## ABSTRACT

The fluctuation (standard deviation), σ, in thermodynamic parameters is treated in Chapters 5 and 6. First, it is shown that for the exact free energy, F, σ(F) = 0, where for an approximate F, σ(F) > 0. This unrecognized result, can lead to the correct F by extrapolating to zero approximate results for F versus the corresponding σ(F). Next, the fluctuation of the energy in the canonical ensemble is derived, where σ^{2}(E) = k
_{B}
T
^{2}
C_{V}
, C_{V}
~ N and thus, σ(E)/E ~ 1/N
^{1/2 }or ~10^{−12 }for a macroscopic system. The fact that σ(E)/E is so small means that for a large system, most of the contribution to <E> comes from “the most probable energy term,” E*(T). This provides an additional option for calculating thermodynamic averages (but not fluctuations) to (2) the standard method based on thermodynamic derivatives and (3) the calculation of statistical averages by the probabilistic approach. This versatility, which is not always emphasized enough, is demonstrated by applying the three methods to a problem of N-independent spins interacting with a magnetic field, H.