Origins of the Combinatorial Basis of Entropy

Abstract

The combinatorial basis of entropy, given by Boltzmann, can be written $H = N^{-1} \ln \mathbb{W}$, where $H$ is the dimensionless entropy, $N$ is the number of entities and $\mathbb{W}$ is number of ways in which a given realization of a system can occur (its statistical weight). This can be broadened to give generalized combinatorial (or probabilistic) definitions of entropy and cross-entropy: $H=κ(ϕ(\mathbb{W}) +C)$ and $D=-κ(ϕ(\mathbb{P}) +C)$, where $\mathbb{P}$ is the probability of a given realization, $ϕ$ is a convenient transformation function, $κ$ is a scaling parameter and $C$ an arbitrary constant. If $\mathbb{W}$ or $\mathbb{P}$ satisfy the multinomial weight or distribution, then using $ϕ(\cdot)=\ln(\cdot)$ and $κ=N^{-1}$, $H$ and $D$ asymptotically converge to the Shannon and Kullback-Leibler functions. In general, however, $\mathbb{W}$ or $\mathbb{P}$ need not be multinomial, nor may they approach an asymptotic limit. In such cases, the entropy or cross-entropy function can be {\it defined} so that its extremization ("MaxEnt'' or "MinXEnt"), subject to the constraints, gives the ``most probable'' (``MaxProb'') realization of the system. This gives a probabilistic basis for MaxEnt and MinXEnt, independent of any information-theoretic justification. This work examines the origins of the governing distribution $\mathbb{P}$.... (truncated)