随机变化的三元表示与熵及其涨落的起源
Ternary Representation of Stochastic Change and the Origin of Entropy and Its Fluctuations
摘要：
随机系统中的变化有三种表示：概率、统计和信息：（i）基于随机变量$u(ω)\to\tilde{u}(ω)$ ;这导致（ii）概率分布$F_u(x)\to F_{\tilde{u}}(x)$、$x\in\mathbb{R}^n$ ;（iii）在相同的可观测$u(ω)$下概率测度$\mathbb{P}\to\tilde{\mathbb{P}}$的变化。在信息表示中，当$x=u(ω)$时，通过Radon-Nikodym导数$\ln\left( \frac{ d \tilde{\mathbb{P}}}{ d\mathbb{P}}(ω)\right)=-\ln\left(\frac{ d F_u}{ d F_{\tilde{u}}}(x)\right)$量化变化。将一个随机变量代入它自己的密度函数，就产生了一个涨落熵，其期望值已经由香农给出。$\mathbb{R}^n$上确定性变换的信息表示揭示了熵和能量项，以及玻尔兹曼和吉布斯的构型熵和柯克伍德的平均力势的概念。相关的$u(ω)$和$\tilde{u}(ω)$的互信息出现，并确定了非平衡热力学熵平衡方程。
英文摘要：
A change in a stochastic system has three representations: Probabilistic, statistical, and informational: (i) is based on random variable $u(ω)\to\tilde{u}(ω)$; 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 $u(ω)$. In the informational representation a change is quantified by the Radon-Nikodym derivative $\ln\left( \frac{ d \tilde{\mathbb{P}}}{ d\mathbb{P}}(ω)\right)=-\ln\left(\frac{ d F_u}{ d F_{\tilde{u}}}(x)\right)$ when $x=u(ω)$. 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 $u(ω)$ and $\tilde{u}(ω)$; and a nonequilibrium thermodynamic entropy balance equation is identified.
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