The quasi-equilibrium phase in nonlinear 1D systems (original) (raw)
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The quasi-equilibrium phase of nonlinear chains
Pramana, 2005
We show that time evolution initiated via kinetic energy perturbations in conservative, discrete, spring-mass chains with purely non-linear, non-integrable, algebraic potentials of the form V (xi − xi+1) ∼ (xi − xi+1) 2n , n ≥ 2 and an integer, occurs via discrete solitary waves (DSWs) and discrete antisolitary waves (DASWs). Presence of reflecting and periodic boundaries in the system leads to collisions between the DSWs and DASWs. Such collisions lead to the breakage and subsequent reformation of (different) DSWs and DASWs. Our calculations show that the system eventually reaches a stable 'quasi-equilibrium' phase that appears to be independent of initial conditions, possesses Gaussian velocity distribution, and has a higher mean kinetic energy and larger range of kinetic energy fluctuations as compared to the pure harmonic system with n = 1; the latter indicates possible violation of equipartition.
Dynamical transitions between equilibria in a dissipative Klein–Gordon lattice
Journal of Mathematical Analysis and Applications
We consider the energy landscape of a dissipative Klein-Gordon lattice with a φ 4 on-site potential. Our analysis is based on suitable energy arguments, combined with a discrete version of the Lojasiewicz inequality, in order to justify the convergence to a single, nontrivial equilibrium for all initial configurations of the lattice. Then, global bifurcation theory is explored, to illustrate that in the discrete regime all linear states lead to nonlinear generalizations of equilibrium states. Direct numerical simulations reveal the rich structure of the equilibrium set, consisting of non-trivial topological (kink-shaped) interpolations between the adjacent minima of the on-site potential, and the wealth of dynamical convergence possibilities. These dynamical evolution results also provide insight on the potential stability of the equilibrium branches, and glimpses of the emerging global bifurcation structure, elucidating the role of the interplay between discreteness, nonlinearity and dissipation. arXiv:1809.07995v2 [nlin.PS] 18 Nov 2018 2 study of transition probabilities. While it is well known that the discrete models admit stationary kink solutions , the breaking of the translational invariance, suggests an important question, concerning the existence of traveling discrete kinks (and of traveling waves solutions in general). The above references, as well as numerous other related works, including (but not limited to) , provide a sense of the interest that these questions have triggered.
CISM International Centre for Mechanical Sciences, 2010
We present an adequate analytical approach to the description of nonlinear vibration with strong energy exchange between weakly coupled oscillators and oscillatory chains. The fundamental notion of the limiting phase trajectory (LPT) corresponding to complete energy exchange is introduced. In certain sense this is an alternative to the nonlinear normal mode (NNM) characterized by complete energy conservation. Well-known approximations based on NNMs turn out to be valid for the case of weak energy exchange, and the proposed approach can be used for the description of nonlinear processes with strong energy exchange between weakly coupled oscillators or oscillatory chains. Such a description is formally similar to that of a vibro-impact process and can be considered as starting approximation when dealing with other processes with intensive energy transfer. At first we propose a simple analytical description of vibrations of nonlinear oscillators. We show that two dynamical transitions occur in the system. First of them corresponds to the bifurcation of anti-phase vibrations of oscillators. And the second one is caused by coincidence of LPT with separatrix dividing two stable stationary states and leads to qualitative change in both phase and temporal behavior of the LPT (in particular, temporal dependence of the amplitude becomes resembling that for vibro-impact vibrations). Next problem under consideration relates to intensive intermodal exchange in the periodic nonlinear systems with finite (n>2) number of degrees of freedom. We consider two limiting cases. If the number of particles is not large enough, the energy exchange between nonlinear normal modes in two-dimensional integral manifolds is considered. When the number of the particles increases the energy exchange between neighbor integral manifolds becomes important that leads to formation of the localized excitations resembling the breathers in the one-dimensional continuum media.
Stationary non-equilibrium states of infinite harmonic systems
Communications in Mathematical Physics, 1977
We investigate the existence, properties and approach to stationary non-equilibrium states of infinite harmonic crystals. For classical systems these stationary states are, like the Gibbs states, Gaussian measures on the phase space of the infinite system (analogues results are true for quantum systems). Their ergodic properties are the same as those of the equilibrium states: e.g. for ordered periodic crystals they are Bernoulli. Unlike the equilibrium states however they are not "stable" towards perturbations in the potential.
Granular chain between asymmetric boundaries and the quasiequilibrium state
Physical Review E, 2014
Some 30 years have passed since we learned that any velocity perturbation develops into a propagating solitary wave in a granular chain, and over a decade has passed since we learned that these solitary waves break and reform upon collision, leaving behind small secondary solitary waves. The production of the latter eventually precipitates the quasiequilibrium state characterized by large energy fluctuations in dissipation-free granular systems. Here we present dynamical simulations on the effects of soft boundaries on solitary wave interaction in granular chains held between fixed walls. We show that at short time scales, a gradient in the distribution of kinetic energy between the boundaries is indeed sustained. At long times, however, such a gradient gets obliterated and there is no measurable difference between the average kinetic energies of the particles adjacent to walls. Our findings suggest that (i) the quasiequilibrium state can effectively erase small gradients of the average kinetic energies of the particles adjacent to walls in a system, (ii) Boltzmann distribution of grain speeds is realized in the system of interest, and (iii) time and space averages yield the same result, thus suggesting that the system is ergodic.
Subdiffusion of nonlinear waves in quasiperiodic potentials
New Journal of Physics, 2012
We study the time evolution of wave packets in one-dimensional quasiperiodic lattices which localize linear waves. Nonlinearity (related to twobody interactions) has a destructive effect on localization, as observed recently for interacting atomic condensates (Lucioni et al 2011 Phys. Rev. Lett. 106 230403). We extend the analysis of the characteristics of the subdiffusive dynamics to large temporal and spatial scales. Our results for the second moment m 2 consistently reveal an asymptotic m 2 ∼ t 1/3 and an intermediate m 2 ∼ t 1/2 law. At variance with purely random systems (Laptyeva et al 2010 Europhys. Lett. 91 30001), the fractal gap structure of the linear wave spectrum strongly favours intermediate self-trapping events. Our findings give a new dimension to the theory of wave packet spreading in localizing environments.
Physical Review Letters, 2006
We introduce a Hamiltonian dynamics for the description of long-range interacting systems in contact with a thermal bath (i.e., in the canonical ensemble). The dynamics confirms statistical mechanics equilibrium predictions for the Hamiltonian Mean Field model and the equilibrium ensemble equivalence. We find that long-lasting quasi-stationary states persist in presence of the interaction with the environment. Our results indicate that quasi-stationary states are indeed reproducible in real physical experiments. PACS numbers: 05.20.Gg, 05.10.-a, 05.70.Ln
Solitary waves in excitable systems with cross-diusion
2000
We consider a FitzHugh-Nagumo system of equations where the traditional diffusion terms are replaced with linear cross-diffusion of components. This system describes solitary waves that have unusual form and are capable of quasi-soliton interaction. This is different from the classical FitzHugh-Nagumo system with self-diffusion, but similar to a predator-prey model with taxis of populations on each other's gradient which we considered earlier. We study these waves by numerical simulations and also present an analytical theory, based on the asymptotic behaviour which arises when the local dynamics of the inhibitor field are much slower than those of the activator field.