Thermodynamics, jets geometry and phase transitions

Abstract: In this talk, we will discuss equations of state and phase transitions in thermodynamics, as well as their relationship to the geometry of jet spaces and differential equations.

The equations of state give a description of the thermodynamic media under consideration and, in the simplest case, are geometrically realized as Legendre manifolds in the contact space. This corresponds to first-order thermodynamics, and phase transitions of the 1st order correspond to the singularities of their projection onto the space of intensive quantities, and thus their study is based on Arnold's theory of the singularities of projections of Legendre and Lagrangian manifolds.

In practice, it is sometimes more convenient to set equations of state by thermodynamic quantities of the second and higher orders, which means a transition from contact geometry to the geometry of the spaces of jets of the second and higher order. This will be discussed in the talk, and the relationship of the equations of state with overdetermined systems of partial differential equations and with integral manifolds of the Cartan distribution is shown.

If time allows, the relationship between second- and higher-order phase transitions and the singularities given by integral manifolds of the Cartan distribution will be considered.

The proposed approach will also be illustrated by examples from thermodynamics.

mathematical physicsanalysis of PDEsdifferential geometry

Audience: researchers in the topic

Geometry of differential equations seminar