Long-term behavior of Hertzian chains between fixed walls is really equilibrium

Abstract

We examine the long-term behaviour of non-integrable, energy-conserved, 1D systems of macroscopic grains interacting via a contact-only generalized Hertz potential and held between stationary walls. Existing dynamical studies showed the absence of energy equipartitioning in such systems, hence their long-term dynamics was described as quasiequilibrium. Here we show that these systems do in fact reach thermal equilibrium at sufficiently long times, as indicated by the calculated heat capacity. This phase is described by equilibrium statistical mechanics, opening up the possibility that the machinery of non-equilibrium statistical mechanics may be used to understand the behaviour of these systems away from equilibrium.

The equilibrium phase in heterogeneous Hertzian chains

“…From equation (12), it is clear that, in the equilibrium phase, δK 2 / K 2 is absent of material dependence. This has been observed previously in MD simulations [52,1,2]. Inverting equation (11) provides one way to calculate the specific heat per grain from an MD simulation.…”

“…This symmetry results in a loss of degrees of freedom in the system, as discussed in Refs. [1] and [2], and the microcanonical specific heat and the PDF of system kinetic energy must be modified to account for this. In this case, it is important to stress that energy is equipartitioned among the independent degrees of freedom in the system at long times.…”

“…An equilibrium value for the specific heat for Hertzian chains in the thermodynamic limit was derived previously [1,2] from an application of Tolman's generalized equipartition theorem [58] to the Hamiltonian, equation ( 1). The result is…”

“…In previous works [1,2], we showed that 1D homogeneous systems of discrete macroscopic grains interacting via a power-law contact potential and held between fixed walls thermalizes an initial solitary wave (SW), ultimately transitioning to an equilibrium phase. The properties of this phase are well-predicted by treating the chain at long times as a 1D gas of interacting spheres in a microcanonical (N V E) ensemble [3].…”

“…Until very recently, the question of whether QEQ is the final phase for such systems was a long-open problem. However, it has now been well-established [57,1,2] that such systems with powerlaw interactions can indeed achieve thermal equilibrium after sufficiently long times, and that the time scale to equilibrium increases with the degree of nonlinearity in the interaction potential.…”