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).
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Garzó, V., Trizac, E. Dissipative homogeneous Maxwell mixtures: ordering transition in the tracer limit. Granular Matter 14, 99–104 (2012). https://doi.org/10.1007/s10035-011-0304-1
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DOI: https://doi.org/10.1007/s10035-011-0304-1