On Higher Order Structures in Thermodynamics

TL;DR

This paper develops a thermodynamic framework based on measurement theory, introducing higher order geometric structures like cubic and fourth order forms to describe skewness and positivity constraints in thermodynamic states.

Contribution

It introduces higher order geometric structures in thermodynamics derived from measurement-based descriptions, including cubic and fourth order forms.

Findings

Higher order forms relate to skewness and state constraints.

Symmetric processes correspond to zero cubic form.

Positivity of the fourth order form imposes additional state conditions.

Abstract

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.