Measuring FPUT thermalization with Toda integrals

TL;DR

This work addresses ergodicity and thermalization in the FPUT-$α$ chain by treating the Toda integral $J$ as an adiabatic invariant that tracks phase-space diffusion. The authors introduce a sigmoid diffusion law for $J(t,ε)$ and quantify two key timescales, $T_d$ (diffusion onset) and $T_{eq}$ (equilibration), finding robust power-law scalings $T_d \sim ε^{-2.33}$ and $T_{eq} \sim ε^{-2.75}$, with a system-size–independent behavior for large $N$ and a nontrivial $ε_c(N) \sim N^{-2}$ crossover signaling a possible KAM-like regime at small $N$. They compare these diffusion times with Lyapunov-based timescales, showing that the Lyapunov time $T_{Λ}$ and its saturation $T_{ΛS}$ are shorter and detect weak chaos earlier, though diffusion ultimately governs equilibration. The results support ergodicity in the thermodynamic limit and reveal a rich near-integrable structure at finite $N$, linking adiabatic invariants to diffusive relaxation in nonlinear many-body systems.

Abstract

We assess the ergodic properties of the Fermi-Pasta-Ulam-Tsingou-$α$ model for generic initial conditions using a Toda integral. It serves as an adiabatic invariant for the system and a suitable observable to measure its equilibrium time. Over this timescale, the onset of action diffusion results in ergodic temporal fluctuations. We compare this timescale with the inverse of the maximum Lyapunov exponent $λ$ and its saturation time, which are systematically shorter. The Toda integral ergodization/equilibrium time is system size independent for long chains, but show dramatic growth when the system size is smaller than a critical one, whose value depends on the energy density. We measure the dependence of energy density on the critical system size and relate this observation to the possible emergence of a Kolmogorov-Arnold-Moser regime. We numerically determine the critical energy density of this regime, finding that it approximately decays as $1/N^2$ with the number of particles N.

Figures (13)

• Figure 1: The Toda integral $J$ from (\ref{['j1']}) along a trajectory of the FPUT model with random initial conditions, together with the sum of momenta products $p_n p_{n+1}$ (blue) and the sum of relative displacement differences $\delta q_n \delta q_{n+1}$ (orange). The system parameters are $N=2047$ particles at the energy level $\varepsilon = 10^{-3}$ for a single random initial condition without time-averages. The (green) dashed line marks the Gibbs average value $J_{eq}$ reached at approximately $t=10^8$.
• Figure 2: Some first Toda integrals along the FPUT dynamics are derived from the traces of even $L$--matrix powers Benet2013. The chosen initial condition is the same as in Fig.\ref{['JandQT']}, i.e. $N=2047$, $\varepsilon = 10^{-3}$ for a single random initial condition. These integrals have been rescaled to start from value zero and reach equilibrium at value one.
• Figure 3: Excitation of the first normal mode in the original FPUT experiment. (a) The blue curves represent the normalized energy spectra $\varepsilon _k$, and the solid red line at 1 the normalized Toda integral $J(t)/J_0$ from (\ref{['j1']}). The two dashed lines about 0.03 with blue and 0.65 with red represent the Gibbs average value for the energy spectra $\varepsilon_k(t)$, and the Toda integral $J(t)/J_0$, respectively. (b) The plot contains the normalized energy spectra and the normalized Toda integral from panel (a), contrasted with the spectral entropy $S(t)$ in (\ref{['spectral']}). The Gibbs value for $S(t)$ is at 3.47, which has not been included in this plot.
• Figure 4: The evolution of the Toda integral at energy density $\varepsilon = 10^{-3}$ and for different system sizes $N=255, 511,1023, 2047, 16383$. $J$-curves correspond to averages derived from 20 realizations. The light-blue area is the energy diffusion window which marks two timescales: the onset of diffusion $T_d=2 \times 10^5$, and the equilibrium time $T_{eq} =10^8$, where $J$ reaches its Gibbs value $J_{eq} \approx 2 \varepsilon$. The coefficients of variation (ratio of the standard deviation to the mean) are $2.4 \%$, $1.8\%$$1.5 \%$$0.9 \%$ and $0.03 \%$ respectively , as shown in Fig.\ref{['std1']} in Appendix.
• Figure 5: Timescales versus the logarithm of the energy density $\varepsilon$. The numerical timescales for $T_d$ and $T_{eq}$ are in orange, and their theoretical prediction derived from Eq.(\ref{['sigmoid']}) in blue. The gap between $T_d$ and $T_{eq}$ increases by lowering the energy density. At very low energies, $T_d$ may remain observable, whereas $T_{eq}$ is only predicted by Eqs.(\ref{['sigmoid']},\ref{['teq']}).