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

Andrea Puglisi 1, Paolo Visco 1, 2, Emmanuel Trizac 2, Frederic van Wijland 1, 3

Physical Review E: Statistical, Nonlinear, and Soft Matter Physics 73 (2006) 021301

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.

  • 1. Laboratoire de Physique Théorique d'Orsay (LPT),
    CNRS : UMR8627 – Université Paris XI - Paris Sud
  • 2. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
    CNRS : UMR8626 – Université Paris XI - Paris Sud
  • 3. Matière et Systèmes Complexes (MSC),
    CNRS : UMR7057 – Université Paris VII - Paris Diderot