Heating Mechanism Affects Equipartition in a Binary Granular System

Abstract

Two species of particles in a binary granular system typically do not have the same mean kinetic energy, in contrast to the equipartition of energy required in equilibrium. We investigate the role of the heating mechanism in determining the extent of non-equipartition of kinetic energy. In most experiments, different species are unequally heated at the boundaries. We show by event-driven simulations that differential boundary heating affects non-equipartition even in the bulk of the system. This conclusion is fortified by studying numerical and solvable stochastic models without spatial degrees of freedom. In both cases, even in the limit where heating events are rare compared to collisions, the effect of the heating mechanism persists.An issue of broad generality for nonequilibrium physics is the assignment of intensive temperature variables to steady states of driven systems. A widely-studied example of practical importance is a dilute system of macroscopic grains. A granular gas of this kind is different from an equilibrium gas in that a continuous supply of energy from an external source is necessary to maintain a steady state and balance dissipation due to inelastic collisions between particles. The energy source determines typical scales of the dynamics in the system, but is presumed not to affect the microscopic constitutive laws such as the equation of state or expressions for transport coefficients [1]. In this Letter we study an externally fluidised granular gas and find that the details of the energising mechanism affect relations between intensive quantities even in the bulk of the system.In a typical real-world situation, energy is delivered at the boundaries of a granular system by vibration, shear or other mechanical means. In equilibrium, a gas placed in contact with a heat bath acquires a uniform temperature and density. However, in the case of a granular system, there are gradients in density and particle motions as a function of distance from the energizing boundary. For dilute gases of inelastic particles there has been considerable progress [2]in describing this inhomogeneous steady state in terms of density and"temperature" fields, where the temperature is a purely kinetic construct, defined as the mean kinetic energy per degree of freedom, by analogy to kinetic theory of a molecular gas.An important conceptual tool in validating the use of nonequilibrium temperatures involves thermal contact between systems [3,4]. One can ask whether a candidate definition for the temperature governs the direction of energy flow and whether the temperatures equalise or evolve to bear a fixed relationship. In this spirit, we consider a granular system with two species of particles. There is no requirement that the species acquire the same kinetic temperature; indeed, experiments [5,6,7] document a violation of equipartition. It is observed that two species a and b acquire temperatures whose ratio γ ≡ T a /T b is affected strongly by the ratio of particle masses, m a /m b , and weakly by their inelasticity. ...

Hydrodynamic character of the nonequipartition of kinetic energy in binary granular gases

“…Note that this effect, cooling of one of the species by the vibrating wall, is not possible when the particles collide with the wall in an elastic way, as v c diverges in this case. On the other hand, it can occur when thermal walls are considered, as made for instance in [7].…”

“…Recently, Wang and Menon [7] reported the results of some event-driven simulations of a granular mixture and reached the conclusion that the heating mechanism affects nonequipartition of energy, even in the bulk of the system. This would imply that the details of the driving used to inject energy into the system can not be ignored in describing mixtures of inelastic gases even far away from the energy source.…”

Granular statistical mechanics: Volume-stress phase space, equipartition and equations of state

“…We reviewed the derivation of an equipartition principle for 2D static granular materials -every degree of freedom, whether SDF or FDF, shares a mean volume of X 0 /2. This result opens again the discussion about the relation between the ensemble statistics and the dynamics of dense granular matter, where an equipartition principle has been shown to be absent, at least in the conventional phase space of positions and momenta [5,6]. However, since static granular systems are the equivalent of 'zero temperature' granular fluids, our result gives hope that an equipartition principle may be found for dense dynamic systems by considering a larger phase space that includes SDF and FDF.…”

“…This is due to: uncertainty over the identities and number of degrees of freedom (DoF), the difficulty to conceive an analog of a thermometer -a 'compactometer', the lack of ergodicity, the difficulty in measuring the compactivity experimentally and recent suggestions of absence of an equipartition principle [5,6] in agitated systems. Here we review a recent argument that the phase space for static systems must include both structural DoF (SDF) [7] and force DoF (FDF) and a derivation of an equipartition principle in 2D.…”

On stress relaxation timescales for dense binary particulate systems

“…Because of the ratio of partial temperature 1 / 2 , the binary species system could lead to anisotropic distribution such as clustering or aggregation after imposing an external shear rate (see, e.g., [5,14,17,20,40,43]). The clustering/jamming itself is a complicated phenomenon in granular ows, which is supposed to interact in much intervened manner, and it is out of the scope of this study.…”