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We investigated the validity of fluctuation-dissipation relations in thenonequilibrium stationary state of fluidized granular media under gravity bytwo independent approaches, based on theory and numerical simulations. Aphenomenological Langevin-type theory describing the fluctuation of center ofmass height, which was originally constructed for a one-dimensional granulargas on a vibrating bottom plate, was generalized to any dimensionality, evenfor the case in which the vibrating bottom plate is replaced by a thermal wall.The theory predicts a fluctuation-dissipation relation known to be satisfied atequilibrium, with a modification that replaces the equilibrium temperature byan effective temperature defined by the center of mass kinetic energy. To testthe validity of the fluctuation-dissipation relation, we performed extensiveand accurate event-driven molecular dynamics simulations for the model systemwith a thermal wall at the bottom. The power spectrum and response function ofthe center of mass height were measured and closely compared with theoreticalpredictions. It is shown that the fluctuation-dissipation relation for thegranular system is satisfied, especially in the high-frequency (short time)region, for a wide range of system parameters. Finally, we describe therelationship between systematic deviations in the low-frequency (long time)region and the time scales of the driven granular system.

Fluctuation-Dissipation Relations for Motions of Center of Mass in Driven Granular Fluids under Gravity