Thermalization of a particle by dissipative collisions

Abstract. -We consider a massive inelastic piston, whose opposite faces have different coefficients of restitution, moving under the action of an infinitely dilute gas of hard disks maintained at a fixed temperature. The dynamics of the piston is Markovian and obeys a continuous Master Equation:however, the asymmetry of restitution coefficients induces a violation of detailed balance and a net drift of the piston, as in a Brownian ratchet. Numerical investigations of such non-equilibrium stationary state show that the velocity fluctuations of the piston are symmetric around the mean value only in the limit of large piston mass, while they are strongly asymmetric in the opposite limit. Only taking into account such an asymmetry, i.e. including a third parameter in addition to the mean and the variance of the velocity distribution, it is possible to obtain a satisfactory analytical prediction for the ratchet drift velocity.Introduction. -Granular materials have been the subject of intense research in the last 20 years in Physics [1]. Most of the non-trivial phenomena that can be observed in a shaken box of sand are due to the inelasticity of collisions among grains [2]. Kinetic energy is dissipated into heat, introducing an intrinsic time irreversibility in the "microscopic" dynamics which can have consequences at a more macroscopic level: for instance species segregation [3], breakdown of energy equipartition [4], apparent Maxwell-demon-like properties such as heat currents against a temperature gradient [5] or mass current against a density gradient [6], ratchet-like net drift of a asymmetrically shaped tracers [7] [8], rectification of thermal fluctuations [9] [10], and so on. It is well known, moreover, that the space asymmetry of an adiabatic piston results in a stationary macroscopic motion [11]. Here we consider, instead, a model of asymmetric granular piston with different coefficients of restitution which, with respect to a previously presented model of granular Brownian ratchet [7], has the advantages of being simpler to be studied analytically as well as realized in the laboratory, and at the same time displays unexpected peculiar properties: in particular we will show how its stationary state is characterized by asymmetric velocity fluctuations, and that this asymmetry becomes crucial when the piston

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“…, that is the same solution already obtained by Martin et al [16]. When α 1 = α 2 = 1 the temperature of the piston is equal to that of the gas.…”

supporting

confidence: 87%

Noise rectification and fluctuations of an asymmetric inelastic piston

2008

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“…This generalizes the elastic fluid result c = (1 − q)/(1 + q) [75,76]. Generally, the impurity and the fluid have different energies, and this lack of equipartition is typical to granular particles [6,77] There are two different regimes of behavior.…”

Section: Model Asupporting

confidence: 68%

The Inelastic Maxwell Model

Ben‐Naim

Krapivsky

2004

The Physics of Granular Media

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“…As already discussed, the result by Martin and Piasecki [10] explains that when the velocity-pdf of the gas is Gaussian, then also the tagged particle has a Gaussian pdf. In particular if the bulk has a temperature T , the tagged particle has a temperature T ′ = α+1 3−α T .…”

Section: A Verification Of Transition Rates Formulamentioning

confidence: 80%

Dynamics of a tracer granular particle as a nonequilibrium Markov process

et al. 2006

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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 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 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 W (τ ) in experiments.

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Thermalization of a particle by dissipative collisions

Cited by 90 publications

References 7 publications

Noise rectification and fluctuations of an asymmetric inelastic piston

Noise rectification and fluctuations of an asymmetric inelastic piston

The Inelastic Maxwell Model

Dynamics of a tracer granular particle as a nonequilibrium Markov process

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