Dissipative homogeneous Maxwell mixtures: ordering transition in the tracer limit

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 α ij ). In the tracer limit (c 1 → 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 $${\mu\equiv m_1/m_2}$$ , there indeed exist two critical values $${\mu_{\rm HCS}^{(-)}}$$ and $${\mu_{\rm HCS}^{(+)}}$$ (which depend on the coefficients of restitution), such that E 1/E = 0 for $${\mu_{\rm HCS}^{(-)}<\mu<\mu_{\rm HCS}^{(+)}}$$ (disordered or normal phase), while $${E_1/E\neq 0}$$ for $${\mu<\mu_{\rm HCS}^{(-)}}$$ and/or $${\mu>\mu_{\rm HCS}^{(+)}}$$ (ordered phase).