Ternary Representation of Stochastic Change and the Origin of Entropy and Its Fluctuations

Abstract

A change in a stochastic system has three representations: Probabilistic, statistical, and informational: (i) is based on random variable $u(\omega )\to \tilde{u}(\omega )$; this induces (ii) the probability distributions $F_u(x)\to F_{\tilde{u}}(x)$, $x\in {\mathbb{R}}^n$; and (iii) a change in the probability measure $\mathbb{P}\to \tilde{\mathbb{P}}$ under the same observable $u(\omega )$. In the informational representation a change is quantified by the Radon-Nikodym derivative $\ln \left(\frac{d\tilde{\mathbb{P}}}{d\mathbb{P}}(\omega )\right)=-\ln \left(\frac{d{F}_u}{d{F}_{\tilde{u}}}(x)\right)$ when $x=u(\omega )$. Substituting a random variable into its own density function creates a fluctuating entropy whose expectation has been given by Shannon. Informational representation of a deterministic transformation on ${\mathbb{R}}^n$ reveals entropic and energetic terms, and the notions of configurational entropy of Boltzmann and Gibbs, and potential of mean force of Kirkwood. Mutual information arises for correlated $u(\omega )$ and $\tilde{u}(\omega )$; and a nonequilibrium thermodynamic entropy balance equation is identified.

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