We study a connection between chemical thermodynamics and information geometry. We clarify a relation between the Gibbs free energy of an ideal dilute solution and an information geometric quantity called an $f$ divergence. From this relation, we derive information geometric inequalities that give a speed limit for the changing rate of the Gibbs free energy and a general bound of chemical fluctuations. These information geometric inequalities can be regarded as generalizations of the Cramér–Rao inequality for chemical reaction networks described by rate equations, where un-normalized concentration distributions are of importance rather than probability distributions. They hold true for damped oscillatory reaction networks and systems where the total concentration is not conserved, so that the distribution cannot be normalized. We also formulate a trade-off relation between speed and time on a manifold of concentration distribution by using the geometrical structure induced by the $f$ divergence. Our results apply to both closed and open chemical reaction networks; thus they are widely useful for thermodynamic analysis of chemical systems from the viewpoint of information geometry.

中文翻译：

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化学热力学的信息几何不等式

我们研究化学热力学和信息几何之间的联系。我们阐明了理想稀溶液的吉布斯自由能与称为的几何信息量之间的关系。$F$ 分歧。从这种关系中，我们得出信息几何不等式，这些几何不等式为吉布斯自由能的变化速率和化学波动的一般界限提供了速度限制。这些信息几何不等式可以看作是速率方程所描述的化学反应网络的Cramér-Rao不等式的推广，其中非标准化浓度分布比概率分布更重要。它们适用于阻尼振荡反应网络和总浓度不守恒的系统，因此分布无法归一化。我们还利用浓度分布诱导的几何结构，在浓度分布的流形上建立了速度与时间之间的权衡关系。$F$分歧。我们的结果适用于封闭和开放的化学反应网络；因此，从信息几何学的角度来看，它们对于化学系统的热力学分析非常有用。