CHAPTER ELEVEN
CLASSICAL STATISTICS
and
THE BOLTZMANN PROBABILITY FUNCTION
Introduction
We are often interested in the behavior of a small (sometimes microscopic) component within a much larger
systemÞ For example, we might want to know the probability that a single atom is in a particular energy state I= ,
or the probability that a particular atom has a velocity @t. This might seem to be an impossible task at first, since
the size of the system, and, therefore, the number of available microstates is quite small, leading one to expect that
fluctuations in the measurements of the various parameters would be quite large. We can get around this problem,
however, by focusing on the much larger system that is in contact with the microscopic system. This larger
system will act something like a reservoir of energy and particles. Since the energy, number of particles, etc., of
the microscopic system and its reservoir are related, and since the much larger system can be treated very
adequately by using statistical methods, we should be able to determine the characteristics of the smaller
microscopic system.
Model System
As a model system, consider the system Eo , shown below, that is made up of two subsystems E and Ew .
We will assume that subsystem E is a small microscopic system, while subsystem Ew is a much larger
macroscopic system. In fact, an even better model than the one diagrammed below might be a small balloon (with
a massless, dimensionless membrane that can allow thermal energy or particles to be exchanged) enclosing a
volume ZE in a much larger room of volume ZEw Þ This “balloon” might enclose a single atom or a large number of
atoms. The room plus balloon composes a composite system E9 Þ
A
A'
ðóóóóóóóóóóóóóóóóóóóóóóóóñóóóóóóóóóóóóóóóóóóóóóóóóò
Ao
Fig. 11.1 A composite system made up of a small microscopic system E in contact with a much larger
system Ew (which is large enough to be considered as a heat and particle reservoir).
In either case, we assume that the system Eo is completely isolated, so that the energy, volume, and number
of particles in Eo is fixed, i.e., .Io œ .Zo œ .Ro œ 0. As in all our statistical work, we actually imagine a large
number (infinite) of similar systems (an ensemble of systems). We are interested in the relative number of
systems within the ensemble which are in the same macrostate, even though their individual microstates may be
different. We want to consider several possible modes of interaction between the microscopic subsystem and the
macroscopic reservoir. Thus, there are several different ensembles of systems to be considered. These are: 1)
the microcanonical ensemble: where no interaction whatsoever are allowed; 2) the canonical ensemble, an
ensemble of systems that allow a transfer of energy (thermal and mechanical) between E and Ew , but not particles
(i.e., the system is “closed")ß and 3) the grand canonical ensemble: an ensemble of systems that are allowed to
transfer energy and particles between E and Ew (an “open" system).
Chapter Eleven: The Boltzmann Probability Function
2
The Probability Function for a Microscopic Subsystem
The number of states accessible to system Eo is given by Ho œ HE HE´, and the probability of finding the
subsystem E in a particular macrostate designated by I= ß Z= ß R= is proportional to the number of states
Ho ÐIo ,I= ,Z9 ß Z= ß R9 ß R= ) œ HE´ÐI9 I= ß Z9 Z= ß R9 R= ÑHE ÐI= ß Z= ß R= Ñ where the reservoir E´ has energy
Io I= ß volume Z9 Z= ß and R9 R= particles.
There are actually two different possibilities to be consider here. First, the subsystem of
interest (system E) may actually be in a single microstate of energy I= (in which case
HE ÐI= Ñ œ ")Þ This might correspond to a single simple harmonic oscillator with energy
=# h #
I= œ Ð= "# Ñh =; or to a particle in an infinite square well with a particular energy I= œ #7P
# ; or
to an atom with a particular kinetic energy I= œ "# 7@=# . If the microscopic system is in a single
microstate (a single quantum-mechanical state), then HE ÐI= Ñ œ ", and we obtain
H9 ÐI9 ß I= Ñ œ HE´ÐI9 I= Ñ
A second possibility is the case where the subsystem might not be in a single quantum
mechanical state. In this case the assumption that HE ÐI= Ñ œ " (i.e., WE œ !) is not valid. We
must, therefore, accept the fact that the microscopic subsystem may have lots of microstates which
give the same macrostate of the microscopic subsystem. The following discussion of a canonical
ensemble is general enough to include both of these cases.
We would like to express the probability of finding the subsystem in a state I= ß Z= ß R= in terms of
measurable macroscopic parameters of the larger system E'. In order to do this we make use of our knowledge of
the entropy of the reservoir Ew ß since W w œ 5 68Hw . A change in the entropy of the reservoir can be expressed by
the equation
.W w œ Œ
`W w
`W w
`W w
.I w Œ
.Z w Œ
.R w



w
w
`I Z w ßR w
`Z I w ßR w
`R w Z w ßI w
(11.1)
Let's try to imagine the small subsystem E as being taken out of Ew (much like allowing the balloon model to
begin with zero volume and increase in size, containing R= particles with a volume Z= and an energy I= . Now,
we can find the change in the entropy of the reservoir required to establish this subsystem. We will let
.I w œ I=
.Z w œ Z=
.R w œ R=
(11.2)
where I= , Z= , and R= are the energy, volume and number of particles removed from system A' in order to establish
the subsystem E. The minus sign arises for each of these quantities because the energy (for example) for the
microscopic subsystem was removed from the resevoir. This means that we can express the change in the entropy
of the reservoir by the equation
.W w œ Œ
`W w
`W w
`W w
I= Œ
Z= Œ
R=



w
w
`I Z w ßR w
`Z I w ßR w
`R w Z w ßI w Ÿ
(11.3)
In the last chapter, we demonstrated that
Œ
`W '
"
œ

`I ' R 'ßZ ' X '
Œ
`W '
T'
œ

`Z ' R 'ßI '
X'
Œ
`W '
.'
œ 
`R ' I 'ßZ '
X'
(11.4)
which means that the change in entropy of the reservoir can be expressed as
.W w œ W w Wow œ "
T w Z=
.w R =
I= w
w
X
X
Xw
(11.5)
where W w is the entropy of the reservoir after the subsystem has been formed, and Wow is the initial entropy of the
system, and where X w , T w , and .w are the temperature, pressure, and chemical potential of the larger reservoir.
Chapter Eleven: The Boltzmann Probability Function
3
Writing the entropy in terms of the number of microstates accessible to the reservoir, we have
5 68Hw 5 68How œ œ
68Œ
or
"
T w Z=
.w R =
I
=

Xw
Xw
Xw
Hw
"
T w Z= .w R=
œ
I
=

œ

Hw!
5X w
5X w
5X w
Hw œ How /B:e" w cI= T w Z= .w R= df
(11.6)
(11.7)
(11.8)
where we have set " w œ "Î5X w . The term "Î5X is so often encountered in statistical physics that we represent this
quantity by the symbol " , which we call the temperature parameter.
Now the probability that a certain configuration occurs is related to the number of microstates associated
with that particular configuration. Thus, the probability of finding our subsystem E with an energy I= , a volume
Z= , and number of particles R= must be given by
T= œ G /" cI= T
w
wZ
= .
wR
=d
(11.9)
where G is some constant independent of the energy, volume and number of particles in the subsystem. In this
expression, the primed quantities explicitly indicate which parameters are associated with the reservoir. In most
texts, however, this distinction is not clearly made - the primes are left off. You are expected to realize that
macroscopic parameters, such as temperature, pressure, and chemical potential, are not associated with a single
atom, or single microscopic region within a gas, but with the much larger universe with which the system is in
contact. Thus, the idea of a single atom having a particular temperature is not correct. Only a large body of
atoms has a well defined temperature. In the rest of this chapter, we will drop the primes and simply remember
that the macroscopic parameters in this last equation are refering to the parameters of the reservoir in contact
with our subsystem, not the subsystem itself.
The Canonical Ensemble
The canonical ensemble describes an ensemble of systems where the number of particles is fixed (i.e., a
closed system). Under these circumstances, the probability of finding the system in a particular state can be
written as
T= œ G /"cI= T Z= d
(11.10)
The Boltzmann Factor
Although this is the most general expression for the canonical ensemble, we will demonstrate later that
the term T Z= is often negligibly small when we consider atomic systemsÞ Likewise, if we consider a fixed
volume, then the term in the exponential containing the volume is just a constant, and can be factored out and
included in the constant G . Thus, for these two cases, we can write the probability function as
T= œ G /"I=
(11.11)
The exponential factor /"I= is known as the “Boltzmann factor”, and the corresponding probability distribution
is called the “Boltzmann” distribution. The Boltzmann distribution gives the probability of finding a small
subsystem (often microscopic) of fixed (or very small) volume with a particular energy I= when the subsystem is
in contact with a larger reservoir at constant temperature X .
Since the sum of all probabilities must equal to unity, the constant of proportionality G , which is
independent of I= , can be determined using this normalization condition. Thus, the constant is given by
"
"
G œ "I œ
/ =
^
(11.12)
=
where the sum is taken over all energy states accessible to the subsystem. This sum is known as the partition
functionß ^ . The partition function is an extremely useful function in statistical mechanics, because, as we will
demonstrate later, a knowledge of the partition function is all that is needed to derive all the pertinent
Chapter Eleven: The Boltzmann Probability Function
4
macroscopic parameters of a system. Thus, the probability of finding a subsystem of fixed volume with a
particular value of the energy can be expressed in terms of the partition function as
T= œ
/"I=
^
(11.13)
This last equation is one of the most powerful equations is statistical physics.
Since T= is the probability that system E has a particular energy I= , we can find the probability that system
E has an energy I within a range between I and I $ I by summing over all the microstates corresponding to
this energy range, or
T ÐIÑ .I œ T= œ G HÐIÑ /"I œ G =I /"I .I
(11.14)
I= œI
where the sum is over all energy states which have energy between I and I $ I . Since the probability T= is
given roughly by the Boltzmann factor times a constant G for each energy state, we see that HÐIÑ œ =ÐIÑ .I is
just the number of different microstates which have an energy in the region between I and I $ I . We call the
number of states within a particular energy range the degeneracy of the system for that particular energy. In many
texts, the symbol for the degeneracy of a particular state is designated by 1ÐIÑß so that the probabilitiy of finding
a system with a particular energy I (within some range .IÑ would be given by
T ÐIÑ .I œ G 1ÐIÑ /"I .I
(11.15)
Again, if we sum over all energies, this must be unity, so we have
" I
.I œ G ( 1ÐIÑ /"I .I œ "
( T ÐIÑ .I œ ( G 1ÐIÑ /
(11.16)
or
Gœ
"
( 1ÐIÑ /
œ
" I
.I
"
^
(11.17)
Note: If the subsystem A has a large number of microstates in the energy range
from I to I $ I , we expect HÐIÑ Òor 1ÐIÑÓ to be a rapidly increasing function of the
energy of the system, just as we demonstrated in an earlier chapter. Since HÐIÑ is a rapidly
increasing function of the energy I and /"I is a rapidly decreasing function of the
energy, we expect T ÐIÑ to be a sharply peaked function of the energy, and for a large
system, with many degrees of freedom, we would expect T ÐIÑ to be negligible except for
µ
µ
values of the energy near I when T ÐIÑ becomes T ÐI Ñ, consistent with arguements
made in an earlier chapter.
As an example of the utility of the Boltzmann distribution, we might ask the probability of finding a single
atom with a particular component of velocity. We can tackle this problem by considering the kinetic energy I of
this atom when it is located within a large collection of atoms in a container with an equilibrium temperature X .
In particular, consider a single monatomic atom in an ideal gas where there is no force of interaction between
particles. The total energy in this case is given by
Iœ
"
"
7@# œ 7Ð@B# @C# @D# Ñ
#
#
(11.18)
so that the probability function is given by
#
T Ð@B ß @C ß @D Ñ .@B .@C .@D œ G/IÎ5X .@ œ G/7@ Î#5X .@B .@C .@D
(11.19)
Chapter Eleven: The Boltzmann Probability Function
5
and the constant can be evaluated using the normalization condition
Gœ
' ' ' @œ∞ /7@# Î#5X .@B .@C .@D
@ϰ
"
œ
' @œ∞ /7@# Î#5X %1@# .@
"
@œ!
œ
"
c#15X Î7d #
$
œ’
7 $#
“
#15X
(11.20)
giving the so-called Maxwell-Boltzmann velocity distribution equation,
0 Ð@B ß @C ß @D Ñ.@B .@C .@D œ ’
7 $# 7@# Î#5X
.@B .@C .@D
“ /
#15X
(11.21)
which we used at the beginning of our study to determine the average kinetic energy for an ideal gas. Remember
that the temperature in this equation is not the temperature of the individual molecule, but the temperature of the
gas in which the molecule resides.
The Negligible Pressure times Volume Factor
To see why it is often possible to use the Boltzmann equation and to ignore the volume of a very small
system let's examine a single atom, say the hydrogen atom. For a single hydrogen atom, we might write the
canonical probability distribution as
T= œ G/"cI= T Z= d
(11.22)
Now the differences in energies within the hydrogen atom for the lowest energy levels are roughly of the order of
electron volts, but let's ask what the contribution of the volume term to the total energy of the system is. If we
assume that the pressure of the surrounding gas is one atmosphere ( ¸ 1 ‚ 10& N/m# ) and assume that the radius
of the atom is roughly 1 ‚ 10"! meters, then the volume term is equivalent to
T Z= ¸ ˆ" ‚ 10& N/m# ‰ˆ1 ‚ 10"! ‰ œ " ‚ "!#& Joules ‚
$
1 eV
œ " ‚ "!' eV
" ‚ "!"*
(11.23)
Thus, the volume term associated with the canonical ensemble for a single atom is practically neglegible in
comparison to the typical energies considered. But even if this were not the case, as long as the volume is held
constant, any volume contribution can be absorbed into the constant G . An interesting question arises, however,
if we actually consider the different possible energy states of the hydrogen atom in more detail. The radius of the
hydrogen atom increases with energy for this quantum system. This means that the system is not, strictly
speaking, a constant volume system. If the energy changes, so does the volume [See Schroeder Problem 6.9].
The Grand Canonical Ensemble
The case where we allow for changes in energy and number of particles, is called the Grand Canonical
Ensemble. In the special case of a fixed volume system (or where the PV term in the exponential is negligible),
the probability of finding a subsystem with R= particles in a state with energy I= is given by
T= œ G e"cI= .R= d
(11.24)
We can again use the normalization condition to determine the constant G . We find that
Gœ
"
e"cI= .R= d
œ
"
m
where m is called the grand partition function or the Gibbs sum. The exponential factor in this last equation is
known as the Gibbs factor
Gibbs Factor œ e"cI= .R= d
The Gibbs factor is most important in applications dealing with quantum statistics - systems where there is a
reasonable probability of finding two or more identical particles in the same quantum state. In these cases, we
concentrate on the probability of occupying a single quantum state of the system. When we do this, the energy I=
associated with this quantum state is just the number of particles in the state times the energy of that state, or
Chapter Eleven: The Boltzmann Probability Function
6
I= œ R= %= . This means that the Gibbs factor can now be written as
Gibbs Factor œ e"c%= R= .R= d œ /"Ð%= .ÑR=
and the probability of finding R particles in a quantum state of energy % is given by
T Ð%ß R Ñ œ
" "Ð% .ÑR
/
m
We find that nature actually restrictions the number of particles that can be in a single quantum state, depending
upon the particle type. It turns out that there are two types of particles found in nature: 1) Fermions (half-integer
spin particles), and 2) Bosons (zero or integral spin particles). The rules of quantum mechanics demand that no
two fermions can occupy the same quantum state, but allow any number of Bosons to occupy the same quantum
state. This difference leads to some interesting consequences which we will consider when we discuss the
quantum nature of statistical physics.