Universality of Energy Equipartition in One-dimensional Lattices

Abstract

We show that a general one-dimensional (1D) lattice with nonlinear inter-particle interactions can always be thermalized for arbitrarily small nonlinearity in the thermodynamic limit, thus proving equipartition hypothesis in statistical physics for an important class of systems. Particularly, we find that in the lattices of interaction potential V (x) = x 2 /2+λx n /n with n ≥ 4, there is a universal scaling law for the thermalization time T eq , i.e., T eq ∝ λ −2 −(n−2) , where is the energy density. Numerical simulations confirm that it is accurate for an even n. A slight correction is needed for an odd n, which is due to the Chirikov overlap occurring in the weakly nonlinear regime between extra vibration modes excited by the asymmetry of potential. Based on this scaling law, as well as previous prediction for the case of n = 3, a universal formula for the thermalization time for a 1D lattice with a general interaction potential is obtained.

Universal law of thermalization for one-dimensional perturbed Toda lattices

“…However, it was also found that for a lattice with asymmetric potential interaction, though T eq still depends on γ in a power law, the exponent deviates from −2. In addition, the numerical result T eq ∼ γ −4.6 for the asymmetric FPU-α model [46] deviates from the conjectured T eq ∼ γ −4 [36] seriously as well.…”

“…It was further conjectured (but not verified) that the nontrivial four-wave resonant interactions would dominate the thermalization process in the thermodynamic limit, leading to T eq ∼ γ −4 and T eq ∼ γ −2 for the FPU-α [36] and FPU-β [37] model, respectively. These conjectures were partially verified in a very recent effort [46] where it was found that in the thermodynamic limit, a universal law, T eq ∼ γ −2 , applies generally to a class of one-dimensional (1D) lattices with interaction potential V (x) = x 2 /2 + λx n /n, where n ≥ 4 is an integer and γ = λε (n−2)/2 is the nonlinearity strength. It also applies to another class of 1D lattices with symmetric interaction potential V (x) = x 2 /2 + λ|x| d /d, where d = m 1 /m 2 > 2 with m 1 and m 2 being two coprime integers and the nonlinearity strength γ = λε (d−2)/2 .…”

“…[34,35], we have revisited the previous studies of thermalization and found that they strongly suggest the FPU-α model (the case of n = 3 in Ref. [46]) be viewed as the perturbed Toda lattice while other models as perturbed harmonic lattices. Given this, we were led to the conclusion that T eq ∼ γ −2 generally applies to the perturbed harmonic lattices in the thermodynamic limit but not to the systems out of this class [46].…”

“…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. Then based on the theoretical results derived from the WT theory [36][37][38]46], the time scale of equipartition is…”

“…[46]) be viewed as the perturbed Toda lattice while other models as perturbed harmonic lattices. Given this, we were led to the conclusion that T eq ∼ γ −2 generally applies to the perturbed harmonic lattices in the thermodynamic limit but not to the systems out of this class [46].…”