Statistical Mechanics - Physics at Oregon State University

24 CHAPTER 2. THE CANONICAL ENSEMBLE
Reservoirs are what we need!
The theoretical way to specify temperature is similar to the experimental setup.
We assume that the system S is in contact with and in thermal equilibrium
with a very large reservoir R with temperature T. Only heat can flow back and
forth between R and S. Volumes do not change, and no work is done between
the system and the reservoir. The number of particles in the reservoir is NR
and is much larger than the number of particles in the system NS. How much
larger?? In thermal equilibrium the fluctuations of the energy per particle in
the reservoir are
∆UR
NR
∝ 1
√ NR
or with UR = ɛRNR we can write this in the form ∆UR = ɛR
the maximum energy of the system is US = NSɛS. If we demand that
NR ≫ N 2 S(ɛS/ɛR) 2
(2.1)
√ NR. Suppose
(2.2)
the system can take all of its energy from the reservoir without noticeably
changing the state of the reservoir! In this case it is really allowed to assume that
the reservoir is at a constant temperature. In practice, this leads to impossibly
large reservoir sizes, but in theory that is, of course, not a problem. For example,
if we assume that a typical reservoir has 10 24 particles, we could only consider
systems of about 10 12 particles. This is too small. In reality, therefore, reservoirs
do lead to finite size effects. These effects are very hard to calculate, and depend
on the nature of the reservoir. We will therefore ignore them, and assume that
the reservoirs are very large, and that their only effect is to set the temperature.
Probabilities.
We will now address the question: what is the probability of finding
the system S in a unique quantum state s if it is in thermal equilibrium
with a reservoir at temperature T? The energy U0 of the total
combination, system plus reservoir, cannot change. Hence if the system is in
a state s with energy ɛs, the reservoir has to have energy U0 − ɛs. The total
number of states available in this configuration for the combination of system
and reservoir is gR(U0 − ɛs). This does not depend on the multiplicity function
of the system, since we specify the state of the system. The multiplicity function
for the combined reservoir and system is
gS+R(U0) =
gR(U0 − ɛs) (2.3)
s
All states of the combination of system and reservoir are equally probably,
and hence the probability of finding the combination system plus reservoir in
the configuration with the system in state s is