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Transcript
```Chapter 7
Statistical physics in equilibrium
7.1
Introduction
The microscopic systems described in thermodynamics consist of many components (atoms,
molecules, electrons, ...) whose behavior can be described in terms of classical or quantum
mechanical equations of motion. Therefore, one expects to derive the laws of thermodynamics in terms of microscopic data. This is the subject of statistical physics.
Since the number of particles in a macroscopic system is huge (∼ 1023 particles in a cubic
centimeter), one cannot expect an exact solution of the equations of motion. Not only
because one would need a supercomputer to deal with 1023 coupled equation, but also
because it is not at all clear how useful it would be to deal with the large amount of data
resulting out of these equations.
Therefore, in statistical physics we will be dealing with average quantities and probability
predictions. Fluctuations around the average value are also possible within the statistical
physics, but for large numbers of particles, N → ∞, these fluctuations will become less
and less important. In order to make rigorous predictions, the statistical physics needs to
have the limit N → ∞.
We can distinguish the following branches of statistical physics.
• Classical statistical physics: the microscopic equations of motion of the particles
are given by classical mechanics.
• Quantum statistical physics: the microscopic equations of motion of the particles
are given by quantum mechanics.
• Statistical physics in equilibrium: the macroscopic variables are time-independent
and the macroscopic world can be described in terms of microscopic average values,
distribution functions or probabilities.
• Statistical physics in non-equilibrium: the macroscopic variables are timedependent. In this case, the microscopic treatment is more complicated.
In this course, we will restrict ourselves to classical and quantum statistical physics in
equilibrium.
73
74
CHAPTER 7. STATISTICAL PHYSICS IN EQUILIBRIUM
7.2
The statistical approach
Starting point: a classical system of N particles with N >> 1.
In classical mechanics, the state of a particle at any instant of time is specified by its
position ~q and momentum p~ → (~p, ~q). Therefore, in the N -particle system, we will
have 6N degrees of freedom:
q1 , . . . , q3N , p1 , . . . , p3N .
The motion of the particles is governed by the Hamiltonian
H(q1 , . . . , q3N , p1 , . . . , p3N ),
and the equations of motion are
∂H
,
∂pi
∂H
,
= −
∂qi
q̇i =
ṗi
i = 1, . . . , 3N.
For the set of N atoms, the Hamiltonian is given by
N
X
p~ 2i
1X
H(~p, ~r) =
U (~ri − ~rj ),
+
2m 2 i6=j
i=1
where U (~r) is the interaction potential, and the equations of motion are
∂H
,
p~˙i = −
∂~ri
∂H
.
~r˙i =
∂~pi
Our objective is to describe the macroscopic properties of this system of N particles in
equilibrium. An important point here is to define macroscopic variables that describe a
system at equilibrium. Such macroscopic variables must fulfill the following requirements
of
1 - time-independence ←→ existence of conserved quantities;
There are only a few possible macroscopic variables that fulfill both requirements for a
system of N particles, which are, for instance,
- total energy E;
- total momentum p~;
7.2. THE STATISTICAL APPROACH
75
~
- total angular momentum L.
Other possible variables (non-dynamical) that fulfill additivity are
- total number of particles N ;
- total volume V ;
- total entropy S.
Additivity means that a closed macroscopic system can be partitioned in a set of (macroscopic) subsystems such that the energy of
the whole system can be obtained as a sum of the energies of the
subsystems,
X
E=
Ei ,
i
with negligible boundary effects. The existence of macroscopic variables that fulfill additivity allows partitioning of a system into independent subsystems.
Imagine that we divide a system into subsystems, change the boundary conditions and put
the system together again. Then, even though the dynamic properties of the subsystems
change because of the change of the boundary conditions, it is still possible to describe
equilibrium with additive variables since these are independent of the boundary effects.
Energy
Energy E is a very special variable since this is the only conserved quantity in a closed
~ are not conserved
system which we usually define in a box. On the contrary, p~ and L
quantities in such a system because of the broken translation- and rotation-invariances.
⋆ Note: in thermodynamics, E means the internal energy U .
Usually, in order to describe an equilibrium state, we will consider variables E, V, N ,
and S. And we will aim at calculating out of the microscopic information the following
quantities:
S = S(E, V, N ),
∂S
1
,
=
T
∂E V,N
P
∂S
=
,
T
∂V E,N
∂S
µ
.
=
−
T
∂N E,V
76
CHAPTER 7. STATISTICAL PHYSICS IN EQUILIBRIUM
7.3
Phase space
In the following sections, we will introduce the main concepts needed in statistical mechanics.
Consider an isolated system with N particles (components). The complete description of
this system is given by the generalized coordinates:
q = (q1 , . . . , q3N ),
p = (p1 , . . . , p3N ).
We define the phase space as follows.
Phase space: 6N -dimensional space (see Fig. 7.1) whose points are given by the 6N
values of (q1 , . . . , q3N , p1 , . . . , p3N ).
Properties of phase space:
• phase space is a cartesian space;
• it is non-metric, i.e., one cannot define invariant distances in the phase space. This
is also the case for the P V -state space.
For N particles, the total numbers of degrees of freedom is 6N , and therefore the total
phase space is 6N -dimensional. The motion of the particles is governed deterministically
by the Hamiltonian
N
X
p~ 2i
1X
H(p, q) =
+
V (~qi − ~qj ),
2m 2 i6=j
i=1
where V (~qi − ~qj ) is the interparticle potential. The equations of motion of the particles
are
∂H
ṗi = −
;
∂qi
∂H
i = 1, . . . , 6N,
(7.1)
q̇i =
∂pi
with certain initial conditions.
The phase space is also called Γ-space. A point (representative point) in this space
corresponds to a state of the N -body system at a given time, i.e, to the microstate of
the system.
A trajectory in the phase space corresponds to the time evolution of the microstate. This
trajectory never intersects with itself since the solution of the system of equations of
motion (7.1) is unique given certain initial conditions (self-avoiding random walk).
If H does not depend explicitly on time, in which case energy is a conserved quantity,
all trajectories in phase space lie on an energy surface which is a hypersurface in Γ-space
(see Fig. 7.1).
77
7.3. PHASE SPACE
Figure 7.1: Representation of the Γ-space.
In Γ-space the history of an N -particle system is represented by one trajectory on
a (6N − 1)-dimensional energy surface.
Figure 7.2: 2-dimesional phase space.
Examples of phase space:
1 - One-dimensional harmonic oscillator (N = 1) with the Hamiltonian
H=
p2
1
+ mω 2 q 2 = E.
2m 2
The trajectory of this system in the Γ-space is given by an ellipse:
2 - One particle in a potential box with ideal reflecting walls:
78
CHAPTER 7. STATISTICAL PHYSICS IN EQUILIBRIUM
Phase space:
In general, for systems with one particle in one dimension, the phase trajectory is a closed
curve. For d > 1, closed trajectories happen only if there are more conserved quantities
than just energy such as, for instance, the Runge-Lenz vector in the Kepler problem.
Most of the examples we will consider in the following will be made on the ideal gas system, which is the low-density limit of the real gas. In this limit, the atomic interactions
contribute very little to the total energy, and therefore the total energy can be approximated by the sum of the energies of the individual atoms. One should note, though, that
interactions cannot be completely ignored since they are responsible for the establishment
of thermal equilibrium (through atoms’ collisions). Therefore, the ideal gas can be viewed
as the limiting case in which the interaction potential approaches zero.
µ-space
An alternative representation of the state of a system of N particles is by specifying the
state of each particle in a 6- dimensional phase space (Fig. 7.3). This space is called
µ-space, and the system is represented by N = 1019 points (number of atoms in the dilute
limit). As time evolves, those points move and collide with one another.
Figure 7.3: µ-space.
79
7.4. THERMAL EQUILIBRIUM FROM THE MICROSCOPIC VIEW
7.4
Thermal equilibrium from the microscopic view
We consider 1 mol of a dilute gas. It occupies a molar volume of
V0 = 2.24 × 104 cm3 at T0 = 273.15 K and P0 = 1 at with R =
P0 V 0
.
T0
The density of the gas is given by
n=
= 2.7 × 1019 atoms/cm3 ,
V0
the number of particles present in a unit volume.
The thermal equilibrium in the gas is reached because of particle collisions. Let us calculate now the relaxation time of the gas towards its thermal equilibrium. To do that, we
describe the scattering cross section, between particles σ (effective area of the collision
region) as
σ = πr02 ,
with r0 ≈ 2 × 10−8 cm being an effective atomic diameter if we consider our particles to
be atoms. We could also consider molecules, etc.
We define the mean free path λ as the average distance traveled by a
particle between two successive collisions. Consider a cylinder containing the gas with cross section A where the length of the cylinder
is of the order of the mean-free path. Then A is approximately equal
to the cross-sectional area of all atoms:
A = (Aλ)(nσ)
→
λ≈
1
∼ 10−5 cm,
nσ
where n is the particle density.
From thermodynamics, we know that the internal energy per particle in an ideal gas is
3
k T . Then, we can obtain the average velocity per particle v as
2 B
1 2 3
mv = kB T
2
2
→
v=
r
3kB T
m
⇐
average velocity.
At T = 300 K, v ≈ 105 cm/s, which has the meaning of the velocity of expansion of a gas
in a free expansion process.
The average time between successive collisions τc is then
τc =
λ
≈ 10−10 s
v
⇐
collision time
and corresponds to the relaxation time needed for the gas to reach local thermal equilibrium.
80
7.5
CHAPTER 7. STATISTICAL PHYSICS IN EQUILIBRIUM
Distribution function
As discussed in the introduction, it is not useful to collect the information about the
behavior of every individual atom; we would rather to calculate statistical properties such
as, for instance, the distribution function
f (p, q, t) .
Let us consider the µ-space, which we divide into cells with
volume ∆η. The cells are 6-dimensional objects and ∆η is
given as
∆η = ∆qx ∆qy ∆qz ∆px ∆py ∆pz .
(7.2)
Each cell is distinguished on a microscopic level but contains
nevertheless a large number of particles. Particles in the ith
p
~2
cell have positions ~qi , momenta p~i , and energy ǫi = 2mi .
We define the occupation number ni as the number of particles
in cell i at time t,
f (~pi , ~qi , t)∆η = ni ,
and then the distribution function as the occupation number per unit volume,
f (~pi , ~qi , t) =
ni
.
∆η
The distribution function has to be normalized so that the conditions
X
ni = N,
i
X
ǫi = E
(7.3)
i
are fulfilled. In the thermodynamic limit (N → ∞, V → ∞), f (~p, ~q, t) is assumed to be a
continuous function, and one also can write
∆η = d3 p d3 q
instead of (7.2). Then, equations (7.3) become
Z
d3 q d3 p f (~p, ~q, t) = N,
p~ 2
= E.
2m
For a uniform particle density, f is independent of ~q and the above expressions are simplified to
Z
N
d3 p f (~p, t) = ,
V
Z
d3 q d3 p f (~p, ~q, t)
81
7.6. ERGODIC HYPOTHESIS
Z
d3 p
p2
E
f (~p, t) = .
2m
V
The distribution function that we have defined, f (~p, ~q, t), evolves in time following the
microscopic equations of motion. When thermal equilibrium is reached, f (~p, ~q, t) should
acquire a time-independent form f (~p, ~q).
⇒ How do we find f (~p, ~q) and relate it to thermodynamic quantities?
We can also define the distribution function in the Γ-space. Let us divide the phase space
into cells with volume
Vγ = d3N q d3N p = dq1 , . . . , dq3N , dp1 , . . . , dp3N .
We define the distribution function (also called probability density) ρ(q, p) as
Wγ = ρ(p, q) d3N q d3N p,
where Wγ is the probability that the system at time t is in volume γ. ρ(q, p) is normalized
as
Z
d3N q d3N p ρ(p, q) = 1.
⇒ How do we find ρ(p, q) and relate it to thermodynamic quantities?
7.6
Ergodic hypothesis
A point in the phase space Γ corresponds to a state of the N -particle system and follows a
trajectory on the energy surface, as a function of time. After a certain time, which should
be long enough compared to the collision time (10−10 s for an ideal gas at the standard
temperature (T = 300 K) and pressure (P = 1 at) conditions), the trajectory reaches a
steady state which corresponds to thermal equilibrium.
From classical mechanics we know that already for N = 2 the motion of the system can
be chaotic (think, for instance, of the double pendulum). Therefore, it is to be expected
that for N = 1023 the motion will be chaotic on the energy surface in phase space and
each point of the energy surface will be reached by the trajectory, i.e., the trajectory in
Γ-space will perform a random walk over all the energy surface. Mathematically this is
stated by the ergodic hypothesis.
Ergodic hypothesis: Given a sufficiently long time, the representative point of an
isolated system will come arbitrarily close to any given point an the energy surface.
82
CHAPTER 7. STATISTICAL PHYSICS IN EQUILIBRIUM
Note: the ergodic hypothesis, nevertheless, does not provide a criterion for ”sufficiently
long time”. Relaxation time is a dynamical property, while the ergodic theory is based
on avoidance of dynamics.
7.7
Fundamental postulate of equal a priori probabilities
We will investigate here the question what the distribution function ρ(p, q) of an isolated
system looks like.
Consider the phase space Γ divided into cells with volume
Vi = d3N q d3N p.
Following Boltzmann, if all the cells in phase space are equally big, each one contains the
same number of points. This is equivalent to saying that
• there are no special points on the energy surface in phase space;
• the paths in phase space are equally distributed;
• the system can be found in any possible microstate (point in phase space) with the
same probability.
7.8
Statistical ensemble
The instruments that are used for measuring processes have a given resolution. They
usually perform averages over time intervals that are small on the macroscopic scale but
large compared to the collision time.
⋆ The basic concept in statistical mechanics is to assume that the time average can
be replaced by an average over a suitably chosen collection of systems called statistical
ensemble.
To see this, let us consider a classical observable O,
O(q, p) = O(q1 , . . . , q3N , p1 , . . . , p3N ),
which could be the kinetic energy,
O = Ekin
N
X
p~ 2i
=
,
2m
i=1
in the simplest case. A measuring process provides a time average of the observable:
Z
1 τ
Ō(t0 ) =
dt O(q(t0 + t), p(t0 + t)),
τ 0
83
7.8. STATISTICAL ENSEMBLE
where t0 is the initial time, and τ is the resolution of the measuring instrument.
For N = 1023 , it is impossible to calculate Ō(t0 ) from the solution of the hamiltonian
equations for the N particles. But, since the trajectories in phase space behave chaotically,
we can use the ergodic hypothesis and obtain a simplified description for Ō. In equilibrium
we write
Z
d3N q d3N p ρ(q, p; t0 , τ ) O(q, p),
Ō(t0 ) =
with
1
ρ(q, p; t0 , τ ) =
τ
Z
τ
dt δ (3N ) (q − q(t0 + t)) δ (3N ) (p − p(t0 + t)),
0
as an average integral by using the δ-Dirac functional. We now assume the fundamental
postulate of the equal a priori probabilities, which we express as
ρ(q, p) = lim ρ(q, p; t0 , τ ).
τ →∞
With this assumption, ρ will be an equally distributed density in phase space. Then,
Ō(t0 ) is independent of t0 and is given by the following average:
hOi ≡
Z
d
3N
q
Z
d3N p ρ(p, q) O(p, q) .
Interpretation 1: hOi describes the physical quantity O of a system in thermodynamical equilibrium where an average on the microscopic state over different times
was performed.
Interpretation 2: hOi describes the average of the physical quantity O over an
ensemble of identical copies of the same system (at a given time and, in our case,
in equilibrium) which do not interact with each other and which are characterized
by a distribution function ρ(p, q) in Γ-space.
Statistical ensemble is defined as an infinite collection of identical copies of the system
characterized by a distribution function ρ(p, q, t) in Γ-space:
ρ(p, q, t) d3N p d3N q = Number of systems in dp dq at time t,
where
(p, q) = (p1 , . . . , p3N , q1 , . . . , q3N ).
In thermal equilibrium, the ensemble evolves into an equilibrium ensemble with ρ(p, q)
independent of time.
84
7.9
CHAPTER 7. STATISTICAL PHYSICS IN EQUILIBRIUM
Liouville theorem
Some important features of the distribution function ρ(q, p, t) are the Liouville equation
and the Liouville theorem.
We consider an ensemble of identical copies of a system, whose state at t = 0 is equally
distributed over the energy surface in phase space. Does the ensemble remain equally
distributed for t > 0? or, equivalently is the function ρ invariant under the hamiltonian
time evolution? The answer is yes, due to the Liouville Theorem.
Let us consider a system with the distribution function ρ(q, p, t) and the velocity of the
phase space ~v given by
~v = (q̇1 , q̇2 , . . . , q̇3N , ṗ1 , . . . , ṗ3N ).
We define the current density ~j as
~j = ρ~v .
~j is equivalent to the electrical current density, but instead of charged particles moving in
real space we have here phase points moving in the Γ-space.
Now, consider a region in the Γ-space G with surface S. The
number of points that cross the surface S per unit time is given
as
Z
~ = dS · ~n),
~ · ~j (dS
dS
S(G)
where ~n is a vector orthonormal to the surface dS. Since there
are no sources or sinks of the ensemble points, the following
equality holds:
Z
Z
∂
~
~
d3N q d3N p ρ(q, p, t).
dS · j = −
∂t
G
S(G)
By making use of the Gauss law we can replace the integral over the surface by a volume
integral:
Z
∂
3N
3N
~ · ~j = 0 ,
ρ(q, p, t) + ∇
(7.4)
d qd p
∂t
G
with
∇≡
∂
∂
∂
∂
,...,
,
,...,
∂q1
∂q3N ∂p1
∂p3N
.
Then,
~ · ~j =
∇
3N X
∂
∂
(ρq̇j ) +
(ρṗj ) .
∂qj
∂pj
j=1
(7.5)
Since eq. (7.4) is valid for all G, we obtain out of eq. (7.4) a continuity equation for the
distribution function ρ:
∂
~ · (~v · ρ(q, p, t)) = 0 ,
ρ(q, p, t) + ∇
∂t
85
7.9. LIOUVILLE THEOREM
which describes the conservation of the total number of points in the ensemble.
By using eq. (7.5), we get
3N 3N X
X
∂
∂ q̇j ∂ ṗj
∂ρ
∂ρ
− ρ=
q̇j
+ ṗj
+
+ρ
.
∂t
∂qj
∂pj
∂qj ∂pj
j=1
j=1
(7.6)
Since all the points in the ensemble are governed by the same Hamiltonian H(q, p), the
second summand in eq. (7.6) is equal to zero and we have
3N
dρ
∂ρ X
=
+
dt
∂t j=1
∂ρ
∂ρ
q̇j +
ṗj
∂qj
∂pj
=0
or
dρ
∂ρ
=0=
+ [ρ, H]
dt
∂t
⇒ Liouville equation
The Liouville equation implies that
ρ (q(t), p(t), t) ≡ ρ (q(0), p(0), 0) .
Liouville’s theorem: the ensemble-”liquid” moves in phase space as an incompressible
liquid (see Fig. 7.9).
Figure 7.4: At any time t the volumes Γ0 and Γt are equal.
When at t = t0 the density is equally distributed, it remains equally distributed at any
time.
In thermal equilibrium
ρ = ρ(q, p)
⇒
and the Liouville equation becomes
[ρ, H] = 0 .
∂ρ
=0
∂t
86
CHAPTER 7. STATISTICAL PHYSICS IN EQUILIBRIUM
It is always fulfilled since in thermal equilibrium
ρ = ρ(H(q, p)).
In what follows we will concentrate on thermal equilibrium and therefore all ensembles
will be
ρ = ρ(H(q, p)).
Note: in quantum mechanics an analog of ρ is the statistical operator ρ̂ (density matrix),
with the continuity equation
i
∂ ρ̂
1 h
ρ̂, Ĥ .
=−
∂t
ih̄
```
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