Driven inelastic Maxwell gases

Mpemba effect in driven granular Maxwell gases

A Mpemba effect refers to the counterintuitive result that, when quenched to a low temperature, a system at higher temperature may equilibrate faster than one at intermediate temperatures. This effect has recently been demonstrated in driven granular gases, both for smooth as well as rough hard-sphere systems based on a perturbative analysis. In this paper, we consider the inelastic driven Maxwell gas, a simplified model for a granular gas, where the rate of collision is assumed to be independent of the relative velocity. Through an exact analysis, we determine the conditions under which the Mpemba effect is present in this model. For monodispersed gases, we show that the Mpemba effect is present only when the initial states are allowed to be nonstationary, while for bidispersed gases, it is present for some steady-state initial states. We also demonstrate the existence of the strong Mpemba effect for bidispersed Maxwell gas, wherein the system at higher temperature relaxes to a final steady state at an exponentially faster rate leading to smaller equilibration time.

Hydrodynamics of granular particles on a line

We investigate a lattice model representing a granular gas in a thin channel. We deduce the hydrodynamic description for the model from the microscopic dynamics in the large-system limit, including the lowest finite-size corrections. The main prediction from hydrodynamics, when finite-size corrections are neglected, is the existence of a steady "uniform longitudinal flow" (ULF), with the granular temperature and the velocity gradient both uniform and directly related. Extensive numerical simulations of the system show that such a state can be observed in the bulk of a finite-size system by attaching two thermostats with the same temperature at its boundaries. The relation between the ULF state and the shocks appearing in the late stage of a cooling gas of inelastic hard rods is discussed.

Velocity distribution of a driven inelastic one-component Maxwell gas

The nature of the velocity distribution of a driven granular gas, though well studied, is unknown as to whether it is universal or not, and, if universal, what it is. We determine the tails of the steady state velocity distribution of a driven inelastic Maxwell gas, which is a simple model of a granular gas where the rate of collision between particles is independent of the separation as well as the relative velocity. We show that the steady state velocity distribution is nonuniversal and depends strongly on the nature of driving. The asymptotic behavior of the velocity distribution is shown to be identical to that of a noninteracting model where the collisions between particles are ignored. For diffusive driving, where collisions with the wall are modeled by an additive noise, the tails of the velocity distribution is universal only if the noise distribution decays faster than exponential.

Driven inelastic Maxwell gas in one dimension

A lattice version of the driven inelastic Maxwell gas is studied in one dimension with periodic boundary conditions. Each site i of the lattice is assigned with a scalar "velocity," v_{i}. Nearest neighbors on the lattice interact, with a rate τ_{c}^{-1}, according to an inelastic collision rule. External driving, occurring with a rate τ_{w}^{-1}, sustains a steady state in the system. A set of closed coupled equations for the evolution of the variance and the two-point correlation is found. Steady-state values of the variance, as well as spatial correlation functions, are calculated. It is shown exactly that the correlation function decays exponentially with distance, and the correlation length for a large system is determined. Furthermore, the spatiotemporal correlation C(x,t)=〈v_{i}(0)v_{i+x}(t)〉 can also be obtained. We find that there is an interior region -x^{*}<x<x^{*}, where C(x,t) has a time-dependent form, whereas in the exterior region |x|>x^{*}, the correlation function remains the same as the initial form. C(x,t) exhibits second-order discontinuity at the transition points x=±x^{*}, and these transition points move away from the x=0 with a constant speed.