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The Enskog kinetic equation is considered to determine the mobility $\lambda$and diffusion $D$ transport coefficients of intruders immersed in a granulargas of inelastic hard spheres (grains). Intruders and grains are in contactwith a thermal bath, which plays the role of a background gas. As usual, theinfluence of the latter on the dynamics of intruders and grains is accountedfor via a viscous drag force plus a stochastic Langevin-like term proportionalto the background temperature $T_\text{b}$. The transport coefficients$\lambda$ and $D$ are determined by solving the kinetic equation by means ofthe Chapman--Enskog method adapted to dissipative dynamics. Both transportcoefficients are given in terms of the solutions of two integral equationswhich are approximately solved up to the second order in a Sonine polynomialexpansion. Theoretical results are compared against numerical solutions of theinelastic 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 thecoefficients $\lambda$ and $D$ allow us to assess the departure of the Einsteinrelation $\epsilon=D/(T_{\text{b}}\lambda)$ from 1. As expected from previousresults for driven granular gases, it is shown that the origin of the deviationof $\epsilon$ from 1 is only due to the non-Maxwellian behavior of referencestate 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 Einsteinrelation. Since the magnitude of $c_0$ is in general very small, deviations ofthe (modified) Einstein relation $\epsilon_0=D/(T_0\lambda)$ from 1 cannot bedetected in computer simulations of dilute granular gases. This conclusionagrees well with previous computer simulation results.

Mobility and diffusion of intruders in granular suspensions. Einstein relation