Thermodynamic Theory of Weakly Excited Granular Systems

We present a thermodynamic theory of weakly excited two-dimensional granular systems from the view point of elementary excitations of spinless Fermion systems. We introduce a global temperature T that is associated with the acceleration amplitude Γ in a vibrating bed. We show that the configurational statistics of weakly excited granular materials in a vibrating bed obey the Fermi statistics. P.A.C.S. numbers 81.35.+k,46.10.+z,05.70.-a,05.20.Dd Granular system is robust to thermal disturbances because its entity is a macroscopic object. [1] For this reason, the granular system is effectively in the ground state at any finite temperature and the excitation may be achieved by subjecting the system to vibration or shaking. Such an external stimulus will inject energy at a constant rate but the energy will be dissipated via collisions, leading the system to reach a steady state. Dynamics of such a steady state are quite complex, where convection [2], density waves [3], segregation [4], anomalous sound propagation [5] and even turbulent behaviors [6] have been observed. There are some indications that fluctuations in physical quantities of granular systems persist over the size of the system [7] and intrinsically nonequilibrium clustering instabilities E-mail address: [email protected] E-mail address: [email protected] 1 appear for particles with large coefficient of restitution [8]. In such cases, we may eventually have to question the validity of the hydrodynamics [9] with the aid of kinetic theory [10], though some attempts have been made to capture some of the essential features of granular convections based on phenomenological hydrodynamics models [11]. In spite of the above negative signs, the validity of the thermodynamics concept has been suggested by several theoretical papers [12,13] and experimental papers [14–17]. In particular, Knight et al [17] have observed a logarithmic relaxation in compaction processes in a three dimensional vibrating bed, which can be understood as the consecutive transitions among the metastable (glassy) configurations. This suggests the validity of the free volume (or hole) theory [18] used for the dense liquid theory as will be shown later. In two dimensions, in particular, the situation is much simpler than in three dimensions, because the particles can form a lattice structure without glassy configurations. For example, the experiment by Clement and Rajchenbach [14] has suggested that nontrivial and distinctive configurational statistics appear to exist for excited granular systems in a vibrating bed. The experiment was conducted with steel balls that have small coefficient of restitution and was monitored carefully to suppress the convection with a suitable choice of the boundary condition. They then observed that the ensemble-averaged density profile obeys a universal function that is independent of the phase of oscillations. The experimental result in Ref. [14] has been recovered by a simulation based on the distinct element method [15] and has been generalized to the case of strong excitions. [16] The existence of such a distinctive configurational statistics, which resembles the problem of packing, appears to be a fairly convincing evidence that kinetic aspects of the vibrating bed might have been decoupled from the statistical configurations averaged over many ensembles and time sequences. Such a simple observation in two dimensional weak dissipation cases enables one to make some progress in characterizing the excitation of vibrating beds in two ways: first, if the kinetics is indeed separated out, then the configurational properties should be determined by the principle of maximum entropy or equivalently the minimization of free energy. Second,