Condensation of Hard Spheres Under Gravity

Starting from Enskog equation of hard spheres of mass m and diameter D under the gravity g, we first derive the exact equation of motion for the equilibrium density profile at a temperature T and examine its solutions via the gradient expansion. The solutions exist only when βμ ≤ μo ≈ 21.756 in 2 dimensions and μo ≈ 15.299 in 3 dimensions, where μ is the dimensionless initial layer thickness and β = mgD/T . When this inequality breaks down, a fraction of particles condense from the bottom up to the Fermi surface. PACS numbers: 05.20-y, 81.35+k, 05.20.Dd, 05.70.Fh Granular materials are basically a collection of hard spheres that interact with each other via hard sphere potential [1]. For this reason, many of the properties of excited granular materials may be understood from the atomistic view of molecular gases, in particular from the view point of kinetic theory [2]. There are, however, several distinctions between molecular gases and granular materials: First, granular materials are macroscopic particles with finite diameter, and thus they cannot be compressed indefinitely. Second, the gravity plays an important role in the collective response of granular materials to external stimuli, largely because of the ordering of grains induced by the gravity. For example, one of the E-mail address: [email protected] 1 notable characteristics of the excited granular materials in a confined system under gravity is the appearance of a thin boundary layer near the surface that separates a fluidized region from a rigid solid region. This has been known for long time as is evidently seen in shearing experiments [3], avalanches [4], and grains subjected to weak excitations [5,6]. In this limit, those grains in a solid region are effectively frozen, and thus do not participate in dynamical processes. Now, the kinetic theory relies on two particle collision dynamics, and has been applied to systems where all the granular particles are in motion colliding with each other. But if a portion of grains are frozen and remain largely motionless, the kinetic theory or in general the continuum theory may pose some problems. Consider for example a system of strongly excited granular particles under gravity, where all the grains undergo collisions and thus the kinetic theory is valid. If we decrease the strength of excitation, then the particles at the bottom will freeze themselves, and the boundary layer will develop at the top. The question we address in this Letter is: How does the kinetic theory describe such process? In a recent paper [6], it has been demonstrated that the granular statistics in the presence of gravity does not follow the usual Boltzmann statistics as in molecular gases, where all the particles are dynamically active, but a new Fermi statistics, where most of the particles are effectively frozen and only a portion of particles near the surface participate in the dynamical process. This is due to the excluded volume effect and the ordering of potential energy by gravity, and the mechanism associated with this Fermi statistics is similar to that of the Fermi gas in a metal. The existence of a thin boundary layer in granular materials should be viewed from such perspective. Our specific objective of this Letter is to use the kinetic theory, in particular the Enskog equation of hard spheres of mass m and diameter D, to explore whether or not the kinetic theory can describe the cross over from Boltzmann to Fermi statistics and if so, under what conditions it occurs. Our particularly interesting discovery is that the prediction of the Enskog equation is only valid when βμ ≤ μo, where μ is the dimensionless initial layer thickness of the granules(or the Fermi energy), β = mgD/T with T the temperature, and