Impurity in a sheared inelastic Maxwell gas

Abstract

The Boltzmann equation for inelastic Maxwell models is considered in order to investigate the dynamics of an impurity (or intruder) immersed in a granular gas driven by a uniform shear flow. The analysis is based on an exact solution of the Boltzmann equation for a granular binary mixture. It applies for conditions arbitrarily far from equilibrium (arbitrary values of the shear rate a) and for arbitrary values of the parameters of the mixture (particle masses m(i), mole fractions x(i), and coefficients of restitution α(ij)). In the tracer limit where the mole fraction of the intruder species vanishes, a nonequilibrium phase transition takes place. We thereby identify ordered phases where the intruder bears a finite contribution to the properties of the mixture, in a region of parameter space that is worked out in detail. These findings extend previous results obtained for ordinary Maxwell gases, and further show that dissipation leads to new ordered phases.