%0 Journal Article %1 keyhere %A Garzo, Vicente %D 2004 %J Physica A: Statistical Mechanics and its Applications %K linearresponse granular driven %P 105--126 %T On the Einstein relation in a heated granular gas %U http://www.sciencedirect.com/science/article/B6TVG-4CJHD90-2/2/a56f15a3d0304a2d0eb850074587b1e3 %V 343 %X Recent computer simulation results Physica A 334 (2004) 513 for granular mixtures subject to stochastic driving have shown the validity of the Einstein relation var epsilonidentical toD/(T0lambda)=1 between the diffusion D and mobility lambda coefficients when the temperature of the gas T is replaced by the temperature of the impurity T0 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 (var epsilonnot equal to1) 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 var 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.

@article{keyhere, abstract = {Recent computer simulation results [Physica A 334 (2004) 513] for granular mixtures subject to stochastic driving have shown the validity of the Einstein relation [var epsilon][identical to]D/(T0[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 T0 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 ([var epsilon][not equal to]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 [var 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.}, added-at = {2007-01-18T16:47:58.000+0100}, author = {Garzo, Vicente}, biburl = {https://www.bibsonomy.org/bibtex/2e685e724b4a98e7ac0d153ee2c12c698/andreapuglisi}, description = {ScienceDirect - Physica A: Statistical Mechanics and its Applications : On the Einstein relation in a heated granular gas}, interhash = {e33195e4c37d549eec7826e1b970abd4}, intrahash = {e685e724b4a98e7ac0d153ee2c12c698}, journal = {Physica A: Statistical Mechanics and its Applications}, keywords = {linearresponse granular driven}, month = {#nov#}, pages = {105--126}, timestamp = {2007-01-18T16:47:58.000+0100}, title = {On the Einstein relation in a heated granular gas}, url = {http://www.sciencedirect.com/science/article/B6TVG-4CJHD90-2/2/a56f15a3d0304a2d0eb850074587b1e3}, volume = 343, year = 2004 }