Sustained strong fluctuations in a nonlinear chain at acoustic vacuum: Beyond equilibrium

Abstract

Here we consider dynamical problems as in linear response theory but for purely nonlinear systems where acoustic propagation is prohibited by the potential, e.g., the case of an alignment of elastic grains confined between walls. Our simulations suggest that in the absence of acoustic propagation, the system relaxes using only solitary waves and the eventual state does not resemble an equilibrium state. Further, the studies reveal that multiple perturbations could give rise to hot and cold spots in these systems. We first use particle dynamics based simulations to understand how one of the two unequal colliding solitary waves in the chain can gain energy. Specifically, we find that for head-on collisions the smaller wave gains energy, whereas when a more energetic wave overtakes a less energetic wave, the latter gains energy. The balance between the rate at which the solitary waves break down and the rate at which they grow eventually makes it possible for the system to reach a peculiar equilibriumlike phase that is characteristic of these purely nonlinear systems. The study of the features and the robustness of the fluctuations in time has been addressed next. A particular characteristic of this equilibriumlike or quasiequilibrium phase is that very large energy fluctuations are possible--and by very large, we mean that the energy can vary between zero and several times the average energy per grain. We argue that the magnitude of the fluctuations depend on the nature of the nonlinearity in the potential energy function and the feature that any energy must eventually travel as a compact solitary wave in these systems where the solitary wave energies may vary widely. In closing we address whether these fluctuations are peculiar to one dimension or can exist in higher dimensions. The study hence raises the following intriguing possibility. Are there physical or biological systems where these kinds of nonlinear forces exist, and if so, can such large fluctuations actually be seen? Implications of the study are briefly discussed.

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

“…This impedes an equal sharing of energy among all grains in the system, hence the long-term dynamics of 1D systems of interacting grains has been described as quasi-equilibrium (QEQ). 40,41,43,44,46,47 To the time scales previously considered in dynamical studies, QEQ was observed to be a general feature of systems with no sound propagation. 40,44 However, recent work has addressed whether QEQ is the final state for such systems.…”

“…40,41,43,44,46,47 To the time scales previously considered in dynamical studies, QEQ was observed to be a general feature of systems with no sound propagation. 40,44 However, recent work has addressed whether QEQ is the final state for such systems. 50,51 These studies found that thermalization is indeed possible after very long times, and that the time scale to equilibrium increases with the degree of nonlinearity.…”

“…Boundary collisions [40][41][42] result in the partial decimation of the original SW and the creation of much smaller magnitude secondary solitary waves 42,45 (SSWs). In contrast, collisions of SW species with each other 43,44 leads to energy being exchanged between waves, and thus a potential for the increase in energy amplitude of one of the waves. Many collisions between SW species in a system with zero energy dissipation therefore leads to both breakdown and buildup processes of SSWs.…”

“…40,41,43,44,46,47 This spatially-symmetric phase is attained when the rates of SSW formation and breakdown balance, and is marked by a large number of SSWs that are equally likely to be moving in either direction. For sufficiently strong and unique perturbations, unusually large and occasionally persistent (rogue) 49 fluctuations in the system's kinetic energy are seen at long times.…”

Soliton Attenuation and Emergent Hydrodynamics in Fragile Matter

“…Nesterenko coined the term sonic vacuum to designate this state in a granular chain and demonstrated the existence of solitonic excitations both experimentally and theoretically [2,7,9]. The fate of these strongly nonlinear excitations in amorphous two-or three-dimensional granular media near the (un)jamming transition or in weakly connected polymer networks is only gradually emerging [8][9][10][11][12][13][14].…”