Appendix A Non-Equilibrium Statistical Mechanics The text has focused on equilibrium statistical mechanics and its wide va- riety of applications. Little has been said about mechanisms and the ap- proach to equilibrium. This appendix presents some simple considerations concerning non-equilbrium statistical mechanics within the framework of the Boltzmann equation and some of its immediate extensions. 1 A.1 Boltzmann Equation A.1.1 One-Body Dynamics Consider a collection of identical, non-localized, randomly prepared sys- tems, which in statistical equilibrium becomes the microcanonical ensemble. We now just do classical mechanics and assume to start with the systems are independent. Place them in their appropriate position in six dimen- sional phase space {p, q}. 2 The distribution function f (p, q,t) is defined in the following manner dN = number of systems in d 3 pd 3 q ≡ f (p, q,t) d 3 pd 3 q (2π) 3 (A.1) The quantity d 3 pd 3 q/(2π) 3 = d 3 pd 3 q/h 3 counts the number of cells in this small volume in phase space. A value f = 1 would then imply that every cell is occupied by one particle. The probability of finding a system in this region of phase space is the probability of picking a member of the ensemble at random, which is dN/N . 1 See [Boltzmann (2011)], and a good reference here is [St¨ ocker and Greiner (1986)]. 2 Here q =(x,y,z) and p =(px,py ,pz ). 335 Introduction to Statistical Mechanics Downloaded from www.worldscientific.com by 186.170.117.73 on 11/07/13. For personal use only.
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June 27, 2011 17:21 WSPC/Book Trim Size for 9in x 6in SMroot
Appendix A
Non-Equilibrium Statistical
Mechanics
The text has focused on equilibrium statistical mechanics and its wide va-
riety of applications. Little has been said about mechanisms and the ap-
proach to equilibrium. This appendix presents some simple considerations
concerning non-equilbrium statistical mechanics within the framework of
the Boltzmann equation and some of its immediate extensions.1
A.1 Boltzmann Equation
A.1.1 One-Body Dynamics
Consider a collection of identical, non-localized, randomly prepared sys-
tems, which in statistical equilibrium becomes themicrocanonical ensemble.
We now just do classical mechanics and assume to start with the systems
are independent. Place them in their appropriate position in six dimen-
sional phase space p,q.2 The distribution function f(p,q, t) is defined
in the following manner
dN = number of systems in d3p d3q
≡ f(p,q, t)d3p d3q
(2π~)3(A.1)
The quantity d3p d3q/(2π~)3 = d3p d3q/h3 counts the number of cells in
this small volume in phase space. A value f = 1 would then imply that
every cell is occupied by one particle.
The probability of finding a system in this region of phase space is the
probability of picking a member of the ensemble at random, which is dN/N .
1See [Boltzmann (2011)], and a good reference here is [Stocker and Greiner (1986)].2Here q = (x, y, z) and p = (px, py, pz).
335
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336 Introduction to Statistical Mechanics
This quantity is used to compute expectation values.
The goal now is to follow the time evolution of the distribution function.
As a function of time, a particle at p0,q0 at time t0 moves to the point
p,q at time t (see Fig. A.1). Let dλ0 = d3p d3q be the phase space
p
0
t d
d0
t
q
Fig. A.1 An ensemble of systems with dN = f(p,q, t) d3p d3q/(2π~)3 members in asix-dimensional phase space volume dλ0 = d3p d3q, at a position p0,q0 at a time t0,is followed along a phase trajectory to a time t.
volume at the time t0
dλ0 = d3p d3q ; phase-space volume at t0 (A.2)
Then with hamiltonian dynamics, one has Liouville’s theorem, which states
that the phase space volume is unchanged along a phase trajectory3
dλ = dλ0 ; Liouville’s theorem (A.3)
Since the number of systems is conserved, the number of systems within
this phase-space volume does not change
dN = dN0 ; number conserved (A.4)
One concludes that the distribution function is unchanged along a phase
trajectory
f [p(t),q(t), t] = f(p0,q0, t0) ; unchanged along phase trajectory
(A.5)
Now write out the total differential of Eq. (A.5), and divide by dt
df
dt=∂f
∂t+∇pf · dp
dt+∇qf · dq
dt= 0 (A.6)
3A proof of Liouville’s theorem can be found in [Walecka (2000)], or [Fetter andWalecka (2006)].
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Non-Equilibrium Statistical Mechanics 337
Hamilton’s equations of motion for a particle state that
dq
dt= v = ∇pH ; Hamilton’s equations
dp
dt= F = −∇qH (A.7)
Equation (A.6) can then be re-written as
∂f
∂t= ∇qH ·∇pf −∇pH ·∇qf = H, fP.B. (A.8)
where the last equality identifies the Poisson bracket of classical mechanics.
In equilbrium, the time derivative of the distribution function at a given
point p,q vanishes
∂f
∂t= 0 ; equilbrium (A.9)
A solution to Eqs. (A.8) and (A.9) is then provided by
f = f(H) ; equilbrium (A.10)
where H = E is a constant of the motion for the particle.
A.1.2 Boltzmann Collision Term
Consider an extension to what in equilbrium is the canonical ensem-
ble, and include zero-range two-particle collisions. The goal here is
to project the exact dynamics of the many-body distribution function
f(p1, · · · , p3N , q1, · · · q3N ; t) down to an approximate equation for the one-
body distribution function f(p,q, t). As the ensemble evolves with time,
particles are now scattered in and out of the phase-space volume dλ (see
Fig. A.2). Assume the r.h.s. of Eq. (A.8) is augmented by a collision term,
so that Eq. (A.6) becomes
df
dt=
(
∂f
∂t
)
collisions
(A.11)
Momentum is conserved in the collisions so that
p1 + p2 = p′1 + p′2 (A.12)
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338 Introduction to Statistical Mechanics
p1
p2
p1
p2
d
Fig. A.2 Collisions take particles in and out of the phase space volume dλ, and mo-mentum is conserved in these collisions.
Detailed balance in a collision states that4
Ratei→f = Ratef→i
or ; σ v12 = σ′ v1′2′ (A.13)
We assume zero-range collisions where σv12, while depending on E1+E2 =
E′1 + E′2, is otherwise independent of the kinematics.5
The number of transitions per unit time in the direction i→ f is given
by(
# of transitions
time
)
i→f
= (incident flux)× σ × (# of target particle)
= (n1v12)σ (n2d3q) (A.14)
4It helps here to think of the quantum mechanical expressions (“Golden Rule”) forthe rate and cross section
Rfi =2π
~δ(Ei − Ef )|〈f |H′|i〉|2
σfi =Rfi
Flux
In these expressions:
• Energy conservation is built in;
• This is for a transition to a given final state. One still needs the number of statesd3p′/(2π~)3 in a large volume V ;
• All factors of V have already been removed from these expressions;
• The hamiltonian is hermitian so that |〈f |H′|i〉|2 = |〈i|H′|f〉|2
However, the calculation is still classical until quantum mechanics later explicitlyappears.
5See Prob. A.3.
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Non-Equilibrium Statistical Mechanics 339
where the particle density n is defined by
n ≡ dN
d3q; particle density (A.15)
Since the interaction is of zero range, everything is evaluated at the same
spatial point q, and in the same spatial volume d3q.
The rate of change of the number of systems in the phase space volume
dλ is then given by the difference of the number of systems scattered in and
the number scattered out per unit time. A combination of Eqs. (A.11)–
(A.15) gives(
∂f
∂t
)
coll
d3p d3q
(2π~)3=
∫
· · ·∫
(σv12)
×[
f(p′1,q; t)d3p′1(2π~)3
] [
f(p′2,q; t)d3p′2 d
3q
(2π~)3
]
d3p
(2π~)3
[
δ(3)(∆p)d3p2
]
−[
f(p,q; t)d3p
(2π~)3
] [
f(p2,q; t)d3p2 d
3q
(2π~)3
]
d3p′1(2π~)3
[
δ(3)(∆p)d3p′2
]
(A.16)
The final factor in each line is just unity. A cancellation of common factors
or ; f(E′1)f(E′2)[1− f(E)− f(E2)] = f(E)f(E2)[1− f(E′1)− f(E′2)]
(A.38)
With the substitution of Eq. (A.37), this becomes
1
D(E′1)
1
D(E′2)
1
D(E)
1
D(E2)[D(E)D(E2)−D(E2)−D(E)]
=1
D(E′1)
1
D(E′2)
1
D(E)
1
D(E2)[D(E′1)D(E′2)−D(E′2)−D(E′1)]
(A.39)
A cancellation of common factors leads to
D(E)D(E2)−D(E2)−D(E) = D(E′1)D(E′2)−D(E′2)−D(E′1)
(A.40)
This equality is now established by direct substitution of Eq. (A.37) and
the use of energy conservation. The l.h.s. is
l.h.s. =[
eβ(µ−E) + 1] [
eβ(µ−E2) + 1]
−[
eβ(µ−E2) + 1]
−[
eβ(µ−E) + 1]
= eβ(2µ−E−E2) − 1
= eβ(2µ−E′1−E
′2) − 1 ; E + E2 = E′1 + E′2
= r.h.s. (A.41)
Thus the one-body Fermi distribution in Eq. (A.37) makes the
Nordheim-Uehling-Uhlenbeck expression in Eq. (A.36) vanish, which leads
to a vanishing of the new collision term. Furthermore, with the one-body
hamiltonian of Eq. (A.19), it is then still true that7
∂f
∂t= ∇qU ·∇pf − v ·∇qf = 0 (A.42)
A.3 Example—Heavy-Ion Reactions
As an application of Boltzmann transport theory, consider a reaction be-
tween two heavy nuclei [Stocker and Greiner (1986)]. Here a program
7See Prob. A.2
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344 Introduction to Statistical Mechanics
applying the Vlasov-Uehling-Uhlenbeck model exists in the literature and
is available to all [Hartnack, Kruse, and Stocker (1993)].
In phase space, the initial configuration is as sketched in Fig. A.3.
p
q
Fig. A.3 Sketch of initial phase-space configuration for two colliding heavy nuclei.
The program proceeds through the following series of steps:
(1) Choose random positions and momenta from the initial Fermi gases in
the two nuclei;
(2) Follow the particles with classical dynamics and zero (short)-range two-
body collisions leading to random final states, while conserving energy
and momentum;
(3) Take a statistical average over many “events” run in parallel to deter-
mine the one-body distribution function8
f(p,q; t) ≡ 1
Nevents
∑
events
f event(p,q; t)
(4) Compute the nuclear density n at time t from f ;
(5) Use a density-dependendent mean-field potential U(n) in determining
the one-body motion;
(6) Take an average of “runs” to get the best f ;
(7) Use this f to compute particle distributions, mean values, etc.
The authors include inelastic N -N processes producing (π,∆) in their pro-
gram, where ∆(1236MeV) with (Jπ , T ) = (3/2+, 3/2) is the first nucleon
resonance. Locate the program and learn to run it. It’s fun.
The experimental results are typically used to study the nuclear equa-
tion of state through U(n).9
8As in an actual experiment.9Compare Probs. A.4–A.5.
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264Bragg-Williams approximation, 258Ising solution (Z=2), 268largest term in sum, 264, 269lattice gas model, 324mean number of pairs, 258nearest neighbors, 257order-disorder transitions, 263properties, 257