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Dynamics of a tracer granular particle as a non-equilibrium Markov process

The dynamics of a tracer particle in a stationary driven granular gas is investigated. We show how to transform the linear Boltzmann equation describing the dynamics of the tracer into a master equation for a continuous Markov process. The transition rates depend upon the stationary velocity distribution of the gas. When the gas has a Gaussian velocity probability distribution function (pdf), the stationary velocity pdf of the tracer is Gaussian with a lower temperature and satisfies detailed balance for any value of the restitution coefficient $α$. As soon as the velocity pdf of the gas departs from the Gaussian form, detailed balance is violated. This non-equilibrium state can be characterized in terms of a Lebowitz-Spohn action functional $W(τ)$ defined over trajectories of time duration $τ$. We discuss the properties of this functional and of a similar functional $\bar{W}(τ)$ which differs from the first for a term which is non-extensive in time. On the one hand we show that in numerical experiments, i.e. at finite times $τ$, the two functionals have different fluctuations and $\bar{W}$ always satisfies an Evans-Searles-like symmetry. On the other hand we cannot observe the verification of the Lebowitz-Spohn-Gallavotti-Cohen (LS-GC) relation, which is expected for $W(τ)$ at very large times $τ$. We give an argument for the possible failure of the LS-GC relation in this situation. We also suggest practical recipes for measuring $W(τ)$ and $\bar{W}(τ)$ in experiments.