  Recent computer simulation results [Barrat {\em et al.}, Physica A 334 (2004)
513] for granular mixtures subject to stochastic driving have shown the
validity of the Einstein relation $\epsilon\equiv D/(T_0\lambda)=1$ between the
diffusion $D$ and mobility $\lambda$ coefficients when the temperature of the
gas $T$ is replaced by the temperature of the impurity $T_0$ in the usual
Einstein relation. This problem is analyzed in this paper by solving
analytically the Boltzmann-Lorentz equation from the Chapman-Enskog method. The
gas is heated by the action of an external driving force (thermostat) which
does work to compensate for the collisional loss of energy. Two types of
thermostats are considered: (a) a deterministic force proportional to the
particle velocity (Gaussian thermostat), and (b) a white noise external force
(stochastic thermostat). The diffusion and mobility coefficients are given in
terms of the solutions of two linear integral equations, which are
approximately solved up to the second order in a Sonine polynomial expansion.
The results show that the violation of the Einstein relation ($\epsilon\neq 1$)
is only due to the non-Maxwellian behavior of the impurity velocity
distribution function (absence of the Gibbs state). At a quantitative level,
the kinetic theory results also show that the deviation of $\epsilon$ from 1 is
more significant in the case of the Gaussian thermostat than in the case of the
stochastic one, in which case the deviation of the Einstein relation is in
general smaller than 1%. This conclusion agrees quite well with the results
found in computer simulations.
