Download Exercises in Statistical Mechanics (2004)

Exercises in Statistical Mechanics (2004)
Prepared by a student based on a course by Doron Cohen, has to be proofed
Department of Physics, Ben-Gurion University, Beer-Sheva 84105, Israel
This exercises pool is intended for a graduate course in “statistical mechanics”. Some of the
problems are original, while other were assembled from various undocumented sources. In particular some problems originate from exams that were written by B. Horovitz (BGU), S. Fishman
(Technion), and D. Cohen (BGU). We thank Tali Farbiash for preparing the printed version.
000... Using random variables
001 Random variables (distance between two particles in a box)
002 Random variables (length of chain molecule)
003 Random variables (fluctuations in number of particles)
005 Random variable x=cos(theta)
006 Microcanonical state of oscillator
007 The ergodic density rho(x)
008 Spreading of free particle
010... The energy spectrum: N(E) and Z(T)
010 The functions N(E) and Z(T) for particle in two level box
011 The functions N(E) and Z(T) for particle in box + gravitation
012 The functions N(E) and Z(T) for N particles in a box
013 The functions N(E) and Z(T) for general dispersion relation
014 What does it mean particle in a 2D box
015 The functions N(E) and Z(T) for N spins
016 Calculation of Z(T) for AB and AA molecules
030... The probability matrix
031* Requirements on rho
032* The rho of spin 1/2
031* Various States of spin 1/2
032* Shanon entropy as a purity measure
033* Spin 1/2 system + spin 1/2 environment
034* Spin 1/2 states in EPR experiment
041* Superposition in Wigner representation
042* Wigner-wyle representation of operators
043* Trace of A*B using Wigner-Wyle
044* Measurements using Wigner-Wyle
045* Wigner function of canonical state
046* Wigner propagator for free particle
047* Wigner propagator for kicked particle
048* The analysis of two slit experiment
100... Canonical formalism
101 State equations derived from Z(T)
102 Fluctuations in energy
103 Helmholtz function
104 Absolute temperature and entropy
105 Shanon definition of entropy
106 Quasi static processes in mesoscopic systems
111* Entropy change due to mixing
112* Entropy change in case of sudden change
113* Sudden versus adiabatic process
114* Two level system, cooling process
115* Adiabatic cooling of liquid He3
2
116* Adiabatic expansion of the universe
150... Grand canonical formalism
151 Boltzmann approximation from canonical ensemble
152 Fluctuations of N in grand canonical ensemble
153 Extensive property of the grand energy
154 Fluctuations in the grand canonical ensemble
200... Application of the canonical formalism
202 Gas in $V(x,y,z)=x^2$ box potential
204 Pressure of gas in a box with gravitation
205 Pressure by particle in a spring-box system
206 Particle on a ring ("rotor") with electric field
207 Polarization of two-spheres system inside a tube
208 State functions of spin 1/2 system
209 Heat capacity of harmonic oscillator
219* Generalization: Debye model.
210 Photon gas as collection of harmonic oscillators
221
222
223
224
225
227
Density of gas inside a rotating box
Classical gas with general dispersion relation
Polarization of classical polar molecules
Magnetization of spin 1 system
Gas in $V(x,y,z)=x^2+y^2+Fz$ box potential
Electron gas in magnetic field
231
232
233
234
235
236
Defects in lattice
Tension for rotating device
Elasticity of a rubber band
Tension of a chain molecule I
Tension of a chain molecule II
The zipper model for DNA molecule
260... Canonical treatment of chemical equilibrium
261 Classical gas in [volume]-[surface] phases equilibrium
262 Two level system with N particles
263* Adhesion A[g] -- A[fermi site]
264* Adhesion A[g] -- A[bose site]
265* The reaction C -- 3A + 2B
266* The reaction photon + photon -- electron + positron
300... Formal
301 Chemical
302 Chemical
303 Chemical
304 Chemical
321 Chemical
322 Chemical
341 Chemical
342 Chemical
343 Chemical
344 Chemical
345 Chemical
approach to chemical equilibrium
equilibrium A--A+e
equilibrium A[volume]--A[surface]
equilibrium A[volume]--A[surface], polar molecules
equilibrium A[volume]--A[polymer]
equilibrium: H2 + D2 -- 2HD
equilibrium: H2[volume] -- 2H[surface]
equilibrium for Fermions in a box
equilibrium for elementary particles
equilibrium for elementary particles
equilibrium for elementary particles
equilibrium for elementary particles
400... Bose gas
401 Heat capacity of ideal Bose gas
402 Bose gas in strong magnetic field
403 Charged Bose gas in box with potential difference
3
404
421
422
423
424
Small oscillations of piston in cylinder with Bose gas
Bose gas for general dispersion relation
Heat capacity of Bose gas
Heat capacity of He4
Bose gas in a uniform gravitational field
450... Fermi gas
451 State equations for ideal Fermi gas
452 Ideal Fermi gas in 2D space
453 Ideal Fermi gas in 2D box
454 Ideal Fermi gas in semiconductor
455 Magnetic properties of T=0 electrons (Pauli)
456 Electrons in $V(x,y,z)=x^2+y^2$ box (effectively 1D)
457 Fermi gas in 2D+3D connected boxes + gravitation
458 Quantum Fermi gas in gravitation field of a star
500... Interacting systems
501 One dimensional hard sphere gas
502 Virial coefficients
503 Two dimensional Coloumb gas
550... phase transitions
553* Mean field: Ferromagnetic Ising model
554* Mean field: Antiferromagnetic Ising model
555* Mean field: Spin 1 version of Ising model
556* Mean field: Unisotropic Heizenberg model
563 Correlation function in 1D Ising model
564 Ising with long range interaction
565 Ising model with disorder
571 Mean field: ferromagnetism with classical spins
572 Mean Field: antiferomagnetism
573 Mean Field: ferroelectricity
574 Mean Field with field h(r)
575 Ferromagnetism for cubic crystal
581 Mechanical model for symmetry breaking
585* Tricritical point
600... Kinetics
601 Effusion from box with Bose gas and magnetic field
602 The system of e404 with hole in one side.
603 Thermionic emission of electrons from a Metal
604 Effusion of electrons from a box in magnetic field
605 Radiation from 1D blackbody fiber
606 Generalize J-incident formula for 1D/2D box
607 Landauer formula for 1D conductance
608 Einstein relation for the conductivity of electrons.
611* Analysis of the spinning radiometer problem
630* Boltzmann equation
700... Stochastic processes (random walk, diffusion, Langevin)
701 Random walk with correlations in time
702* Random walk with variable steps, Levi flights
703* Diffusion modes in 1D box
703* The competition between diffusion and drift
704* Langeving and Milikan experiment
705* Tunneling with activation (Kremer)
706* Stochastic resonance (driven double well)
4
800... Theory of response (FD relation, Onsager)
801 Fluctuations for stationary process
802 Fluctuations of harmonic oscillator
803 Diffusion of Brownian particle
804 Nyquist theory for ring/resistor/RL-circuit
805 Measurements of current using galvanometer
900... Master equations (Pauli, Bloch, Fokker-Plank)
901* Particle in a double well, damped oscillations
902* Bloch Eq: Spin Resonance
903* Master Eq for damped classical particle
====== [1] E001: Random variables (distance between two particles in a box)
In a one dimensional box, with length L, two particles are turning around. The particles don’t know about each
other. The probability function for finding a particle in a specific place in the box is uniform.
Let r be the relative location of the particles.
Find f (r) dr = p (r < r̂ < r + dr) and also hr̂i and σr , in two ways.
====== [2] E002: Random variables (length of chain molecule)
Molecule can be described as an N monomers chain.
The probability that a monomer will be horizontal is p and then it’s length is a, otherwise it’s length is b. Let L be
the molecule’s length.
Find f (n) ≡ p (L = na + (N − n) b). calculate hLi and σL , in two ways.
What is the behavior of σL /hLi as a function of N ?
a
b
====== [3] E003: Random variables (fluctuations in the number of particles)
N0 particles in a closed tank with volume V0 are given. we’ll focus on an area with volume V which we’ll pronoun
”The system”. The number of the particles in the system is a random variable N .
(a) Find hN i and σN using the probability theory.
(guideline: define random variables X̂n which determines if a certain particle is in the system. i.e. Xn = 0 or
1).
(b) Find the probability function f (N ).
(c) Assume |V /V0 − 12 | << 1 and treat N as a continuous random variable. Find the probability function f (N ) in
this assumptions framework.
5
====== [4] E005: Changing random variables x = cos(θ)
Assume that the random phase θ has a uniform distribution. Define a new random variable x = cos (θ). What is the
probability distribution of x ?
====== [5] E006: Oscillator in a microcanonical state
Assume that a harmonic oscillator is prepared in a microcanonical state with energy E. Write down what is ρ (x, p).
Find the projected probability distribution ρ (x). [This involves an integral over a delta function]. Show that you get
the same distribution as in E005.
====== [6] E007: The ergodic density ρ (x)
Find an expression for ρ (x) of a particle which is confined by a potential V (x), assuming that the its state is
microcannonical with energy E. Distinguish the special cases of d = 1, 2, 3 dimensions. In particular show that
in the in the d = 2 case the density forms a step function. Contrast your results with the canonical expression
ρ (x) ∝ exp (−βV (x)).
====== [7] E008: Spreading of a free particle
2
p
Given a free classic particle H = 2m
, that has been prepared in time t = 0 in a state represented by the probability
function
2
2
ρt=0 (X, P ) ∝ exp −a (X − X0 ) − b (p − p1 )
(a) Normalize ρt=0 (X, P ).
(b) Calculate hXi, hP i, σX , σP , E
(c) Express the random variables X̂t , P̂t with X̂t=0 , P̂t=0
(d) Express ρt (X, P ) with ρt=0 (X, P ). (Hint: ’variables replacement’).
(e) Mention two ways to calculate the sizes appeared in paragraph b in time t. use the simple one to express
σx (t) , σp (t) with σx (t = 0) , σp (t = 0) (that you’ve calculated in b).
====== [8] E010: The functions N (E) and Z (T ) for particle in two level box
Given particle in a well H =
p2
2m
+ V (x).
(a) Draw in the phase space the possible trajectories of the particle in the well.
(b) Calculate N (E) and the energy levels in the semy-classic proximity.
(c) Calculate Z (β) and show that
Z (β) =
m
2πβ
12
L cosh
1
βE
2
6
V(x)
ε/2
ε/2
L\2
======
tation
[9]
L\2
E011: The functions N (E) and Z (T ) for particle in a box + gravi-
Find the distribution function Z (β) of a particle in a three dimensional box with a gravitation field along axis −Z.
Assume the box dimensions are L × L × (Zb − Za )
Guideline: write the hamiltonian and calculate
Z Z Z
dxdpx dydpy dzdpz −βH(x,y,z,px py pz )
Z (β) =
e
zπ
2π
2π
Z
Zb
gravitation
Za
====== [10] E012: The functions N (E) and Z (T ) for particle in a box
In this question one must evaluate Z (β) using the next equation
Z (β) =
X
e−βEn =
Z
g (E) d (E) e−βE
n
P3 pαi
(a) Particle in a three dimensional space H = i=1 2m
Calculate g (E) and through that evaluate Z (β)
Guideline: for calculating N (E) one must evaluate some points (n1 n2 n3 )- each point represents a state - there’s
in ellipse En1 n2 n3 ≤ E
(b) N particles with equal mass in a three dimensional space. assume that it’s possible to distinguish between those
3N
particles. Prove: N (E) = const · E 2
Find the const. use Dirichlet’s integral (private case) for calculating the ’volume’ of an N dimensional Hyper-ball:
Z
Z
N
π2
... Πdxi = N RN
2 !
P 2
xi ≤ R 2
Calculate g (E) and from there evaluate Z (β)
7
======
tion
[11]
E013:
The functions N (E) and Z (T ) for general dispersion rela-
Find the states density function g (E) and the distribution function Z (β) for a particle that moves in a d dimensional
space with volume V = Ld .
Assume the particle has dispersion relation
ν
case a’ E = C|P
p |
2
case b’ E = m + p2
Make sure that you know how to get a result also in the ”quantal” and the ”semiclassical” way.
====== [12] E014: What does it mean particle in a 2D box
What is two dimensional gas?
Given gas in a box with dimensions (L << L) L × L × L.
Determine what are the energies of the uniparticle states. Show that there’s an energy range 0 < E < Emax where
it’s possible to relate the gas as a gas in a 2 − D space with a states density function
g (E) = A
m
0≤E
2π
A ≡ L2
====== [13] E015: The functions N (E) and Z (T ) for N spins
Given an N spin system
Ĥ =
N
X
ε (α)
σ̂z
2
α=1
Calculate ZN (β) in two different ways:
(a) The short way - Calculate ZN (β) by factoring the sum, similar to what we did in the class for N particle system.
(b) The long way - write the En energy levels of the system (mark with n = 0 the basic level and with n = 1, 2, 3, ...
the following levels). Give a physical meaning to the index n. find the degeneration gn of every level. Calculate
the distribution function
X
ZN (β) =
gn e−βEN
n
note: in this question n is the index of the energy levels and not of the state energies. therefore, there’s a need to
explicitly conclude gn in the sum.
====== [14] E016: Calculation of Z (T ) for AB and AA molecules
Calculate the distribution function of diatomic molecule AB. Assume it’s possible to relate the molecule, like two
”balls” with m mass and S spin each one, attached with a spring length r0 . Assume it’s a hard spring so it’s possible
to assume that the molecule is in the lowest strip of the vibration. In other words, we relate the molecule like a rigid
body (”rotor”). Relate separately in case of molecule AA composed of identical atoms. Relate specifically in case
of identical atoms with spin 0 and identical atoms with spin 21 . in the last case, determine what is the probability
to find the molecule in triplet condition and distinguish between the borderline cases of low temperature and high
temperature.
====== [15] E101: State equations derived from Z (T )
8
Make sure you’r well aware of the basic equations of the canonical ensemble, and knows how to prove those equations
for the state functions.
X
(∗) Z (β, X) ≡
e−βEr
r
E=−
y=
∂ ln Z
∂β
1 ∂ ln Z
β ∂X
F (T, X) ≡ −
S=−
1
ln Z (β, X)
β
∂F
∂T
More definitions
(Heat capacity) Cx ≡ ∂E
∂T |X
∂y
(Generalized susceptibility) χ ≡ ∂X
(*) for a classical particle
Z
X
dxdp
Er 7→ H (XP )
7→
2π
r
====== [16] E102: Fluctuations in energy
2
= T 2 CX
Prove that σE
2
Where σE ≡ hH 2 i − hHi2
and CX ≡ ∂E
∂T |X
2
Guideline: Express σE
by the distribution function and use the result we got for E in order to get the requested
expression.
====== [17] E103: Helmholtz function
We define F ≡ − β1 ln Zβ
Prove:
(∗)E = F + T S
(∗∗)
S = − ∂F
∂T
∂F
y = − ∂X
It’s possible to rely on the expressions that express X, S, y by the distribution function and by the F definition.
====== [18] E104: Absolute temperature and entropy
In a general quasi static process we defined
∂E
∂E
dQ ≡ dE + dW =
dβ +
+ y dX
∂β
∂X
9
We expressed E and y by the distribution function Z (β)
Using differential equations technic, for an integration factor to the non precise differential dQ,you learned in the
course, which is solely, a function of β . Show that the integration factor you get is
T −1 (β) = β
therefore It’s possible to write dQ = T dS where dS is a precise differential.
Find the function S and show
1
∂
S = − − ln Z (β)
β
∂ 1
β
By definition, S is an entropy function, and T is called in an absolute temperature.
====== [19] E105: Shanon definition of entropy
Show that it’s possible to write the entropy function expression as:
X
S=−
pr ln (pr )
r
It’s possible to acount this equation as a definition when we talk about other ensembles.
====== [20] E106: Quasi Static processes in a mesoscopic system
Write the basic level energy of a particle with mass m, which is in a box with final volume V . (Take boundary
conditions zero in the limits of the box). In temperature zero, β −1 = 0 , calculate explicitly the pressure caused by
the particle. Use the equation
X ∂Er p=
pr −
∂V
r
Compare it to the equation developed in class for general temperature
P =
1 −1
β
V
and explain why in the limit β −1 → 0 we don’t get the result you calculated. (Hint - notice the title of this question).
====== [21] E151: Boltzmann approximation from canonical ensemble
Given N particle gas with uniparticle state density function g (E).
In the grand canonical ensemble, in Boltzman proximity, the results we get for the state functions N (βµ) , E (βµ) are
Z ∞
N (βµ) =
g (E) dE f (E − µ)
0
Z
E (βµ) =
g (E) dE E · f (E − µ)
Where f (E − µ) = e−β(E−µ) is called the Boltzman occupation function.
In this exercise you need to show that you get those equations in the frame of the proximity ZN ≈ N1 ! Z1N .
For that, calculate Z, that you get from this proximity for ZN and derive the expressions for N (βµ) , E (βµ).
====== [22] E152: Fluctuations of N in grand canonical ensemble
10
Show that
2
h4N i =
1 ∂
β ∂µ
2
ln Z = T
∂N
∂µ
TV
From that prove the equation
T
∂V
h4N 2 i
=
−
hN i2
V 2 ∂p N,T
The last step demands manipulation of equations in thermodynamics
Hints:
−1
∂N
∂µ
=
∂µ T,V
∂N T V
dµ = νdp + SdT, ν =
V
S
,S=
N
N
∂p
∂µ
|T = ν |T
∂ν
∂ν
====== [23] E153: Extensive property of the grand energy
Explain why F, Ω are extensive functions in the thermodynamic limit, so that
F (β; λV, λN ) = λF (β; V, N )
Ω (βµ; λV ) = λΩ (βµ; V )
Guideline: Note that if you split the system, then in neglecting ”surface” interaction, the functions Z, Z will be
factorized.
Z ≈ ZA + ZB, F ≈ F A + F B
Result: therefore,
Ω (βµ; V ) = V Ω (βµ; 1)
Prove that from here, we can conclude that
Ω (βµ; V ) = −V ∗ p (βµ)
remark: Generalization of considerations such these were written by Euler.
A
B
====== [24] E154: Fluctuations in the grand canonical ensemble
A fluid in a volume V is held (by a huge reservoir) at a temperature T and chemical potential µ. Do not assume
3
an ideal gas. Find the relation between h(E − hEi) i and the heat capacity CV (T, z) at constant fugacity z. Find
3
the relation between h(N − hN i) i and the isothermal compressibility χT (V, µ) = − (∂v/∂µ) |V,T where v = V /hN i.
11
[Hint: Evaluate 3rd derivatives of the grand canonical partition function.] Find explicitly results in case of a classical
ideal gas.
====== [25] E202: Gas in V (x, y, z) = x2 box potential
Given N classical particles in potential
V (x, y, z) =
1
2
2 ax
∞
0 ≤ x, 0 ≤ y ≤ L, 0 ≤ Z ≤ L
else
We need to find the pressure on the side x = 0.
The potential given in the problem:
Definition of a new potential, which is a generalization of the potential given in the problem.
X =0
X
X
X
====== [26] E204: Pressure of gas in a box with gravitation
A gas given in a gravitation field. Use the former exercise to write the distribution function ZN (β, Za , Zb ).
Calculate the forces (in their absolute value) that work on the lower sidewall (Fa ) and on the upper sidewall (Fb ) of
the box.
What is the difference between those forces?! explain the result you get.
====== [27] E205: Pressure by particle in a spring - box system
Mass m is connected to a spring (in the middle) with natural length L and a constant K. X is a given parameter.
the system is in equilibrium in temperature T .
(a) Write the hamiltonian (careful!!!).
(b) Find Z (β, X).
(b) Calculate the tention F as a function of the parameter X.
12
Find explicit expressions in the limit of high temperature, low temperature and explain the results you get.
F
m
X
====== [28] E206: Particle on a ring (”rotor”) with electric field
Particle with mass m and charge e is free to move along the perimeter of a ring placed on x − y plain, (with a radius
R). A uniform, electric field ε̄ = x̂ε in the system, and it has a thermic balance in temperature T .
(a) Write the Hamiltonian H (pθ , θ) of the particle.
(b) Calculate the distribution function Z (β, ε) ; β −1 = kB T
(c) What is the probability function ρ (θ) of the angled coordinate θ?
(d) Calculate the average place of the particle in a cartesian coordinates (meaning hxi, hyi)
(e) What is the probability function ρ (x) of the coordinate x? add a schematic drawing.
(f) Express the polarization P (ε). For a weak ε develope P (ε) up to first order for - ε: P (ε) = p0 + χε + 0 ε2
and find χ.
Use the next equations:
Z 2π
1
e±z ln θ dθ = I0 (Z)
2π 0
I01 (Z) = I1 (Z)
1
1
I0 (Z) = 1 + Z 2 + Z 4 + ...
4
64
y
θ
x
R
====== [29] E207: Polarization of two-spheres system inside a tube
13
Given two balls in a very long, hollow tube, with length L. The mass of each ball is m, The charge of one ball is −q
and the charge of the other one is +q. The ball’s radius is negligible, and the electrostatic attraction between the
balls is also negligible. The balls are rigid and can’t pass through each other. The balls are attached to a drop, whose
surface tension causes it’s gravity constant γ to work on the balls toward each other. (The force does not depend on
the distance between the balls). The system is in an external electric uniform field ε̄ = εx̂ and in thermic balance in
temperature T .
(a) Write the hamiltonian of the system H (p1 , p2 , x1 , x2 ) = Ek + V (x) when Ek is the kinetic energy. Define
properly V (x) when x = x2 − x1 and write a diagram of V (x).
(b) Calculate the distribution function Z (β, ε).
(c) Find the probability function of x, ρ (x) and the average distance hxi between the balls. Express again ρ (x) by
hxi.
(d) Find the polarization p as a function of ε. Use the distribution function.
(e) Develop p (ε) up to first order in the field: P (ε) = p0 + χε + O ε2 .
This development is valid in a weak field, Define what is a weak field. Express your answers with L, m, q, γ, T, ε.
ε
−q
+q
x1
====== [30] E208: State functions of spin
x2
1
2
system
Find the state functions E (T, B) , M (T, B) , S (T, B) for N spins system:
H = −γB
N
X
σta
a=1
Write the results for a weak magnetic field γB << T . Especially find the magnetic susceptibility χ and S (B → 0)
====== [31] E209: Heat capacity of harmonic oscillator
Find the energy and the heat capacity of the next system:
Ĥ =
p̂2
1
+ mw2 X̂ 2
2m 2
This system can be a model to a particle adsorbed to the surface of a solid. Compare to the result you get in a
classical treatment.
====== [32] E219: Heat capacity, Debye model
Find expressions for the heat capacity in Debye model. Make the calculation for general spectral distribution with
high frequency cutoff, and discuss d = 1, 2, 3.
14
====== [33] E210: Photon gas as collection of harmonic oscillators
p̂2
1
+ mw2 x̂2
2m 2
p
we define n̂ = a+ a, a = w2 x̂ − i √12w p̂, and then it’s possible to write H = wn̂ + const. If const = 12 w, n̂ is an
operator with self states |n > and the matching eigenvalues are n = 0, 1, 2.... therefore, the stationary states of the
oscillator are
H=
|n >, En = nw
The next reinterpretation is acceptable for the states |n >: |n > is a state where n particles (”bozons”) occupies
uniparticle level, with energy w. When the system, described as H is in equilibrium with the environment
Pn =
1 −βEn
1 −βwn
=
e
e
Z (β)
Z (β)
Prove Z (β) =
hni =
1
1−e−βw
1
≡ f (w)
eβw − 1
f (w) is called Bose’s occupation function.
The hamiltonian that describes the electromagnetic field, if we use ’normal coordinates’, is a collection of independent
oscillators.
X
H=
wkα nˆkα
kα
We give the hamiltonian the next reinterpretation: |k, α > is a uniparticle state of photon with k momentum and α
polarization. Periodic boundary conditions are ~k → 2π
L (m1 , m2 , m3 ). Two transversal polarization directions α = 1, 2.
The photon energy in the state |k, α > is wkα = |~k|. The state |...nkα ... > is a multiparticle occupation state with
nkα photons, settelling the uniparticle state |kα >. The occupation states |...nkα ... > are the stationary states of the
system, and we get
X
E(...nkα ...) =
nkα wkα
kα
In thermal equilibrium of the electromagnetic field, described with H, with the environment
P (...nkα ...) =
1 −βE(...nkα ...)
e
Z
(a) Calculate the average number of photons N (T, V ). The temperature and the volume of the medium are given.
(b) Calculate the distribution function Z (β, V ). Use factorization and integration in parts.
(c) Calculate the thermic energy of the field in two ways. show that you get
E (T, V ) =
π2
V T4
15
(d) Find the radiation pressure. Is it possible to use the results from exercise 17 with ν = 1. Show that any way,
Pressure= 13 VE .
To solve paragraphs (a) - (d) use the next hints. always remember that photon gas = electromagnetic field = oscillator’s
collection, so there’s no need to panic...
Hints to solve (a) - (d)
15
(a) The photon’s number N (T, V ) =
X
→2+S
kα
Z
∞
0
L
2π
2
P
kα hnkα i
d2 k, hnkα i = f (wkα )
x2 dx
= 2ζ (3) = 2
ex − 1
1
1
1
+ 3 + 3 ...
13
2
3
(b)
ln Z (β) =
X
ln Z (k,α) (β) = ...
kα
Z
0
∞
x2 dx
π4
=
6ζ
(4)
=
ex − 1
15
(c)
E (T, V ) =
X
wkα hnkα i = ...
kα
E (T, V ) = −
∂ ln Z
= ...
∂β
(e)
P (T, V ) =
1 ∂ ln Z
= ...
β ∂V
Additional materials about photon gas will be learned in the frame of the grand canonical ensemble. Formally, photon
gas is like Bose gas with chemical potential µ = 0.
====== [34] E221: Density of a gas inside a rotating box
A cylinder of radius R and height H rotates about its axis with a constant angular velocity Ω. It contains an ideal
classical gas at temperature T . Find the density distribution as function of the distance r from the axis. Note that
the Hamiltonian in the rotating frame is H 0 (p, q; Ω) = H (p, q) − ΩL (p, q) where L (p, q) is the angular momentum.
It is conceptually useful to realize that it is formally the Hamiltonian of a charged particle in a magnetic field
(=”Coriolis force”) plus centrifugal potential V (r). Explain how this formal equivalence can be used in order to make
a ”shortderive” in the above calculation.
====== [35] E222: Classical gas with general dispersion relation
Consider a gas of noninteracting particles in d dimensional box with kinetic energy of the form E (p) = C|p|s , and
define γ = 1 + (s/d). Find the partition function
and the equations of state. In particular prove the following results:
The energy is E =
heat capacity ratio
NT
(γ−1) . The entropy
Cp
is Cv
=γ .
is S =
N
(γ−1)
ln (P V γ ) + f (N ). In an adiabatic process P V γ = const. The
====== [36] E223: Polarization of classical polar molecules
Find the polarization P̃ (ξ) and the electric susceptibility χ for gas of N classical molecules with dipole moment µ,
The system’s temperature is T .
16
====== [37] E224: Magnetization of spin 1 system
Find the magnetic susceptibility for gas of N , uncharged, particles . with spin 1. The system’s temperature is T .
H1 =
P2
− γBSz
2m
====== [38] E225: Gas in V (x, y, z) = x2 + y 2 + Fz box potential
N particles with no spin and mass m were inserted to a space between two parallel, horizontal surfaces. in order to
prevent the particles from ”running away”, a two dimensional harmonic potential was created. i.e. the particles see
the potential
V (x, y, z) =
1
2
2 mw
∞
x2 + y 2
t1 < z < z 2
else
therefore, we’ll mark L = Z2 − Z4
This question has two independent parts. If you did’t succeed the first one, skip to the next one. express your answers
through the data N, m, b, w, c, ε, T only. attached useful equations
Heat capacity (”Classical” versus ”quantal”):
(a) Calculate the classical distribution function
Z 2 2
d αd β −βH1
Z1 =
e
, ZN = Z1N
L2
(It’s possible to use the natural units h = 2π). find the heat capacity c (x) of the gas.
(b) Calculate the quantal distribution function, in the limit where L is very big. Define what is a big L.
Guideline: What are the stationary states |N > of a singular particle with potential V ?
P
Calculate: Z2 = r e−βEr , ZN ' N1 ! Z1N
To calculate Z1 use the (pastorizations) of the sum.
(c) Find the heat capacity c (x) of the gas , through the distribution function you calculated in paragraph b’. check
the heat capacity behavior in high temperatures - define what is a high temperature and check if we get the
classical result you calculated in paragraph a’.
(d) Calculate the forces F2 , F1 that the particles operate on the lower and upper side of the ”box”,respectively.
Electrical field:
→
−
We add an electrical field E = εẑ. Assume the particles have charge e.
(e) Write the uni-particle hamiltonian and calculate the classical distribution function Z1 (p : Z2 , Z1 , ε)
Show that in the limit ε → 0 we get the expression, for Z1 you found in paragraph (a).
(f) Find the forces F2 , F1 that the particles operate on the lower and upper sides of the ”box”,respectively. What
is the equivalent force that operates on the system?
(g) Find the polarization E of the electron gas as a function of the electrical field (The polarization E is defined
from the equation dW = Edε)
(h) Write E (γ) = 21 E
for a week ε field. Define what is a week field.
2
E (γ) = χε · 0 ε , and through that, find the susceptibility χ
bring the expression to the form
17
Equations:
The distribution function of a free particle in a one dimensional space, in the limit where the space ”volume” L is
very large
Zqm = Zu = L
m
2πβ
12
The distribution function of a one dimensional harmonic oscillator
1
1
, Zqm =
βw
2 sin h 12 βw
Zu =
====== [39] E227: Electron gas in magnetic field
The interaction of the electron with the electromagnetic field is described by the hamiltonian (e = −|e|):
Ĥ =
2
1 p̂ − e − rB · σ̂
2m
For an homogenic magnetic field along axis Z it’s possible to choose (”Landau scaling”):
→
−
A = (0, Bx, 0)
→
−
B = ∇ × A = (0, 0, B)
Notice that the degree of freedom of the movement along axis Z doesn’t depend on both degrees of freedom of the
orbital movement in the plane vertical to axis Z. The self states of the hamiltonian are
Ĥ|p2 nrσ2 >= Ep2 nrσ2 |p2 nrσ2 >
Ep2 nrσ2
wc ≡
p2
= 2 +
2m
1
+ n wc − γBσ2
2
eB
, −∞ ≤ p2 ≤ ∞
m
n = 0, 1, 2..., σ2 = ±1
r = 1, 2...
L2
eB (Landau’s direction)
2π
Calculate the distribution function Z (β) for the electron gas, assuming that it’s possible to relate to the electrons
like Boltzman’s particles.
ZN =
1 N
Z
N! 1
From here, derive expressions for the magnetization M̃ and for the susceptibility χ. Notice the expressions you got
includes two terms
(a) Diamagnetic term because of the ”orbital” interaction of the electron with the field.
b Paramagnetic term because of the interaction of the electron spin with the field.
18
Diamagnetism is a quantal effect. To prove this argument, take the hamiltonian Ĥ omitting the spin’s degree of
freedom and calculate the distribution function by the definition of the classical statistical mechanics.
Z 3 2
d xd p −β
Z1 (β, B) ≡
H (~x, p~)
3 e
(2π)
Exercise appendix
This appendix includes a simple way to get to the expression for the self energies of the Hamiltonian Ĥ given in the
exercise.
Ĥ = H Ẑ P̂2 ; ŷ p̂y , X̂ P̂X ; σ̂2
We’ll define a new set of coordinates
dx̂
= mi[Ĥ, x̂] = p̂x − eÂx
Π̂x = m
dt
Π̂y = m
dŷ
= mi[Ĥ, ŷ] = p̂y − eÂy
dt
X̂ ≡ X̂ +
1
Π̂y
mw2
1
Π̂x
mw2
Classical Πx Πy is the mechanical momentum of the electron, and XY are the expressions for the center of the helix.
note: [Π̂x , Π̂y ] = imw2
h
i
1
X̂, Ŷ = −i
mw2
Ŷ ≡ ŷ +
1
π̂y is the conjugate of π̂x and mwc X̂ is the conjugate of Ŷ .
X, Y commute with π̂x , π̂y , therefore, mw
c
We’ll write the hamiltonian through the canonical new coordinates.
1
π̂y ; Ŷ , mwc X̂; σ̂z =
Ĥ = H ẑ, p̂z ; π̂x ,
mwc
p̂2
= 2 +
2m
π̂22
1
+ mwc2
2m 2
π̂
mwc
2 !
− γB σ̂z
notice that the freedom degree X̂ Ŷ doesn’t appear in Ĥ (as expected) because X̂ Ŷ are movement constants.
immediately we see that Ĥ includes transeltore freedom degree (direction z), harmonically freedom degree, spin’s
freedom degree and degenerated X̂ Ŷ freedom degree. There for the self energies of the hamiltonian are:
1
p2
+ n wc − rBσz
Ep2 nrσ1 = 2 +
2m
2
2
It’s possible to set Landau’s degeneration r = 1, 2... L
2π eB from the semy classical equation
Z Z
d (mwc X) dY
N =
2π
z
19
====== [40] E231: Defects in Lattice
A perfect lattice is composed of N atoms on N sites. If M of these atoms are shifted to interstitial sites (i.e. between
regular positions) we have an imperfect lattice with M defects. The number of available interstitial sites is N 0 and
is of order N . The energy needed to create a defect is w. Assume
that T << w, and show that the leading order
estimate for the typical number of defects is M = N N 0 exp −w
.
Evaluate
also the contribution of the defects to the
2T
entropy, and to the specific heat in the same level of approximation.
====== [41] E232: Tension of rotating device
The system in the drawing is in balance (Temperature T ). Find Tension F in the axis.
m
L\2
L\2
F
X
====== [42] E233: Elasticity of a rubber band
The elasticity of a rubber band can be described by a one dimensional model of a polymer, involving N molecules of
length a each, linked together end-to-end. The angle between successive links can be taken as 0 &ring; or 180 &ring;
and the joints can turn freely. The distance between the end points is x, and the temperature is T . Find the force
(tension) f which is necessary to maintain the distance x. Does the polymer try to expand or to contract? You can
use the canonical formalism in order to solve the problem. Explain why the effect is of ”entropic” nature.
====== [43] E234: Tension of a chain molecule I
A three dimensional chain molecule consists of N units, each having length a. The units are joined so as to permit
free rotation about the joints. The distance between the two ends is L, and the temperature is T . Find the tension
f acting between both ends of the chain molecule.
====== [44] E235: Tension of chain molecule II
N monomeric units are arranged along a straight line to form a chain molecule. Each unit can be either in a state α
(with length a and energy Eα ) or in a state β (with length b and energy Eβ ). Derive the relation between the length
L of the chain molecule and the tension f applied between at the ends of the molecule. Find the compressibility
20
χT = (∂L/∂f )T . Plot schematically L and χT as a function of
fa
kBT
and interpret the shape of the plots.
====== [45] E236: The zipper model for DNA molecule
The DNA molecule forms a double stranded helix with hydrogen bonds stabilizing the double helix. Under certain
conditions the two strands get separated resulting in a sharp ”phase transition” (in the thermodynamic limit). As
a model for this unwinding, use the ”zipper model” consisting of N parallel links which can be opened from one
end. If the links 1, 2, 3, ..., p are all open the energy to open to p + 1 link is ε and if the earlier links are closed the
energy to open the link is infinity. The last link p = N cannot be opened. Each open link can assume G orientations
corresponding to the rotational freedom about the bond. Construct the canonical partition function. Find then the
average number of open links hpi as function of x = Gexp [−ε/T ]. Plot hpi as function of x (assuming N very large).
What is the value of x at the transition? Study hpi near the transition: what is its slope as N − − > ∞? Derive the
entropy S. What is it at the transition region and at the transition? Do the same for the heat capacity. What is the
order of the transition?
x
L
====== [46] E261: Classical gas in [volume]-[surface] phases equilibrium
An ideal gas composed of point particles with mass m, moves between parallel boards of a capacitor. The surface of
each one of them is A and the distance between them is L, as described in the figure.
Force f~ operates on the particles, in vertical direction to the boards, which pushes the particles to the lower board.
particles can be adsorbed to the boards. the adsorbed particles move over them freely, and adsorbed potential −E
operates on them (when E > 0)in addition to force f~.
The system is in balance, in temperature T . Moreover, It’s given that the average number of the particles that move
between the boards and are not adsorbed over them is N , and their average density is n̄.
Assume that the gas particles maintain Maxwell-Boltzman statistics and therefore it’s possible to carry out the
calculations in the classical statistical mechanics frame.
Express all of your answers with E, L, n̄, T, f = |f~|, m and through physical and mathematical constants only.
(a) Calculate n (x, y, z), The density of the particles per volume unit in some point between the boards. Define the
coordinate system you use.
(b) Calculate the ratio
Φ+
Φ−
between the flow that hits the upper board and the flow that hits the lower board.
(c) Calculate and which are the densities of the particles adsorbed over the upper board and the lower board
respectively. Moreover,calculate the ratio .
Guideline: It’s possible to make the calculation through the chemical potentials of the gas between the boards
and over them.
21
A
L
f
====== [47] E262: Two level system with N particles
Consider N particles in a two level system. The one-particle energies of the two levels are E1 and E2. Consider
separately the two following cases: (i) The particles can be distinguished; (ii) The particles are identical Bosons. Find
the expectation values n1 and n2 of the occupation numbers. Discuss the special limits N = 1 and N infinity. Explain
the connection with Fermi/Bose occupation statistics.
====== [48] E301: Chemical equilibrium A-A+e
N atoms type A were put in a tank with volume V . The system got to a state of thermochemical equilibrium
AA+ + e−
Express the percent of the ionized atoms through the mass me , mA ,ionization energy ε, temperature T , and gas density
N
V .
====== [49] E302: Chemical equilibrium A[volume]-A[surface]
Consider a tank with water volume V ,and over it oil is floating.The surface contact between the water and the oil is
S. In the water and over the contact surface between the water and the oil, large molecules with mass m are moving.
Assume that the potential energy of each molecule is E1 when it’s in the water, and E2 when it’s on the boundary
between the water and the oil (E2 > 0, E1 > 0) E2 − E1 = E0 > 0.
Assume that the large molecules are classical ideal gas (which means there’s no interaction between the large
molecules).What is is the system’s temperature T ?
a Calculate the chemical potential µl of the large molecules in the water.
b Calculate the chemical potential µs of the large molecules on the boundary between the water and the oil.
c What is the ratio between the large molecules density in the water, and their density on the boundary between
the water and the oil in equilibrium?
d What is the total energy of the large molecules?
====== [50] E303: Chemical equilibrium A[volume]-A[surface], polar molecules
Consider thin gas (can be considered as an ideal classical gas) of atoms A with mass m floating over absorbed surface,
the density of the atoms is N
V and the system’s temperature is T . on the surface M absorbtion centers B. Each
22
absorbtion center can connect at most one of the gas atoms, The binding energy is E. When an atom connects to
an absorption center, a polar molecule AB, with electrical dipole d~ is created, which is free to be in one of the five
states: d~ = ±dix̂, ±diŷ, diẑ (see drawing attached). in one of the states the direction of the dipole is vertical to the
surface and in the other states it’s parallel to the surface (see drawing).An electrical field ~ε vertical to the surface is
~
operated. The interaction energy between the field and the dipole is −~ε · d.
(a) Find the coverage rate of the surface
hni
M
in two ways:
The long way - Canonical ensemble (like page 51).
The short way - Grand canonical ensemble (like page 52).
~ of the surface. The polarization is defined from dW = −D
~ · d~ε. Explain why
b Calculate the polarization D
~
~ = Phdi.
D
A
A
z
x
non adsorped
atoms
A
A
absorption
surface
A
A
y
B
adsorption where
d=di y
A
B
XB
empty
absorption site
adsorption where
d=di z
====== [51] E304: Chemical equilibrium A[volume]-A[polymer]
Consider a polymer composed with M monomers. The polymer is in a gas with temperature β and chemical potential
µ. The gas molecules can absorb the polymer’s monomers. The connection energy of the gas molecule to the monomer
is ε. The natural length of a monomer is a, when a gas molecule is absorbed to it, it’s length is b.
(a) Calculate ZN for the polymer, and from that, calculate Z.
(b) Calculate Z by the factorization.
Guideline: in paragraph b’ write the polymer’s states in this form |nr (r = 1...M ) > when nr = 0, 1. Accordingly,
if there is or there is no absorption. Write N(nr ) E(r) , and show the sum you need to calculate for Z is factorized.
(c) Calculate the average length L of the polymer.
Guideline: Express L̂ through N̂ . Calculate N ≡ hN̂ i in two ways:
Way I - to derive from Z (page
Way II - Express N̂ through n̂r and then use the probability theory and the result for hn̂r i.
====== [52] E321: Chemical equilibrium: the law of mass action
23
Write the law of mass action for the reaction H2 + D2 − − > 2HD. Determine the equilibrium constant K (T ) in
terms of the masses mH , mD andω0 the vibrational frequency of HD. Assume temperature is high enough to allow
classical approximation for the rotational motion. Show that K (∞) = 4.
====== [53] E322: Chemical equilibrium: H2 [3D] − 2H[2D]
An H2 molecule (mass 2mH ) decomposes into H atoms when it is absorbed upon a certain metallic surface with an
energy gain ε per H atom due to binding on the surface. This binding is not to particular sites on the surface: the
H atoms are free to move on the surface. Consider the H2 as an ideal gas, and express the surface density of the H
atoms as a function of the H2 pressure.
====== [54] E341: Chemical equilibrium for Fermions in a box
p2
Fermions are locked in a box of volume V . The energy of a single Fermion is Ep = 2m
+ mc2 . Pretending that there is
no conservation law for the number of Fermions, calculate the average energy density and the average particle density.
Discuss the consequences of having conservation law for the number of Fermions.
====== [55] E342: Chemical equilibrium for elementary particles
Consider the reaction F − − > A + B, where F and A are spin 21 Fermions, and B are spin 0 Bosons. The masses
are mF , mA , mB respectively, and the energy gain of the reaction is ε. Given N , and assuming ideal gases at high
temperature T , write the equations that determine the densities nF , nA , nB in equilibrium. Write the equations at
T = 0 and plot (qualitatively) the densities as functions of ε.
====== [56] E343: Chemical equilibrium for elementary particles
The reaction γ + γ. → .e+ + e− occurs inside a star, where γ is a photon and e are the positron and electron (mass m).
In general e are relativistic. Assume overall charge neutrality, and that the system is in equilibrium at temperature
T . Find an expression for the densities of e. Also find these densities in the limit T << mc2 . Solve the same problem
for the reaction γ + γ.. →→ Π+ + Π− where Π are bosons with mass M . Can these bosons become Bose-condensed
if the temperature is sufficiently lowered? Explain the result physically.
====== [57] E344: Chemical equilibrium for elementary particles
Consider the reactionγ + γ. → .e+ + e− , whereγ is a photon ande are the positron and electron (mass m). Assume a
constant density difference n0 = n− − n+ , and that the system is in equilibrium at temperature T . Derive equations
from which the densities n− andn+ can be determined in terms of n0 .
(Hint: Find first an expression for the product n− n+ ). Find the Fermi momentum pF at T = 0 in the nonrelativistic
case. Specify the condition on n0 that allows a nonrelativistic limit. Extend the results to the finite temperature
pF 2
2
2m << T << mc
====== [58] E345: Chemical equilibrium for elementary particles
In a certain medium, there were at the beginning N neutrons per volume unit. Some of them decomposed according
to
np + e− + ν̄
All of the particles are fermions with spin 21 . Consider masses mn , mp , me , mν = 0 and Consider temperature T . Mark
with N 0 the neutron’s density in a thermo chemical state of equilibrium. Write the equilibrium equation for N 0 in
four cases, defined by the criterions
(I) The particles are nonrelativistic (except nitrino) The particles are hyper relativistic (negligible mass).
(II) The temperature is zero.
24
The temperature is high (Boltzman proximity).
Define the conditions for the assumption’s validity. write the equations using the data only! there’s no need to solve
the equations.
====== [59] E401: Heat capacity of ideal Bose gas
N bosons with spin 1 and mass m are in a tank with volume V . The gas is in thermic equilibrium in temperature T .
Write explicit expression using N, m, V, T for Tc , and also explicit expressions for the chemical potential, the energy
and the pressure in the boundaries Tc << T, T < Tc . Get an explicit expression for Cv in the temperatures boundaries
0 ≤ T ≤ Tc and complete the missing details in the graph.
Cv (T = Tc ) ≥=?
Cv (Tc << T ) =?
CV
Tc
====== [60] E402: Bose gas in strong magnetic field
Consider gas of N bosons with spin 1 in a box with volume V . A strong magnetic field B is present. Assume that
the temperature T is low, so the gas is in a state of condensation. Find the occupation of the spin states
(N−1 + N0 + N+1 = N )
Define what is ”strong magnetic field”.
H1 =
p2
− γBSz
2m
====== [61] E403: Charged Bose gas in box with potential difference
Consider N bozons with mass m, positive charge e and spin 0. The particles are in a tank in thermic equilibrium,
and temperature T . The tank has two zones A and B, The volume of each zone is L3 .A battery creates potential
difference V between the zones. The potential in every zone is homogenous.
Find the condition on N , so if V = 0 then there’s no condensation, but if V = ∞ then there’s condensation.
23
23
answer: L3 ζ 23 · mT
< N < 2L3 ζ 32 · mT
2π
2π
Below, assume that the particles in zone A are in a condensation state and the particles in zone B can be described
in the Boltzman proximity frame.
(a) What is the number of the particles in zone B. What is the condition for V , so that Boltzman proximity will
be valid
(b) What is the number of the particles in zone A. How many of them are in condensation state?
(c) Show that the condensation in zone A as long as Vc < V . Find an explicit expression for Vc .
25
remark: This problem is formally identical to the ’bozons’ problem with spin
[zone B] down, [zone A] up, and potential difference eV γB.
B
isolation
ring
1
2
in magnetic field.
A
====== [62] E404: Small oscillations of piston in cylinder with Bose gas
Consider a cylindric tank with length L and basis area A, divided to two parts by a partition with mass m,that is
free to move. On one side of the of the partition there are Na identical bozons with mass ma and from the other side
Nb identical bozons with mass mb
(*) Assume that gas a can be described in the Boltzmann proximity frame.
(**) Assume that gas b is in condensation state.
(a) Find the location of the partition χ in an equilibrium state when the temperature is T under the assumptions
(*) (**).
(b) Write the condition that allow the assumption (*) to be valid in a state of equilibrium.
(c) Write the condition on temperature T so the assumption (**) is indeed valid.
(d) Find the frequency of the partition’s small oscillations around the equilibrium point.
Use only M, T, Mb , Nb , ma , Na , A, L.
L
x
A
Na
ma
N b mb
====== [63] E421: Bose gas for general dispersion relation
Consider an ideal Bose gas in d dimensions whose single particle spectrum is given by ε = C|p|s . Find the condition on s, d for the ence of Bose-Einstein condensation. In particular show that for nonrelativistic particles
in
two dimensions (s = d = 2) the system does not exhibit Bose-Einstein condensation. Show that P = ds VE ) and
CV (T − − > ∞) = ds N
====== [64] E422: Heat capacity of Bose gas
2
CP
1 (z) g 5 (z) /2
3 (z)
Consider an ideal Bose gas and show that the ratio C
=
3g
g
where z is the fugacity. Why is
V
2
2
2
CP − − > ∞ in the condensed phase?
CP
Find γ in the adiabatic equation of state. Note that in general γ 6= C
.
V
26
====== [65] E423: Heat capacity of He4
The specific heat of He4 at low temperatures has the form Cv = AT 3 + B (T ) exp
−4
T
. What can you deduce about
the excitations of the system? (assume that the density of excitation modes has the form g (ω) ω p 0. What would be
the form of Cv for a similar system in a two dimensional world?
====== [66] E424: Bose gas in a uniform gravitational field
Consider an ideal Bose gas of particles with mass m in a uniform gravitational
field of acceleration
h
ig. Show that the
πmgL 8
3
critical temperature for the Bose-Einstein condensation is Tc = Tc0 1 + 9 1/ζ 2
1/2 , where L is the
Tc0
height of the tank, mgL << kBTc0 and Tc0 = Tc0 (g = 0).
1
Hint: g 32 (z) = g 23 (1) − 2 (−π ln (z)) 2 + O (ln (z)).
Show that the condensation
is accompanied by a discontinuity in the specific heat at Tc with the result
3 πmgL 9
4CV = − 16π ζ 2 N T c0 1 .
2
∂z
Hint: 4CV is due to discontinuity in ∂T
N,V
====== [67] E451: State equations for ideal Fermi gas
N fermions with 21 spin and mass m are in a tank with volume V . The gas is in thermic equilibrium in temperature
T.
Assume it’s possible to relate to the temperature as a low one, and find explicit expressions, up to second order in
temperature, for the state equations
N
µ = µ T;
V
E = E (T, V ; N )
P = P (T, V ; N )
Define what is a low temperature. Use only N, m, V, T . Write expressions also for the heat capacity Cv and the
compressibility KT .
1 ∂V
KT ≡
V ∂P T
Guideline: Write an expression for N = N (βµ) and find µ (β, N/V ) while keeping terms up to O T 2
Similar to the calculation of N (βµ) it is possible
to calculate E (βµ) up to second order in temperature.
Now there’s to place the expression for µ T ; N
you found earlier, and write the result as a development of T while
V
keeping terms op to second order only! This is the ”trickiest” phase..., You’ll have to use the development
α
(1 + χ) = 1 + αχ +
α (α − 1) 2
χ + θ χ3
2
several times and to make sure not to losing the first and the second order terms during the algebra process.
====== [68] E452: Ideal Fermi gas in 2D space
N fermions with mass m and spin 21 are in a two dimensional box with area A.
Below, we’ll use the usual calculation strategy to find the gas state equations. Show that:
µ
m
N (β, µ) = A T ln 1 + e T
π
27
Guideline: Use variables exchange X = eβ(E−µ) and use the integral
N = Am
π µ. Explain that result intuitively by looking at
energy EF ≡ µ (T → 0). Show that in low temperatures
µ (T ) ≈ EF − T e−
R∞
dx
x(x+1)
= ln
1+X1
X1
. Take T = 0 and find
g (E). Find the chemical potential µ T, N
A and the fermi
1
Er
T
While in high temperatures we get a result consistant with the general expression you developed in the Boltzman’s
proximity frame.
Find E (β, µ) and P (β, µ) in temperature zero, and by placing the expression for µ T = 0, N
A . Show that the
following results
2
2
π 1 N
π 1 N
E=A
,P =
m2 A
m2 A
Clarify yourself why in zero temperature P ∝
1
A2
while in high temperature P ∝
1
A.
====== [69] E453: Ideal Fermi gas in 2D box
N fermions with mass m and spin 21 are in a box , it’s dimensions are L × L × γ, (γ << L). The system is kept in low
temperature T . Find the pressure on the box walls and the heat capacity of the gas in conditions where it’s possible
to consider as 2 dimensional. The one particle states are |npx py i
2
p2y
p2
1
π
Enpx py =
R + x +
2m γ
2m 2m
It’s possible to consider the gas as a 2 dimensional as long as the occupation of the levels 1 < n is negligible.
2
2
m 1
π
1
π
g (E) = 2 · A ,
<E <4·
2π 2m γ
2m γ
2
1
π
It’s convenient to take in to calculation the E = 2m
like an attribute level to the uniparticles state energy, but
γ
there’s a need to be careful when you calculate the pressure on the upper and lower walls.
E
π
1
2
2m ( l )
0
n=1
n=2
n=3
n=4
====== [70] E454: Ideal Fermi gas in semiconductor
Given electron gas in a metal. The state uniparticle density function is
g (E) = gv (E) + gc (E)
Assume that near the gap exists area the proximity
gc (E) ' L
1
1
V
· (2mc ) 2 (E − Ec ) 2
(2π)
28
gv (E) ' L
1
1
V
· (2mv ) 2 (Ev − E) 2
(2π)
We define an occupation function of the holes in the valence band
P k (E − µ) ≡ 1 − f (E − µ)
Assume that the metal is attached to the reservoir with temperature β and chemical potential µ. Assume that that
Ev + T << µ << Ec − T and justify the boltzman’s approximation in the conductance band f (E − µ) ' e−β(E−µ) ,
and in the valence band f k (E − µ) ' eβ(E−µ) . Now, find in the usual way the functions Nc (β, µ) , Nch (β, µ). The
result is
m 32 3
µ − Ec
c
T 2 exp
Nc (β, µ) = 2 · V
2π
T
m 32
v
Nv (β, µ) = 2 · V
2π
µ − Ev
T exp −
T
3
2
Now, assume you begin with closed system in temperature zero. It’s given that then, the valence band is full and the
conductive band is empty. now you raise the temperature (The system is empty!). Assume mc = mv and show that
µ (T ) =
Nc =
Ec − Ev
2
Nvh
Eg /2
∝ T ∝ exp −
T
3
2
E
E=E r
µ
gap
E=Ev
E=0
====== [71] E455: Magnetic properties of T = 0 electrons (Pauli)
N electrons are in a box as follows: (I) two-dimensional with area A; (II) three dimensional with volume V . The
temperature is zero. We create a magnetic field B. The electrons behave like an ideal fermi gas. The one particle
hamiltonian is
→
−
p2
H1 =
− rBσz
2m
(a) Show a schematic drawing of the uniparticle states density function. Distinguish between a spin up conditions
and a spin down ones.
(b) Determine which of the following graphes describes the magnetization M (B) in each one of the cases (I) (II)
and complete the missing details (Ms =?, Bc =?, χ =?).
Use only γ, m, V, A, N .
29
Ms
χ
M
χ
s
Bc
Bc
χ
Ms
Bc
====== [72] E456: Electrons in V (x, y, z) = x2 + y 2 box (effectively 1D)
Given N electrons kept between capacitor’s boards by potential
1
mw2 x2 + y 2 0 ≤ z ≤ L
2
V (x, y, z) =
∞
else
The system is in thermic equilibrium state, in temperature T ≈ 0. Find the force the works on the capacitor’s boards
assuming it’s possible to relate to the gas as a uni dimensional. Explain this assumption and cast condition on N so
that assumption will be valid.
Guideline: Find first the one particle states system and demonstrate it using a schematic drawing similar to what has
been done in the class. Use the data N, L, m, w only.
====== [73] E457: Fermi gas in 2D + 3D connected boxes + gravitation
Given a ’mezoscopic’ box with L × L × `. L is a macroscopic size while ` is a mezoscopic size and s ` << L. We insert
N fermions with spin 12 and mass m in to a box. in paragraphs a-d assume the temperature is T = 0. (mezoscopic
size is very small in relation to a large macroscopic size, in relation to microscopic size).
(a) Describe the uniparticle states density and note what is the energy field which the uniparticle states density in
it are like a particle in a two dimensional box.
(b) Find the fermi level EF in condition it’s possible to relate to the box as a two dimensional one. What is the
maximum number of Nmax fermions it’s possible to insert in to the box in this condition.
(c) For N < Nmax , as above, find the pressure P on the sided walls of the box, and the force F on the horizontal
walls. The box, as above, is attached to a tank with the dimensions L × L × L which is D higher from the box.
consider gravitation. (assume a very strong gravitation field g, so the question will be reasonable from the order
of magnitude).
(d) In the conditions of paragraph b’, i.e. N = Nmax , What is the minimum hight Dmin to place the tank so all of
the fermions will stay in the box?
(e) The temperature of the system was raised a little bit. Assume temperature T and also N = Nmax , D = Dmin .
as a result, some of the particles that were in the mezoscopic box, transferred to the tank. Estimate the number
N 0 of these particles. For that, assume that the chemical potential of the system is almost not changing as a
result of raising the temperature. Express your answers using T, g, m, L, ` only.
given:
Z
0
∞
1
x2
dx = Γ
x
e +1
3
1
3
1− √
ζ
= 0.678
2
2
2
30
tank
L
L
D
L
box
L
l
L
====== [74] E458: Quantum Fermi gas in gravitation field of a star
Consider a neutron star as non-relativistic gas of non-interacting neutrons of mass m in a spherical symmetric equiG
librium configuration. The neutrons are held together by a gravitational potential −mM
of a heavy object of mass
r
M and radius r0 at the center of the star (G is the gravity constant and r is the distance from the center).
Assume that the neutrons are classical (Boltzmann) particles at temperature T and find their density n (r) at r > r0 .
Is the potential confining? [By definition, for a confining potential there a solution with n (r) − − > 0 at r − − > ∞].
Next, consider the neutrons as fermions at T = 0, and find n (r). Is the potential confining? Extend your result to
the case T 6= 0, and discuss the connection with the Boltzmann limit.
====== [75] E501: One dimensional hard sphere gas
N beads with diameter a are threaded over a wire, length L. Assume 1 << N, N a << L. The system is in thermic
equilibrium, temperature T . Find the force F that operates on the edges of the wire. Write the result in the shape
T
F = N Lef
. Express Lef f using the data and explain it’s physical meaning.
f
Hints:
(a) While calculating the distribution function, notice that if the beads permutation were permitted, it was causing
Z → N !Z.
(b) Assume that a typical distance between two beads is much bigger than a.
(c) To calculate a product from the shape A = ΠN
n=1 an look at the sum ln A and use reasonable proximities to
calculate them.
31
====== [76] E502: Virial coefficients
Find the second virial coefficient for: Ideal Bose gas; Ideal Fermi gas; Classical hard sphere gas.
====== [77] E503 : 2D coulomb gas
N ions of positive charge q, and N ions with negative charge −q are constrained to move in a two dimensional square
of side L. The interaction energy of charge qi at position ri with another charge qj at rj is −qiqjln|ri − rj| where
2
qi, qj = q. Prove that Z (β, L) = L(N (4−βq ) f (β).
2
Estimate f (β) for the case N = 1, and explain what happens if β1 < q2 . Discuss now the case N >> 1. Explain
2
what happens if β1 < q4 .
Hint: The partition function is in general a monotonic increasing function of the volume. It follows, for this particular
model system, that f (β) = ∞.at low temperatures. The N = 1 case can be used in order to illuminate the reason
for this divergence. Explain what is the additional ingredient that is required in order to stabilize the physics of this
model.
====== [78] E563: Correlation function in 1D Ising model
Consider the Ising model in one dimension with periodic boundary condition and with zero external field. Evaluate
hσi i by using the transfer matrix method. What is hσi i at T = 0?
Find the correlationfunction
G (r) = hσ1 σ1 + ri and show that in the limit of infinite sample (N − − > At∞) it has
the form G (r) exp
−r
ξ
. what temperature ξ diverges and what is its significance?
====== [79] E564: Ising with long range interaction:
Consider the
Ising model of magnetism with long range interaction: the energy of a spin configuration is given by
J
+ Σi,j si sj − hΣi si where J > 0, and the sum is on all i and j, not restricted to nearest neighbors. The
E = − 2N
energy E in terms of m = ΣNi si can be written as E (m, h) = − 12 JN m2 − hN m.
J
Explain why N is included in the definition of the coupling N
.
Evaluate the free energy F (m; T, h) assuming that it is dominated by a single m. Hence m becomes a variational
parameter. By minimizing F find m (h, T ) and a critical temperature Tc . Plot qualitatively m (h) above and below
the transition. Plot qualitatively F (m) for T > Tc and T < Tc with both h = 0 and h 6= 0. Explain the meanings of
the various extreme. Expand F (m; T, h = 0) up to order m4 . What is the meaning of the m2 coefficient?
====== [80] E565: Ising model with disorder
Consider a one dimensional Ising model of spins σi = 1, where i = 1, 2, 3, ..., N and σN +1 = σ1 . Between each two
spins there is a site for an additional atom, which if present changes the coupling J to J (1 − λ).
The Hamiltonian is then H = −JΣi σi + 1 (1 − λni ), where ni = 0 or 1. There are N 0 = Σni atoms, so that N 0 < N .
Evaluate the partition sum by allowing all configurations of spins and of atoms. If the atoms are stationary impurities
one needs to evaluate the free energy F for some given random configuration of the atoms: Then one can average F
over all configurations. Evaluate the averaged F . Find the entropy difference between the two results and explain its
origin.
32
====== [81] E571: Mean field: ferromagnetism with classical spins
Apply the mean field approximation to the classical spin-vector model H = −JΣsi sj − hΣsi where si is a unit vector
and i, j are neighboring sites on a lattice. The lattice has N sites and each site has z neighbors. Find the magnetization
M = hcosθi i in the mean field approximation, where θi is the angle relative to an assumed orientation of M . Find the
transition temperature Tc by solving for M at h = 0. Find M (T ) for T < Tc to lowest order in Tc − T and identify
β
the exponent β in M (Tc − T ) . Of what order is the transition? Find the susceptibility χ (T ) at T > Tc and identify
−γ
the exponent γ in χ (Tc − T ) .
====== [82] E572: Mean field: antiferromagnetism
Antiferromagnetism is a phenomenon akin to ferromagnetism. The simplest kind of an antiferromagnet consists of
two equivalent antiparallel sublattices A and B such that terms of A have only nearest neighbors in B and vice versa.
Show that the mean field theory of this type of (Ising) antiferromagnetism yields a formula like the Curie-Weiss law
−1
for the susceptibility χ (Tc − T ) , except that T − Tc is replaced by T + Tc where Tc is the transition temperature
into antiferromagnetism (Neel’s temperature). Below Tc of an antiferromagnet drops again. Show that in the above
?the susceptibility χ immediately below T mean field theory the rate of increase of χc is twice the rate of decrease
immediately above. Assume that the applied field is parallel to the antiferromagnetic orientation.
====== [83] E573: Mean field: ferroelectricity
Consider electric dipoles on sites of
along one of the crystal axes ph100i. The
a simple cubic lattice which point
interaction between dipoles is U = p1 p2 − 3 (p1 r) (p2 r) /r2 / 4πr3 where r is the distance between the dipoles and
r = |r|. Assume nearest neighbor interactions and find the ground state configuration. Consider either ferroelectric
(parallel dipoles) or anti-ferroelectric alignment (anti-parallel) between neighbors in various directions. Develop a
mean field theory for the ordering and for the average polarization P (T ) at a given site at temperature T . Find the
critical temperature Tc and the susceptibility χ at T > Tc assuming an electric field in the h100i direction.
====== [84] E574: Mean field with field h (r)
Consider a ferromagnet with magnetic moments m (r) on a simple cubic lattice interacting with their nearest neighbors.
The ferromagnetic coupling is J and the lattice constant is a. Extend the mean field theory to the situation that the
magnetization is not uniform but is slowly varying: Find the mean field equation in terms of m (r) and its gradients
(to lowest order). Assume an external magnetic field h (r), that in general can be a function of r. Consider T > Tc
where Tc is the critical temperature, so that only lowest order in m(r) is needed. For a small h(r) find the response
m (r), and evaluate it explicitly in two limits: (I) The response is characterized by the susceptibility for uniform h,
and (II) the response is characterized by the correlation function (why?) for h (r) δ 3 (r). In the latter case identify
the correlation length.
====== [85] E575: Ferromagnetism for cubic crystal
A cubic crystal which exhibits ferromagnetism at low temperatures, can be described near the critical temperature
Tc by an expansion of a Gibbs free energy G (H, T ) = G0 + 21 rM 2 + uM 4 + vΣMi4 − HM , where H = (H1, H2, H3)
is the external field, and M = (M 1, M 2, M 3) is the total magnetization, and r = a (T − Tc ). The other parameters,
namely G0 and a > 0 and u > 0 and v, are independent of H and T . The constant v is called the cubic anisotropy,
and can be either positive or negative. At H = 0 find the possible solutions of M which minimize G, and the
corresponding expressions for G (0, T ). These solutions are characterized by the magnitude and direction of M .
Show that the region of stability is u + v > 0. Determine the stable equilibrium phases when T < Tc for the different
cases v > 0 and −u < v < 0. Show that there is a second order phase transition at T = Tc , and determine the critical
indices α, β and γ for this transition. These are defined by the expressions CV, H = 0 |T − Tc |−α for both T > Tc
β
and T < Tc , and |M |H = 0 (Tc − T ) for T < Tc and χij = (∂Mi /∂∂Hj ) δij |T − Tc |−γ for T > Tc .
33
====== [86] E581: Mechanical model for symmetry breaking
The following mechanical model illustrates the symmetry breaking aspect of second order phase transitions. An
air tight piston of mass M is inside a tube of cross sectional area a. The tube is bent into a semicircular shape
of radius R. On each side of the piston
there is an ideal gas of N
atoms at a temperature T . The volume to
the right of the piston is aR π2 − ϕ while to the left is aR π2 + ϕ . The free
energy of the system has the form
π
π
3
3
F (T, ϕ) = M gR cos ϕ − N T 2 + ln aR 2 − ϕ /N λ + ln aR 2 + ϕ /N λ .
Explain the terms in F . Interpret the minimum condition for F (ϕ) in terms of the pressures in the two chambers.
Expand F up to 4th order in ϕ. Show that there is a symmetry breaking transition and find the critical temperature
Tc . Describe what happens to the phase transition if the number of atoms on the left and right of the piston is
N (1 + δ) and N (1 − δ) respectively. Note that it is sufficient to consider |δ| << 1 and include a term ϕδ in the
expansion. At a certain temperature the left chamber (containing N (1 + δ) atoms) is found to contain a droplet of
liquid coing with its vapor. Which of the following statements may be true at equilibrium:
(a) The right chamber contains a liquid coming with its vapor;
(b) The right chamber contains only vapor;
(c) The right chamber contains only liquid.
M
φ
R
a
====== [87] E601: Effusion from box with Bose gas and magnetic field
Bosons with mass m, spin 1 and magnetic moment r in a condensation state, in temperature T are in a box. The box
is in strong magnetic field B. We drill in the box a hole which it’s area is δA.
By a Shtrelen - grelech tool, we separate the emitting particles flow from the hole to three sheafs, then we turn in to
three tanks.
The gathering of the particles duration time in the tanks is t. How many particles will be in each tank?
express your answer using only m, r, B, T, δA, t.
Guideline: Write the hamiltonian of a single particle. Explain to yourself what are the uniparticle states. find what
is the velocity function NS (0) of the particles with spin St = 1, 0, −1. Define what is a strong magnetic field. Use the
following immediate integrals:
Z ∞
Z ∞
2
1
π2
x3
x3 e−x dx = ,
dx =
2
x
2 0 e −1
12
0
====== [88] E602: The system of e404 with hole in one side
Given the system that was presented in exercise e404. relate to the two next cases:
case I: A little hole, with area δA was pierced in the left basis of the box.
case II: A little hole, with area δA was pierced in the right basis of the box.
In each case, answer the next questions.
(a) What is the velocity function N (v) of the particles emitting from the hole?
(b) What is the flow of the particles going out from the hole per unit of time?
(c) Is the partition going to move? (be careful with your answer!). If so, what will be it’s velocity? (assume the
hole area is small so the process is quasi static and in every moment and moment, there’s an equilibrium).
34
A useful integral:
R∞
0
xdx
ex −1
=
π
6
====== [89] E603: Thermionic emission of electrons from a metal
g (E) = gmetal (E) + gvaccum (E)
Find the rate of the emitting electrons from a metal with work function W , when the temperature is ’low’ and equal
to T .
Metal
vacuum
E
extended states
in the vacuum
V(x)
µ
localized states
in the metal
====== [90] E604: Effusion of electrons from a box in magnetic field
Electrons are in a given box, in the presence of a magnetic field B. We drill a hole in the box and measure the spin
p2
4Nt
using m, r, B, βµ. Assume H1 = 2m
− rBσz .
of the emitting electrons, with velocity v < v 0 < v + dv. Find α ≡ 4N
v
====== [91] E605: Radiation from 1D balckbody fiber
Given an optic fiber with length L and it’s section area is A. The fiber is in thermic equilibrium in temperature T .
Assume the fiber is a one dimensional medium for the electromagnetic field (relate to the magnetic radiation as a
photon gas).
(a) What is the condition (on temperature)so the one dimensional proximity will be valid?
35
(b) What is the electromagnetic energy density per length unit?
(c) What is the radiation pressure on the fiber edges?
(d) Assuming that the radiation can emit from the boundary of the fiber, what is the emitting radiation flow (energy
per time unit)?
(e) What is the spectral distribution J (w) of the emitting radiation flow?
(f) What is the entropy and what is the heat capacity of the system?
Given:
Z
∞
0
ex
x
π2
dx =
−1
6
====== [92] E606: Generalize J-incident formula for
1D
2D
box
Generalize the equation for J incident for the cases of two dimensional gas and one dimensional gas. in each case,
note what is the ’volume’, what are the units of J and especially, what is the geometric factor in the equation.
====== [93] E607: Landauer formula for 1D conductance
1D design with passage coefficient |t2 | is attached to electromotive force ϕ by leads which are an ideal 1D conductors,
as described in the following drawing. Show that the conductance of the sample G,which is defined through the
equation I = Gϕ (I = current, ϕ =electromagnetic force) is given by the Landauer formula.
G=
e2 2
|t |
2π
(low temperatures).
Left
lead
Sample
Right
lead
emf=
====== [94] E608: Einstein relation for the conductivity of electrons
Given a metal design. We mark with ϕ (x) the electrical potential in the sample and with N (x) the spacial density
of the electrons in the design. According to the kinetic theory s
J~ (x) = −σ∇% − eD∇N
36
σ is the conductivity and D is the diffusion coefficient. In an equilibrium state J~ (x) ≡ 0, especially in a state of
σ
equilibrium that we get in the presence of outer field ϕ (x) 6= const and therefore has to : eD
= ∇N
∇ϕ .
Use the principles of the statistical mechanics to show that from here derives
Z
σ
= −e2 dE g(E)f 0 (E − µ)
D
σ
= e2 g (Er )
low temperatures D
2
σ
High temperatures D
= N eT
g (E) is the uniparticles states density per volume unit. Hint - notice that
Z
N (x) = g (E − eV (x)) dEf (E − µ)
====== [95] E701: Random walk with correlations
The total displacement of a particle is a sum over steps X (t), where t is discrete. If we define the velocity as
dx
v (t) = X(t)
τ 0 , where τ 0 is the time between steps, then the random walk is described by the equation dt = v (t).
(a) Given the velocity-velocity
correlation function c (t2 − t1) = hv (t1) v (t2)i, write down an expression for the
rh
i
2
spreading S (t) =
h(x (t) − x (0)) i .
(b) Find an expression the diffusion coefficient, assuming that c (τ ) is short range.
(d) More generally, show that
dS(t)
dt
is equal to the [−t, t] integral of c (τ ).
(e) Assume that c (τ ) has zero integral and power law tails c (τ ) =
S (t) depending on the value of α.
−c0
τα .
Determine the sub-diffusive behavior of
====== [96] E801: Fluctuations for stationary process
Prove that the Fourier components of a stationary noisy signal have a variance which is proportional to the time of
the measurement.
Show that the coefficient of proportionality is just the power spectrum (defined as the Fourier transform of the
correlation function).
====== [97] E802: Fluctuations of harmonic oscillator
A particle (x, p) of mass m is bounded by a harmonic potential of frequency Ω, and experiences a damping with a
coefficient eta. It is subject to an external force f (t).
(a) Write the generalized susceptibility that describes the response of x to the driving by f (t).
(b) Using the F D relation deduce what is the power spectrum of the x fluctuations.
(c) What are the fluctuations of the velocity?
(d) Show that in the limit η − − > 0 the second moments hx2 i and hv 2 i are as expected from the canonical formalism.
====== [98] E803: Diffusion of Brownian particle
Brownian motion is formally obtained as the Ω − − > 0 limit of the previous problem.
(a) Calculate the velocity-velocity correlation function of the Brownian particle in the limit of high temperature.
37
(b) Show that it is an exponential function, and identify the correlation time.
p
(c) Write the relation between the dispersion [h(x (t) − x (0))i2] and the velocity correlation function.
(d) Deduce that the particle diffuses in space and write the expression for the diffusion coefficient.
(e) Show that in the limit of zero temperature the velocity-velocity correlation function has a zero integral and
power law tails (recall Exe.701).
(f) In the latter case deduce that instead of diffusive spreading one should observe slow logarithmic growth of the
variance.
====== [99] E804: Nyquist theory for a ring / RL-circuit
Derive the Nyquist expression for the current-current correlation function in a closed ring, taking into account its
inductance. Use the following procedure:
(a) Write the R-L circuit equation for the current I, where the flux Phi(t) through the ring is the driving parameter.
(b) Identify the generalized susceptibility, and observe that it is formally the same expression as in the problem of
Brownian motion.
(c) Calculate the current-current correlation function taking the classical / high temperature limit.
====== [100] Exam2001A
Given two balls in a very long horizontal tube with length L. Each ball’s mass is m.
One ball has charge q and the other −q. The ball’s radius is negligible and the electrostatic attraction between them
is negligible.
The balls are rigid and can’t move through one each other. The balls are attached to a drop of fluid with face tension
causing a constant gravitation force between them (The force is not dependent on the distance between the balls).
The system is in an electric field. The field is horizontal (parallel to the tube). The temperature is T .
Write the hamiltonian H (x1, x2, p1, p2) of the system.
Draw schematically the potential V (r), where r = x2 − x1.
Calculate the distribution function Z (β, E).
Assume the tube is very long. Define this assumption.
Find the distribution function ρ (r) of the distance between the balls.
What is the average distance hri between the balls?
Find the polarization P (E) of the system as a function of the electrical field (Use the distribution function for help).
Open the polarization up to first order P (E) = P0 + χE . (Write P0 and χ).
This development is valid in a weak field. Define what is a weak field.
====== [101] Exam2001B
N Bozon’s molecules B are inserted in to a box with volume V . Each molecule is composed from two atoms A.
The atom’s mass is m and the binding energy of the molecule is ε.
The system’s temperature is T . Some of the molecules are falling apart. Assume there are molecules in condensation
state. Assume the un connected atoms can be treated in the Boltzman’s proximity frame.
How many (un connected) atoms are in e box?
What is the condition for the Boltzman’s proximity to be valid for the atoms?
How many molecules are ed in arousing states?
What is the condition for the condensation assumption to be valid?
What is the pressure on the box’es walls?
Who pressure’s the most (The molecules ? the atoms ?).
38
====== [102] Exam2001C
A large number M >> 1 of adsorption sites are ordered along the length of a ring.
Between every two adsorption sites a spin σi = ±1 is located. The ring is soaked in a gas temperature T and it’s
chemical potential. At most, one particle gas can be adsorbed to a given site ni = 0, 1.
The adsorption energy is ε > 0 if the two adjacent spins are in the same direction.
The adsorption energy is −ε if the adjacent spins are in the opposite direction.
Assume positive adsorption energy that gives ”priority” to the promagnetical order.
Write the expression for the energy E [σi , ni ] of a given configuration.
Write the transfer matrix T that is shown in the calculation of the grand canonical distribution function Z (β, µ) of
the system.
(Guideline: carry out the sum over the occupation options. Define T for the remaining sum over she spins).
Find the self values of the transfer matrix.
Write expressions for the basic function F (T, µ) and for the adsorbed particles N = Σhni i.
Write an expression for the correlation length ξ that characterize the arrangement of the spins in the system.
====== [103] Exam2002A: Classical Canonical Formalism
Particle of mass m and charge e is free to move on a ring of radius R. The ring is located in the x − y plan. The
position of the particle on the ring is x = R cos (θ) and y = R sin (θ). There is an electric field E is the x direction.
The temperature is T .
(a) Write the Hamiltonian H (θ, p) of the particle.
(b) Calculate the partition function Z (β, E).
(c) Write an expression for the probability distribution ρ (θ).
(d) Calculate the mean position hxi and hyi.
(e) Write an expression for the probability distribution ρ (x). Attach a schematic plot.
(f) Write an expression for the polarization. Expand it up to first order in E, and determine the susceptibility.
1
2π
Z
2π
exp (z cos (θ)) dθ = I0 (z)
0
I00 (z) = I1 (z)
I0 (z) = 1 +
1
1
z2 +
z 4+...
4
64
====== [104] Exam2002b: Quantum Bose Gases
A cylinder of length L and cross section A is divided into two compartments by a piston. The piston has mass M
and it is free to move without friction. Its distance from the left basis of the cylinder is denoted by x. In the left side
of the piston there is an ideal Bose gas of Na particles with mass ma . In the right side of the piston there is an ideal
Bose gas of Nb particles with mass mb . The temperature of the system is T .
(*) Assume that the left gas can be treated within the framework of the Boltzmann approximation.
(**) Assume that the right gas is in condensation.
(a) Find the equilibrium position of the piston.
(b) What is the condition for (*) to be valid?
(c) Below which temperature (**) holds?
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(d) What is the frequency of small oscillations of the piston.
Express your answers using L, A, Na , Nb , ma , mb , T, M .
====== [105] Exam2002C: Phase transition of Ising cluster
Consider a cluster of N spins si = ±1. The interaction between any two spins is −si sj , with > 0.
PThe interaction
of each spin with the external magnetic field H is −Hsi . The total magnetization is defined as m =
si . The inverse
temperature is β.
P
(a) Show that The partition function can be written as Z (β, H) = m g (m) exp 21 Bm2 + hm . Express g (m) and
B and h using (N, , H, β).
b
(b) Assume that
P B = N , define the magnetization as M =
Z (b, h) = M exp (−N ∗ A (M )).
m
N,
and write the partition function as
Write the expressions for A (M ) and for its derivatives A0 (M ) and A00 (M ).
(c) Determine the critical temperature Tc , and write an equation for the mean field value of M . Make a qualitative
plot of A (M ) below and above the critical temperature.
(d) Write an approximation for A (M ) up to order M 4 . On the basis of this expression determine the temperature
range where mean filed theory cannot be trusted. Hint: you have to estimate the variance hM 2 i in the Gaussian
approximation. What happens with this condition in the thermodynamic limit (N → ∞)?
3
1
1
2 ln ((1 + x) (1 − x)) ≈ x + 3 x + ...