Abstract
Disagreements over the meaning of the thermodynamic entropy and how it should be defined in statistical mechanics have endured for well over a century. In an earlier paper, I showed that there were at least nine essential properties of entropy that are still under dispute among experts. In this paper, I examine the consequences of differing definitions of the thermodynamic entropy of macroscopic systems.
Two proposed definitions of entropy in classical statistical mechanics are (1) defining entropy on the basis of probability theory (first suggested by Boltzmann in 1877), and (2) the traditional textbook definition in terms of a volume in phase space (also attributed to Boltzmann). The present paper demonstrates the consequences of each of these proposed definitions of entropy and argues in favor of a definition based on probabilities.
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Notes
I am defining macroscopic systems as in Ref. [1]. A macroscopic systems contains a sufficiently large number of particles, so that the statistical fluctuations of extensive variables is smaller than the resolution of experimental measurements.
I use the standard definition of “distinguishability” from quantum mechanics. Two particles are distinguishable if exchanging them produces a different microscopic state. Since exchanging two classical particles produces a different point in phase space, classical particles are intrinsically distinguishable.
References
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Acknowledgements
I would like to thank Jan Tobochnik for very useful comments and suggestions. I would also like to thank Dennis Dieks for an interesting discussion. Finally, I would like to thank Erwin Frey and the members of the Arnold Sommerfeld Center for Theoretical Physics in Munich for their gracious hospitality during this work.
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Swendsen, R.H. Choosing a Definition of Entropy that Works. Found Phys 42, 582–593 (2012). https://doi.org/10.1007/s10701-012-9627-y
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DOI: https://doi.org/10.1007/s10701-012-9627-y