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arXiv · 0709.3124

Combinatorial Entropy for Distinguishable Entities in Indistinguishable States

Robert K. Niven
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Abstract

The combinatorial basis of entropy by Boltzmann can be written $H= {N}^{-1} \ln \mathbb{W}$, where $H$ is the dimensionless entropy of a system, per unit entity, $N$ is the number of entities and $\mathbb{W}$ is the number of ways in which a given realization of the system can occur, known as its statistical weight. Maximizing the entropy (``MaxEnt'') of a system, subject to its constraints, is then equivalent to choosing its most probable (``MaxProb'') realization. For a system of distinguishable entities and states, $\mathbb{W}$ is given by the multinomial weight, and $H$ asymptotically approaches the Shannon entropy. In general, however, $\mathbb{W}$ need not be multinomial, leading to different entropy measures. This work examines the allocation of distinguishable entities to non-degenerate or equally degenerate, indistinguishable states. The non-degenerate form converges to the Shannon entropy in some circumstances, whilst the degenerate case gives a new entropy measure, a function of a multinomial coefficient, coding parameters, and Stirling numbers of the second kind.

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Robert K. Niven. 2007-09-21. Combinatorial Entropy for Distinguishable Entities in Indistinguishable States. https://doi.org/10.1063/1.2828767

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