热力学形式主义中的非平衡：第二定律、气体和信息几何
Nonequilibrium in Thermodynamic Formalism: the Second Law, gases and Information Geometry
摘要：
在非平衡态热力学和信息论中，相对熵（或KL散度）起着非常重要的作用。考虑一个Hölder Jacobian $J$和Ruelle（转移）算子$\mathcal{L}_{\log J}.$两个平衡概率$μ_1$和$μ_2$，可以通过离散时间{ \it热力学运算}相互作用，该运算由作用{ \it的对偶Ruelle算子} $ \mathcal{L}_{\log J}^*$描述。我们认为，法律$μ\to \mathcal{L}_{\log J}^*(μ)$，产生非平衡，可以被看作是一个热力学操作后，表明它是热力学第二定律的表现。我们还证明了相对熵的变化满足$$ D_{K L} (μ_1,μ_2) - D_{K L} (\mathcal{L}_{\log J}^*(μ_1),\mathcal{L}_{\log J}^*(μ_2))= 0.$$.此外，我们给出了在$J,μ_1$上得到$h(\mathcal{L}_{\log J}^*(μ_1))\geq h(μ_1)$的充分条件，其中$h$是熵.回顾一个自然的黎曼度量在Banach流形的霍尔德平衡概率，我们展示了二阶泰勒公式的无穷小切线变化的KL散度;一个重要的估计在信息几何。我们引入热、功、体积、压力和内能等概念，这些概念在这里起着气体热力学中类似概念的作用。我们简要介绍了MaxEnt方法。
英文摘要：
In Nonequilibrium Thermodynamics and Information Theory, the relative entropy (or, KL divergence) plays a very important role. Consider a Hölder Jacobian $J$ and the Ruelle (transfer) operator $\mathcal{L}_{\log J}.$ Two equilibrium probabilities $μ_1$ and $μ_2$, can interact via a discrete-time {\it Thermodynamic Operation} described by the action {\it of the dual of the Ruelle operator} $ \mathcal{L}_{\log J}^*$. We argue that the law $μ\to \mathcal{L}_{\log J}^*(μ)$, producing nonequilibrium, can be seen as a Thermodynamic Operation after showing that it's a manifestation of the Second Law of Thermodynamics. We also show that the change of relative entropy satisfies $$ D_{K L} (μ_1,μ_2) - D_{K L} (\mathcal{L}_{\log J}^*(μ_1),\mathcal{L}_{\log J}^*(μ_2))= 0.$$ Furthermore, we describe sufficient conditions on $J,μ_1$ for getting $h(\mathcal{L}_{\log J}^*(μ_1))\geq h(μ_1)$, where $h$ is entropy. Recalling a natural Riemannian metric in the Banach manifold of Hölder equilibrium probabilities we exhibit the second-order Taylor formula for an infinitesimal tangent change of KL divergence; a crucial estimate in Information Geometry. We introduce concepts like heat, work, volume, pressure, and internal energy, which play here the role of the analogous ones in Thermodynamics of gases. We briefly describe the MaxEnt method.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
