Mean-field model of free-cooling inelastic mixtures (original) (raw)
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Steady-state properties of a mean-field model of driven inelastic mixtures
PHYSICAL REVIEW E, 2002
We investigate a Maxwell model of inelastic granular mixture under the influence of a stochastic driving and obtain its steady-state properties in the context of classical kinetic theory. The model is studied analytically by computing the moments up to the eighth order and approximating the distributions by means of a Sonine polynomial expansion method. The main findings concern the existence of two different granular temperatures, one for each species, and the characterization of the distribution functions, whose tails are in general more populated than those of an elastic system. These analytical results are tested against Monte Carlo numerical simulations of the model and are in general in good agreement. The simulations, however, reveal the presence of pronounced non-Gaussian tails in the case of an infinite temperature bath, which are not well reproduced by the Sonine method.
EPL (Europhysics Letters), 2014
Distribution of granular temperatures in granular gas mixtures is investigated analytically and numerically. We analyze space uniform systems in a homogeneous cooling state (HCS) and under a uniform heating with a mass-dependent heating rate Γ k ∼ m γ k . We demonstrate that for steep size distributions of particles the granular temperatures obey a universal powerlaw distribution, T k ∼ m α k , where the exponent α does not depend on a particular form of the size distribution, the number of species and inelasticity of the grains. Moreover, α is a universal constant for a HCS and depends piecewise linearly on γ for heated gases. The predictions of our scaling theory agree well with the numerical results.
Dissipative homogeneous Maxwell mixtures: ordering transition in the tracer limit
Granular Matter, 2011
The homogeneous Boltzmann equation for inelastic Maxwell mixtures is considered to study the dynamics of tracer particles or impurities (solvent) immersed in a uniform granular gas (solute). The analysis is based on exact results derived for a granular binary mixture in the homogeneous cooling state (HCS) that apply for arbitrary values of the parameters of the mixture (particle masses m i , mole fractions c i , and coefficients of restitution α ij). In the tracer limit (c 1 → 0), it is shown that the HCS supports two distinct phases that are evidenced by the corresponding value of E 1 /E, the relative contribution of the tracer species to the total energy. Defining the mass ratio µ ≡ m 1 /m 2 , there indeed exist two critical values µ (−) HCS and µ (+) HCS (which depend on the coefficients of restitution), such that E 1 /E = 0 for µ (−) HCS < µ < µ
Velocity distribution of inelastic granular gas in a homogeneous cooling state
Physical review. E, Statistical, nonlinear, and soft matter physics, 2003
The velocity distribution of inelastic granular gas is examined numerically on a two-dimensional hard disk system in nearly elastic regime using molecular dynamical simulations. The system is prepared initially in the equilibrium state with the Maxwell-Boltzmann distribution, then after several inelastic collisions per particle, the system falls in the state that the Boltzmann's equation predicts with the stationary form of velocity distribution. It turns out, however, that due to the velocity correlation the form of the distribution function does not stay time independent, but gradually returns to the Maxwellian immediately after the initial transient till the clustering instability sets in. It shows that, even in the homogeneous cooling state (Haff state), where the energy decays exponentially as a function of collision number, the velocity correlation in the inelastic system invalidates the assumption of molecular chaos and the prediction of the Boltzmann's equation fails.
Coarse-grained dynamics of the freely cooling granular gas in one dimension
Physical Review E, 2011
We study the dynamics and structure of clusters in the inhomogeneous clustered regime of a freely cooling granular gas of point particles in one dimension. The coefficient of restitution is modeled as r0 < 1 or 1 depending on whether the relative speed is greater or smaller than a velocity scale δ. The effective fragmentation rate of a cluster is shown to rise sharply beyond a δ dependent time scale. This crossover is coincident with the velocity fluctuations within a cluster becoming order δ. Beyond this crossover time, the cluster size distribution develops a nontrivial power law distribution, whose scaling properties are related to those of the velocity fluctuations. We argue that these underlying features are responsible behind the recently observed nontrivial coarsening behaviour in the one dimensional freely cooling granular gas.
On the Boltzmann Equation for Diffusively Excited Granular Media
Communications in Mathematical Physics, 2004
We study the Boltzmann equation for a space-homogeneous gas of inelastic hard spheres, with a diffusive term representing a random background forcing. Under the assumption that the initial datum is a nonnegative L 2 (R N ) function, with bounded mass and kinetic energy (second moment), we prove the existence of a solution to this model, which instantaneously becomes smooth and rapidly decaying. Under a weak additional assumption of bounded third moment, the solution is shown to be unique. We also establish the existence (but not uniqueness) of a stationary solution. In addition we show that the high-velocity tails of both the stationary and time-dependent particle distribution functions are overpopulated with respect to the Maxwellian distribution, as conjectured by previous authors, and we prove pointwise lower estimates for the solutions.
Deviation from Maxwell distribution in granular gases with constant restitution coefficient
Physical Review E, 2000
We analyze the velocity distribution function of force-free granular gases in the regime of homogeneous cooling when deviations from the Maxwellian distribution may be accounted only by the leading term in the Sonine polynomial expansion, quantified by the second coefficient a 2. We go beyond the linear approximation for a 2 and find three different values ͑three roots͒ for this coefficient which correspond to a scaling solution of the Boltzmann equation. The stability analysis performed showed, however, that among these three roots only one corresponds to a stable scaling solution. This is very close to a 2 , obtained in previous studies in a linear with respect to a 2 approximation.
Ordering Phenomena in Cooling Granular Mixtures
Physical Review Letters, 2004
We report two phenomena, induced by dynamical correlations, that occur during the free cooling of a two-dimensional mixture of inelastic hard disks. First, we show that, due to the onset of velocity correlations, the ratio of the kinetic energies associated with the two species changes from the value corresponding to the homogeneous cooling state to a value approximately given by the mass ratio m1/m2 of the two species. Second, we report a novel segregation effect that occurs in the late stage of cooling, where interconnected domains appear. Spectral analysis of the composition field reveals the emergence of a growing characteristic length.
International Journal of Modern Physics C, 2007
We present a universal description of the velocity distribution function of granular gases, f(v), valid for both, small and intermediate velocities where v is close to the thermal velocity and also for large v where the distribution function reveals an exponentially decaying tail. By means of large-scale Monte Carlo simulations and by kinetic theory we show that the deviation from the Maxwell distribution in the high-energy tail leads to small but detectable variation of the cooling coefficient and to extraordinary large relaxation time.