Dissipative homogeneous Maxwell mixtures: ordering transition in the tracer limit

TL;DR

This paper investigates the energy distribution and phase transitions of tracer particles in a granular gas using the homogeneous Boltzmann equation, revealing two distinct phases depending on particle mass ratios and restitution coefficients.

Contribution

It provides exact results for inelastic Maxwell mixtures in the homogeneous cooling state, identifying phase transitions in tracer energy contributions based on mass ratios.

Findings

Two phases identified: disordered and ordered.

Critical mass ratios depend on restitution coefficients.

Tracer energy contribution is zero or non-zero depending on phase.

Abstract

The homogeneous Boltzmann equation for inelastic Maxwell mixtures is considered to study the dynamics of tracer particles or impurities (solvent) immersed in a uniform granular gas (solute). The analysis is based on exact results derived for a granular binary mixture in the homogeneous cooling state (HCS) that apply for arbitrary values of the parameters of the mixture (particle masses $m_i$, mole fractions $c_i$, and coefficients of restitution $\alpha_{ij}$). In the tracer limit ($c_1\to 0$), it is shown that the HCS supports two distinct phases that are evidenced by the corresponding value of $E_1/E$, the relative contribution of the tracer species to the total energy. Defining the mass ratio $\mu = m_1/m_2$, there indeed exist two critical values $\mu_\text{HCS}^{(-)}$ and $\mu_\text{HCS}^{(+)}$ (which depend on the coefficients of restitution), such that $E_1/E=0$ for $\mu_\text{HCS}^{(-)}<\mu<\mu_\text{HCS}^{(+)}$ (disordered or normal phase), while $E_1/E\neq 0$ for $\mu<\mu_\text{HCS}^{(-)}$ and/or $\mu>\mu_\text{HCS}^{(+)}$ (ordered phase).