Equilibrium distribution functions: Another look

Abstract

The purpose of this study is to provide students with an alternate "derivation" of the equilibrium distribution, one which is based on discrete examples which are then, by simple mathematical induction, extrapolated to the limits of large numbers of particles.

Comments on the Extensivity of the Boltzmann Entropy

“…Importantly, as we shall see for lim N→∞ (lnN! -NlnN + N) )= ∞ [10,11,15], which means that at the thermodynamic limit this form of the Stirling approximation can not be used to prove the extensivity of the Boltzmann entropy. The consequence is that the Boltzmann entropy, as shown in the following for the ideal gas and other simple systems, is in general not extensive and thus not an extensive state function irrespective on the number of particles N present.…”

Energy Distributions in Small Populations: Pascal versus Boltzmann

“…A number of attempts have been made to derive the Boltzmann distribution law without using Lagrange's method or Stirling's approximation (5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17). Some of these articles take a thermodynamic approach wherein the Helmholtz energy is minimized (5)(6)(7)(8) or the entropy is maximized (9); others (10)(11)(12)(13)(14) use discrete substitutions to eliminate constant N and constant E constraints, where N is the total number of particles and E is the total energy of a thermodynamic system; others eliminate Stirling's approximation by a series substitution (15,16). A unique approach was recently offered by Friedman and Grubbs (17) who used a geometrical argument to predict Boltzmann probabilities from hyperplanes in N-dimensional state space, a line of reasoning that also led to the binomial coefficients of Pascal's triangle.…”

“…A number of attempts have been made to derive the Boltzmann distribution law without using Lagrange’s method or Stirling’s approximation − . Some of these articles take a thermodynamic approach wherein the Helmholtz energy is minimized − or the entropy is maximized ; others − use discrete substitutions to eliminate constant N and constant E constraints, where N is the total number of particles and E is the total energy of a thermodynamic system; others eliminate Stirling’s approximation by a series substitution , .…”