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Although the Gamma distribution is known within statistical mechanics to describe the probability that a classical ideal gas at a fixed temperature occupies a given energy state it appears to have been all but forgotten. In this paper we present a new and much simpler derivation starting from the Maxwellian velocity distribution. The analysis is extended to open systems in which the number of particles fluctuates. Computer simulations using a hard‐sphere model of a classical ideal gas are used to support the theoretical considerations and show how the Gamma distribution applies in practice. The resulting probability distributions are used to calculate the Shannon information entropy which is then compared with the thermodynamic entropy. We point out an important proof in information entropy that also appears to have been overlooked in statistical thermodynamics and argue on the basis of the work presented here that information theoretic entropy and thermodynamic entropy, whilst obviously related, are not necessarily identical.

Clausius? original papers covering the re-working of Carnot?s theory and the subsequent development of the concept of the entropy of a body are reviewed critically. We show that Clausius? thinking was dominated by the then prevalent idea that a body contained heat and argue that his concept of aequivalenzwerth, the forerunner of entropy, was intended as a measure of the ability of the heat in a body to be transformed into work. This view of heat had been rejected by the mid 1870s but the concept of the entropy of a body, the successor to aequivalenzwerth, not only survived but went on to dominate thinking in thermodynamics. We show that the support lent to the idea of entropy by the statistical approaches of first Maxwell and later Boltzmann shifted the emphasis in thermodynamics away from heat engines and cycles to microscopic phenomena, with the consequence that many of the flaws and contradictions in Clausius? early work appear to have been overlooked. We cite some modern authors who questioned Clausius? concept of entropy and show that even though Clausius? ideas have been decisively rejected in the past his thinking on entropy and reversibility is nonetheless still accepted by many.