The study of the deviation of the velocity distribution function of fluidized granular matter from the equilibrium Maxwell-Boltzmann (MB) distribution is a subject of high current interest from the experimental, simulation, and theoretical points of view.  Our paper was among the first ones to measure the kurtosis, as well as the high-energy tail, of the velocity distribution of a fluid of inelastic hard spheres by means of Monte Carlo simulations of the Enskog-Boltzmann equation.  The simulation results validated theoretical estimates made by van Noije and Ernst in 1998.

  Does it describe a new discovery or new methodology that's useful to others?

Actually, the methodology is not new, since the simulation method is a straightforward extension to inelastic collisions of the DSMC method already developed by Graeme Bird in the sixties.  On the other hand, we took special care to numerically evaluate the quantities of interest.  For instance, the fourth collisional moment was computed indirectly from the simulation values of the second collisional moment, as well as directly.  Both routes should give equivalent results in the steady state and we found an agreement better than 99.9% in all the cases.  On the theoretical side, we pointed out that a certain degree of ambiguity is present when estimating the kurtosis by neglecting nonlinear terms.  We also showed that the scaling solution in the homogeneous cooling state is equivalent to the steady-state solution corresponding to a driving force proportional to the velocity.  To illustrate that the form of the high-energy tail may depend on the type of driving mechanism, we considered a deterministic force parallel to the particle velocity but constant in magnitude and found an underpopulated high-energy tail with respect to the Maxwell-Boltzmann distribution.

  Could you summarize the significance of your paper in layman's terms?

Quoting Campbell (1990), a granular material is a collection of a large number of discrete solid particles.  In the rapid flow regime, instead of moving in many-particle blocks, each particle moves freely except for quasi-instantaneous binary collisions with other particles.  In this regime, the velocity of each particle can be decomposed into a sum of the mean velocity of the bulk material and an apparently random component.  The mean square value of the random velocities is referred to as the "granular temperature," by analogy between the random motion of the granular particles and the thermal motion of molecules in a gas.  On the other hand, while collisions between molecules are elastic, the solid grains are macroscopic (or mesoscopic) objects and thus the collisions are inelastic; i.e. part of the kinetic energy is dissipated by collisions into the internal degrees of freedom of the granular particles.  Therefore, left to itself, the granular temperature would quickly dissipate to nothing.  To maintain the granular temperature, energy must be continually injected into the random components of the velocity to balance that which is lost to the dissipative collisions. In real experiments, the granular system is usually maintained in a fluidized state by rapidly shaking the container. Collisions of the particles with the walls of the container provide the "heating" mechanism to compensate for the interparticle collisional "cooling," so a steady state is achieved.  At a theoretical level, it is preferable to assume that the external energy injection acts uniformly on all the particles of the system.  An interesting way of modeling the heating effect of a vibrating container consists of assuming that each particle experiences a large number of (weak) random "kicks" between two successive collisions.  This stochastic force acting uniformly on all the particles has the properties of a Gaussian white noise.  Regardless of the type of external driving, what is important is that the velocity distribution in the steady state differs from the equilibrium MB distribution.  This deviation is commonly characterized by the kurtosis of the distribution (essentially measuring whether the distribution is flatter or less flat than the MB distribution) and by the high-energy tail (typically being overpopulated with respect to the MB distribution). The simplest theoretical model of a granular fluid consists of a system of smooth inelastic hard spheres with a constant coefficient of normal restitution.  By assuming that the incoming velocities of two colliding particles are statistically uncorrelated one can derive an extension to inelastic collisions of the Enskog-Boltzmann equation.  Even in the isotropic and spatially homogeneous case, the Enskog-Boltzmann equation is a nonlinear integro-differential equation extremely difficult to solve.  Nevertheless, by assuming that the steady state solution is close to the MB distribution and neglecting nonlinear terms in the deviation, van Noije and Ernst were able to obtain an estimate for the kurtosis in the case of a granular gas heated with a white noise force.  They also got an overpopulated high-energy tail in the form of a stretched exponential.  The main aim of our paper was to check these theoretical predictions by means of Monte Carlo simulations of the Enskog-Boltzmann equation.

  How did you become involved in this research?

Our group had a long experience in the kinetic theory of gases far from equilibrium, from both the theoretical and computer simulation viewpoints. In the last few years we have been applying the tools of kinetic theory to granular flows, which, on the other hand, exhibit a rich phenomenology quite different from that of normal gases.  The research of this particular paper was suggested by conversations with Prof. M. H. Ernst, who kindly provided us with a preprint of his paper with van Noije [Gran. Matt. 1, 57 (1998)].  We already obtained our simulation results by May 1998, even before van Noije and Ernst's paper was published, but we did not write a paper at that moment because we had other commitments and did not think the results were pressing enough. In fact, Ernst had to refer to our work as "unpublished results." At last, we wrote the paper in June 1999 and submitted it to Gran. Matt.  We are glad to know that our paper has had an impact much larger than we anticipated.