14
CHAPTER 2. REVIEW OF STATISTICAL MECHANICS
2.1
Principles of statistical mechanics
2.1.1
Macroscopic and microscopic states
A macroscopic system is made of a huge number of particles. The power of thermodynamics
is that such many particles can, with a great deal of precision, be described by just a few pa-
rameters, the thermodynamic quantities and the relations between them. By giving a few inde-
pendent parameters, say, the total energy, volume, and number of particles, the thermodynamic
state of a macroscopic system is uniquely determined. Such a state is called a macroscopic
state. However, there could be still a tremendously large number of di
ff
erent configurations of
the particles which have the same total energy and occupy the same volume. A specific configu-
ration is called a microscopic state. The number of microscopic states of the same macroscopic
parameters is called the degree of degeneracy.
Example:
The macroscopic state of 1000 coins with half of them up has 1000!
/
(500!500!)
microscopic states.
Exercise
: Think about how many microscopic states would be if there were 10
23
coins, which
could be made, for example, of spin-1
/
2’s.
2.1.2
Ergodic hypothesis
Consider a classical system of
N
particles, the microscopic state is a point at the 6
N
-dimensional
space (
R
,
P
). The evolution is governed by the Hamiltonian. Due to the collisions (interaction)
between di
ff
erent particles, the trajectory is very sensitive to the initial condition. From two
points close to each other in the phase space, after a short period of time, the trajectories would
deviate from each other exponentially far away. Note that a trajectory can never cross into itself
in the phase space. So, eventually a trajectory would have passed almost the neighborhood of
all points in the phase space. Here comes the
Ergodic hypothesis
:
Given su
ﬃ
cient time, a trajectory in the phase space will come arbitrarily close to
any
accessible
points.
Because of the sensitivity of the trajectory to the initial condition, a small uncertainty of the
initial condition would eventually lead to an almost totally random distribution of points in the
phase space.
In statistical mechanics, we consider an ensemble of systems instead of just one system.
Suppose the relative probability of a system in the ensemble being at the state
s
is
ρ
(
s
,
t
) at time
t
, the expectation value of a physical quantity
O
is
⟨
O
(
t
)
⟩
=
∑
s
O
(
s
)
ρ
(
s
,
t
)
∑
s
ρ
(
s
,
t
)
.
(2.1)
Starting from an initial distribution of the ensemble in the phase space
ρ
(
s
,
0), the distribution
would eventually evolve to a random distribution. Such an evolution or relaxation usually oc-
curs with di
ff
erent timescales. One is the short-timescale dynamics by short-range collisions
between particles. For example, for electrons in metals, the collisions take only a few femtosec-
ond (10
−
15
sec). So, in an extremely short time, the system would reach local equilibrium in