Mobility and diffusion of intruders in granular suspensions. Einstein relation

The Enskog kinetic equation is considered to determine the mobility $\lambda$ and diffusion $D$ transport coefficients of intruders immersed in a granular gas of inelastic hard spheres (grains). Intruders and grains are in contact with a thermal bath, which plays the role of a background gas. As usual, the influence of the latter on the dynamics of intruders and grains is accounted for via a viscous drag force plus a stochastic Langevin-like term proportional to the background temperature $T_\text{b}$. The transport coefficients $\lambda$ and $D$ are determined by solving the kinetic equation by means of the Chapman--Enskog method adapted to dissipative dynamics. Both transport coefficients are given in terms of the solutions of two integral equations which are approximately solved up to the second order in a Sonine polynomial expansion. Theoretical results are compared against numerical solutions of the inelastic Enskog equation by means of the direct simulation Monte Carlo (DSMC) method. Good agreement between theory and simulations is in general found, specially in the case of the second Sonine approximation. The knowledge of the coefficients $\lambda$ and $D$ allow us to assess the departure of the Einstein relation $\epsilon=D/(T_{\text{b}}\lambda)$ from 1. As expected from previous results for driven granular gases, it is shown that the origin of the deviation of $\epsilon$ from 1 is only due to the non-Maxwellian behavior of reference state of intruders (measured by the cumulant $c_0$) when the bath temperature $T_\text{b}$ is replaced by the intruder temperature $T_0$ in the Einstein relation. Since the magnitude of $c_0$ is in general very small, deviations of the (modified) Einstein relation $\epsilon_0=D/(T_0\lambda)$ from 1 cannot be detected in computer simulations of dilute granular gases. This conclusion agrees well with previous computer simulation results.