Statistical Mechanics - Physics at Oregon State University

3.5. OVERVIEW OF CALCULATION METHODS. 55
occur easily, it is also easy to induce a change in the average by external means,
e.g. changing µ. Finally, the fluctuations ∆N are proportional to √ N. In the
thermodynamic limit the fluctuations in the number of particles are infinitely
large, but the fluctuations per particle become infinitesimally small.
3.5 Overview of calculation methods.
The development of the ideas used in statistical mechanics starts with the discussion
of a closed system, with extensive parameters V, U, and N (and possibly
more), which can be set at specified values. These extensive parameters are the
independent parameters describing the system. The link between the microscopic
world and thermodynamics is obtained by calculating the entropy. There
are several ways of doing so (more about that in a later chapter), but here we
start with the multiplicity function, g(U, V, N). This function gives the number
of microscopic states available to the system at given U, V, and N. Next, we
define the entropy analogue by S(U, V, N) = kB log(g(U, V, N)). We then take
the thermodynamic limit, making the system infinite, and all variables become
continuous. The thermodynamic limit is always needed, no matter which formulation
of statistical mechanics one is using. Sometimes people try to hide it, and
calculate statistical mechanical functions for small systems. But those results
are suspect, because only in the thermodynamic limit can we prove that the
entropy analogue defined above is equivalent to the entropy defined in thermodynamics.
Once we have shown that the entropy analogue is indeed equivalent
to the thermodynamical entropy, we can use all results from thermodynamics,
and we are in business.
The path followed above is a good path for theoretical development, but
often not a good way for practical calculations. Using the temperature T as an
independent variable makes life much easier (most of the time). In our model
system we can now choose the following combinations of independent variables:
(T, V, N), (T, V, µ) , and (T, p, N). Note that we need at least one extensive
variable. Other combinations are possible to, but do not lead to useful free
energies (see thermodynamics). Next, we construct the Hilbert space of all
possible quantum states which are consistent with the values of the extensive
parameters in the combination. This space is, for the three cases above, S(V, N)
, S(V ) , and S(N), respectively. Next, we calculate the partition sum for these
situations:
Z(T, V, N) =
Z(T, V, µ) =
s∈S(V,N)
s∈S(V )
ζ(T, p, N) =
s∈S(N)
e −βɛs (3.62)
e −β(ɛs−µns)
e −β(ɛs+pvs)
(3.63)
(3.64)