We investigate freely cooling systems of rough spheres in two and three dimensions. Simulations using an event driven algorithm are compared with results of an approximate kinetic theory, based on the assumption of a generalized homogeneous cooling state. For short times t, translational and rotational energy are found to change linearly with t. For large times both energies decay like t −2 with a ratio independent of time, but not corresponding to equipartition. Good agreement is found between theory and simulations, as long as no clustering instability is observed. System parameters, i.e. density, particle size, and particle mass can be absorbed in a rescaled time, so that the decay of translational and rotational energy is solely determined by normal restitution and surface roughness.

We study the exchange of kinetic energy between translational and rotational degrees of freedom for inelastic collisions of rough spheres. Even if equipartition holds in the initial state it is immediately destroyed by collisions. The simplest generalisation of the homogeneous cooling state allows for two temperatures, characterizing translational and rotational degrees of freedom separately. For times larger than a crossover frequency, which is determined by the Enskog frequency and the initial temperature, both energies decay algebraically like t −2 with a fixed ratio of amplitudes, different from one.Kinetic theory of inelastically colliding particles has become a subject of growing research activity, motivated partly by renewed interest in granular materials. A Boltzmann equation and the Enskog variant of it have been formulated for inelastically colliding particles with normal and tangential restitution [1][2][3][4][5][6]. Whereas the derivation of the kinetic equation is relatively straightforward, the methods of solution which were developped for elastic collisions cannot be taken over to the inelastic case because there is no simple stationary or local equilibrium distribution, around which one could expand the nonlinear kinetic equation. One simple distribution is the homogeneous cooling state (HCS) [7,8], which depends on time only implicitly via the average kinetic energy T (t). The latter is predicted to decay like t −2 for large times. The homogeneity assumption is certainly violated when clustering occurs and, in particular, if inelastic collapse happens. Nevertheless we have found recently [9] in a model of inelastically colliding rods that the kinetic energy follows on average a t −2 behaviour, even if clustering occurs. This suggests that the above scaling law may be a useful approximate description, even when the assumptions of HCS break down.In this note we show that rotational and translational energy relax with different rates in general. Once friction is included, HCS with a single time dependent temperature is no longer consistent with the time evolution of translational and rotational energy separately. Instead one has to introduce two temperatures which characterize translational and rotational degress of freedom separately. Both are found to fall off like t −2 with the ratio approaching a constant value, which is determined by the coefficients of normal and tangential restitution.We briefly recall the collision dynamics of hard spheres with normal and tangential restitution. These results can for example be found in Cerginani [6]. We consider two spheres of equal diameter a, mass M and moment of inertia I. The unit-vector from the center of the first sphere to the center of the second is denoted byn and velocities and angular velocities before collision by v 1 , v 2 , ω 1 and ω 2 . The relative velocity of the contactpoint before collision is given by V = v 2 + a 2n × ω 2 − v 1 + a 2n × ω 1 . Normal and tangential restitution determine the relative velocity after collision according ton...

The time dependence of deviations from the Gaussian state in a freely cooling homogeneous system of smooth inelastically colliding spheres is investigated by kinetic theory. We determine the full time dependence of the coefficients of an expansion around the Gaussian state in Generalized Laguerre polynomials. Approximating this system of equations to sixth order, we find that the asymptotic state, where the mean energy T follows Haff's law with time independent cooling rate, is reached within a few collisions per particle. Two-dimensional molecular dynamics simulations confirm our results and show exponential behavior in the high-energy tails.

We investigate the free cooling of inelastic rough spheres in the presence of Coulomb friction. Depending on the coefficients of normal restitution ǫ and Coulomb friction µ, we find qualitatively different asymptotic states. For nearly complete normal restitution (ǫ close to 1) and large µ, friction does not change the cooling properties qualitatively compared to a constant coefficient of tangential restitution. In particular, the asymptotic state is characterized by a constant ratio of rotational and translational energies, both decaying according to Haff's law. However, for small ǫ and small µ, the dissipation of rotational energy is suppressed, so that the asymptotic state is characterized by constant rotational energy while the translational energy continues to decay as predicted by Haff's law. Introducing either surface roughness for grazing collisions or cohesion forces for collisions with vanishing normal load, causes the rotational energy to decay according to Haffs law again in the asymptotic long-time limit with, however, an intermediate regime of approximately constant rotational energy.

We have developed a kinetic theory of hard needles undergoing binary collisions with loss of energy due to normal and tangential restitution. In addition, we have simulated many particle systems of granular hard needles. The theory, based on the assumption of a homogeneous cooling state, predicts that granular cooling of the needles proceeds in two stages: An exponential decay of the initial configuration to a state where translational and rotational energies take on a time independent ratio (different from unity), followed by an algebraic decay of the total kinetic energy of approximately t(-2). The simulations support the theory very well for low and moderate densities. For higher densities, we have observed the onset of the formation of clusters and shear bands.