Chapter 6
Ergodicity and the Approach to
Equilibrium
6.1
Equilibrium
Recall that a thermodynamic system is one containing an enormously large number of
constituent particles, a typical ‘large number’ being Avogadro’s number,
N
A
= 6
.
02
×
10
23
.
Nevertheless, in
equilibrium
, such a system is characterized by a relatively small
number of thermodynamic state variables. Thus, while a complete description of a (classical)
system would require us to account for
O
(
10
23
)
evolving degrees of freedom, with respect
to the physical quantities in which we are interested, the details of the initial conditions
are effectively forgotten over some microscopic time scale
τ
, called the collision time, and
over some microscopic distance scale,
ℓ
, called the mean free path
1
. The equilibrium state
is timeindependent.
6.2
The Master Equation
Relaxation to equilibrium is often modeled with something called the
master equation
. Let
P
i
(
t
) be the probability that the system is in a quantum or classical state
i
at time
t
. Then
write
dP
i
dt
=
summationdisplay
j
(
W
ji
P
j
−
W
ij
P
i
)
.
(6.1)
Here,
W
ij
is the rate at which
i
makes a transition to
j
. Note that we can write this equation
as
dP
i
dt
=
−
summationdisplay
j
Γ
ij
P
j
,
(6.2)
1
Exceptions involve quantities which are conserved by collisions, such as overall particle number, mo
mentum, and energy. These quantities relax to equilibrium in a special way called
hydrodynamics
.
1
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CHAPTER 6.
ERGODICITY AND THE APPROACH TO EQUILIBRIUM
where
Γ
ij
=
braceleftBigg
−
W
ji
if
i
negationslash
=
j
∑
′
k
W
jk
if
i
=
j ,
(6.3)
where the prime on the sum indicates that
k
=
j
is to be excluded. The constraints on the
W
ij
are that
W
ij
≥
0 for all
i,j
, and we may take
W
ii
≡
0 (no sum on
i
). Fermi’s Golden
Rule of quantum mechanics says that
W
ji
=
2
π
planckover2pi1
vextendsingle
vextendsingle
(
i

ˆ
V

j
)
vextendsingle
vextendsingle
2
ρ
(
E
i
)
,
(6.4)
where
ˆ
H
0
vextendsingle
vextendsingle
i
)big
=
E
i
vextendsingle
vextendsingle
i
)big
,
ˆ
V
is an additional potential which leads to transitions, and
ρ
(
E
i
) is
the density of final states at energy
E
i
.
If the transition rates
W
ij
are themselves timeindependent, then we may formally write
P
i
(
t
) =
(
e
−
Γ t
)
ij
P
j
(0)
.
(6.5)
Here we have used the Einstein ‘summation convention’ in which repeated indices are
summed over (in this case, the
j
index). Note that
summationdisplay
i
Γ
ij
= 0
,
(6.6)
which says that the total probability
∑
i
P
i
is conserved:
d
dt
summationdisplay
i
P
i
=
−
summationdisplay
i,j
Γ
ij
P
j
=
−
summationdisplay
j
parenleftBig
summationdisplay
i
Γ
ij
parenrightBig
P
j
= 0
.
(6.7)
Suppose we have a timeindependent solution to the master equation,
P
eq
i
. Then we must
have
Γ
ij
P
eq
j
= 0
=
⇒
P
eq
j
W
ji
=
P
eq
i
W
ij
.
(6.8)
This is called the condition of
detailed balance
. Assuming
W
ij
negationslash
= 0 and
P
eq
j
= 0, we can
divide to obtain
W
ji
W
ij
=
P
eq
i
P
eq
j
.
(6.9)
6.2.1
Example: radioactive decay
Consider a group of atoms, some of which are in an excited state which can undergo nuclear
decay. Let
P
n
(
t
) be the probability that
n
atoms are excited at some time
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 Fall '11
 staff
 Thermodynamics, Hamiltonian mechanics, Dynamical systems, Ergodic theory, phase space

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