From Statistical Thermodynamics to (C-)Integrability and back

Abstract: Phase transitions in macroscopic systems at the equilibrium have shown to be intimately related to the theory of nonlinear conservation laws. Examples range from simple systems like van der Waals fluids, (uniaxial) Nematic Liquid Crystal models and spin systems to complex systems as Random Matrix models. In such models, order parameters fulfil suitable nonlinear PDEs with prescribed initial conditions and phase transitions are explained in terms of shock waves travelling in the space of control parameters (e.g. thermodynamic variables). Restricting our analysis to simple systems, many paradigmatic models in Statistical Thermodynamics turn out to be C-Integrable, that is there exists a nonlinear transformation which maps a non-linear PDE of hydrodynamic type associated with the physical problem to a linear one. Conversely, from a C-Integrability ansatz applied to first principles of Thermodynamics one can retrieve families of models which often generalise known ones, providing insights of physical relevance.

The talk aims at discussing the connection between Statistical Thermodynamics and C-Integrability in the context of fluid systems, with a special focus on recent results on biaxial Nematic Liquid Crystals (doi: 10.1098/rspa.2023.0701). We will see that the occurrence of distinct thermodynamic phases in certain degenerate domains of thermodynamic variables is identified by reductions of the underlying system of conservation laws. This highlights the importance of developing solid mathematical and theoretical tools to extend simple thermodynamic models and construct more general ones.

Mathematics

Audience: researchers in the topic

Fluids and Structures Seminar @ UEA