We present the development of the approach to thermodynamics based on measurement. First of all, we recall that considering classical thermodynamics as a theory of measurement of extensive variables one gets the description of thermodynamic states as Legendrian or Lagrangian manifolds representing the average of measurable quantities and extremal measures. Secondly, the variance of random vectors induces the Riemannian structures on the corresponding manifolds. Computing higher order central moments, one drives to the corresponding higher order structures, namely the cubic and the fourth order forms. The cubic form is responsible for the skewness of the extremal distribution. The condition for it to be zero gives us so-called symmetric processes. The positivity of the fourth order structure gives us an additional requirement to thermodynamic state.

中文翻译：

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热力学中的高阶结构


我们介绍了基于测量的热力学方法的发展。首先，我们回想一下，将经典热力学视为广泛变量的测量理论，可以将热力学状态描述为表示可测量量和极值测量的平均值的勒让德或拉格朗日流形。其次，随机向量的方差导出相应流形上的黎曼结构。计算高阶中心矩，可以得到相应的高阶结构，即三次和四阶形式。三次形式决定了极值分布的偏度。它为零的条件给了我们所谓的对称过程。四级结构的正性给我们对热力学状态提出了额外的要求。