Statistical Analogues of Thermodynamic Extremum Principles

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European Journal of Physics

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As shown by Jaynes, the canonical and grand canonical probability distributions of equilibrium statistical mechanics can be simply derived from the principle of maximum entropy, in which the statistical entropy $S=-\,{k}_{{\rm{B}}}{\sum }_{i}{p}_{i}\mathrm{log}{p}_{i}$ is maximised subject to constraints on the mean values of the energy E and/or number of particles N in a system of fixed volume V. The Lagrange multipliers associated with those constraints are then found to be simply related to the temperature T and chemical potential μ. Here we show that the constrained maximisation of S is equivalent to, and can therefore be replaced by, the essentially unconstrained minimisation of the obvious statistical analogues of the Helmholtz free energy F = E − TS and the grand potential J = F − μN. Those minimisations are more easily performed than the maximisation of S because they formally eliminate the constraints on the mean values of E and N and their associated Lagrange multipliers. This procedure significantly simplifies the derivation of the canonical and grand canonical probability distributions, and shows that the well known extremum principles for the various thermodynamic potentials possess natural statistical analogues which are equivalent to the constrained maximisation of S.