Phase Transition in a Vlasov-Boltzmann Binary Mixture

TL;DR

This paper proves the existence and stability of phase transitions in a kinetic model of a binary mixture with long-range interactions, showing how homogeneous states become unstable and segregated phases emerge at low temperatures.

Contribution

It introduces a kinetic Vlasov-Boltzmann model for a binary mixture and rigorously proves bifurcation phenomena and phase transition stability at varying temperatures.

Findings

Homogeneous Maxwellian equilibrium becomes unstable below critical temperature.

Segregated phases emerge as stable non homogeneous solutions.

Bifurcation phenomena are proven for any Knudsen number.

Abstract

There are not many kinetic models where it is possible to prove bifurcation phenomena for any value of the Knudsen number. Here we consider a binary mixture over a line with collisions and long range repulsive interaction between different species. It undergoes a segregation phase transition at sufficiently low temperature. The spatially homogeneous Maxwellian equilibrium corresponding to the mixed phase, minimizing the free energy at high temperature, changes into a maximizer when the temperature goes below a critical value, while non homogeneous minimizers, corresponding to coexisting segregated phases, arise. We prove that they are dynamically stable with respect to the Vlasov-Boltzmann evolution, while the homogeneous equilibrium becomes dynamically unstable.