Dissipative homogeneous Maxwell mixtures: ordering transition in the tracer limit

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).