Tracer diffusion coefficients in a sheared inelastic Maxwell gas

We study the transport properties of an impurity in a sheared granular gas, in the framework of the Boltzmann equation for inelastic Maxwell models. We investigate here the impact of a nonequilibrium phase transition found in such systems, where the tracer species carries a finite fraction of the total kinetic energy (ordered phase). To this end, the diffusion coefficients are first obtained for a granular binary mixture in spatially inhomogeneous states close to the simple shear flow. In this situation, the set of coupled Boltzmann equations are solved by means of a Chapman-Enskog-like expansion around the (local) shear flow distributions for each species, thereby retaining all the hydrodynamic orders in the shear rate $a$. Due to the anisotropy induced by the shear flow, three tensorial quantities $D_{ij}$, $D_{p,ij}$, and $D_{T,ij}$ are required to describe the mass transport process instead of the conventional scalar coefficients. These tensors are given in terms of the solutions of a set of coupled algebraic equations, which can be \emph{exactly} solved as functions of the shear rate $a$, the coefficients of restitution $\alpha_{sr}$ and the parameters of the mixture (masses and composition). Once the forms of $D_{ij}$, $D_{p,ij}$, and $D_{T,ij}$ are obtained for arbitrary mole fraction $x_1=n_1/(n_1+n_2)$ (where $n_r$ is the number density of species $r$), the tracer limit ($x_1\to 0$) is carefully considered for the above three diffusion tensors. Explicit forms for these coefficients are derived showing that their shear rate dependence is significantly affected by the order-disorder transition.