Coexistence of shocks and rarefaction fans: complex phase diagram of a simple hyperbolic particle system

TL;DR

This paper explores the complex hydrodynamic behaviors, including shocks and rarefaction fans, in a simple asymmetric particle system, revealing a phase diagram with changing convexity and mixed wave solutions.

Contribution

It provides a rigorous analysis of the hydrodynamic flux and entropy solutions, demonstrating how the phase diagram varies with model parameters in a simple hyperbolic particle system.

Findings

Hydrodynamic flux changes convexity in certain parameter regions.

The phase diagram includes coexistence of shocks and rarefaction fans.

Entropy solutions exhibit complex wave interactions depending on initial conditions.

Abstract

This paper investigates the non-equilibrium hydrodynamic behavior of a simple totally asymmetric interacting particle system of particles, antiparticles and holes on $\mathbb{Z}$. Rigorous hydrodynamic results apply to our model with a hydrodynamic flux that is exactly calculated and shown to change convexity in some region of the model parameters. We then characterize the entropy solutions of the hydrodynamic equation with step initial condition in this scenario which include various mixtures of rarefaction fans and shock waves. We highlight how the phase diagram of the model changes by varying the model parameters.