On Higher Order Structures in Thermodynamics.

Valentin Lychagin,Mikhail Roop

doi:10.3390/e22101147

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.

Highlights

The geometrical interpretation of thermodynamic systems in equilibrium goes back already to the 19th century [1] and is reflected recently in Reference [2]
This paper presents some new results in equilibrium thermodynamics that arise from the measurement approach to thermodynamics
Components of which in thermodynamics are specific energy and volume, one gets a Legendrian manifold representing the average of random vector

Summary

Introduction

The geometrical interpretation of thermodynamic systems in equilibrium goes back already to the 19th century [1] and is reflected recently in Reference [2]. Legendrian manifolds represent averages of measurable quantities (or extremal probability distributions) that are extensive thermodynamic variables, while the Riemannian structure is their variance, i.e., contact and Riemannian structures arise from the first two central moments of random vectors. We develop the geometrical approach to thermodynamic states and extend it by considering the third and the fourth order moments and corresponding symmetric forms of the third and the fourth order on Legendrian manifolds of two types of gases, ideal and van der Waals. The third order symmetric form represents the skewness of the extremal probability distribution, as well as thermodynamic processes along which the skewness is equal to zero, we, call symmetric We elaborate such processes for ideal gases explicitly, and, in the case of real gases represented by the van der Waals model, we show that there are domains on their Legendrian manifold where there are either three types of such processes, or one. In Reference [9,10], the central moments are used to construct scalar differential invariants of Aff(W )

Third Central Moment σ3

Ideal Gas

Fourth Central Moment σ4

Discussion