Statistical Mechanics - Physics at Oregon State University

4 CHAPTER 1. FOUNDATION OF STATISTICAL MECHANICS.
have to be able to define an outside heat reservoir. Therefore, even though in
statistical mechanics we will be able to evaluate the entropy analogue for very
small systems or of the largest possible system, the universe, we cannot equate
this calculated quantity with the entropy used in thermodynamics.
Microscopic origin of model parameters.
In thermodynamics one postulates relations between all the parameters (equations
of state) and formulas for different forms of the energy (state functions).
Typical equations of state are:
for an ideal gas, or the van der Waals form
pV = NRT (1.1)
2 N
(p + a
V 2 )(V − Nb) = NkBT (1.2)
Based on macroscopic considerations, the parameters a and b in the second
equation are simply parameters which could be fit to the experimental results.
They can only be given a definite meaning in a discussion of the microscopic
aspects of the models. For example, using experimental data one derives a
value for b of about 1 m3 /atom. This is not a problem in thermodynamics,
but it is in statistical mechanics, where the parameter b is equal to the volume
of an atom. Having this statistical mechanics background available, one can
exclude certain ranges of parameter values as unrealistic. Next, the equations
of thermodynamics are used together with these equations of state to describe
the properties of the system. Phase transitions are possible when using the van
der Waals equation, but not with the ideal gas law.
1.2 Program of statistical mechanics.
What do we do?
Statistical mechanics aims to derive all these macroscopic relations from a
microscopic point of view. An atomic model is essential in this case; thermodynamics
can be derived without any reference to the atomic nature of matter.
Since the number of particles is very large in many cases (we always have to
take the limit N → ∞ ), one can only discuss average quantities. The program
of statistical mechanics can therefore be described by the following four steps:
1. Assume an atomic model.
2. Apply the equations of motion (Newton, Schrödinger, etc).
3. Calculate average quantities.