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贝叶斯热力学第二定律

The Bayesian Second Law of Thermodynamics

Anthony Bartolotta, Sean M. Carroll, Stefan Leichenauer, Jason Pollack

备注：
40 pages. Additional information and animations at http://preposterousuniverse.com/research/bsl/
发表信息：
Phys. Rev. E 94, 022102 (2016)
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摘要：

我们推导出一个广义的热力学第二定律，使用贝叶斯更新明确纳入系统的测量在其演变的某个点的影响。通过允许实验者的知识被测量过程更新，这个公式解决了统计系统的熵有时会向下波动的事实与随机演化系统的知识随时间推移而退化的信息论思想之间的紧张关系。贝叶斯第二定律可以写为$ΔH(ρ_m, ρ) + \langle \mathcal{Q}\rangle_{F|m}\geq 0$，其中$ΔH(ρ_m, ρ)$是原始相空间概率分布$ρ$和测量更新分布$ρ_m$之间的交叉熵的变化，并且$\langle \mathcal{Q}\rangle_{F|m}$是从系统流出的广义热流的期望值。我们还推导出第二定律的改进版本，该第二定律将熵增加从下面约束为非负数，以及Jarzynski等式的贝叶斯版本。我们证明了使用简单的分析和数值例子的形式主义。

英文摘要：

We derive a generalization of the Second Law of Thermodynamics that uses Bayesian updates to explicitly incorporate the effects of a measurement of a system at some point in its evolution. By allowing an experimenter's knowledge to be updated by the measurement process, this formulation resolves a tension between the fact that the entropy of a statistical system can sometimes fluctuate downward and the information-theoretic idea that knowledge of a stochastically-evolving system degrades over time. The Bayesian Second Law can be written as $ΔH(ρ_m, ρ) + \langle \mathcal{Q}\rangle_{F|m}\geq 0$, where $ΔH(ρ_m, ρ)$ is the change in the cross entropy between the original phase-space probability distribution $ρ$ and the measurement-updated distribution $ρ_m$, and $\langle \mathcal{Q}\rangle_{F|m}$ is the expectation value of a generalized heat flow out of the system. We also derive refined versions of the Second Law that bound the entropy increase from below by a non-negative number, as well as Bayesian versions of the Jarzynski equality. We demonstrate the formalism using simple analytical and numerical examples.

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