On Higher Order Structures in Thermodynamics

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.

Flexible Lagrangians

We introduce and discuss notions of regularity and flexibility for Lagrangian manifolds with Legendrian boundary in Weinstein domains. There is a surprising abundance of flexible Lagrangians. In turn, this leads to new constructions of Legendrians submanifolds and Weinstein manifolds. For instance, many closed n-manifolds of dimension n > 2 can be realized as exact Lagrangian submanifolds of $T^{\ast }S^n$ with possibly exotic Weinstein symplectic structures. These Weinstein structures on $T^{\ast } S^n$, infinitely many of which are distinct, are formed by a single handle attachment to the standard 2n-ball along the Legendrian boundaries of flexible Lagrangians. We also formulate a number of open problems.