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

Abstract

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 $\alpha$. 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 $W(\tau )$ defined over trajectories of time duration $\tau$. We discuss the properties of this functional and of a similar functional $\bar{W}(\tau )$ 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 $\tau$, 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 $W(\tau )$ at very large times $\tau$. We give an argument for the possible failure of the LS-GC relation in this situation. We also suggest practical recipes for measuring $W(\tau )$ and $\bar{W}(\tau )$ in experiments.

16 pages, 3 figures, submitted for publication

References in corpus (6)

• An Extension of the Fluctuation Theorem
• A fluidized granular medium as an instance of the Fluctuation Theorem
• On the Application of the Gallavotti-Cohen Fluctuation Relation to Thermostatted Steady States Near Equilibrium
• Fluctuations of internal energy flow in a vibrated granular gas
• Granular gases: dynamics and collective effects
• Injected power and entropy flow in a heated granular gas

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