Universal route to thermalization in weakly-nonlinear one-dimensional chains

Abstract

We apply Wave Turbulence theory to describe the dynamics on nonlinear one-dimensional chains. We consider α and β Fermi-Pasta-Ulam-Tsingou (FPUT) systems, and the discrete nonlinear Klein-Gordon chain. We demonstrate that resonances are responsible for the irreversible transfer of energy among the Fourier modes. We predict that all the systems thermalize for large times, and that the equipartition time scales as a power-law of the strength of the nonlinearity. Our methodology is not limited to only these systems and can be applied to the case of a finite number of modes, such as in the original FPUT experiment, or to the thermodynamic limit, i.e. when the number of modes approach infinity. In the latter limit, we perform state of the art numerical simulations and show that the results are consistent with theoretical predictions. We suggest that the route to thermalization, based only on the presence of exact resonance, has universal features. Moreover, a by-product of our analysis is the asymptotic integrability, up to four wave interactions, of the discrete nonlinear Klein-Gordon chain.

Coexistence of Ballistic and Fourier Regimes in the β Fermi-Pasta-Ulam-Tsingou Lattice

“…In this Letter, we use concepts of wave-kinetic theory to investigate the low-temperature regime of the β Fermi-Pasta-Ulam-Tsingou model (β-FPUT) [34][35][36][37], paradigmatic anharmonic 1D lattice. In the thermodynamic limit, the mechanism of thermalization at the mesoscopic scale is related to four-wave resonant interactions [22]. We give evidence from direct numerical simulations that the system splits into two independent sets of modes: the low-k modes, with the mean free path exceeding what we call the "mesoscopic" scale λ, and the remaining modes that, interacting resonantly, relax to local thermodynamic equilibrium.…”

“…The approach based on the wave kinetic equation, i.e., the phonon Boltzmann equation of solid state physics and main object of wave turbulence theory [15], has recently opened an important perspective in this field [16][17][18][19]. The wave kinetic equation concerns phonons that interact with each other through resonant n-wave collisions, providing an effective relaxation mechanism toward thermodynamic equilibrium [19][20][21][22]. Although the heat conductivity can be computed rigorously [17,18,23] when a pinning onsite potential is introduced, for unpinned anharmonic lattices energy conduction appears nontrivially anomalous [8,14,16,[24][25][26].…”

“…In the wave-turbulence formalism [15,17,22,23] we use the normal variable a k ≔ ðω k q k þ ip k Þ= ffiffiffiffiffiffiffiffi 2ω k p , with ω k ¼ 2j sinðk=2Þj, and the wave-action spectral density associated with a k , n k ¼ nðk; x; tÞ. One can derive, although not fully rigorously, the following wave kinetic equation [15][16][17][18][19][20]26]:…”

“…(4) predicts ballistic advection of wave action carried by harmonic excitations with speed v k . The rhs of (4) is the four-wave collision integral, and represents the effective mechanism of relaxation to equilibrium [22]. Note that, for finite N and small nonlinearity, six-wave resonant interactions dominate [19,20,38], but we made sure that N in our simulations is large enough for the Fourier space to be sufficiently dense, thereby making four-wave interactions dominate [22].…”

Five-Wave Resonances in Deep Water Gravity Waves: Integrability, Numerical Simulations and Experiments

“…The quintet resonances obtained represent 3 → 2 processes (transforming three waves into two waves), and the interacting wavenumbers must contain positive and negative elements from the set {±K 1 , ±K 2 , ±K 3 }. Notably, the fact that minimal 5-wave resonances can be based on non-resonant triads was demonstrated by one of us in another one-dimensional resonant system (also with subadditive dispersion relation) of historical importance: The Fermi-Pasta-Ulam-Tsingou system [34,35].…”