Dynamic Properties Of Polydisperse Granular Gases Driven By Gaussian White Noise

In this paper, we studied dynamic properties of polydisperse granular gases driven by Gaussian white noise in one and two dimensions, focuing on the evolution of average energy and the steady-state dynamic behaviors (include the global granular temperature, global pressure, non-Gaussian velocity distribution, spatial density distribution, spatial correlations of density and velocities, distributions of path lengths and free times between collisions, collision rate). The particles employed in our study are assigned granularity with fractal size distribution and quasi Gaussian size distribution. For the polydisperse granular systen with fractal size distribution, the inhomogeneity of the particle size distribution is characterized by a fractal dimension D . The higher D implies more inhomogeneity in the particle size distribution. However, for the polydisperse granular systen with quasi Gaussian size distribution, the inhomogeneity of the particle size distribution can be measured by the standard deviationσat the same mean valueμ. The larger value ofσindicates greater inhomogeneity in the particle size distribution.First, we presented a dynamic model of a quasi one-dimensional polydisperse granular mixture with fractal size distribution, in which the particles are driven by Gaussian white noise, and studied the steady-state dynamic properties of the sysytem. Firstly, we define the partial and global granular temperature and global pressure of the mixture. By Monte Carlo simulations, we found that, with the increase of D , the global granular temperature and the kinetic pressure decrease, the velocity distribution deviates more obviously from the Gaussian one ( such as the higher kurtosis, the larger fourth cumulant a2 and the fatter tails), and distribution of interparticle spacing deviates more obviously from the elastic form, i.e., the particles cluster more pronouncedly at the same value of the restitution coefficient e (0 < e< 1). On the other hand, as the restitution coefficiente decreases, the dynamic behavior has the similar evolution as above at the fixed D .Second, we presented a dynamic model of a one-dimensional granular gas with quasi Gaussian size distribution, in which the rods are thermalized by a viscosity heat bath. By Monte Carlo simulations, the effect of the dispersion of the quasi Gaussian size distribution on dynamic behavior of the system is investigated in the same inelasticity case. When the typical relaxation timeτof the driving Brownian process is longer than the mean collision timeτ_c, the average energy of the system decays exponentially with time towards a stable asymptotic value, and the energy relaxation timeτBto a nonequilibrium steady state becomes shorter with increasing values ofσ. In the steady state, asσincreases, the velocity distribution deviates more obviously from the Gaussian one, such as the higher kurtosis and the fatter tails, the spatial density distribution becomes more clusterized, the statistical entropy H_M/H_M~+ of the system decreases, the spatial correlations of density and velocities become stronger (such as the two-particle correlation function C ( x )shows higher peak and is a power-law form decay near the origin, the average velocity Cv ( x )is smaller and a power-law form increase for small x ).Finally, we present a dynamical model of two-dimensional polydisperse granular gases with fractal size distribution, in which the smooth hard disks are engaged in a two-dimensional horizontal rectangular box and driven by standard white noise. By Monte Carlo simulations, we find the inhomogeneity of the disk size distribution has great influence on the steady-state dynamic properties. With the increase of the fractal dimension D , the distributions of path lengths and free times between collisions deviate more obviously from expected theoretical forms for elastic spheres and have an overpopulation of short distances and time bins. The collision rate increases with D , but it is independent of time. Meanwhile, the tails of the velocity distribution functions rise more significantly above a Gaussian as D increases, but the non-Gaussian velocity distribution functions do not demonstrate any apparent universal form for any value of D. The spatial velocity correlations are apparently stronger with the increase of D. The perpendicular correlations are about one-half of the parallel correlations, and the two correlations are a power-law decay function of dimensionless distance and are long range. Moreover, the parallel velocity correlations of postcollisional state at contact are more than twice as large as the precollisional correlations, and both of them show almost linear behavior of the fractal dimension D.In this paper, the study indicates that the energy dissipation due to the more inhomogeneity of the particle size distribution or the smaller restitution coefficient e in the inelastic collisions causes a variety of very peculiar phenomena as above.