In Nonequilibrium Thermodynamics and Information Theory, the relative entropy (or, KL divergence) plays a very important role. Consider a H\"older Jacobian $J$ and the Ruelle (transfer) operator $\mathcal{L}_{\log J}.$ Two equilibrium probabilities $\mu_1$ and $\mu_2$, 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 $\mu \to \mathcal{L}_{\log J}^*$, 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 $$ D_{K L} (\mu_1,\mu_2) - D_{K L} (\mathcal{L}_{\log J}^*(\mu_1),\mathcal{L}_{\log J}^*(\mu_2))= 0.$$ Furthermore, we describe sufficient conditions on $J,\mu_1$ for getting $h(\mathcal{L}_{\log J}^*(\mu_1))\geq h(\mu_1)$, where $h$ is entropy. Recalling a natural Riemannian metric in the Banach manifold of H\"older 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.
翻译:在 Nonequiliblium 热力学和信息理论中, 相对的恒温值( 或, KL 差值) 扮演着非常重要的角色 。 我们认为H\" older Jacobian $J$ 和Ruelle( 传输) 操作员$\ mathcal{L ⁇ log J} 。 $ 2 平衡概率 $\ mu_ 1 和 $\ mu_ 2$ 。 可以通过离散时间 热力操作进行互动 。 由Ruelle 操作的双倍动作所描述 。 $\ mathcal { calsal{ { { lig $} 。 我们认为, 法律 $\ cocoupal ral {\\ logal J} $。 在显示热力学第二定律的表达式后, $@ kK} ( umo) etropylex $, ral_ bl_ ma\\\ lax a modeal deal demodealmental_ $.