Statistical Mechanics
Z. Suo
The Boltzmann Distribution
A system in thermal contact with a reservoir of energy. A reservoir of energy is a
large system. Its number of states,  R , is a smooth function of its energy U R . The temperature
of the reservoir, T , is defined by
 log  R 1
 .
U R
T
The temperature of the reservoir, T, does not change when the reservoir is in thermal contact with
another, smaller, system, because the energy transferred between the reservoir and the smaller
system is much smaller than the total amount of energy in the reservoir.
In what follows we will call the smaller system simply the system. The system and the
reservoir are in thermal contact, interacting weakly. All other modes of interactions are blocked:
no exchange of molecules, or volume, or anything else. The system can be in any one of a set of
states,  1 ,  2 ,  3 ,...,  s ,...
When the system equilibrates with the reservoir, the system still
fluctuates among all these states. What is the probability for the system to be in each state?
Dissect the set all the states of the composite into a family of configurations
according to the state of the small system. To apply the fundamental postulate, we need to
construct an isolated system. The composite of the system and the reservoir is an isolated system.
We now speak of three systems: the reservoir, the small system (which we will call the system),
and their composite. A state of the composite is specified by a state of the system and a state of
the composite. When the system is in a specific state  s , the reservoir can be in a large number
of states. All the states of the composite when the system is in the state  s constitute a
configuration of the composite. We want to know the probability for the composite to be in such
a configuration.
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Statistical Mechanics
Z. Suo
Let U tot be the energy of the composite of the reservoir and the system. The system can
be in any one of a set of states,  1 ,  2 ,  3 ,...,  s ,... These states need not be at the same energy
level, because the system can draw energy from the reservoir. When the system is in state  s , the
energy of the system is U s , and the energy of the reservoir is U R  U tot  U s . Because the
temperature of the reservoir, T, is independent of the energy in the reservoir, log R is linear in
U R , namely,
log  R U tot  U s   log  R U tot  
Us
.
T
Take exponential of both sides of the equation, and we obtain that
 R U tot  U s    R U tot exp  U s / T  .
Note that  R U tot  is the number of states of the reservoir at energy U tot . Upon losing energy
U s to the system, the reservoir reduces its number of states by a factor exp  U s / T  , known as
the Boltzmann factor.
Note that  R U tot  U s  is also the number of states of the composite in the configuration
that the small system is in state  s . Thus, the total number of states of the composite is
tot   R Utot  U s   R Utot Z .
The sum is taken over all states of the system. The sum over all states,
Z   exp  U s / T  ,
is known as the partition function.
The probability for a system to be in a specific state.
In thermal contact with a
reservoir of energy held at temperature T, a small system fluctuates from one state to another.
What is the probability for the small system in any of its state? As an isolated system in
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Statistical Mechanics
Z. Suo
equilibrium, the composite is equally probable in each of the tot states. Consequently, the
probability to find the composite with the small system in state  s is Ps   R U tot  U s  / tot ,
namely,
Ps 
exp  U s / T 
.
Z
The probability depends on the energy of every state of the small system and the temperature of
the reservoir, but is independent of any other characteristics of the reservoir. This probability
distribution is known as the Boltzmann distribution.
The probability for a system to have a certain amount of energy. When the system is
held at a specific energy, U s , the system becomes an isolated system with a specific set of states.
Denote the number of states of the system held at energy U s by U s  . The system is equally
probable to be in any one of these states.
On the other hand, the system has different
probabilities to be in any two states with different energies. The relative probability for the
system to have the two different energies, U s and U r , is
PU s  U s exp  U s / T 

.
PU r  U r exp  U r / T 
The system in thermal contact with the reservoir forms a family of isolated systems with various
amounts of energy. Usually the number of states is an increasing function of the energy. Say
U s  U r , then U s  > U r  . Given the two energies, the ratio of the numbers of states of the
system is U s  / U r  , and the ratio of the numbers of states of the reservoir is
exp  U s / T  / exp  U r / T  . The two ratios have the opposite trends. At a high temperature, the
ratio of the system prevails, and the system is more likely to have a high energy. At a low
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Statistical Mechanics
Z. Suo
temperature, the ratio of the reservoir prevails, and the system is more likely to have a low
energy.
The probability for a system to be in a configuration. A subset of the states of a
system is called a configuration of the system. Let configuration A consists of a set of states:
A   i ,  j ,  s ,.... The partition function of configuration A is
ZA 
 exp  U
 s A
s
/T .
The sum is taken over all the states that constitute the configuration A. The probability for the
system to be in configuration A is
PA 
ZA
.
Z
Thermal fluctuation of an RNA molecule. Consider a system consisting of a RNA
molecule in a liquid. Our experimental observation shows that the RNA molecule fluctuates
between two conformations: a chain and a loop. Thus, all the states of the system are divided
into two subsets: one corresponding to each conformation. The fraction of time that the RNA
molecule spends in each conformation is the probability of the conformation.
Divide water molecules between wine and cheese. Say our composite system of wine
and cheese is in thermal contact with a reservoir held at temperature T. The partition of water
molecules in the wine and cheese is an internal variable on the system. For a given partition, the
system has a partition function. When the seal is punctured, the most likely partition of water
maximizes the total number of states of the combined system of the reservoir, the wine and the
cheese. According to the above analysis, this fundamental postulate is equivalent to maximizing
the partition function.
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Statistical Mechanics
Z. Suo
The mean energy. In thermal contact with a reservoir, a system rapidly switches from
one state to another, so that the energy of the system fluctuates from one level to another. If we
take many measurements, or if our instrument is so slow that a reading takes a time much longer
than the system fluctuates from one state to another, we will get the mean value of the energy,
defined by
U   PsU s .
Inserting Ps  Z 1 exp  U s / T  into this definition, we can readily confirm that
U T2
 log Z
.
T
The mean energy of the system is also a function of the temperature of the reservoir.
The variance of energy. The variance of energy is
Var U   T 2
U
.
T
For a system of many degrees of freedom f, the mean energy scales as U  f .
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