detailed balance consideration implies that
Jemitted (ω) dω = aJincident (ω) dω
(286)
where a = a(ω) is the emissivity. From here we get the Planck formula
1
Jemitted (ω) = a(ω) 2
4π
ω3
eβω − 1
1 3
T
4π 2
=
ν3
eν − 1
(287)
where ν = ω/T is the scaled frequency. Note that the peak of a blackbody radiation is at ν ≈ 3 which is known as
Wein’s law. Upon integration the total blackbody radiation is
Z
Jemitted =
∞
Jemitted (ω)dω =
0
1
4π 2
π4
14
T4
(288)
which is know as Stephan-Boltzmann Law.
Je (w)
υ
39
[7] Systems with interactions
ε = interaction strength
T = the temperature
ε T perturbative treatment (cluster expansion)
T << ε models of collective excitations
ε ∼ T phase transition
Models for phase transition:
lattice gas
Ising model (1D,2D)
Landau-Ginzburg
Treatment approaches:
Analytic (Yang-Lee, Onsager)
Mean field theory (heuristic, Bragg-Williams)
Criticality and the scaling hypothesis
Field theory: renormalization procedure
====== [7.1] The virial expansion
Define the fugacity as
z ≡ eβµ
(289)
and note that
∂
1 ∂
=z
β ∂µ
∂z
(290)
The grand canonical partition function in Gibbs-Boltzmann approximation is:
Z(β, µ) =
X
ZN z N
≈ e Z1 z
(291)
N
Hence for z 1
ln Z (z; β) =
V
λ3T
z
(292)
40
For non ideal gas, as well as for quantum gas we would like to go beyond the Gibbs approximation:
ln Z (z; β) =
V
λ3T
X
∞
bn (T )z n
(293)
n=1
where b1 = 1. We would like to find a procedure to determine the other coefficients. Once they are known we get the
state equations from
∂
ln Z
∂z
T
P =
ln Z
V
N = z
(294)
(295)
leading to
N
=
V
P
=
T
1
λ3T
1
λ3T
X
∞
n=1
X
∞
nbn (T )z n
(296)
bn (T )z n
(297)
n=1
It is customary to eliminate z from the first equation and to substitute the result into the second equation so as to
get
n−1
∞
X
PV
3 N
=
a` (T ) λT
NT
V
(298)
a1 = b1 = 1
a2 = −b2
a3 = 4b22 − 2b3
(299)
(300)
(301)
`=1
with
====== [7.2] Beyond the Gibbs approximation
On the one hand we have
Z(z) =
∞
X
ZN z N
(302)
N =0
On the other hand
ln Z(z) =
∞
X
B` z `
(303)
`=1
The relation between the two set of coefficients is the same as in probability theory where B` are the moments of the
distribution. Using the expansion ln(1 + x) = x − (1/2)x2 + (1/3)x3 + ... we get
B1 = Z1
B2
1
= Z2 − Z12
2
1
B3 = Z3 − Z1 Z2 + Z13
3
(304)
(305)
(306)
41
We can use this relation is two ways. One the one hand we can evaluate a few ZN s so as to get the leading order
B` s. On the other hand once the leading order B` s are known we can generate from them a generalized Gibbs
approximation for all(!) the ZN s via a simple exponentiation.
The classical partition functions, taking the interactions into account, can be written as
ZN
1
=
N!
1
λ3T
N
(307)
QN
where QN is the configuration space integral. Note that it has the dimensions of V N . It equals V N if there are no
interaction. If there are short range interactions the corrections are proportional to V . It follows that the B` s are
proportional to V , so we write
B` =
V
λ3T
b`
(308)
with b1 = 1 and
b2 =
1
2
1
V λ3T
Z
dx1 dx2 e−βu(x1 −x2 ) − 1
(309)
Note that quantum mechanically b2 = ±2−5/2 + 23/2 (ζ − ζ 0 ). To find the other bs one should a adopt a diagrammatic
approach which is explained in the next section.
====== [7.3] The cluster expansion
What represents graph? Constants in pertovative developments appear in the shape
1
(n!)
Z
f (x2,3 ) , f (x3,4 ) , f (x5,6 ) , f (x6,7 ) , f (x2,3 ) , f (x7,8 ) , f (x2,3 ) , f (x8,9 ) , f (x9,10 ) h (x1 ) ...h (x10 ) dx1 ....dx(310)
10
we can represent this type of constant diagramicaly, with the ’graph’
3
1
6
2
1+
45
+
7
+
+
9
11
8
10
+
+3
+3
+
+3
+
+...
+...
+
reducibility N = 11, K = 5
N = n1 + ....nk = 1 + 3 + 3 + 2 + 2 = 11
[...] =
(11!)
(1!) (3!) (3!) (2!) (2!)
(311)
(312)
42
different graphs identical represent numerical equal constants. The number of the graphs, identical to this graph
(including this graph itself) is NS! . The symmetry factor is
S = S0 + S1 ...SS = 2 ∗ 1 · 2 · 3 · 6 · 2 · 2
(313)
Diagrammatic summation:
∞
X
Z≡
(graphs sum with N angles) = 1
(314)
(graphs sum with n angles)
(315)
N =0
F ≡
∞
X
n=1
The basic Lemma:
Z
= eF
(316)
The proof:
e
F
∞
X
1
≡ 1+
k!
= 1+
= 1+
k=1
∞
X
∞
X
!k
(317)
graphs sum with n angles
n=1
∞
X
k=1 n1 =1
∞
X
∞
X
...
(318)
(graphs sum with N angles N = n1 + ...nk )
nk =1
(graphs sum with N angles) ≡ Z
(319)
N =1
remark: The number of the identical evologically graphs in the first sum is
this number is NS!
H=
N
X
p~i 2 X
+
u (~xi − ~xj )
2m i<j
i=1
ZN (β, V ) =
1
N!
m
2πβ
3N
Z
2
k!
S0
∗
nk !
n1 !
S1 ... Sk
where as in the second sum,
(320)
dN ~xe−β
P
i<j
u(xil )
(321)
we define
e−βυ(x) ≡ 1 + f (x)
ZN (β, V ) =
1
N!
m
2πβ
(322)
3N
Z
2
dx~1 ...dx~N
Y
i<j
(1 + f (xij ))
(323)
43
∞
X
Z (β, µ, V ) =
eβµN ZN (β, V ) = diagrammatic sum
(324)
N =0
where
m
2πβ
32
eβµ
(325)
f (xij ) ≡ e−βυ(xij ) − 1
(326)
h (xi ) ≡
with the help of the basic Lemma we are able to find the expansion coefficients in the expression
1
ln Z (z, T, V ) =
V
mT
2π
32 X
∞
bn Z n
(327)
n=1
leading to
bn =
mT
2π
32 (n−1)
1 X
[irreducible diagrams with n nodes]
V
(328)
with h (x) := 1 and f (x) := e−βv(x) − 1 . For example
b1 =
1
V
Z
mT
2π
23
mT
2π
3
b2 =
b3 =
(329)
d~x = 1
1
1
·
V (2!)
Z
dx~1 dx~2 f (x1,2 )
1
1
·
[3]
V (3!)
(330)
(331)
the classic example is calculating the coefficients for rigid ball gas in the limit V → ∞
note: limV →∞ bn (T, V ) ≡ bn (T ) < ∞
====== [7.4] From gas with interaction to Ising problem
Consider classical gas with interactions:
H=
N
X
p2α
+ U (r~1 , ..., r~N )
2m
α=1
(332)
The N particle partition function is
ZN
1
=
N!
1
λT
3N Z
d3N r e−βU (r1 ,...,rN )
(333)
44
We got reduction to the static gas problem. So for simplicity let us consider a lattice version:
H = U (r~1 , ..., r~N ) =
1X
u (x, x0 ) n (x) n (x0 )
2 0
(334)
x,x
The grand partition function is

Z=
X


X
X
1
exp −β 
u (x, x0 ) n (x) n (x0 ) − µ
n (x)
2 0
x
(335)
x,x
n(·)
where n(x) = 0, 1. Or we can write
n (x) =
1 + σ (x)
2
(336)
where σ (x) = ±1. Then we get

Z=
X

exp −β 

X
ε (x, x0 ) σ (x) σ (x0 ) − h
σ (x) + const
(337)
x
hx,x0 i
σ(x)
X
where ε = u/4. We represent graphically the interaction ε (x, x0 ) between two sites x and x0 by “bonds”. The notation
hx, x0 i means from now on summation over all the bonds without double counting. In the simplest case there are
interactions only between near-neighbor sites.
We see that the calculation of Z for static lattice gas is formally the same as calculation of Z for an Ising model. The
following analogies should be kept in mind
occupation N
chemical potential µ
fugacity z = eβµ
grand canonical Z(β, µ)
←→
←→
←→
←→
magnetization M = 2N − N
magnetic field h
define z = e2βh
canonical Z(β, h)
(338)
(339)
(340)
(341)
From now on we refer to Ising model, but for the formulation of some theorems in the next section it is more convenient
to use the lattice gas language for heuristic reasons. Note also that N is more convenient than M because it does not
skip in steps of 2.
====== [7.5] Yang and Lee theorems
The probability for a given spin configuration is
PR ∝ e−β(ER −hMR )
(342)
or, disregarding re-definition of the energy floor, it can be written as
PR ∝ e−βER z N
(343)
where ER is the energy of the interactions. The partition function is
Z (z; β) =
N
X
N =0
ZN (β)z N
(344)
45
The Helmholtz function is
F (z; β) = −
1
ln Z(z; β)
β
(345)
The expectation value of the total magnetization is
hN i = −βz
∂
F (z; β)
∂z
(346)
As we increase z we expect the magnetization hN i to grow. We expect hN i/N to have a well defined value in the
limit N → ∞. Moreover, below some critical temperature we expect to find a phase transition. In the latter case
we expect hN i to have a jump at zero field (z = 1). The Yang and Lee theorems formulate these expectations in a
mathematically strict way.
Given N it is clear that we can write the polynomial Z as a product over its roots:
Z (z) = const ×
N
Y
(z − zr )
(347)
r=1
Consequently
F (z) = −
N
1X
ln(z − zr ) + const
β r=1
(348)
and
hN i = z
N
X
r=1
1
z − zr
(349)
note that there is a strict analogy here with the calculation of an electrostatic field in a 2D geometry.
In the absence of interactions (infinite temperature) we get that all the roots are at z = −1. Namely,
Z (z; β) =
N
X
N N
CN
z = (1 + z)N
(350)
N =0
So we do not have phase transition since the physical axis is 0 < z < 1, where this function is analytical. The questions
is what happens to the distribution of the roots as we increase the interaction (lower the temperature), and what is
the limiting distribution in the thermodynamics limit (N → ∞). There are three statements that give answers to
these questions due to Yang and Lee. The first statement is regarding the existence of the thermodynamics limit:
F (z)
= exists
N →∞ N
f (z) = lim
(351)
The second statement is that all the roots are lying on the circle |zr | = 1. The third statement is that below the
critical temperature the density of roots at z = 1 becomes non-zero, and hence by Gauss law hN i/N has a jump at
zero field. This jump is discontinuous in the thermodynamics limit:
46
M
T<Tc
h
N
µ
Ζ
Τ>>Τ c
T<Tc
Z
h<0
h>0
Z
h<0
h>0
47
[8] Phase Transitions
====== [8.1] The Ising model
The energy of a given Ising model configuration state is
E[σ (·)] = −
X
ε (x, x0 ) σ (x) σ (x0 ) −
X
(352)
h (x) σ (x)
x
hx,x0 i
The canonical state is
p[σ (·)] =
1 −βE[σ(·)]
e
Z
(353)
where the partition function is
 
Z[h (·) , β] =
X
exp β 

X
0
ε (x, x ) σ (x) σ (x ) +
hx,x0 i
σ(·)
0
X
h (x) σ (x)
(354)
x
We expand the Helmholtz function as
F [h (·) , T ] = F0 (T ) −
1 X
g (x, x0 ) h (x) h (x0 ) + O h4
2T
0
(355)
x,x
From here we get
hσ (x)i = −
∂F
1X
g (x, x0 ) h (x0 )
=
∂h (x)
T 0
(356)
x
and
∂F
= g (x, x0 )
hσ (x) σ (x )i0 = −T
∂h (x) ∂h (x0 ) 0
0
(357)
for an homogeneous field we replace F [h (·) , T ] by F (h, T ) and we get
M̃ = −
∂F
= N χh + O h3
∂h
(358)
where
χ=
1X
g (r)
T r
(359)
we also have the usual expressions
E (T ) = F0 (T ) + T S (T )
S (T ) = −
∂F (T )
∂T
(360)
(361)
48
C (T ) = T
∂S
dE
=
∂T
dT
(362)
====== [8.2] The correlation function g (r)
The experimental measurement of g (r) is based on the following. Given a configuration σ(x) the intensity of the
scattering is
Z
2
−i~
q ·~
x
I (q) ∝ σ (x) e
d~x
(363)
If we average over configurations we get
Z
I (q) ∝
0
dxdx0 hσ (x) σ (x0 )ie−iq·(~x−~x )
(364)
Thus the FT of the correlation function is a measurable quantity.
We shall see that Landau’s approach in the harmonic approximation leads to the Ornstein-Zernike expression
g (k) ∝
1
2
k + ξ2
(365)
This leads to
g (r) ∼ exp(−r/ξ)
g (r) ∼ 1/rd−2
if ξ < ∞
if ξ = ∞
(366)
(367)
The information about order-disorder is in g (r). If ξ < ∞ there is no long range order, and we get
χ=
1X
g (r) < ∞
T r
(368)
as ξ → ∞ the susceptibility diverges which suggests spontaneous magnetization as phase transition. Below the critical
temperature one should define the correlation function differently:
g̃ (r) ≡ hσ (x) σ (x0 )i0 − hσ (x)i20
(369)
====== [8.3] Critical behavior and the scaling hypothesis
phase diagrams (parameters space)
h
T0
T
49
for 2D Ising model with near neighbor interactions Tc = 2.27. The qualitative plots of the state equations are:
Tc
E
T
C(T)
C~ t
−α
T
M(h)
Tc <T
h
M~ χ h
−α
χ∼ t
M(T)
M~
h=01
t β
M(h)
T
M~ h
T=Tc
1/γ
h
This defines the exponents α, β.γ, δ. Two other exponents ν and η are defined via the critical behavior of the correlation
function:
g (r) ∼ exp(−r/ξ)
g (r) ∼ 1/rd−2+η
with ξ ∼ |T − Tc |−ν
at T = Tc
(370)
(371)
below Tc the behavior is similar to T > Tc provided the correlation function is re-defined.
The scaling hypothesis is
F0 (s1/ν t) = sd F0 (t)
1/ν
g(r/s, s
d−2+η
t) = s
g(r; t)
(372)
(373)
where t = |T − Tc |. From here it follows that
F s1/ν t, s(d+2−η)/2 h = sd F (t, h)
(374)
from here we derive the critical behavior
C ∼ |t|−α
M ∼ |t|β
χ ∼ t−γ
M ∼ |h|1/δ
α = 2 − νd
β = (d − 2 + η)ν/2
γ = (2 − η)ν
δ = (d + 2 + η)/(d − 2 + η)
(375)
(376)
(377)
(378)
The so called “classical” mean-field exponents are
ν
η
α
β
γ
δ
=
=
=
=
=
=
1/2
0
0
1/2
1
3
These are independent of dimensionality and do not satisfy the above equalities!
====== [8.4] Solution of the 1D Ising Model
(379)
(380)
(381)
(382)
(383)
(384)
50
Assuming only near neighbor interactions
X
E [σ] = −ε
σi σj −
X
(385)
hi σ i
i
hiji
The partition function is
Z[h, β] =
X
e−βE[σ]
(386)
2 cosh (βhi )
(387)
σ(·)
For ε = 0 we get
Z[h, β] =
N
Y
i=1
and hence
F [h, T ] = −T
N
X
N
hi
1 X 2
ln 2 cosh
≈ −N T ln (2) −
h
T
2T i=1 i
i=1
(388)
The correlation function is
g(r) = −T
∂F
= δij = δr,0
∂hi ∂hj
(389)
and hence the susceptibility is
χ=
1
1X
g (r) =
T r
T
(390)
The magnetization is
∂F
M̃ = −
= N tanh
∂h
h
≈ N χh + O h3
T
(391)
We turn now to the case ε 6= 0 and assume homogeneous field h. This allows to use the transfer matrix method for
solutions. We
E[Σ] = −ε
X
hiji
σi σj − h
X
σi
(392)
i
Let us define the scaled parameters
ε̃ ≡ βε
(393)
h̃ ≡ βh
(394)
and the matrix
Tσ0 σ00
1
0 00
0
00
≡ exp ε̃σ σ + h̃ (σ + σ ) =
2
eε̃+h̃ e−ε̃
e−ε̃ eε̃−h̃
!
(395)
51
The eigenvalues of this matrix are
r
λ± = eε̃ cosh h̃ ± e−ε̃ 1 + e4ε̃ sinh2 h̃
(396)
The partition function can be calculated as
N
Z (β, h) = trace T N = λN
+ + λ−
(397)
and hence for very large N we get
F (T, h) = −N T ln (λ+ )
(398)
Expanding we get
ε 1 exp 2 ε T
− N
h2
F (T, h) ≈ −N T ln 2 cosh
T
2
T
(399)
Hence
χ=
ε
1
exp 2
T
T
(400)
Now we would like to calculate the correlation function at zero field.
g (π) ≡ hσ0 σr i =
1 X
σ0 Tσr0 σr σr TσNr σ−r
0
Zσσ
0
(401)
r
We have
Tσ0 σ00 =
√1
2
√1
2
√1
2
− √12
!
λ+ 0
0 λ−
√1
2
− √12
√1
2
√1
2
!
(402)
with
λ+ = 2 cosh (ε̃)
λ− = 2 sin h (ε̃)
(403)
(404)
so we get
1
g (r) = trace
Z
1 0
0 −1
T
r
1 0
0 −1
T
N −r
=
−r
−r
λr+ λN
+ λr− λN
−
+
N
λN
+ + λ−
(405)
and hence for very large N we get
λ−
λ+
r
= e−r/ξ
(406)
h ε i−1
ξ = ln coth
T
(407)
g (r) =
where
52
Note that as expected
∞
X
ε
g (r) = exp 2
T
−∞
(408)
====== [8.5] Solution of the 2D Ising model
The full details of the Onsager solution for this problem is in Huang. Also here the transfer matrix approach is used.
Recall that the zero field solution of the 1D model is
1
ln Z = ln (2) + ln (cosh (ε̃))
N
(409)
The 2D solution is
1
1
ln Z = ln (2) +
N
2
Z Z
dθdθ0
2
(2π)
h
i
2
ln (cosh (2ε̃)) + sinh (2ε̃) (cos θ + cos θ0 )
(410)
High temperatures (ε̃ → 0) is the weak interaction mean field limit. As we increase ε̃ we get to the critical point
ε̃c = 2.27. The temperature regimes are determined by the parameter
κ≡
2 sinh (2ε̃)
2
(cosh (2ε̃))
≤1
(411)
At the critical point κ attains its maximal value. The following plot shows how κ depends on ε.
h
k
====== [8.6] Mean field theory - heuristic approach
The standard Ising Hamiltonian is
H = −ε
X
σi σj − h
X
hiji
σk
(412)
k
Let us define an effective Hamiltonian for the spin at site i
"
H
(i)
#
=− h+ε
X
hσj i σi = − [h + γεhσi] σi
(413)
neighbors
hence we get the self-consistent equation
hσi = tanh
1
(h + γεhσi)
T
(414)
53
This equation should be solved for hσi.
slope
rE / T
1
<σ>
h
rE
By inspection of the plot we observe that for h = 0 the condition for getting a non trivial solution is γε/T > 1.
Therefore Tc = γε. If we want to explore the behavior in the critical region it is convenient to re-write the equation
in the following way:
h = T tanh−1 hσi − Tc hσi
(415)
and to approximate it as
1
h = (T − Tc ) hσi + Tc hσi3
3
(416)
for T > Tc we get the Curie-Weiss law
1
h
T − Tc
hσi =
(417)
hence we get the critical exponent γ = 1. For T = Tc we get
hσi =
3
h
Tc
13
(418)
hence we get the critical exponent δ = 3. For T < Tc we get
hσi =
Tc − T
3
T
12
(419)
hence we get the critical exponent β = 1/2.
====== [8.7] The anti-ferromagnetic phase transition
The Ising Hamiltonian with ε 7→ −ε
H=ε
X
hiji
σi σj − h
X
σk
(420)
k
We mark the magnetization of the two sub lattices by Ma and Mb . We define
1
(Ma + Mb )
2
1
=
(Ma − Mb )
2
M =
Ms
(421)
(422)
54
Without the magnetic field, the problem is the same as the ferromagnetic one with Ms as the order parameter. The
heuristic mean field equations are
1
(h − Tc Mb )
T
1
= tanh
(h − Tc Ma )
T
Ma = tanh
(423)
Mb
(424)
The standard development gives in the critical region, after addition and subtraction of the two equations,
1
(T − Tc ) Ms + Tc 3M 2 Ms + Ms3 = 0
3
1
(T + Tc ) M + Tc 3Ms2 M + M 3 = h
3
(425)
(426)
From here it follows that
Ms = 0
or
2
3M +
Ms2
= 3
Tc − T
T
(427)
1
h
(2 + Ms2 )M + M 3 =
3
Tc
(428)
Μ1
Μ
h
h=rE
T
C=rE
χ
Tc
T
55
If magnetic field h is strong enough, it can destroy the order and cause Ms = 0. that’s not surprising because especially
in temperature 0
1
1
γ − h
E (↑↓↑↓) = N · − γ , E (↑↑↑↑) = N ·
2
2
(429)
hence, the phase diagram is there’s no line beyond order I phase. In the region T ∼ Tc h ∼ 0 we get that the
magnetization is

M = χh = 

1
(430)
 h
2
Tc + T 1 + 16 Ms (T )
we can get a better general expression for all of the temperature range by blocked differentiation of the heuristic
equations
χ=
1
2
Tc + T cosh
Tc
T Ms
(431)
(T )
remark: in the region T ∼ Tc substitution of Ms (T ) gives
χ=
1
Tc +T
1
4Tc −2T
Tc < T
T < Tc
(432)
====== [8.8] Mean field theory - a variational approach
We look for a solution to the variation problem
F [ρ] ≡ hHi − T S [ρ] = minimum
(433)
The canonical state is assumed to be well approximated by
ρ (σ1 ...σN ) =
#
"
1
b
N
(2 cosh (βb))
exp βb
X
σk
(434)
k
where b is the affective field. For the calculation we use the identity F [ρ] = F0 [ρ] + hH − H0 i, where H0 = −b
leading to
"
2
#
b
1
b
b
b
− γ tanh
− (h − b) tanh
F ρ = N −T ln 2 cosh
T
2
T
T
P
i
σi ,
(435)
The variational equation for b is as expected
b
b = h + γ tanh
T
(436)
Hence, we get the variational free energy
"
F (T, h) = N −T ln 2 cosh
b (h)
T
2 #
1
b (h)
+ γ tanh
2
T
(437)
56
and the magnetization
∂F ρb ∂b
∂F ρb
b (h)
∂F (T, h)
=−
−
= N tanh
M̂ = −
∂h
∂b ∂h
∂h
T
(438)
To make calculation it is convenient to transform to
A T, M̄ ≡ F (T, h) + h + M̃
(439)
such that
dA = −SDT + hdM̃
(440)
We notice that
b = T tgh−1 (M ) =
1
T ln
2
1+M
1−M
(441)
and
1
h = T ln
2
1+M
1−M
− γM
(442)
Thus
1
1
1+M
1
A (T, M ) = N −T ln 2 + T ln 1 − M 2 + T M ln
− γM 2
2
2
1−M
2
1+M
1
1+M
1−M
1−M
= NT
− N γM 2
ln
+
ln
2
2
2
2
2
(443)
(444)
From this expression it is convenient to derive explicit expressions for the state equations. Especially for the heat
capacity we get
1
C (T ) = N T ln
2
1+M
1−M
·
∂M
|h=0
∂T
(445)
hence α = 0. This result can be obtained directly from C (T ) = dhHi/dT where hHi = −N 12 γεM 2 .
C (Τ)
T
====== [8.9] Mean field theory - Bragg Williams formulation
57
Consider an Ising model with N sites. Given a spin configuration define
M̃ = total magnetization
(446)
M = M̃ /N
1
1
N+ = (N + M̃ ) = N (1 + M )
2
2
1
1
N− = (N − M̃ ) = N (1 − M )
2
2
N+− = number of bonds connecting spins with opposite direction
(447)
(448)
(449)
(450)
It follows that
X
X
(451)
σi = M̃
σi σj =
hiji
1
γN − 2N+−
2
(452)
The Bragg Williams approximation is
N+− ∼
1
γN × 2
2
N+
N
N−
N
(453)
Assuming that it holds for typical configurations we approximate the energy functional as
E[σ] ≈ −N ×
1
εγM 2 − hM
2
(454)
The number of configuration with magnetization M is
gM =
2
4
1
1
N!
≈ const e−N ( 2 M + 12 M +...)
(N+ )! (N −)!
(455)
In order to derive the latter approximation note that the derivative of ln(gM ) is of the form
1
1
1
(ln(1 + x) − ln(1 − x)) ≈ x + x3 + x5 + ...
2
3
5
(456)
With this approximation we get
Z=
XX
e−βE[σ] ≈
M σ∈M
X
M
gM e−βE(M ) =
X
e−A(M )
(457)
M
where
1
1
(1 − βγε) M 2 + M 4 − βhM
A (M ) = N ×
2
12
(458)
The mean field equation is A0 (M ) = 0. Long range order is implied if
lim hM i =
6 0
h→+0
(459)
58
Let us say that T > Tc , then we have only one minimum. One can ask whether is is allowed to ignore the quartic
term. This should be checked self consistently. The dispersion in the harmonic approximation is
δM =
p
1
hM 2 i = p
N (1 − (Tc /T ))
(460)
If we what the quartic term to be negligible we get the condition N 1/(1 − (Tc /T )).
M
M
T>Tc
====== [8.10] Landau’s approach
In the spirit of Bragg-Williams approach we would like to take into account the spatial fluctuations of the magnetization
ϕ(x) in the calculation of the partition function. We therefore introduce the following field theory model:
Z[h] =
X
e−A[ϕ(·)]
(461)
ϕ(·)
where
Z
1
1
1
2
α (∇ϕ) + uϕ2 + uϕ4 − hϕ + const
2
2
4
Z
1
dk
2
2
(αk
+
r)|
ϕ̃(k)|
+
...
=
(2π)d 2
A[ϕ(·)] =
dx
(462)
(463)
By convention we set α = 1 which fix the dimensions of ϕ and of the other model parameters. We write these
dimensions as Ld , accordingly
2−d
2
2+d
= −
2
= −2
= d−4
dϕ =
(464)
dh
(465)
dr
du
(466)
(467)
Note that there is a possibility to reduce Ising model to Landau-Ginzburg model by using a weight function
X
σ(·)
→
Z Y
dϕ(x) e− 4 u(ϕ(x)
1
2
2
−1)
(468)
x
and realizing that the ferromagnetic interaction term −σ(x)σ(x0 ) corresponds to the gradient terms (ϕ(x) − ϕ(x0 ))2
in the Landau-Ginzburg model.
====== [8.11] The mean field approximation
We define the mean field ϕ̄ via the equation A (ϕ) = minimum. The mean field for an homogeneous field h (x) = h is
obtained from the equation
rϕ + uϕ3 = h
(469)
59
In particular for h = 0+ we get
(
ϕ̄0 = 0 0 < t
1
−r 2
t<0
u
(470)
To discuss symmetry breaking. More generally the mean field equation is
−∇2 + r ϕ + uϕ3 = h (x)
(471)
The solution up to first order in h (x) is
Z
ϕ̄ (x) = ϕ̄0 +
g (x − x0 ) h (x0 ) dx0
(472)
where
Z
0
g (x − x ) =
0
eik(x−x )
dk̄
(473)
2
d
k 2 + (1/ξ)
(2π)
and
(
ξ=
1
0<t
r− 2
−1
(−2r) 2 t < 0
(474)
Hence ν = 1/2 and η = 0.
====== [8.12] The Gaussian approximation
We expand
¯ + δϕ (x)
ϕ (x) = ϕ (x)
A [ϕ] = A [ϕ̄] + δ 2 A [δϕ] + ...
(475)
(476)
In what follows we use the Gaussian integral
Z
Z=
dϕe
−A(ϕ)
≈e
−A(ϕ̄)
Z
dϕe
− 12 A00 (ϕ̄)(ϕ−ϕ̄)2
=e
−A(ϕ̄)
− 12
1 00
A (ϕ̄)
2π
(477)
We get
1
A [~
ϕ] = − λϕ̄40 −
4
Z
h (x) ϕ0 dx −
1
2
Z
g (x − x0 ) h (x) h (x0 ) dxdx0
(478)
and
0
δA [ϕ ] =
Z
dx
1
1
2
(∇ϕ0 ) +
2
2
2 !
1
ϕ2
ξ
(479)
60
We also note that
Z
Dϕ e
−
R
“
“
” ”
2
dx 12 ϕ −∇1 +( ξ1 ) ϕ
Z
=
Dϕ e
−
1
Ld
P
k
“
”
2
ϕ(k) k2 +( ξ1 ) ϕ(k)
= const ×
Y
k
2 !− 12
1
k +
ξ
2
(480)
Where ϕ̄ is the mean field.
Higher terms in the expansion of of A (ϕ) around ϕ̄ are neglected, leading to Ginzburg’s Criterion. Namely, in the
calculations we are going to neglected the non-Gaussian terms. This is justified if u ξ d−4 . Optional formulation
for r < 0 is g (ξ) ϕ̄2 . In any case we get the condition
|T − Tc | const u2/(4−d)
(481)
This condition defines borders of the critical region. Within the critical region the Gaussian approximation breaks
down because of non-Gaussian fluctuations.
T<Tc
Tc <T
T~Tc
Let us calculate the free energy in the Gaussian approximations.
F [h (·) , T ] = FM F + FG
(482)
where
"
FM F
1 a2
2
= − Lα T · (T − T0 ) −
4
λ
−t
λ
12 Z
#
h (x) dx
t<o
Z
1
g (x − x0 ) h (x) h (x0 ) dxdx0
+ −
2
t≤0
(483)
and
2
1
1 X
2
ln k +
F0 = − T
2
ξ
k
!
1
= − L2 T
2
Z
2 !
1
ln k +
d
ξ
(2π)
dk
2
(484)
from here we can derive expression for the specific heat where T ∼ T0
C = CM F + CG
(485)
where
CM F =
0
1 2 a2 2
2 L λ Tc
0<t
t<0
(486)
and
CG ∼ |t|
d−4
2
====== [8.13] Regularization and Renormalization Group
(487)
61
In the field theory model, we should perform the sum over all the field configurations ϕ (x) or equivalently over all of
the Fourier components ϕ̃ (k). In order not get an infinite result we should in general use two cutoff parameters:
(488)
(489)
L = linear size of the model
Λ = largest momentum scale
For the regularization free energy we get a finite result
F (X; Λ, L) = −T ln Z(X; Λ, L)
(490)
where X are the parameters of the model. There is always a trivial “engineering” scaling relation
F (sdX X; Λ/s, sL) = F (X; Λ, L)
(491)
while the thermodynamic limit implies
F (X; Λ, sL) = sd F (X; Λ, L)
(492)
On top perturbation theory allow to integrate the high Fourier components within a shell δΛ leading to
F (X; Λ, L) = F (X − δX; Λ − δΛ, L)
δΛ/Λ
= F (X − δX; Λ/e
, L)
h
i−dX
(X − δX); Λ, L/s)
= F ( eδΛ/Λ
h
i−d h
i−dX
= edΛ/Λ
F ( eδΛ/Λ
(X − δX); Λ, L)
(493)
(494)
(495)
(496)
Suppressing the cutoffs, and changing notations we have deduced the scaling relation
F (X(Λ)) = s−d F (X(Λ/s))
(497)
and we can get for X(Λ) an equation
dX
= R(X, Λ)
dΛ
(498)
That can be re-scaled as
dx
= R̂(x)
dτ
(499)
Near a fixed point one can deduce
F (X(Λ)) = s−d F (sDX X)
(500)
For the correlation function one can deduce
g(X(Λ)) = s−2d−2Dh g(sr, sDX X)
====== [8.14] Digression: scaling
(501)
62
A function of one variable has a scaling property if
F (sx) = sDF F (x)
(502)
where DF is the scaling exponent. It follows that F (x) = const xDF . For example F (x) = x2 has the scaling exponent
DF = 2. If we have say two variables then the more general definition is
F (sDx x, sDy y) = sDF F (x, y)
(503)
Note that the scaling exponents can be multiplied by the same number, and still we have the same scaling relation.
It follows that there is a scaling function such that
F (x, y) = y DF /Dy f
x
y Dx /Dy
(504)
For example F (x, y) = x2 + y 3 has the scaling exponents Dx = 1/2, Dy = 1/3, DF = 1. More generally any “physical”
function has an “engineering” scaling property that follows trivially from dimensional analysis.
Consider the flow equation
dX
= F (X, t)
dt
(505)
Assuming that F (X, t) has a scaling property we can rescale X and redefine the time variable t such as to get an
equation for a time independent flow:
dx
= f (x)
dτ
Obviously the same can be done if x represents several variables.
(506)
63
Special Topics
[9] Ergodicity, entropy, irreversibility and dissipation
The notion of ergodicity for closed chaotic system is explained. The definition of thermodynamic entropy and the
third law are discussed. Distinction is made between the notions of ”irreversibility” and ”dissipation”. The classical
and the quantum mechanical case are treated on the same footing.
====== [9.1] Representation of states
The state of the system is represented by the probability function ρ(Q, P ), or by the density matrix ρnm . In the
quantum mechanical case the choice of basis is arbitrary. It is also possible to use Wigner function in order to have
a transparent analogy with the classical case.
Given a complete basis for representation, we obtain a complete set of Wigner functions. Each Wigner function
”occupies” a ”Planck cell” whose volume is (2π~)d , where d is the dimensionality. Thus a choice of basis is the
quantum mechanical analogue of phase space partitioning.
Of particular interest is the ”energy representation”, where the basis consists of stationary eigenstates, which are
determined via diagonalization of the Hamiltonian. The corresponding partitioning of phase space is into ”energy
shells” of volume (2π~)d . This leads to Wyle law: The number of states up to energy E is equal to the corresponding
phase space volume divided by (2π~)d .
====== [9.2] Evolution and Ergodization
Assuming closed system, the state after time t is Λ(t)ρ, where the propagator Λ(t) is determined by the Hamiltonian.
The propagator Λ(t) has an inverse Λ(t)−1 . By operating with the inverse we get back the initial distribution. Thus
the evolution which is induced by the Hamiltonian is always reversible, irrespective of chaoticity.
For sake of later discussion we also define an operation Λerg that we call ”ergodization”. This operation associates a
stationary state Λerg ρ with a given state ρ. The definition in the quantum mechanical case is
[Λerg ρ]nm = δnm × ρnm
(507)
where n and m labels basis states in the ”energy representation”. Obviously the results of ”ergodization” is a
stationary state. It is also clear that Λerg does not have an inverse, and therefore ”ergodization” is an irreversible
operation. In the classical case the corresponding definition is
[Λerg ρ](Q, P ) = normalization ×
Z
δ(H(Q, P ) − H(Q0 , P 0 )) ρ(Q0 , P 0 )dQ0 dP 0
(508)
As in the quantum mechanical case the ”ergodization” results in a stationary state, that is smeared uniformly over
the energy surfaces, and has the same energy distribution as that of the original state.
====== [9.3] coarse grain similarity
A key concept in the formulation of irreversibility, entropy, and the third law of thermodynamics is ”coarse grain
similarity”. In order to address classical mechanics and quantum mechanics on the same footing, we adopt the
following definition.
We pre-define a set of observables {O}. In the classical case these are the observables that define location in phase
space up to a give resolution . A corresponding definition in the quantum mechanical case is assumed. We also
assume that (2π~)d .
64
We say that ρA ∼ ρB if for any observable {O} we have tr(ρA O) = tr(ρB O). This means that for any practical purpose
ρA and ρB are indistinguishable.
====== [9.4] The ergodic time
There are three important time scales that should be distinguished:
terg
tH
τϕ
=
=
=
ergodic time
Heisenberg time
”decoherence” time
(509)
(510)
(511)
We assume that the system is chaotic. The ergodic time is the time that it takes for Λ(t)ρ to become similar (in the
coarse grain sense) to the stationary distribution Λerg ρ. The ergodic time is determined by the resolution , and by
the chaoticity of the motion (Lyapunov exponent). This is further explained below.
The instability of chaotic motion is characterized by an exponential factor exp(−λt), where λ is the Lyapunov
exponent. This implies the following expression for the ergodic time:
terg ≈ ln() ×
1
λ
(512)
(513)
We see the the dependence on is very weak. As a practical estimate we can simply write terg ∼ 1/λ.
It is important to realize that unlike the classical dynamics, the quantum dynamics of closed time-independent systems,
is always quasi-periodic. The Heisenberg time is the time it takes to resolve individual energy levels. It is related to
(2π~)d . We assume the typical situation of having terg tH . For t tH we have quantum mechanical recurrences.
The issue of these recurrences is further discussed in the next section, where we also define the notion of ”decoherence
time”.
====== [9.5] The ”decoherence” time
We assume later on that the system is perfectly isolated. However, if the system is not isolated, we can define, both
classically and quantum mechanically, a notion of ”decoherence” time. It is the time that it takes for Λ(t)ρ to become
stationary. Note the similarity with the definition of ergodic time. It is important to realize that a closed system
never becomes stationary. It is only ”decoherence” that can lead to a ”stationary” state. Hamiltonian evolution can
lead only to a stationary-like state (in the coarse grain similarity sense).
Let us assume that the environment induces an extremely small noise into the motion of the system. The intensity
of this noise is denoted by ν. A similar reasoning as in the previous section leads to the estimate:
τϕ ≈ ln(ν) ×
1
λ
(514)
This estimates holds classically (in the quantum mechanical case it looses validity if the noise is extremely faint).
Again we see that the dependence on ν is very weak. For any practical purpose we can use the estimate τϕ ∼ 1/λ,
hence τϕ ∼ terg .
For the later discussion of irreversibility we should add either of two assumptions. The first possibility: The system is
perfectly isolated and we limit the discussion to the time period t tH . The second possibility, which is in fact more
”realistic”, is to assume that τϕ ∼ terg tH . Either way we assume that quantum mechanical recurrences are not an
issue. Else the system should be regarded as microscopic, and the notion of ”irreversibility” becomes irrelevant.
====== [9.6] ergodicity and entropy
65
Ergodicity means that Λ(t)ρ becomes similar (in the coarse grain sense) to the stationary distribution Λerg ρ,
Λ(t)ρ ∼ Λerg ρ
for t > terg
(515)
It is important to realize that by operating on the left hand side with Λ(t)−1 we can get back the initial state. This
cannot be done with the right hand side because the ”information” on the initial state had been lost.
A popular measure for quantification of the approach to ergodic-like state is ”entropy”. The ”Shanon”, or ”VonNeumann” entropy of a state is defined as
S0 [ρ] = −
X
pr ln(pr )
(516)
r
In the quantum mechanical case the {pr } are the eigenvalues of ρ. In the classical case we should assume partitioning
of phase space into Planck cells and we define pr as the probability which is located in a give Planck cell.
In the quantum mechanical case it is evident that S0 [ρ(t)] for an isolated system is a constant of the motion. For
example, if the initial state is a pure state, then it remain always a pure state, and we get S0 [ρ(t)] = 0 for any time.
In the classical case there is an increase in S0 [ρ(t)] once the distribution develops fine details on the scale of Planck
cell partitioning. Up to this point in time, S0 [ρ(t)] is a constant of motion, as in the quantum mechanical case. This
is because Lioville theorem implies that the probabilities {pr } are just permuted, as in a ”musical chair game”.
Thus, the definition above of S0 [ρ] is not satisfactory from conceptual point of view. A meaningful ”thermodynamic”
definition of entropy is required, that can be applied both in the classical context and also in the quantum mechanical
context. The obvious definition is
S[ρ] = maximum(S0 [∼ρ])
(517)
This notation implies the following: Rather than considering ρ, we consider a class of states ∼ρ that are similar (in
the coarse grain sense) to ρ. For each we calculate S0 , but eventually we take the maximum. Loosely speaking we can
say that the thermodynamic entropy S[] is the S0 [] entropy of the coarse grained distribution. [The latter phrasing is
used in many textbooks, whenever a classical definition of entropy is discussed].
The advantage of the above definition is quite clear. In case of S0 [ρ(t)] we have entropy that does not change in time,
while in the classical case it has an eventual phase-space-partitioning-related increase. In case of S[ρ(r)] there is an
increase in the entropy once t > terg , and there is a complete correspondence between the classical and the quantum
mechanical definitions.
Note that the increase in the entropy in case of S0 [ρ(t)] is ~ related, while in case of S[ρ(t)] it is related. It is ,
not ~, that is relevant to the issue of irreversibility. Moreover, recall that in ”reality” the role of the mathematically
motivated is taken by the physically-motivated ν.
====== [9.7] Irreversibility
Consider Hamiltonian H(Q, P ; x) that depends on the parameter x = x(t). Let us assume that initially x = x1 , and
the system is in stationary state. Then, at t = 0, we make a sudden change x1 7→ x2 . After a while (t > terg ), the
system become stationary-like (in the coarse grain sense). The process is of coarse reversible. But in fact, any small
”decoherence” effect or any small inaccuracy in the determination of Λ(t) makes it practically impossible to reverse
the evolution process.
The distribution ρ(P, Q), just after the sudden change, occupies some phase space volume. Due to the chaotic evolution
it becomes smeared uniformly over the (new) energy surfaces. This is reflected in an increase of the thermodynamic
entropy.
The simplest example of such irreversible process is ”free expansion”: Consider a cylinder with one-particle gas, and
a movable piston. Assume that the piston is abruptly moved from an interior position x = x1 to some outer position
x = x2 . Say that the accessible volume becomes twice larger. This means that the new energy surface is twice larger.
After ergodization the increase in the thermodynamic entropy is ln(2).
66
====== [9.8] Dissipation
”Free expansion” is not a typical irreversible process, because energy change is not associated. A variation of ”free
expansion” that reflects the the physics of generic processes is as follows: Consider one-particle gas inside a 3D smooth
well. For simplicity let us assume that the initial state is microcanonical. At t = 0 the walls are abruptly deformed
outward. The motion along the deformed energy surfaces leads to ergodic-like distribution within a relatively thick
energy shell. This leads to increase in entropy, and in general also to change in the energy of the system.
Consider the same system with periodic driving: The (smooth) walls are deformed periodically. As a result the
thickness of the occupied energy shell becomes larger and larger with time. In fact it can be argued that the result is
diffusion in energy space. Further argumentation implies that this diffusion is biased, leading to a steady increase in
the average energy. This effect is called dissipation.
The notion of ”dissipation” is typically associated with irreversibility. Still these are two different concepts. In order
to have dissipation extra assumptions should be satisfied. The standard ”free expansion” experiment is an example
for irreversibility with which dissipation is not associated.
====== [9.9] Related concepts
One should be careful not to make confusion between the following concepts:
• Irreversibility of the evolution process.
• Lack of time-reversal invariance.
• Irreversibility of a driving process.
The first concept has been discussed in the previous sections. For simplicity of the following discussion let us assume
that the system is perfectly isolated (ν = 0). This means that the operator Λ(t) has an inverse Λ(t)−1 . Moreover, we
assume that is arbitrarily small. This means that Λ(t)−1 can be realized (at least in the mathematical sense), and
therefore the evolution process should be regarded as reversible.
The notion of ”time-reversal invariance” implies a stronger statement. There should exist an operation T , (which in
the standard example is the reversal of the velocity of the particle), such that Λ(t)−1 = Λ(t)T for any t. If there are
electric field E and magnetic field B then there exist T such that Λ(t; E, B)−1 = Λ(t; E, −B)T . Therefore we have time
reversal invariance only if B = 0
The notion of ”irreversibility of a driving process” is a synonym for ”lack of adiabaticity”. It appears in the context
of dissipation studies: If we move a piston outside, and then back inside, we do not reach exactly to the same state,
unless the driving process is strictly adiabatic. Rather, the system is ”heated up”. This is called ”friction effect”.
Another example is the heating of a conductor due to a driving with electromotive force (”resistance effect”).
The mathematical formulation of the latter notion of ”irreversibility” is as follows. Let Λ[xA ] be the propagator for a
system that undergoes a driving process x = xA (t). The question is whether there exist a driving process x = xB (t),
such that Λ[xA ]−1 = Λ[xB ]. This means that we can ”undo” the effect of the driving simply by ”reversing” the driving.
This is indeed the case in the strict adiabatic (quasi-static) limit.
Thus the notion of ”reversible evolution” is not in contradiction with having ”irreversible dissipation process”. The
former notion implies that the evolution can (formally) be reversed, while the latter notion implies that it is not
possible to reverse it merely by a simple manipulation of the driving scheme.
====== [9.10] Do we need many particles?
In the above formulation of irreversibility one can have in mind ”one-particle-gas” system. On the other hand, it is
quite clear that in order to ”see” irreversibility (eg in a free expansion experiment), we need ”many-particle-gas”. Do
we have a contradiction here?
The answer is that there is no contradiction. One should distinguish between the ”probability theory” framework and
the ”statistical theory” framework. Let say that we have a coin. It has 50% probability to fall face-down, and 50%
67
probability to fall face-up. The formulation of this claim concerns one coin. But obviously, if we want to ”see” the
truth in this claim, we should do statistics over many identical coins.
Thus, in the formulation of a free expansion experiment, it is essential to think about the ”state” of the particle using
a probability theory point of view. The state of the particle is represented by a probability function ρ(Q, P ). In order
to realize a free expansion experiment we need many particle, else we will have ”bad statistics” (for one particle we
will have no statistics at all).
====== [9.11] The fallacy of Gibbs paradox
In many textbooks the discussion of Gibbs paradox is associated with the indistinguishbility of particles in quantum
mechanics. This is obviously a misleading point of view. As we are going to explain Gibbs paradox has nothing to
do with this indistinguishbility. In fact there is no paradox at all...
Before we explain the mathematical side of this issue, let us emphasize the physical side. We can create a virtual
classical world (on the computer) where all the particles are distinguishable. Any ”simulation” of any chemical
reaction in this virtual world will give results that are consistent with our real-world experiments. There is no need
to define in the simulation software whether the particles are distinguishable or not.
Now to the mathematics. If the particles are distinguishable then the N -particle partition function is Z[N ] = (Z1 )N .
If they are indistinguishable then we write ZN = Z[N ] /N !. Obviously the results that we get for the entropy and for
the chemical potential are very different. Does it constitute a problem?
In order to answer this question consider a box divided into regions ”a” and ”b”. A particle A can be in either in the
”a” or in the ”b” region, which can be regarded formally as a chemical equilibrium A[a] A[b]. The probability to
have n particles in region ”a” and N − n particles in region ”b” is
n
p(n) = CN
a
b
Z[n]
Z[N
−n]
ab
Z[N
]
=
b
Zna ZN
−n
ab
ZN
(518)
The first equality reflects the point of view that the particles are ”distinguishable”, while the second equality reflects
the point of view that they are ”indistinguishable”. Obviously the two points of view are equivalent.
The usual derivation of the ”law of mass action” is obtained by looking for p(n) = maximum. Using the point of view
that the particles are indistinguishable we get the equation µa (n) = µb (N − n), where µ is the chemical potential,
and from it we get the ”law of mass action” for this particular ”reaction”. If we insist on the non-conventional point
of view, which regards the particles as ”distinguishable”, then we get different expressions for µ, and also a more
complicated version of chemical equilibrium condition. But the final result (the law of mass action) is of course the
same.
====== [9.12] entropy of many particle gas
It follows from the previous section that the indistinguishbility of particles is not an issue in the analysis of chemical
equilibrium. In fact it is also not an issue in the definition of entropy. whether there is an entropy increase due to the
mixing of two ”identical gases”, depends on our practical ability to make a distinction between the particles of the
two gases.
The mathematical formulation of the above statement is as follows: We should specify in advance the set of observables
{O} which define the ”coarse grain similarity”. The usual assumption is that we are not able to distinguish between
particles. This implies, in case of ”many-particle-gas”, a relatively short ergodic time. However, if we insist that it is
possible to monitor the motion of individual particles, then the ergodic time is formally extremely large: It becomes
larger than the time that it takes to ”diffuse” all the way from one side of the container to the other side of the
container.
It is important to realize that both points of view are correct. The determination whether there is an entropy increase,
or what is the ergodic time, depends on pre-specified assumption regarding our ability to distinguish between different
states of the system. Different definitions of ”coarse-grain similarity” lead to different (but still meaningful) results
for the entropy increase and for the ergodic time.
68
====== [9.13] Is the entropy extensive quantity?
The Shanon entropy S0 [] is in general not an ”extensive” quantity. If we prepare independently two subsystems A
and B, then the Shanon entropy of AB satisfies S0 [AB] = S0 [A] + S0 [B]. The converse is not true. Say that AB is a
singlet state of two spin 1/2 particles. In such case S0 [AB] = 0 while S0 [A] = S0 [B] = ln(2). I think that in general
it can be proved that S0 [A] + S0 [B] ≥ S0 [AB].
The ”thermodynamic entropy” is an extensive quantity. The term ”thermodynamic” hides extra assumptions. In
the thermodynamic context it is assumed that the state of any system is ”canonical”. This brings us back to the
discussion of ”Gibbs paradox”. Consider the mixing of two ”identical gases”. If the particles are indistinguishable, we
have no entropy increase, and the entropy after the mixing equals the total entropy before the mixing. If the particles
are distinguishable, then there is entropy increase: In such case the entropy after the mixing is larger than the total
entropy prior to the mixing. It is important to realize that in the latter case the final state of the system is quite
different from the initial state, whereas in the former case there is ”coarse-grain” similarity.
69
[10] Work and Heat
====== [10.1] Irreversibility and Dissipation
Assume an isolated system with Hamiltonian H(X), where X is a set of control parameters that determine the “fields”.
For simplicity assume that at t = 0 the system is in a stationary state. A driving process means that X = X(t)
is changed in time. In particular a cycle means that X(tfinal ) = X(t=0). A driving process is called reversible is we
can undo it. In the latter case the combined process (including the ”undo”) is a closed cycle, such that at the end
of the cycle the system is back in its initial state. Generally speaking a driving cycle becomes reversible only in the
adiabatic limit. Otherwise it is irreversible.
One should not confuse reversibility with micro-reversibility. The latter term implies that the mechanical evolution
has time reversal symmetry (TRS). This TRS implies that if we could reverse that state of the system at some
moment (and also the magnetic field if exists) then ideally the system would come back to its initial state. This
is called Lodschmit Echo. In general it is impossible to reverse the sate of the system, and therefore in general
micro-reversibility does not imply reversibility!
The irreversibility of typical systems is related to chaos. The simplest example is free expansion. In this example X
is the location of a piston. At t = 0 the system is prepared in an ergodic state, say a microcanonical state on the
energy surface H(XA ) = E. The piston is moved outwards abruptly form XA to XB . After some time of ergodization
the system will become ergodic on H(XB ) = E. There is no way to reverse this process.
The more interesting scenario from our point of view is a slow cycle. Using the the assumption of chaos it can
be argued that at the end of the cycle the state will occupy a√shell around H(XA ) = E. If the system is driven
periodically (many cycles), the thickness of this shell grows like DE t with DE ∝ Ẋ 2 . This diffusion in energy space
implies (with some further argumentation) monotonic increase of the average energy. Thus irreversibility implies
dissipation of energy: The system is heated up on the expense of the work which is being done by the driving source.
Below we discuss both the strict adiabatic limit (which is reversible), and also the case of slow non-adiabatic driving.
In the latter case linear response theory allows to calculate the rate of energy absorption.
====== [10.2] The calculation of work
Consider some Hamiltonian, for example of a particle (or particles) that is (are) confined by a potential
H = H(r, p; X)
(519)
where X is some parameter. We define the generalized force F which is associated with the parameter X as
F = −
∂H
∂X
(520)
Define the energy of the system as
E = hHi = trace(Hρ)
(521)
If we change X in time then from
dE
=
dt
∂H
∂t
= −hFit Ẋ
(522)
it follows that
Z
W
≡ work done on the system = Efinal − Einitial = −
hF it dX
(523)
70
This is an exact expression. Note that hFit is calculated for the time dependent (evolving) state of the system. Within
he frame work of linear response theory
hFit ≈ hFiX − η Ẋ
(524)
The first terms is the conservative force, which is a function of X alone. The subscript implies that the expectation
value is taken with respect to the instantaneous adiabatic state. The second term is the leading correction to the
adiabatic approximation. It is the “friction” force which is proportional to the rate of the driving. The net conservative
work is zero for a closed cycle while the “friction” leads to irreversible dissipation of energy with a rate
Ẇirreversible = η Ẋ 2
(525)
More generally it is customary to write
W
= −W + Wirreversible
(526)
where the first term is the conservative work, or so to say “the work which is done by the system”
Z
W
=
Z
hFiX dX
XB
=
y(X) dX
(527)
XA
where in the last equality we have defined the state function y = hFiX , so as to have a more tangible expression.
====== [10.3] Examples
Examples for generalized forces are:
parameter
generalized force
wall displacement (or change of volume) Newtonian force (or pressure)
homogeneous electric field
total polarization
homogeneous magnetic field
total magnetization
magnetic flux through a ring
electric current
The last item deserves some more explanation. First we note that the current in a ring (times the area) is like
magnetization. The direct identification of hFi as the current is based on the following argumentation: If we make a
change dX of the flux during a time dt, then the electro motive force (EMF) is −dX/dt, leading to a current I in the
ring. The energy increase of the ring is the EMF times the charge, namely dE = (−dX/dt) × (Idt) = −IdX.
Example 1:
X
= position of a wall element (or scatterer)
Ẋ = wall (or scatterer) velocity
hFi = Newtonian force
−η Ẋ = friction law
η Ẋ 2 = rate of heating
71
Example 2:
X = magnetic flux through the ring
−Ẋ = electro motive force
hFi = electrical current
−η Ẋ = Ohm law
η Ẋ 2 = Joule law
x(t)
x(t)
====== [10.4] Definition of heat
If we distinguish between system and environment that we can define the notion of heat flow. The total Hamiltonian
is
Htotal = H(r, p; X(t)) + Hint + Henv
(528)
The environment is characterized by its temperature. It is also convenient to assume that both the initial and the
final states are ”ergodic”. This means that at the end of the driving process there is an extra waiting period that
allows the system to equilibrate with the bath. It is implied that both the initial and the final states of the system
72
are canonical. Now we define
hHtotal iB − hHtotal iA
Q = heat ≡ − hHenv iB − hHenv iA
(529)
Efinal − Einitial ≡ hHiB − hHiA = Q + W
(531)
W = work ≡
(530)
It is important to emphasize that the definition of work is the same as in the previous section, because we regard
Htotal as describing an isolated driven system. However, E is redefined as the energy of the system only, and therefore
we have the additional term Q in the last equation.
====== [10.5] Quasi static process
In general we have the formal identity:
dE
X
=
dpr Er +
X
pr dEr
(532)
r
We would like to argue that in the adiabatic limit we can identify the first term as the heat dQ and the second term
is the work −dW . One possible scenario is having no driving. Still we have control over the temperature of the
environment. Assuming a quasi-static process we have
dX = 0
X
dE =
dpr Er = T dS
(533)
dW = 0
dQ = T dS
(535)
(536)
(534)
A second possible scenario is having an isolated systems going through an adiabatic process:
dpr = 0
X
pr dEr = −ydX
dE =
(537)
(538)
r
dQ = 0
dW = ydX
(539)
(540)
Any general quasi-static process can be constructed from small steps as above, leading to
Z
B
Q =
T dS
(541)
y(X)dX
(542)
A
Z B
W =
A
73
T
B
A
X
T
rigid
isolated
X
dW=0
dE=dQ
dQ=0
dE=dW
====== [10.6] More on heat flow
S
T
dQ = T dS
C≡T
(543)
∂S
∂T
(544)
For X = const process
Cx = T
∂S dE =
∂T X
dT X
> 0
(545)
74
====== [10.7] More on work
y
x
En
X
dW = ydX
χ≡
∂y
∂X
(546)
(547)
Assuming the levels are ”crowded” 0 < dX in adiabatic process, causes lower temperature, 0 < dX in isothermal
process, causes heat absorption
75
====== [10.8] cooling/work cycle
X
dQ2
T2
S=S 1
S=S 2
T1
dQ1
X
dQ1 = T1 ∆S
(548)
dQ2 = −T2 ∆S
(549)
levels getting closer ⇒ the system absorbs heat (dQ1 > 0)
levels going up ⇒ work should be invested (dW > 0)
levels getting apart ⇒ the system emits heat (dQ2 < 0)
levels going down ⇒ the system is doing work (dW < 0)
76
[11] Fluctuations and Dissipation
In order to know the expectation value of an operator we need only spectral information which is present in g(E, X)
or in Z(β, X). Note that these functions contains only spectral information about the system (no information on
the dynamics). Still it is enough for the calculation of the conservative force. For example, in case of a canonical
preparation
X ∂En ∂H
1 ∂ ln(Z)
−
=
pn −
=
∂x
∂x
β ∂X
n
hFi0 =
(550)
In contrast to that, the fluctuations of F − hFi0 requires knowledge of the dynamics, and cannot be calculated from
the partition function.
The essence of the fluctuation dissipation relation is to relate the response to the fluctuations in equilibrium. At first
stage we would like to cite its simplest version, which is obtained for one-parameter DC driving of a system that had
been prepared in a canonical equilibrium. This version is also known as Nyquist formula. Later on we shall discuss
various generalizations.
====== [11.1] The notion of power spectrum
In this section we discuss the mathematical notion of fluctuations. Consider a stationary stochastic classical variable
y(t), and define its correlation function as
C(τ ) = hy(τ )y(0)i
(551)
The power spectrum C̃(ω) is the Fourier transform of C(τ ).
In practice a realization of y() within time interval 0 < t0 < t can be Fourier analyzed as follows:
t
Z
0
y(t0 )eiωt dt0
yω =
(552)
0
and we get
h|yω |2 i = C̃(ω) × t
(553)
where we assume that t is much larger compared with the correlation time.
====== [11.2] The notion of linear response
Let us assume that X(t) is an input signal, while y(t) is the output signal of some black box. Linear response means
that the two are related by
Z
∞
y(t) =
α(t − t0 ) X(t0 ) dt0
(554)
−∞
The response kernel α(t − t0 ) can be interpreted as the output signal that follows a δ(t − t0 ) input signal. We assume
a causal relationship, meaning that α(τ ) = 0 for τ < 0.
The linear relation above can be written in terms of Fourier components as:
yω = χ(ω)Xω
(555)
77
where χ(ω) is called the generalized susceptibility. Because of causality χ(ω) is analytic in the upper complex plane.
Consequently its real and imaginary parts are inter-related by the Hilbert transform:
Z
∞
Re[χ(ω)] =
−∞
Im[χ(ω 0 )] dω 0
ω0 − ω
π
(556)
(the reverse Hilbert transform goes with an opposite sign).
The imaginary part of χ(ω) is the sine transforms of α(τ ), and therefore it is proportional to ω for small frequencies.
Consequently it is useful to define
∞
Z
χ0 (ω) ≡ Re[χ(ω)] =
α(τ ) cos(ωτ )dτ
(557)
0
µ(ω) ≡
Im[χ(ω)]
=
ω
Z
∞
α(τ )
0
sin(ωτ )
dτ
ω
(558)
and to write
yω = χ0 (ω)Xω − µ(ω)Ẋω
(559)
====== [11.3] The fluctuation dissipation relation
Coming back to the physical problem let us assume that there is a linear response relation between δX(t) = X(t) − X0
and hFit − hFi0 . For DC driving we regard χ0 and µ as constants (the latter we call η) and write
hFit = hFi0 + χ0 (X − X0 ) − η Ẋ
= hFiX − η Ẋ
(560)
The in-phase response gives the conservative effect, and it has been absorbed into the first term. The out-of-phase
response gives the dissipation. The Nyquist formula expresses the dissipation coefficient as follows:
η =
1
νT
2T
(561)
The intensity of the fluctuations at temperature T is defined as
Z
∞
hF(τ )F(0)iT dτ
νT =
(562)
−∞
Note that in the above writing it is convenient to assume that the equilibrium value of the fluctuating force is hFiT = 0,
else F should be re-defined so as to have a zero average.
Applications: Referring to the two examples in Section II we have the following consequences: (1) If a moving
object experiences friction −η Ẋ then it is associated with experiencing a fluctuating force with intensity ν = 2ηT .
(2) If we have a closed circuit with resistor whose conductance is G = η, then the fluctuations of the current are
characterized by intensity ν = 2GT . This is called thermal Nyquist noise.
78
====== [11.4] The phase-space formulation of the FD relation
The illuminating derivation of the FD relation is based on the observation that energy absorption is related to having
diffusion in energy space. To simplify the presentation we use below classical language.
We can deduce the diffusion in energy space from the relation
Z
t
F(t0 )dt0
E(t) − E(0) = −Ẋ
(563)
0
leading to
h(E(t) − E(0))2 i = Ẋ 2
Z tZ
0
t
hF(t0 )F(t00 )idt0 dt00
(564)
0
where the averaging assumes a microcanonical preparation. Thus we get
δE 2 (t) = 2DE t
(565)
where the leading order estimate for the diffusion is
DE
=
1 2
Ẋ
2
Z
∞
hF(τ )F(0)iE dτ
=
−∞
1
νE Ẋ 2
2
(566)
On long times we assume that the probability distribution ρ(E) = g(E)f (E) of the energy satisfies the following
diffusion equation:
∂ρ
∂
=
∂t
∂E
g(E)DE
∂
∂E
1
ρ
g(E)
The energy of the system is hHi =
Ẇ =
d
hHi = −
dt
Z
R
Eρ(E)dE. It follows that the rate of energy absorption is
∞
dE g(E) DE
0
(567)
∂
∂E
ρ(E)
g(E)
(568)
For a microcanonical preparation ρ(E) = δ(E − E) we get
Ẇ =
d
1 d
hHi =
[g(E) DE ] dt
g(E) dE
E=E
(569)
This diffusion-dissipation relation reduces immediately to the microcanonical version of the fluctuation-dissipation
relation:
η =
1 1 d
[g(E)νE ]
2 g(E) dE
(570)
The canonical result is obtained if we use ρ(E) = (1/Z)g(E)e−βE . It is also useful to mention that if we have a Fermi
occupation of non-interacting particles, then we can relate the dissipation to the one-particle calculation as follows:
ηtotal =
1
g(EF ) νEF
2
(571)
This is obtained by re-interpretation of f (E) as the Fermi occupation function (with total occupation N ), rather than
probability distribution.
79
====== [11.5] The Einstein-Langevin route to the FD relation
In this approach one assumes that F acts on a test particle, leading to stochastic dynamics which is described by
Langeving equation mẍ = F. The FD relation is deduced from the requirement of having thermal equilibrium as the
long time limit.
It is convenient to isolate the average (= “friction”) term from F, and accordingly to redefine F as a stochastic
variable (= “noise”) that has zero average. Consequently the Langevin equation is written as
mẍ = −η ẋ + F(t)
(572)
where F is a stochastic variable that satisfies
hF(t)i = 0
hF(t2 )F(t1 )i = C(t2 − t1 )
(573)
(574)
It is assumed that C(τ ) has a short correlation time. We are interested in the dynamics over larger time scales (we
have no interest to resolve the dynamics over very short times). We also note that if F were a constant force, then
the particle would drift with velocity (1/eta)F. The coefficient (1/η) is called mobility.
The equation for the velocity v = ẋ can be written as
d (η/m)t
1 (η/m)t
e
v(t) =
e
F(t)
dt
m
(575)
leading to the solution
v(t) =
1
m
Z
t
0
dt0 e−(η/m)(t−t ) F(t0 )
(576)
−∞
We assume above that t τη where τη = m/η is the damping time. This means that the initial velocity at t = has be
already “forgotten”, and thus the average velocity is zero. We turn now to calculate the velocity-velocity correlation
for this steady state solution:
1
hv(t2 )v(t1 )i = 2
m
Z
t1
−∞
t2
Z
0
00
dt0 dt00 e−(η/m)(t1 +t2 −t −t ) C(t0 − t00 )
(577)
−∞
we can treat C(t0 − t00 ) like a delta function. Then it is not difficult to find that
hv(t2 )v(t1 )i =
1 −(η/m)|t2 −t1 |
e
2ηm
Z
∞
C(τ )dτ =
−∞
1
m
ν
2η
e−|t2 −t1 |/τη
(578)
The displacement x(t) − x(0) of the particle is the integral over its velocity v(t0 ). On the average it is zero, but the
second moment is
Z tZ
2
h(x(t) − x(0)) i =
0
t
dt0 dt00 hv(t00 )v(t0 )i =
0
ν
× t ≡ 2Dt
η2
(579)
Hence we have diffusion in space. From the above we deduce the Einstein relation
D
mobility
=
ν
2η
= Temperature
(580)
The last equality is in fact an FD relation. It is deduced from the velocity velocity correlation function which should
coincide for t1 = t2 = t with the canonical result h 21 mv 2 i = 12 T .
80
[12] Quantum fluctuations (I)
====== [12.1] Fluctuations of a single observable
The non-symmetrized correlation function of F is defined as
S(τ ) = hF(τ )F(0)i0
(581)
This function is complex, but its Fourier transform is real and can be calculated as follows:
Z
∞
S̃(ω) =
C(τ )e
iωτ
dτ =
−∞
X
n
pn
X
m
Em − En
|Fmn | 2πδ ω −
~
2
(582)
By this convention ω > 0 corresponds to absorption of energy (upward transitions), while ω < 0 corresponds to
emission (downward transitions). It is a straightforward algebra to show that for a canonical preparations there is a
detailed balance relation:
S̃(ω) = S̃(−ω) exp
~ω
T
(583)
This implies that if we couple to the system another test system (e.g. a two level “thermometer”) it would be driven
by the fluctuations into a canonical state with the same temperature.
====== [12.2] Fluctuations of several observables
Given an Hamiltonian H(X) that depends on several control parameters Xj we define a set of generalized forces Fj .
Assuming that the system is prepared in a stationary state the fluctuations of the generalized forces are characterized
by
S kj (τ ) = hF k (τ )F j (0)i
(584)
We use the common interaction picture notation F k (τ ) = eiHt F k e−iHt , where H = H(X) with X = const. The
expectation value assumes that the system is prepared in a stationary state. It is also implicitly assumed that the
result is not sensitive to the exact value of X. It is convenient to write S kj (τ ) as the sum
~
S kj (τ ) = C kj (τ ) − i K kj (τ )
2
(585)
where
E
1D k
F (τ )F j (0) + F j (0)F k (τ )
2
E
iD k
[F (τ ), F j (0)]
~
C kj (τ ) =
K kj (τ ) =
(586)
(587)
kj
kj
It is important to realize that both kernels have a good classical
We
limit.
∗ use the notations S̃ (ω),kjand C̃ (ω),kjand
jk
kj
kj
K̃ (ω) for their Fourier transforms. It follows from S (τ ) = S (−τ ) , and from the reality of C (τ ) and K (τ )
that the following reciprocity relations hold:
S̃ kj (ω) =
h
i∗
S̃ jk (ω)
C̃ kj (ω) = C̃ jk (−ω)
kj
jk
K̃ (ω) = −K̃ (−ω)
(588)
(589)
(590)
81
====== [12.3] The Onsager reciprocity relations
For systems where time reversal symmetry is broken due to the presence of a magnetic field B, we can formulate some
reciprocity relations. First we mention that by definition we have the following trivial identities:
C kj (−τ, B) = +C jk (τ, B)
K kj (−τ, B) = −K jk (τ, B)
(591)
(592)
Taking time reversal into account we have also
C kj (τ, −B) = +[±]C kj (−τ, B)
K kj (τ, −B) = −[±]K kj (−τ, B)
(593)
(594)
where the plus (minus) applies if the signs of F k and F j transform (not) in the same way under time reversal.
Combining the the above equalities we get
C kj (τ, −B) = [±]C jk (τ, B)
K kj (τ, −B) = [±]K jk (τ, B)
(595)
(596)
The Kubo formula that we discuss in the next section implies that the same reciprocity relations applies to the
response kernel αkj , to the susceptibility χkj and to the DC conductance Gkj . These are called Onsager reciprocity
relations Note that in the absence of magnetic field the elements of the conductance matrix have definite symmetry,
meaning that each of them belongs either to the symmetric part η kj or to the antisymmetric part B kj .
====== [12.4] Generalized Detailed balance relations
It is easily verified that
S̃ kj (ω) =
X
n
pn
X
m(6=n)
Em − En
k
j
Fnm
Fmn
2πδ ω −
~
(597)
It is convenient to write pn = f (En ). The corresponding expressions for C̃ kj (ω) and K̃ kj (ω) contain two terms due to
the non-comutativity of the operators. We can simplify these expressions by interchanging the n,m dummy indexes
in the second term. Thus we get
X f (En ) − f (Em ) X
Em −En
K̃ (ω) = −iω × 2π
t
ω−
E
−
E
~
n
m
n
m
X
X
Em −En
k
j
C̃ kj (ω) = π
(f (En ) + f (Em ))
Fnm
Fmn
δ ω−
t
~
m
n
kj
k
j
Fnm
Fmn
δ
(598)
(599)
Assuming canonical equilibrium we can prove the additional relation
kj
jk
S̃ (ω) = S̃ (−ω) exp
~ω
T
(600)
or the equivalent relation
2
K̃ (ω) = iω ×
tanh
~ω
kj
~ω
2T
C kj (ω)
(601)
82
Note that for a canonical state
f (En )−f (Em ) = tanh((En −Em )/(2T )) × (f (En )+f (Em ))
(602)
The Kubo formula that we discuss in the next section is expressed using K kj (τ ). But it is more convenient to
use C kj (τ ). The above relation between the two is the basis for the FD relation and its generalizations.
====== [12.5] Fluctuations of a bath
The eigenstates of the ”free” environment are n. In particular in case of a ”bath” we use the standard Fock basis:
|ni = |{nα }i = |n1 , n2 , n3 , ...i
(603)
with eigen-energies
En =
X
ω α nα
(604)
α
The energy spectrum is characterized by the smoothed density of states
g(E) =
X
δ(E − En )
(605)
n
We assume that the bath is prepared in a stationary state, and define its equilibrium temperature as
1
TEQLB
=
d
ln(g(E))
dE
(606)
The interaction of a system with the bath, for small x displacements, can be written as:
Htotal = = H0 (x, p) + E + xB
(607)
A matrix element Bnm is non-zero only for “one-photon” excitations. For such excitations |Em −En | = ~ωα Consequently Bnm is a sparse banded matrix, with
bandwidth = ~ωc
(608)
Once we know the system-bath Hamiltonian we can calculate the fluctuations spectrum:
*
S̃(ω) =
X
m(6=n)
Em − En
|Bmn |2 2πδ ω −
~
+
(609)
where the average is over n. The average can be either microcanonical or canonical, accordingly we sometimes use
the notations S̃E (ω) or S̃T (ω) respectively.
We note that in case of interaction with chaos there is a non-universal scale δx = ` beyond which the mixing is no
longer growing. This is in analogy with the DLD model. Note that the results of the RMT analysis is in agreement
with DLD. Still it is not clear if S(q, ω) can be defined in general if the interaction cannot be treated as a weak
perturbation.
83
====== [12.6] The fluctuations of an Ohmic bath
Sometimes it is convenient to characterize the system by its response, and from this to deduce the power spectrum of
the fluctuations. So we regard K̃(ω) as the input and write
~
C̃(ω) =
coth
2
~ω
2T
h
i
Im K̃(ω)
C̃(ω)
exp
cosh(~ω/2T )
S̃(ω) =
~ω
2T
(610)
(611)
The so called Ohmic power spectrum is
(612)
K̃ohmic (ω) = i2ηω
C̃ohmic (ω) = η~ω coth
S̃ohmic (ω) = η~ω
~ω
2T
(613)
2
1 − e−~ω/T
(614)
for which the friction coefficient is µ = η = const, and the intensity of fluctuations is ν = 2ηT .
====== [12.7] The Ohmic harmonic bath
We define F
F=
X
cα Qα =
X
α
α
cα
~
2mα ωα
1/2
(aα + a†α )
(615)
The asymmetrized correlation function of F is denoted S(τ ) while the symmetrized one id denoted C(τ ). Their FTs
S̃(ω) and C̃(ω) describe the power spectrum of the fluctuations. For preparaion in state n
S̃(ω) =
XX
α
c2α |hnα ±1|Qα |nα i|2 2πδ(ω ∓ ωα )
(616)
±
Using
hnα −1|Qα |nα i =
hnα +1|Qα |nα i =
~
2mα ωα
~
2mα ωα
1/2
1/2
√
√
nα
(617)
1 + nα
(618)
we get
S̃(ω) =
X
α
h
i
~
2πc2α (1+nα )δ(ω − ωα ) + nα δ(ω + ωα )
2mα ωα
(619)
For canonical preparation hnα i = f (ωα ) where (from here on ~ = 1)
1
eβω − 1
1
f (−ω) = −
= −(1 + f (ω))
1 − e−βω
f (ω) =
(620)
(621)
84
Thus we get
(
1
(1 + f (ω))
S̃(ω) = 2J(|ω|) ×
= 2J(ω)
1 − e−βω
f (−ω)
(622)
where we define
π X c2α
δ(ω − ωα )
2 α mα ωα
J(ω) =
(623)
with anti-symmetric continuation. For Ohmic bath J(ω) = ηω, with some cutoff frequency ωc .
====== [12.8] The Ohmic spin bath
We define F
F=
X
cα Qα =
α
X
cα (aα + a†α )
(624)
α
Thus Qα is the first Pauli matrix. Its non-trivial matrix elements are
√
n
hnα −1|Qα |nα i =
√ α
hnα +1|Qα |nα i =
1 − nα
(625)
(626)
In complete analogy we get
S̃(ω) =
X
h
i
2πc2α (1−nα )δ(ω − ωα ) + nα δ(ω + ωα )
(627)
α
For canonical preparation hnα i = f (ωα ) where (from here on ~ = 1)
1
+1
1
f (−ω) =
= 1 − f (ω)
1 + e−βω
f (ω) =
eβω
(628)
(629)
Thus we get
(
1
(1 − f (ω))
S̃(ω) = 2J(|ω|) ×
= 2J(ω)
1 + e−βω
f (−ω)
(630)
C̃(ω) = J(ω)
(631)
and
where we define
J(ω) = π
X
c2α δ(ω − ωα )
α
with symmetric continuation. For Ohmic bath J(ω) = ν, with some cutoff frequency ωc .
(632)
85
====== [12.9] Chaotic environment
H = H(Q, P ; x)
∂H
F(t) = −
(Q(t), P (t); x)
∂x
(633)
(634)
The correlation function:
SE (τ ) = hF(τ )F(0)iE
(635)
The power spectrum of the fluctuations (assuming preparation in the nth state):
S̃E (ω) =
X
m
Em −En
2
|Fmn | 2πδ ω −
= g(E+ω) σ(E+ω|E)2
~
(636)
One can define TFLCT , and we can verify the validity of the FD relation. It is convenient to characterize the fluctuations
by the symmetrized correlation function, and to regard the temperature as an independent parameter.
2π~σ 2
S̃(ω) = 2πσ δ(ω) +
R
∆
2
~ω
ω
~ω
G
exp
∆
ωcl
2TFLCT
(637)
where the first term reflects static noise (correlation time infinite) due to the RMS of the diagonal matrix elements.
Conside the AB interferene paradigm. Assume that one arm is in a mixture state. So |1 + eiφ |2 should be averaged
over the non dynamical φ. This type of depasing does not diminish time reversal symmetry, hence weak localization
effect survives.
G() = semiclassical envelope (bandprofile)
R() = lower cutoff function (level repulsion)
∆b =
~
= ~ωc = bandwidth
τcl
(638)
86
[13] Quantum fluctuations (II)
In this section we describe fluctuations in space using the form factor S̃(q, ω). Then we discuss the fluctuations of the
current in a conductor and the associated fluctuations of the electrostatic potential in a bulk metal.
====== [13.1] The form factor
Form theoretical point of view is is reasonable to characterize the fluctuations, and then to get from Kubo/FD the
dissipation coefficient. We define
h
i
S(q, ω) = FT hU(x2 , Q(t2 ))U(x1 , Q(t1 ))i
(639)
where the expectation value assumes that the bath is in a stationary state of its unperturbed Hamiltonian. The forceforce correlation function is obtained via differentiation. In particular the local power spectrum of the fluctuating
force is
Z
S̃(ω) =
dq 2
q S(q, ω)
2π
(640)
and the intensity of the fluctuations at a given point in space is
Z
ν ≡ S̃(ω=0) =
dq 2
q S(q, ω=0)
2π
(641)
We split S(ω) into symmetric and antisymmetric part
~
S̃(ω) = C̃(ω) − i K̃(ω)
2
(642)
====== [13.2] The form factor for the DLD/ZCL models
In the standard one dimensional DLD model we assume
hU(x2 , Q(t2 ))U(x1 , Q(t1 ))i = w(x2 − x1 ) S(t2 − t1 )
(643)
such that
S(q, ω) = w̃(q)S̃(ω)
(644)
Note that this way of writing implies w00 (0) = −1 else S(τ ) would not be the force-force correlation function. For
example we may assume
1 r 2
w(r) = ` exp −
2 `
2
(645)
If the spatial correlation distance is very large we get ZCL model:
1
w(r) = const − r2
2
(646)
87
leading to
S(q, ω) =
2π
δ(q) S̃(ω)
q2
(647)
This means the the power spectrum of the force(!) is is δ(q), which means that the force-force correlation function
does not decay in space.
====== [13.3] DLD bath
UDLD = −
X
cα Qα u(x−xα )
(648)
α
Taking into account that the oscillators are independent of each other we get
hU(x2 , t2 )U(x1 , t1 )i =
X
c2α hQα (t2 )Qα (t1 )iu(x2 −xα )u(x1 −xα )
(649)
α
"
Z
=
dx
#
X
c2α hQα (t2 )Qα (t1 )iδ(x
− xα ) u(x2 −x)u(x1 −x)
(650)
α
Z
= SZCL (t2 − t1 )
u(x2 −x)u(x1 −x) dx
= w(x2 − x1 ) SZCL (t2 − t1 )
(651)
(652)
Where we have assumed homogeneous distribution of the oscillators, and SZCL (τ ) is defined implicitly by the above
equality. With the convention w00 (0) = −1 it can be identified as the local force-force correlation function. Consequently we get
h
i
S(q, ω) = FT hU(x2 , Q(t2 ))U(x1 , Q(t1 ))i = w̃(q) SZCL (ω)
(653)
====== [13.4] The fluctuations of the current in a ring
For one particle we can define the microcanonical fluctuation spectrum CE (ω) and the canonical fluctuation spectrum
CT (ω). If we have many body occupation by N particles, we can express the additive K(ω) using the one-particle
correlation functions. (The commutator is additive and does not care about the Pauli exclusion principle). We can
refer to it as Einstein relations:
(
g(EF ) C̃EF (ω)
K̃(ω) = iω ×
(N/T ) C̃T (ω)
degenerated Fermi occupation
dilute Boltzman occupation
(654)
Once we know K̃(ω) we can calculate the dissipation coefficient using the Kubo formula:
µ(ω) =
1
K̃(ω)
i2ω
(655)
For hard chaos one can use the white noise approximation leading to µ(ω) ≈ const ≡ G. The fluctuations of the many
body current are deduced from the canonical FD relation:
~
C̃(ω) =
coth
2
~ω
2T
(
i
~ω
~ω
(T /∆) C̃EF (ω)
Im K̃(ω) =
coth
2T
2T
N C̃T (ω)
h
(656)
88
Note that for the non-symmetryized power spectrum S̃(ω) we have the same expression with
coth
~ω
2T
7−→
2
1 − e−~ω/T
(657)
It is important to observe that the intensity ν of the current fluctuations is not simply the one-particle ν times the
number of particles. Rather, for low temperature Fermi occupation the effective number of particles that contribute
to the noise is T /∆ where ∆ is the mean level spacing. Needless to say that the result for S̃(ω) can be confirmed by
a direct calculation utilizing Fock space formalism.
====== [13.5] The fluctuations of the electrostatic potential
So far we have ignore the spatial aspect of the fluctuations. For a metal we can use FD realation in order to relate
their spati-temporal power spectrum S(q, ω) to the conductivity:
S(q, ω) =
e2 1
~ω coth
σ q2
~ω
2T
for ω <
1
1
, |q| <
τc
`
(658)
or in case of the asymetrized version:
coth
~ω
2T
7→
2
1 − e−~ω/T
(659)
The ohmic behavior is cut-off by the Drude collision frequency 1/τc , and the elastic mean free path is ` = vF τc , where
vF is the Fermi velocity.
The details of the derivation are as follows. We have to figure out the response of the electrostatic potential U to an
external perturbation Hint = ρext U . Coulomb law is
U=
1 4πe2
ρext
ε(q, ω) q 2
(660)
where ε(q, ω) is the dielectric constant. Hence from the FD theorem we get
S(q, ω) =
4πe2
−1
~ω
Im
~
coth
2
q
ε(q, ω)
2T
(661)
Given the conductivity the dielectric constant is found as follows. From ∂ρ/∂t = −∇J and J = −σ∇U − D∇ρ
we deduce (iω − Dq 2 )ρ = σq 2 U . This we substitute into ∇2 U = −4π(ρ + ρext ) to get q 2 U = (4π/ε)ρext with the
idntification
ε(q, ω) = 1 +
4πσ
−iω + Dq 2
(662)
which gives
4πσω
1
=−
Im
2
ε(q, ω)
(Dq + 4πσ)2 + ω 2
(663)
89
[14] Linear response theory
====== [14.1] The adiabatic picture in brief
We find the adiabatic basis |n(X)i with eigenenergies En (X) and define
Ajnm
∂
= i~ n(X) m(X)
∂Xj
Bnij = ∂i Ajn − ∂j Ain
(664)
(665)
We use the notation Ajn = Ajnn , and note the following identities:
Ajnm
=
Bnij
=
j
−i~Fmn
[n 6= m]
Em −En
i
j
X 2~Im Fnm Fmn
(Em − En )2
(666)
(667)
m(6=n)
If we have 3 control variables it is convenient to use the notations:
~ −
X
7 → (X1 , X2 , X3 )
~n −
A
7 → (A1nn , A2nn , A3nn )
~ n 7−→ (B 23 , B 31 , B 12 )
B
(668)
(669)
(670)
~ is the rotor of A.
~
such that B
The Schrodinger equation is
d
i
|ψi = − H(X(t)) |ψi
dt
~
(671)
We expand the state as
|ψi =
X
an (t) |n(X(t))i
(672)
n
and get the equation
dan
i
iX
= − (En −Ẋ · An )an −
Wnm am
dt
~
~ m
(673)
where
Wnm ≡ −
X
Ẋj Ajnm
for n6=m, else zero
(674)
j
In the adiabatic limit the first order response of a system that has been prepared in the nth adiabatic state is
hF k i = −
X
Bnkj Ẋj
(675)
j
This corresponds to the geometric part of the response in the Kubo formula. The Kubo formula contains an additional
non-adiabatic (dissipative) term that reflects FGR transitions between levels.
90
====== [14.2] The Kubo formula
Given an Hamiltonian that depends on several control parameters
H = H(r, p; X1 (t), X2 (t), X3 (t))
(676)
we define generalized forces in the usual way:
Fk = −
∂H
∂xk
(677)
The expectation values hF k i is related to the driving Xj (t) by the causal response kernel αkj (t − t0 ).
k
hF it =
XZ
∞
αkj (t − t0 ) δXj (t0 )dt0
(678)
−∞
j
The Kubo expression for this response kernel is
αkj (τ ) = Θ(τ ) K kj (τ )
(679)
The Fourier transform of αkj (τ ) is the generalized susceptibility χkj (ω). It has analytic properties which are implied
by causality. Useful expressions are:
χkj (ω) =
Z
∞
−∞
iK̃ kj (ω 0 ) dω
ω − ω 0 + i0 2π
=
X
pn
X
m
n
k
j
j
k
−Fnm
Fmn
Fnm
Fmn
+
~ω−(Em −En )+i0 ~ω+(Em −En )+i0
(680)
For one parameter driving K(ω) is pure imaginary and we get
µ(ω) ≡
Im[χ(ω)]
ω
=
1
K(ω) ≡
i2ω
1
[S(ω) − S(−ω)]
2~ω
(681)
This result can be deduced directly from the FGR picture. For the purpose of such derivation one assumes that the
system is driven by noisy field whose power spectrum is |Xω |2 . Then one expresses the net energy absorption as the
difference of upward and downward transitions.
====== [14.3] The canonical FD relation
It is possible to relate K̃ kj (ω) to C̃ kj (ω) provided extra information is give about the preparation. We recall that for
canonical state
2
K̃ (ω) = iω ×
tanh
~ω
kj
~ω
2T
C kj (ω)
(682)
For one parameter driving we get the FD relation
µ(ω) ≡
Im[χ(ω)]
ω
=
1
K(ω) =
i2ω
1
tanh
~ω
~ω
2T
C̃(ω)
(683)
Exercise: Calculate the fluctuations of a harmonic oscillator Ω in thermal equilibrium; Deduce the velocity-velocity
correlation; Take the Ω → 0 limit. In the latter case the integral over the velocity-velocity correlation function gives
the diffusion coefficient for Brownian motion.
91
====== [14.4] The generalized conductance matrix
In the DC limit the linear relation between the generalized forces and the rate of the driving can be written as a
generalized Ohm law:
hF k i = −
X
Gkj Ẋj
(684)
j
The Kubo formula reduces to the following expression for the conductance matrix:
Im[χkj (ω)]
=
ω→0
ω
Gkj = lim
Z
∞
K kj (τ )τ dτ
(685)
0
We can split Gkj into symmetric and anti-symmetric components:
η kj
=
B kj
=
1 kj
1
Im[K̃ kj (ω)]
(G +Gjk ) =
lim
2
2 ω→0
ω
Z ∞
Re[
K̃ kj (ω)]
1 kj
dω
(G −Gjk ) = −
2
ω2
−∞ 2π
(686)
(687)
The following expressions are useful:
η kj
= −π~
X f (En ) − f (Em )
n,m
B
kj
=
X
n
f (En )
En − Em
X
m(6=n)
k
j
Fnm
Fmn
δΓ (Em − En )
k
j
2~Im Fnm
Fmn
(Em −En )2 + (Γ/2)2
=
X
n,m
(f (En ) − f (Em ))
(688)
k
j
−i~Fnm
Fmn
2
(Em − En ) + (Γ/2)2
(689)
In the above expression we have introduced a smearing scale Γ which formally replaces the infinitesimal i0 that
appears in the Kubo expression for the generalized susceptibility. This is like giving the energy levels a “width”. If
we set Γ = 0 we get the strictly adiabatic limit, where the dissipation is zero (η kj = 0). In order to get dissipation we
should assume Γ larger than the level spacing. This assumption can be justified if the driving rate is non-adiabatic,
or if there is some coupling to an external bath.
92
====== [14.5] The Born-Oppenheimer picture
Linear response theory is the leading formalism to deal with driven systems. Such systems are described by a
Hamiltonian
H = H(r, p; X(t))
(690)
where (Q, P ) is a set of canonical coordinates (in case that the Hamiltonian is the outcome of ”quantization”), and
X(t) is a set of time dependent classical parameters (”fields”). For example, X can be the position of a piston. In
such case Ẋ is its velocity. More interesting is the case where X is the magnetic flux through a ring. In such a case
Ẋ is the electro motive force. The Kubo formula allows the calculation of the response coefficients. In the mentioned
examples these are the “friction coefficient” and the “conductance of the ring” respectively.
In the limit of a very slow time variation (small Ẋ), linear response theory coincides with the “adiabatic picture”. In
this limit the response of the system can be described as a non-dissipative “geometric magnetism” effect (this term was
coined by Berry and Robbins). If we increase Ẋ beyond a certain threshold, then we get Fermi-golden-rule transitions
between levels, leading to absorption of energy (“dissipation”). Then linear response theory can be regarded as a
generalization of “Ohm law”.
The Born-Oppenheimer picture is the leading formalism to deal with Hamiltonians of the type
Htotal = H0 (X, P ) + H(r, p; X)
(691)
The standard textbook example is the study of diatomic molecules. In such case X is the distance between the nuclei.
It is evident that the theory of driven systems is a special limit of this problem, which is obtained if we treat X as a
classical variable.
Htotal =
basis:
1 X 2
p + H(Q, P ; x)
2M j j
(692)
|x, n(x)i = |xi ⊗ |n(x)i
|Ψi =
X
Ψn (x) |x, n(x)i
(693)
n,x
hx, n(x)|H|x0 , m(x0 )i = δ(x−x0 ) × δnm En (x)
(694)
hx, n(x)|pj |x0 , m(x0 )i = (−i∂j δ(x−x0 )) × hn(x)|m(x0 )i = −i∂j δ(x−x0 )δnm − δ(x−x0 )Ajnm (x)
(695)
hence: pj 7→ −i∂j − Ajnm (x).
Htotal =
1 X
(pj − Aj (x))2 + E(x)
2M j
Hinteraction ≈ −
X
j
ẋj Aj (x)
(696)
(697)
93
[15] The Kubo formula for non-interacting Fermions
====== [15.1] Generalizations of the FD relation
For canonical state we can use the relation
f (En )−f (Em ) = tanh((En −Em )/(2T )) × (f (En )+f (Em ))
(698)
while in case of Fermi occupation
1
f (En ) − f (Em ) ≈ − [δT (En − EF ) + δT (Em − EF )] × (En − Em )
2
(699)
Note that it is essential to keep the proper symmetry with respect to En ↔ Em permutation. We conclude that
kj
K̃Fkj (ω) = iω × g(E) C̃E
(ω)
F
(700)
with implicit thermal average over E ≈ EF . If the above relations hold as a global equality then we can write an
equivalent time domain relation. In case of Fermi occupation we get
KFkj (τ ) = −g(EF )
∂
C̃ kj (τ )
∂τ EF
(701)
where the temperature is regarded as regularization.
Im[χkj (ω)]
=
ω→0
ω
Gkj = lim
Z
0
∞
KFkj (τ )τ dτ = g(EF )
Z
0
∞
kj
CE
(τ )dτ
F
(702)
Note optional expressions:
Gkj = −ig(EF )
Z
∞
−∞
kj
C̃E
(ω) dω
1
kj
F
= g(EF )C̃E
(ω ∼ 0) + g(EF )
F
ω − i0 2π
2
Z
∞
−∞
kj
−iC̃E
(ω) dω
F
= η kj + B kj
ω
2π
(703)
In the absence of magnetic field C kj (τ ) has definite symmetry with respect to τ 7→ −τ . This can be exploited in order
to simplify the handling of the FD relation. A good example are Fermions in a ring. The current I transforms with
minus sign under time reversal, while all the other “forces” that are associated with the scalar potential transforms with
plus sign. Consequently the current-current and the force-force correlations are symmetric, while the force-current
correlations are antisymmetric. Respectively we get
1
kj
g(EF ) C̃E
(ω ∼ 0)
F
2
#
Z ∞ " kj
C̃EF (ω) dω
= g(EF )
=
ω
2π
−∞
Gkj
= η kj =
Gkj
= B kj
if C(τ ) is symmetric
if C(τ ) is antisymmetric
(704)
(705)
94
====== [15.2] Expressions using Green functions
Using the definitions:
i
Im[G(E)] ≡ − (G+ − G− ) = −πδ(E − H)
2
X
C kj (E 0 , E) = 2π
(706)
k
j
Fnm
δ(E 0 − Em )Fmn
δ(E − En )
(707)
nm
2
trace F k Im[G(E 0 )] F j Im[G(E)]
π
=
(708)
we can write
kj
CE
(ω) =
~ kj
C (E + ~ω, E) + C jk (E − ~ω, E)
2g(E)
1
χ (ω) =
2π
kj
η
kj
=
≈
=
=
kj
B
Z Z
f (E)dEdE
−C kj (E 0 , E)
C jk (E 0 , E)
+
~ω − (E 0 − E) + i(Γ/2) ~ω + (E 0 − E) + i(Γ/2)
Z Z
~
f (E) − f (E 0 )
dEdE 0 C kj (E 0 , E) δΓ (E 0 − E)
−
2
E − E0
1
kj
g(EF ) C̃E
(ω ∼ 0)
[FD version]
F
2
~
trace F k Im[G(EF )] F j Im[G(EF )]
π
π~ trace F k δ(EF − H) F j δ(EF − H)
Z Z
~
= −i
2π
Z
(f (E) − f (E 0 ))dEdE 0
∞
≈ g(EF )
−∞
=
0
kj
−iC̃E
(ω) dω
F
ω
2π
(709)
(710)
(711)
(712)
(713)
(714)
C kj (E 0 , E)
(E 0 − E)2 + (Γ/2)2
(715)
[FD version]
(716)
~
trace F k G+ (EF ) F j G− (EF ) − F k G− (EF ) F j G+ (EF )
4π
(717)
Expressions with Γ included should be based on
kj
kj
CE
(τ ) 7→ CE
(τ )e−(Γ/2~)τ
kj
G
Z
∞
= ~g(EF )
−∞
kj
−iC̃E
(ω) dω
F
~ω − i(Γ/2) 2π
(718)
(719)
95
====== [15.3] Kubo and Thouless
There are several ways in which the Kubo expression for the Ohmic conductance is written. The most popular version
is
G = π~ g(EF )2 |Inm |2
(720)
where the bar indicates that an average should be taken over the near diagonal matrix elements. For chaotic systems
|Inm |2 is equal, up to a symmetry factor, to the dispersion of the the matrix elements along the diagonal. Note that
Inn = −∂En /∂Φ. It is common to use the notation φ = (e/~)Φ. Hence we get the Thouless expression
e2
1
G = factor ×
× 2
~
∆
∂En 2
∂φ (721)
where the numerical factor depends on symmetry considerations, and ∆ is the mean level spacing at the Fermi energy.
There is a more refined relation by Kohn.
====== [15.4] Conductivity and Conductance
Consider a ring geometry, and assume that the current is driven by the flux Φ. In order to have a better defined
model we should specify what is the vector potential A(r) along the ring. We can regard the values
of A at different
H
points in space as independent parameters (think of tight binding model). Their sum (meaning A(r)·dr) should be
Φ. So we have to know how Φ is ”distributed” along the ring. This is not just a matter of ”gauge choice” because the
electric field E(r) = −Ȧ(r) is a gauge invariant quantity. So we have to know how the voltage is distributed along
the ring. However, as we explain below, in linear response theory this information is not really required. Any voltage
distribution that results in the same electro-motive force, will create the same current.
In linear response theory the current is proportional to the rate in which the parameters are being changed in time.
Regarding the values of A at different points in space as independent parameters the basic linear response relation
takes the form
Z
hJ(r)i =
σ(r, r 0 )E(r 0 )dr0
(722)
where σ(r, r0 ) is called the conductivity matrix. The current density has to satisfy the continuity equation ∇·hJ(r)i =
0. From here it follows that if we replace A by A + ∇Λ(r), then after integration by parts we shall get the same
current. This proves that within linear response theory the current should depend only on the electromotive force −Φ̇,
and not on the way in which the voltage is distributed. Note that A 7→ A + ∇Λ(r) is not merely a gauge change: A
gauge transformation of time dependent field requires a compensating replacement of the scalar potential (which is
not the case here).
In the following it is convenient to think of a device which is composed of ”quantum dot” with two long leads, and
to assume that the two leads are connected together as to form a ring. We shall use the notation r = (r, s), where r
is the coordinate along the ring, and s is a transverse coordinate. In particular we shall distinguish a left ”section”
r = rB and a right section r = rA of the two leads, and we shall assume that the dot region is described by a scattering
matrix Sab .
We further assume that all the voltage drop is concentrated across the section r = rB , and we measure the current IA
through the section r = rA . With these assumptions we have two pairs of conjugate variables, which are (ΦA , IA ) and
(ΦB , IB ). Note that the explicit expression for the current operator is simply
1
IA = e (v δ(r − rA ) + δ(r − rA )v)
2
(723)
where v is the r component of the velocity operator. We are interested in calculating the conductance, as define
96
through the linear response relation hIA i = −GAB Φ̇B . The Kubo expression takes the form
GAB =
~
trace [IA Im[G] IB Im[G]]
π
(724)
This is yet another version of the Kubo formula. Its advantage is that the calculation of the trace involves integration
that is effectively restricted to two planes, whereas the standard version (previous section) requires a double integration
over the whole ”bulk”.
Before we go on we recall that it is implicit that for finite system Im[G] should be ”smeared”. In the dot-leads setup
which is described above, this smearing can be achieved by assuming very long leads, and then simply ”cutting” them
apart. The outcome of this procedure is that G± is the Green function of an open system with outgoing wave (ingoing
wave) boundary conditions.
====== [15.5] From the Kubo formula to the Landauer formula
Before we go on we recall that it is implicit that for finite system Im[G] should be ”smeared”. In the dot-leads setup
which is described above, this smearing can be achieved by assuming very long leads, and then simply ”cutting” them
apart. The outcome of this procedure is that G± is the Green function of an open system with outgoing wave (ingoing
wave) boundary conditions. As customary we use a radial coordinate in order to specify locations along the lead,
namely r = ra (r), with 0 < r < ∞. We also define the channel basis as
hr, s|a, ri = χa (s) δ(r − ra (r))
(725)
The wavefunction in the lead regions can be expanded as follows:
|Ψi =
X
Ca,+ eika r + Ca,− e−ika r
|a, ri
(726)
a,r
We define projectors P + and P − that project out of the lead wavefunction the outgoing and the ingoing parts
respectfully. It follows that P + G+ = G+ , and that P − G+ = 0, and that G− P − = 0 etc. We define the operator
ΓA =
X
|a, rA i~va ha, rA |
(727)
a∈A
= δ(r − rA ) ⊗
X
|ai~va ha|
(728)
a∈A
where va = (~ka /mass) is the velocity in channel a. The matrix elements of the second term in Eq.(728) are
ΓA (s, s0 ) =
X
χa (s) ~va χ∗a (s0 )
(729)
a∈A
The operator ΓB is similarly defined for the other lead. Note that these operators commute with the projectors P ± .
It is not difficult to realize that the current operators can be written as
IA = (e/~)[−P + ΓA P + + P − ΓA P − ]
IB = (e/~)[+P + ΓB P + − P − ΓB P − ]
(730)
(731)
Upon substitution only two (equal) terms survive leading to the following version of Kubo formula:
GBA =
e2
trace ΓB G+ ΓA G−
2π~
(732)
97
There is a well known expression (Fisher-Lee) that relates the Green function between plane A and plane B to the S
matrix. Namely:
G+ (sB , sA ) = −i
X
χb (sB ) √
a,b
1
1
Sba √
χ∗ (sA )
~vb
~va a
(733)
Upon substitution we get
GBA =
e2 X X
|Sba |2
2π~
(734)
a∈A b∈B
This is the Landauer formula. Note that the sum gives the total transmission of all the open channels.
====== [15.6] From the Kubo formula to the BPT formula
It should be emphasized that the original derivations of the Landauer and the BPT formulas are based on a scattering
formalism which strictly applies only in case of an open system (= system with leads which are connected to reservoirs).
In contrast to that Kubo formula is derived for a closed system. However, it can be shown that by taking an appropriate
limit it is possible to get the BPT formula from the Kubo formula. Namely,
~
trace F k Im[G+ ] F j Im[G+ ]
π
†
~
∂S ∂S
=
trace
4π
∂xi ∂xj
η kj =
i~
trace F 3 (G+ +G− ) F j Im[G+ ]
2π
e
∂S † ∂S †
=
S −
S
+ intrf
trace PA
4πi
∂xj
∂xj
B 3j = −
(735)
(736)
(737)
(738)
So the sum is
G3j =
e
∂S †
S
trace PA
2πi
∂xj
For more details see Phys. Rev. B 68, 201303(R) (2003).
(739)
98
[16] Scattering approach to mesoscopic transport
The most popular approach to transport in mesoscopic devices takes the scattering formalism rather than the Kubo
formalism as a starting point, leading to the Landauer and the BPT formulas. We first cite these formulas and then
summarize their common derivation. This should be compared with the Kubo-based derivation of the previous section.
====== [16.1] The Büttiker-Prétre-Thomas-Landauer formula
We assume without loss of generality that there are three parameters (x1 , x2 , x3 ) over which we have external control,
where x3 = Φ is the AB flux. The expression for the current IA that goes out of lead A, assuming DC linear response,
can be written as
IA = −
X
G3j ẋj
(740)
j
where −ẋ3 = −Φ̇ is the EMF, and therefore G33 is the conductance in the usual sense. The Büttiker-Prétre-ThomasLandauer formula for the generalized conductance matrix is
G3j =
e
∂S †
trace PA
S
2πi
∂xj
(741)
In particular for the Ohmic conductance we get the Landauer formula:
G33 =
e2
trace(tt† )
2π~
(742)
In order to explain the notations in the above formulas we consider a two lead system. The S matrix in block form
is written as follows:
S=
rB tAB e−iφ
tBA eiφ
rA
!
(743)
where r and t are the so called reflection and transmission (sub) matrices respectively. We use the notation φ = eΦ/~.
In the same representation, we define the left lead and the right lead projectors:
PA =
!
0 0
,
0 1
PB =
1 0
0 0
!
(744)
The following identity is important in order to obtain the Landauer formula from the BPT formula:
dS
e
e
e
= i (PA SPB − iPB SPA ) = i (PA S − SPA ) = −i (PB S − SPB )
dΦ
~
~
~
(745)
Another important identity is
trace(PA SPB S † ) = trace(tt† ) =
XX
|tab |2
a∈A b∈B
The trace() operataion is over the channel indexes.
(746)
99
====== [16.2] Floque theory for periodically driven systems
The solution of the Schrodinger equation
i
dψ
= Hψ
dt
(747)
with time independent H is
|ψ(t)i =
X
e−iEt |ψ (E) i
(748)
E
where the stationary states are found from
H|ψ (E) i = E|ψ (E) i
(749)
Consider now the more complicated case where H depends periodically on time. Given that the basic frequency is ω
we can write
H(t) =
X
H(n) e−inωt
(750)
The solution of the Schrodinger equation is
|ψ(t)i =
∞
X X
E
e−i(E+nω)t |ψ (E,n) i
(751)
n=−∞
where the Flouqe states are found from
X
0
0
H(n−n ) |ψ (E,n ) i = (E + nω)|ψ (E,n) i
(752)
n0
and E is defined modulo ω.
====== [16.3] The Floque scattering matrix
In scattering theory we can define a Flouqe energy shell E. The solution outside of the scattering region is written as
|ψ(t)i =
∞
X
e−i(E+nω)t
n=nfloor
X
1 +ikan r
1 −ikan r
e
− Ban √
e
⊗ |ai
Aan √
van
van
a
(753)
where van and kan are determinedPby the available energy E + nω. The current in a given channel is time dependent,
but its DC component is simply n (|Ban |2 − |Aan |2 ). Therefore the continuity of the probability flow implies that
we can define an on-shell scattering matrix
Bbnb =
X
Sbnb ,ana Aana
(754)
ana
We can write this matrix using the following notation
Sbnb ,ana
nb −na
≡ Sb,a
(E + na ω)
(755)
100
Unitarity implies that
X
|Sbnb ,ana |2 =
X
bnb
X
n
|Sb,a
(E)|2 = 1
(756)
n
|Sb,a
(E + nω)|2 = 1
(757)
bn
|Sbnb ,ana |2 =
X
ana
an
If the driving is very slow we can use the adiabatic relation between the incoming and outgoing amplitudes
X
Bb (t) =
Sba (X(t)) Aa (t)
(758)
a
where Sba (X) is the conventional on-shell scattering matrix of the time independent problem. This relation implies
that
n
Sb,a
(E) =
ω
2π
ω/2π
Z
Sba (X(t)) einωt dt
(759)
0
For sake of later use we note the following identity
X
n 2
n|Sb,a
| =
n
i
2π
Z
2π/ω
dt
0
Sba (X(t))
Sba (X(t))
∂t
(760)
====== [16.4] Current within a channel
Consider a one dimensional channel labeled as a. Let us take a segment of length L. For simplicity assume periodic
boundary condition (ring geometry). If one state is occupied the current is
ia (E) =
e
va
L
(761)
If several states are occupied we should integrate over the energy
Ia =
X
E
Z
fa (E)ia (E) =
dk
ia (E) =
(2π/L)
Z
Z
L e e
fa (E)dE
va dE =
2πva L
2π
(762)
For fully occupied states withing some energy range we get
ia =
e
e2
(E2 − E1 ) =
(V2 − V1 )
2π
2π
If we have a multi channel lead, then we have to multiply by the number of open channels.
(763)
101
====== [16.5] The Landauer formula
Consider a multi channel system which. All the channels are connected to a scattering region which is described by
an S matrix. We use the notation
gba = |Sba |2
(764)
Assuming that we occupy a set of scattering states, such that fa (E) is the occupation of those scattering states that
incident in the ath channel, we get that the outgoing current at channel b is
e
Ib =
2π
Inserting 1 =
e
Ib =
2π
"
Z
!
X
dE
gba fa (E)
#
− fb (E)
(765)
a
P
a gba
in the second term we get
"
Z
dE
#
X
gba (fa (E) − fb (E))
(766)
a
Assuming low temperature occupation with
fa (E) = f (E − eVa ) ≈ f (E) − f 0 (E)eVa
(767)
we get
Ib = −
e2 X
gba (Vb − Va )
2π a
(768)
which is the multi channel version of the Landauer formula. If we have two leads A and B we can write
"
#
e2 X X
IB = −
gba (VB − VA )
2π
(769)
b∈B a∈A
Form here it follows that the conductance is
G=
e2
e2 X X
gba =
trace(PB SPA S † )
2π
2π
b∈B a∈A
where PA and PB are the projectors that define the two leads.
(770)
102
====== [16.6] The BPT formula
Assuming that the the scattering region is periodically driven we can use the Floque scattering formalism. The
derivation of the expression for the DC component Ib of the current in channel b is very similar to the Landauer case,
leading to
Ib
e
=
2π
Z
e
=
2π
Z
e
=
2π
Z
e
≈
2π
Z
"
!
X
dE
n
|Sba
(E
2
− nω)| fa (E + nω)
#
− fb (E)
(771)
a,n
"
#
X
dE
n
|Sba
(E − nω)|2 (fa (E − nω) − fb (E))
(772)
a,n
"
#
X
dE
n
|Sba
(E)|2 (fa (E)
− fb (E + nω))
(773)
a,n
"
#
X
dE
n
nω|Sba
(E)|2 (−fa0 (E))
a,n
#
"
e X
n
2
=
nω|Sba (E)|
2π a,n
(774)
In the last two steps we have assumed very small ω and zero temperature Fermi occupation. Next we use an identity
that has been mentioned previously in order to get an expression that involves the time independent scattering matrix:
Z 2π/ω
e X ω
Sba (X(t))
Ib = i
dt
Sba (X(t))
2π a 2π 0
∂t
(775)
which implies that the pumped charge per cycle is
Q=i
e
2π
I
dX
X X Sba (X)
b∈B
a
∂X
I
Sba (X) ≡ −
G(X)dX
(776)
with
e X X Sba (X)
e
∂S †
G(X) = −i
Sba (X) = −i trace PB
S
2π
∂X
2π
∂X
a
(777)
b∈B
Note: since S(X) is unitary it follows that the following generator is Hermitian
H(X) = i
∂S †
S
∂X
The trace of a product of two hermitian operators is always a real quantity.
(778)
103
====== [16.7] BPT and the Friedel sum rule
If only one lead is involved the BPT formula becomes
dN = −i
1
trace
2π
∂S †
S
∂X
dX
(779)
where dN is the number of particles that are absorbed (rather than emitted) by the scattering region due to change
dX in some control parameter. A similar formula known as the Friedel sum rule states that
dN = −i
1
trace
2π
∂S †
S
∂E
dE
(780)
where N (E) counts the number of states inside the scattering region up to energy E. Both formulas have a very
simple derivation, since they merely involve counting of states. For the purpose of this derivation we close the lead
at r = 0 with Dirichlet boundary conditions. The eigen-energies are found via the equation
det(S(E, X) − 1) = 0
(781)
Let us denote the eigenphases of S as θr . We have the identity
i
X
δθr = δ[ln det S] = trace[δ ln S] = trace[δSS † ]
(782)
r
Observing that a new eigenvalue is found each time that one of the eigenphases goes via θ = 0 we get the desired
result.
104
[17] The modeling of the environment
====== [17.1] The model Hamiltonian
Htotal = H0 (x, p) + H(Q, P ; x)
(783)
Interaction with a bath:
Htotal = H0 (x, p) + U(x, Qα ) + Hbath (Qα , Pα )
1 2
p + V (x)
2M
H0 (x, p) =
Hbath (Qα , Pα ) =
X
(785)
X Pα 2
α
UZCL = −x
(784)
1
+ mα ωα2 Qα 2
2mα
2
(786)
cα Qα
(787)
α
UDLD = −
X
cα Qα u(x−xα )
(788)
α
For an interaction with a general chaotic environment we write
Htotal = = H0 (x, p) + E + xB
(789)
x
dissipation
fluctuations
driving source
"slow" DoF
"system"
Q
driven system
"fast" DoF
"environment"
105
V
x
V
V
V
V
(Q,P)
106
====== [17.2] The notion of temperature
Inspired by the Kubo formula we define “noise” and “friction” parameters
ν ≡
h
i
S̃(ω)
ω∼0
i
 h
Im
K̃(ω)
1

2
ω
µ ≡
(790)
=
ω∼0
1 d
S̃(ω)
~ dω
ω∼0
(791)
Next we can define a fluctuations temperature via
µ=
1
ν
2TFLCT
(792)
This definition is well define both classically and quantum mechanically, but in the latter case we have the optional
expression
1
TFLCT
1 d
2
S̃(ω)
=
~ S̃(ω) dω
ω∼0
(793)
In case of a canonical preparation we can use the expression for S(q, ω) using Bnm to prove the ”detailed balance”
property,
S̃(ω) = S̃(−ω) exp
~ω
T
(794)
which is equivalent to:
2
K̃(ω) = i tanh
~
~ω
2T
C̃(ω)
(795)
For small frequencies the latter implies
µ=
1
ν
2T
(796)
and hence for canonical preparation TFLCT = T . If we have general preparation (not canonical) then we can generalize
the DC version of the FD relation as follows:
1
1 d
[g(E) νE ]
(797)
µ=
2 g(E) dE
Hence we get
1
TFLCT
=
1
hνE i
1 d
[νE g(E)]
g(E) dE
which is similar but not identical with
1
1 d
=
[g(E)]
TEQLB
g(E) dE
(798)
(799)
The two expressions coincide if νE is independent of temperature. They also coincide if the energy of the environment
is canonically distributed.
107
[18] The Feynman Vernon formalism
====== [18.1] Feynman Vernon formalism
A
X
K(xAt , xBt |xA0 , xB0 ) =
F [xA , xB ] ei(S[x
]−S[xB ])
(800)
xA ,xB
F [xA , xB ] =
D
E
E
D
ψ (E) U [xB ]−1 U [xA ] ψ (E)
=
U [xB ] ψ (E) U [xA ] ψ (E)
i
U [x] = T exp − (E + x(t)B)
~
(801)
(802)
The dephasing factor
A
1
|F [xA , xB ]| ≡ e− ~2 SN [x
,xB ]
(803)
ZCL expression:
SN [xA , xB ] =
1
2
Z tZ
1
2
Z tZ
0
t
dt0 dt00 C(t0 −t00 ) (xA (t0 )−xB (t0 ))(xA (t00 )−xB (t00 ))
(804)
dt0 dt00 C(t0 −t00 ) [w(xA (t0 )−xA (t00 )) + w(xB (t0 )−xB (t00 ) − 2w(xA (t0 )−xB (t00 ))]
(805)
0
DLD expression:
SN [xA , xB ] =
0
t
0
The same is obtained from c-number fluctuating field.
====== [18.2] FV noise term derivation
We can easily derive SN [xA , xB ] for a c-number noisy field. The amplitude which is contributed to the path integral
by a pair of paths is
Z t
Z t
0 1
A 2
A
0
0 1
B
0
B 2
˙
FV amplitude = exp i
dt ( m(ẋ ) − U (x , t )) −
dt ( m(x ) − U (x , t ))
2
2
0
0
Z t
Z t
= exp i
dmṘṙ −
dτ (U (xA , t0 ) − U (xB , t0 ))
0
(806)
(807)
0
Averaging over realizations and using
hei
R
dt0 f (t0 )
1
i = e− 2
we get the desired result.
R
dt0 dt00 hf (t0 )f (t00 )i
(808)
108
====== [18.3] FV in phase space
The propagator of the Wigner function
Z
R,P
K(R, P |R0 , P0 ) =
Z
d[R]
d[r] exp
R0 ,P0
i
1
S [R, r] − 2 SN [R+(r/2), R−(r/2)]
~ eff
~
(809)
At high temperatures, disregarding friction and external fields:
Z t
Z
i t
ν
[w(0) − w(r(t0 ))]dt0
d[r] exp −
mR̈(t0 )r(t0 )dt0 − 2
~ 0
~ 0
R0 ,P0
Z t
Z r,p
Z
Z t
ν
i
0
0
0
0
0
= FT
mR(t )r̈(t )dt − 2
[w(0) − w(r(t ))]dt
d[r] d[R] exp −
~ 0
~ 0
r0 ,p0
Z r
m
m
1
0
0
= FT δ(p − (r − r0 )) δ(p0 − (r − r0 )) exp −ν w(0) −
w(r )dr
t
t
r−r0 r0
Z r
Z
1
m
0
0
w(r )dr
=
drdr0 exp i (r − r0 )(R − R0 ) − iP r + P0 r0 − ν w(0) −
t
r−r0 r0
Z
R,P
K(R, P |R0 , P0 ) =
Z
d[R]
(810)
If we integrate over R we get
Z
K(P |P0 ) =
dr exp [−i(P − P0 )r − ν(w(0) − w(r))]
(811)
109
====== [18.4] Master equation
Propagation in High Temperature Environment:
hU(x0 , t0 )U(x00 , t00 )i = νδ(t0 −t00 ) w(x0 −x00 )
(812)
with
1 r 2
w(r) = ` exp −
2 `
2
(813)
The ZCL master equation:
∂ρ
=
∂t
− ∂R
1
η
∂2
P + ∂P
P + ν
m
m
∂P 2
ρ
(814)
The DLD master equation:
∂ρ
=
∂t
− ∂R
1
η
P + ∂P
GF ? P + νGN ?
m
m
ρ
(815)
The friction kernel is defined as follows
GF (P − P 0 ) ≡ FT
w0 (r)
r
=
1
Ĝ
~/`
P −P 0
~/`
(816)
and the noise kernel is
GN (P − P 0 ) ≡
1
FT (w(r)−w(0)) =
~2
2 `
1
P −P 0
Ĝ
− δ(P −P 0 )
~
~/`
~/`
The high temperature dephasing rate:
dephasing rate =
1 2
ν`
~2
(817)
110
====== [18.5] FV and fidelity
Given a driving scheme we define
P(t) = |F [xA , xB ]|2
(818)
Assuming |xA −xB | = d we get
Z Z
1 A B2 t t
0
00
0 00
P(t) = exp − 2 |x −x |
C(t −t ) dt dt
~
0 0
(819)
Substitution of
C̃(ω) = 2πσ 2 δ(ω) +
2π~σ 2
R
∆
~ω
∆
G
~ω
∆b
(820)
leads to:
P(t) = exp(−ct2 )
P(t) = exp(−γt)
P(t) = exp(−(σvt/~)2 )
(821)
(822)
(823)
with:
2
d
C(τ ∼ 0)
~
2
1
d
C̃(ω ∼ 0) = 2 νd2
γ =
~
~
c =
(824)
(825)
which is identified as the Γ/~ in case of a two level system.
The FV perturbative expression for P(t) holds for limited time. There may be non-prt crossover if its breakdown
happens before P(t) becomes vanishingly small or gets to its ergodic value.
xA(t)
xB(t)
(Q,P)
111
====== [18.6] FV and dephasing
The importance of time reversal symmetry if the correlation time is long.
The other extreme is extremely short correlation time. Then for well separated paths we easily recover the high
temperature result.
Let us assume short but finite correlation time, and let us try to take into account that the paths are in general not well
separated. We assume that the interfering paths are long and have well defined statistical properties. Consequently
Z tZ
0
t
dt0 dt00 C(t0 − t00 ) w(x(t00 ) − x(t0 )) =
Z
Z
dτ C(τ )
dr P (r; τ ) w(r) × t
(826)
0
hence
dephasing rate =
1
~2
Z
dq
2π
Z
∞
0
dω
w̃(q)C̃(ω) P̃cl (q, ω)
π
(827)
Note that in general we have
Z
0
∞
dω
P̃cl (q, ω) = 1
π
Z
dq
w̃(q) = `2
2π
(828)
(829)
Restrictions on QCC. To explain the fallacy of the semiclassical approximation (no back reaction, closed channels
cannot be excited). Ad-hoc improvement
dephasing rate =
1
~2
Z
dq
2π
Z
∞
−∞
dω
S(q, ω) P̃qm (q, ω)
2π
(830)
For diffusive trajectories with available energy E
P̃cl (q, ω) = Θ(E − ~ω)
2Dq 2
(Dq 2 )2 + ω 2
(831)
or assuming thermal state maybe the many body version?
P̃qm (q, ω) =
2Dq 2
~ω/T
e~ω/T − 1 (Dq 2 )2 + ω 2
Our purpose is to justify such a procedure in a clean way. Hence we turn to a scattering approach.
(832)
112
[19] The dissipative dynamics of a two level system
Pre-draft version
====== [19.1] Modeling
Here x = vσ3 = position in the double well. Nuclear physics application: boundary may have either of two shapes
H0 = (~Ω/2)σ1
(833)
The Hamiltonian of a nearby chaotic system:
H(Q, P ; x) =
1 2
(P +P22 + Q21 +Q22 ) + (1+x) · Q21 Q22
2 1
H = E + xB
(834)
(835)
It can be argued that Bnm is a banded matrix.
bandwidth = ∆b = ~/τcl
Htotal =
1
~Ωσ1 + E + vσ3 B
2
"
Htotal
E+vB Ω/2
=
Ω/2 E−vB
(836)
(837)
#
(838)
F = −Q21 Q22
(839)
B = {−Fnm }
(840)
Simulations:
|Ψ(t = 0)i =
basis:
1
√ (| ↑i + | ↓i) ⊗ |ψ (E) i
2
(841)
|νi ⊗ |ni
ρν,ν 0 (t) =
X
Ψν,n (t)∗ Ψν 0 ,n (t)
(842)
n
"
#
1
1
1 + M3 M1 − iM2
~
ρ(t) = (1 + M · ~σ ) 7→
2
2 M1 + iM2 1 − M3
(843)
113
~ ·M
~
S(t) = (2 trace(ρ(t)2 ) − 1) = M
(844)
Pedagogical remark: Given a basis ν for the representation of the spin, the wavefunction Ψ can always be written as
|Ψi =
X
|νi ⊗ |ψ (ν) i
(845)
ν
where the unnormalized wavefunction ψ (ν) is called the relative state of the environment. With this notation the
elements of the reduced probability matrix are:
0
ρν,ν 0 = hψ (ν) |ψ (ν ) i
(846)
Hence the overlap of the relative states determines the purity of the spin state. In particular orthogonality of the
relative states implies a maximally mixed spin state.
====== [19.2] The parameters of the theory, Regimes
Htotal = H0 + E + xB
parameter
∆ ∝ ~d
∆b ∝ ~
T
dT
Γ
significance
environment mean level spacing
environment bandwidth
environment temperature
environment heat capacity
system energy
strength of coupling
dT =
dT
dE
−1
= heat capacity ∼ d
(847)
Assuming that x performs motion with amplitude A and velocity V , then Γ is related to (σA)2 and (σV )2 .
Γ
∆
= minimum
!
2 ~σ 2/3
A ,
V
∆
∆2
σ
(848)
The parameter Γ
Hnm = En δnm + xBnm
(849)
∆ = mean level spacing
∆b = bandwidth
|Bnm | ∼ σ for |En −Em | < ∆b
Assume a small constant perturbation |x| = δx
Γ(δx) ≈
σδx
∆
2
∆
(850)
114
Note that Γ/∆ is the number of levels that are mixed non-perturbatively, as implied by perturbation theory (to
infinite order).
Re-write the Hamiltonian in the adiabatic basis:
Hnm = En δnm + ẋ
i~Bnm
En −Em
(851)
Assume a slow variation ẋ
Γ(ẋ) ≈
~σ ẋ
∆2
2/3
∆
(852)
limits:
Parameters: T, dT , ∆ ∝ ~d , ∆b ∝ ~, , Γ
The thermodynamic Limit. Mathematical definition of the Thermodynamic Limit: d → ∞ (infinitely many degrees
of freedom). Physical definition of the Thermodynamic Limit: /dT T (having well defined temperature). In case
of two level (spin) system: Ω dT × T In case of d0 dimensional system: d0 dT
The High Temperature Limit. Mathematical definition of the high temperature limit: T → ∞ (vanishing friction
effect in this limit). Note: high temperature does not imply a lot of noise! Physical definition of the high temperature
limit: ~ω/T 1 for the physically relvevnat frequencies. Sufficient condition: T ∆b In case of the Spin-Boson
model: T Ω, Γ.
K
=
1
16π
Γ
= Kondo Parameter
T
(853)
The Semiclassical Limit. Mathematical definition of the semiclassical limit: ~ → 0 (scaled planck constant). Physical
definition of the semiclassical limit: Γ > ∆b (the non-perturbative regime). This should be contrasted with: Γ ∆b
(Fermi golden rule regime). Note that the adiabatic / standard-prt regime is: Γ < ∆ (Born-Oppenheimer regime).