Divergence law of the equilibration time of a weakly anharmonic Hamiltonian oscillator chain as the anharmonicity tends to zero

For the one-dimensional Hamiltonian chain of anharmonic oscillators with Hamiltonian $H=\sum_i[p_i^2/2+q_i^2/2+(q_{i+1}-q_i)^2/2+gq_i^4/4]$ (a discrete $\phi^4$, or discrete nonlinear Klein-Gordon, chain), simulation shows that Boltzmann-Gibbs equilibrium is always reached for any $g>0$, but the equilibration time diverges as the anharmonicity $g\to 0$. Parisi's 1997 study, which established the absence of an ergodicity threshold in this chain, could not determine the form of this divergence: its relaxation-time estimates were fit equally well by perturbative power laws ($\tau\propto g^{-5}$, or $g^{-4.2}$ with modified initial conditions; the naive perturbative expectation being $g^{-2}$) and by exponential-type laws ($\tau\propto\exp(0.3/g)$ over an intermediate range, or a stretched exponential $\tau\sim\exp(Ag^{-0.3})$), two behaviors with different theoretical implications (perturbative versus nonperturbative mechanism). Later work has sharpened both branches — wave-turbulence and quasi-resonance theory predict and numerically support $T_{\rm eq}\propto g^{-2}$ in the thermodynamic limit, while rigorous results prove exponentially long stability times and other numerics find exponential-type growth at fixed small system size or low nonlinearity — but the asymptotic law itself, its system-size dependence, and the reconciliation of the two regimes remain unsettled; a 2023 review states that establishing the transition between the power-law and exponential predictions in terms of nonlinearity and system size is an open problem of paramount importance, and a 2026 review of universal thermalization laws states that whether the hierarchy of thermalization-time scaling regimes persists asymptotically, and whether a finite thermalization threshold exists, remain open questions. The problem: determine the precise asymptotic dependence of the equilibration time on $g$ as $g\to 0$, including its dependence on chain length and the crossover between the two regimes.

Background

In 1955 Fermi, Pasta, Ulam and Tsingou simulated a chain of oscillators coupled by weakly nonlinear springs, expecting to observe equipartition — the state in which energy is shared equally among the normal modes (the independent Fourier-mode oscillations of the linearized chain) — and instead found persistent recurrences. Whether, and on what time scale, a weakly nonlinear Hamiltonian lattice approaches the Boltzmann-Gibbs equilibrium distribution has remained a central question of nonequilibrium statistical mechanics ever since.

A natural gapped variant of their chain is the one-dimensional discrete $\phi^4$ chain (also called the discrete nonlinear Klein-Gordon chain): unit-mass oscillators with harmonic nearest-neighbor coupling and an on-site harmonic plus quartic potential,

$$H = \sum_{i=1}^{N}\left[\frac{p_i^2}{2} + \frac{q_i^2}{2} + \frac{(q_{i+1}-q_i)^2}{2} + \frac{g\,q_i^4}{4}\right],$$

where the anharmonicity $g$ measures the nonlinear coupling relative to the harmonic part. Unlike the acoustic Fermi-Pasta-Ulam-Tsingou (FPUT) chain (whose linear mode frequencies extend down to zero), its harmonic spectrum $\omega^2(k)=1+4\sin^2(\pi k/N)$ lies between 1 and 5, so all modes have frequencies of order one. In a 1997 Europhysics Letters paper, G. Parisi simulated this chain with $N=8192$ (with checks at $N=2048$ and $N=16384$ showing no significant difference), initial conditions populating only low-wavenumber modes with random phases, and monitored the real-space equilibration indicator $\Delta(t)=\langle\sum_i q_iq_{i+1}\rangle/\langle\sum_i q_i^2\rangle$, which is close to one initially and vanishes in the Boltzmann-Gibbs state. He concluded that there is no ergodicity threshold: local equilibrium is reached for every $g>0$, but the equilibration time $\tau$ diverges as $g\to 0$. Estimating $\tau$ from the logarithmic derivative $D(t)=\partial\log\Delta/\partial t$ by two different definitions, he found power-law fits ($\tau\propto g^{-5}$, or $g^{-4.2}$ with a modified self-consistent initial condition; a naive perturbative estimate gives $g^{-2}$) and exponential-type fits ($\tau\propto\exp(0.3/g)$ over an intermediate range, or a stretched-exponential form $\tau\sim\exp(Ag^{-\gamma})$ with $\gamma\approx 0.3$, which diverges faster than any power but slower than $\exp(A/g)$) to be equally good, and stated that the precise form of the divergence was not determined.

The two behaviors carry different theoretical weight. A power-law divergence is a perturbative effect — visible order by order in $g$ — consistent with resonant mode coupling. An exponential-type divergence would be nonperturbative, in line with the theory of nearly integrable Hamiltonian systems: the Kolmogorov-Arnold-Moser (KAM) theorem guarantees that most invariant tori of an integrable system survive a small perturbation, but in the thermodynamic limit ($N\to\infty$ at fixed energy density) resonances become dense and KAM does not apply; Nekhoroshev-type estimates nevertheless suggest that times to leave near-integrable behavior can grow faster than any power of $1/g$, and an early argument of Fucito et al. bounds $\tau$ above by $\exp(A/g)$. In the opposite direction, wave turbulence theory — the statistical description of weakly nonlinear dispersive waves, leading to kinetic equations for the mode energies — predicts for one-dimensional chains a power law $T_{\rm eq}\propto g^{-2}$ driven by four-wave quasi-resonances: groups of modes whose frequency detuning is smaller than the nonlinear frequency broadening $\Omega$ (which grows with $g$ and with $N$), rather than exactly resonant. At finite $N$, exact four-wave resonances are absent or isolated in gapped chains, so equilibration must proceed through higher-order (e.g. six-wave) interactions with steeper power laws; the Chirikov overlap criterion (frequency broadening comparable to the spacing between adjacent mode frequencies) locates the crossover. Numerical work has since given support to both branches — power laws including the $g^{-2}$ law in or near the thermodynamic limit, and exponential-type growth at fixed small $N$ or low nonlinearity — and rigorous results prove exponentially long stability times for the Klein-Gordon lattice, uniformly in the thermodynamic limit. What is missing is the double-scaling structure of $\tau(g,N)$ that would reconcile the branches: the asymptotic law of $\tau$ as $g\to 0$ and the crossover between the regimes as a function of $N$.

Problem Statement

For the one-dimensional Hamiltonian chain of anharmonic oscillators (the discrete $\phi^4$, or discrete nonlinear Klein-Gordon, chain)

$$H = \sum_{i=1}^{N}\left[\frac{p_i^2}{2} + \frac{q_i^2}{2} + \frac{(q_{i+1}-q_i)^2}{2} + \frac{g\,q_i^4}{4}\right]$$

with periodic boundary conditions ($q_{N+1}\equiv q_1$), at fixed energy density, and with initial conditions in which only low-wavenumber Fourier modes are populated with random phases (Parisi's convention), determine the precise asymptotic dependence of the equilibration time $\tau$ (the time after which the chain is statistically indistinguishable from the Boltzmann-Gibbs equilibrium state, operationally measured through indicators such as Parisi's $\Delta(t)$) on the anharmonicity as $g\to 0$, including its dependence on the chain length $N$: (i) in the thermodynamic limit, does $\tau(g)$ follow the perturbative power law $\tau\propto g^{-2}$ predicted by four-wave quasi-resonance (wave turbulence) theory for arbitrarily small $g$, or does it eventually grow faster than any power? (ii) at fixed $N$, does $\tau(g)$ diverge exponentially in $1/g$ (an exponential or stretched-exponential law, as in the Nekhoroshev-type or Arnold-diffusion picture — Arnold diffusion being the extremely slow drift of energy along the dense web of resonances of a nearly integrable system), or through a sequence of power-law regimes $\tau\propto g^{-n}$ with $n=4,6,8,\dots$ increasing as $g$ decreases, and with what ultimate behavior? (iii) where is the crossover $g_c(N)$ (equivalently $N_c(g)$) between these regimes — that is, what is the double-scaling structure of $\tau(g,N)$, and does the equilibration time remain a finite power law in the thermodynamic limit for all $g>0$? A claim that a specific law holds must state the equilibration-time definition and initial-condition convention used (different definitions and initial-condition classes have produced different apparent exponents).

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