Radon–Nikodym (RN) derivative between two measures arises naturally in the affine structure of the space of probability measures with densities. Entropy, free energy, relative entropy, and entropy production as mathematical concepts associated with RN derivatives are introduced. We identify a simple equation that connects two measures with densities as a possible mathematical basis of the entropy balance equation that is central in nonequilibrium thermodynamics. Application of this formalism to Gibbsian canonical distribution yields many results in classical thermomechanics. An affine structure based on the canonical representation and two divergences are introduced in the space of probability measures. It is shown that thermodynamic work, as a conditional expectation, is indicative of the RN derivative between two energy representations being singular. The entropy divergence and the heat divergence yield respectively a Massieu–Planck potential based and a generalized Carnot inequalities.

两种度量之间的Radon-Nikodym（RN）导数自然出现在具有密度的概率度量空间的仿射结构中。介绍了熵，自由能，相对熵和熵产生等与RN导数相关的数学概念。我们确定了一个简单的方程，该方程将两个度量与密度联系起来，作为熵平衡方程的可能数学基础，而熵平衡方程是非平衡热力学的中心。这种形式主义在吉布斯正则分布中的应用在经典热力学中产生了许多结果。在概率测度的空间中引入了基于典范表示和两个发散的仿射结构。结果表明，热力学功是有条件的期望，表示两个能量表示形式之间的RN导数是奇异的。熵发散和热发散分别产生基于Massieu–Planck势和广义卡诺不等式的信息。