Thermalization in the discrete nonlinear Klein-Gordon chain in the wave-turbulence framework

Abstract

We study the time of equipartition, Teq, of energy in the one-dimensional Discrete Nonlinear Klein-Gordon (DNKG) equation in the framework of the Wave Turbulence (WT) theory. We discuss the applicability of the WT theory and show how this approach can explain qualitatively the route to thermalization and the scaling of the equipartition time as a function of the nonlinear parameter , defined as the ratio between the nonlinear and linear part of the Hamiltonian. Two scaling laws, Teq ∝ −2 and Teq ∝ −4 , for different degrees of nonlinearity are explained in terms of four-wave or six-wave processes in the WT theory. The predictions are verified with extensive numerical simulations varying the system size and the degree of nonlinearity.

Universal law of thermalization for one-dimensional perturbed Toda lattices

“…Comparing with previous studies, the resultant thermalization law (T eq ∝ò −2 ) provides a unified and consistent picture for thermalization of 1D nonlinear chains. Finally, we notice that thermalization in the Klein-Gordon lattice [46,48] also follows this law, though this lattice belongs to another class that possesses on-site potential. This implies that such a law exists beyond the models studied here.…”

“…We assume that the exact nontrivial n-wave scattering processes caused by the nth-order perturbation, in the thermodynamic limit, dominate the thermalization process of the perturbed system, and the strength of the scattering processes is characterized by ò n . Then based on the predictions of the WT theory [44][45][46][47][48], the time scale of equipartition is…”

“…Recently,in the wave turbulence (WT) framework [37][38][39][40][41][42][43], the power-law relationship is theoretically affirmed [44][45][46]. This theory [44] assumes that exact multi-wave resonances govern the long time dynamics of a weak nonlinear lattice, and the irreversible energy mixing among the Fourier modes is achieved by the nontrivial resonances.…”

Nonintegrability and thermalization of one-dimensional diatomic lattices

“…It has been shown that the key to identifying the universal exponent −2 is to select a suitable reference integrable system so that the perturbation strength can be defined accurately. We noted that the thermalization in the Klein-Gordon lattice [23,26] also follows this law, though this lattice belongs to another class that possesses on-site potential.…”

Experimental Evidence of a Hydrodynamic Soliton Gas

“…The major question in this field is again the understanding of the statistical properties of an interacting ensemble of nonlinear waves, described by integrable equations, in the presence or not of randomness; the latter may arise from initial conditions which evolve under the coaction of linear and nonlinear effects, [9][10][11][12][13][14][15][16][17]. In contrast to many nonintegrable closed wave systems that reach a thermalized state characterized by the equipartition of energy among the degrees of freedom (Fourier modes) [18,19], integrable equations are characterized by an infinite number of conserved quantities and their dynamics is confined on special surfaces in the phase space. This prevents the phenomenon of classical thermalization and it opens up the fundamental quest on what is the large time state of integrable systems for a given class of initial conditions.…”