Driven inelastic Maxwell gases

Abstract

We consider the inelastic Maxwell model, which consists of a collection of particles that are characterized by only their velocities, and evolving through binary collisions and external driving. At any instant, a particle is equally likely to collide with any of the remaining particles. The system evolves in continuous time with mutual collisions and driving taken to be point processes with rates ${\tau }_c^{-}1$ and ${\tau }_w^{-1}$ respectively. The mutual collisions conserve momentum and are inelastic, with a coefficient of restitution $r$. The velocity change of a particle with velocity $v$, due to driving, is taken to be $\Delta v=-(1+r_w)v+\eta$, mimicking the collision with a vibrating wall, where $r_w$ the coefficient of restitution of the particle with the "wall" and $\eta$ is Gaussian white noise. The Ornstein-Uhlenbeck driving mechanism given by $\frac{dv}{dt}=-\Gamma v+\eta$ is found to be a special case of the driving modeled as a point process. Using both the continuum and discrete versions we show that while the equations for the one-particle and the two-particle velocity distribution functions do not close, the joint evolution equations of the variance and the two-particle velocity correlation functions close. With the exact formula for the variance we find that, for $r_w\ne -1$, the system goes to a steady state. On the other hand, for $r_w=-1$, the system does not have a steady state. Similarly, the system goes to a steady state for the Ornstein-Uhlenbeck driving with $\Gamma \ne 0$, whereas for the purely diffusive driving ($\Gamma =0$), the system does not have a steady state.

9 pages, 4 figures

References in corpus (6)

• Dynamics of Freely Cooling Granular Gases
• Energy decay in three-dimensional freely cooling granular gas
• Quasi-elastic solutions to the nonlinear Boltzmann equation for dissipative gases
• Transition from a dissipative to a quasi-elastic system of particles with tunable repulsive interactions
• High-Energy Tail of the Velocity Distribution of Driven Inelastic Maxwell Gases
• Velocity fluctuations in a one dimensional Inelastic Maxwell model

Cited by in corpus (10)

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• Asymptotic behavior of the velocity distribution of driven inelastic gas with scalar velocities: analytical results
• Hydrodynamics of granular particles on a line
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• Driven inelastic Maxwell gas in one dimension
• Asymptotic velocity distribution of a driven one dimensional binary granular Maxwell gas
• Dynamics of a randomly kicked particle