Thermalization of Weakly Nonintegrable FPUT and Toda Dynamics: A Lyapunov Spectrum Perspective

Abstract

We study the thermalization slowing down of Fermi-Past-Ulam-Tsingou (FPUT) chains and of Toda chains with nonintegrable boundaries. We focus on the transition from FPUT to harmonic chains, from FPUT to Toda chains with fixed boundaries, and from nonintegrable open boundary Toda to integrable fixed boundary Toda. We compute the Lyapunov spectrum and analyze its scaling properties upon approaching integrable limits. We analyze the scaling of the largest Laypunov exponent, the rescaled Lyapunov spectrum, and the Kolmogorov-Sinai entropy. Using additional analytic arguments we demonstrate evidence that all three cases are operating in the regime of a Long Range Network of nonintegrable perturbations.

Figures (5)

• Figure 1: Fixed boundary condition FPUT-$\alpha$ with $N = 32$. We perform the calculations starting from an initial condition with all lattice displacements set to zero, while the lattice momenta are randomly drawn subject to the constraint that the energy density equals 0.1. As the integrable limit $H_\textrm{lin,FBC}$ is approached by decreasing $\alpha$, the maximum Lyapunov characteristic exponent (mLCE) $\Lambda_1$ steadily decreases, as shown in panel (a). In panel (b), we plot the rescaled Lyapunov spectra (LS) $\mkern 1.5mu\overline{\mkern-1.5mu\Lambda\mkern-1.5mu}\mkern 1.5mu(\rho) \equiv \Lambda_i/\Lambda_1$ as a function of $\rho = i/N$, with $i = 1, \ldots, N$, for values of $\alpha$ equally spaced between 0.1 and 0.3. The area under these spectra appears to saturate to a finite value, consistent with the behavior of the rescaled Kolmogorov–Sinai entropy $\kappa$ shown in panel (c). This scaling of the Lyapunov spectra indicates that thermalization slows down according to the long-range network (LRN) scenario.
• Figure 2: We interpolate between the nonintegrable FPUT-$\alpha$ chain and the integrable Toda chain by tuning the parameter $\varepsilon$, imposing fixed boundary conditions in both cases. The potential is given by Eq. \ref{['V_Mixed_FBC']} with $N = 32$, and the initial condition is the same as in Fig. \ref{['Fig:FPUT_FBC']}. The parameter $\varepsilon$ is sampled at logarithmically spaced values between 0.1 and 1.0. As in Fig. \ref{['Fig:FPUT_FBC']}, panels (a), (b), and (c) show the maximal Lyapunov characteristic exponent (mLCE) $\Lambda_1$, the rescaled Lyapunov spectrum $\mkern 1.5mu\overline{\mkern-1.5mu\Lambda\mkern-1.5mu}\mkern 1.5mu(\rho) \equiv \Lambda_i/\Lambda_1$ as a function of $\rho = i/N$$(i = 1, \ldots, N)$, and the rescaled Kolmogorov–Sinai (KS) entropy $\kappa$, respectively. As before, the results indicate that thermalization slows down in accordance with the long-range network (LRN) scenario.
• Figure 3: Mixed boundary condition Toda chain with $N = 32$. The potential is taken as in Eq. \ref{['V_Toda_MBC']}, whose integrable limit corresponds to the fixed-boundary Toda chain. The initial condition is identical to that used in Fig. \ref{['Fig:FPUT_FBC']}. The parameter $\varepsilon$ is varied over logarithmically spaced values in the interval $0.1 \leqslant \varepsilon \leqslant 1.0$. Panels (a), (b), and (c) display the maximal Lyapunov characteristic exponent (mLCE) $\Lambda_1$, the rescaled Lyapunov spectrum $\mkern 1.5mu\overline{\mkern-1.5mu\Lambda\mkern-1.5mu}\mkern 1.5mu(\rho) \equiv \Lambda_i/\Lambda_1$ as a function of $\rho = i/N$$(i = 1, \ldots, N)$, and the rescaled Kolmogorov–Sinai (KS) entropy $\kappa$, respectively. The results again support the picture that thermalization becomes progressively slower, consistent with the long-range network (LRN) mechanism.
• Figure 4: We plot $\log_{10}\Lambda_1$, where $\Lambda_1$ denotes the maximal Lyapunov characteristic exponent (mLCE), as a function of inverse system size $1/N$ for three models: the fixed-boundary FPUT-$\alpha$ chain (blue circles), a linear chain with a single FPUT-$\alpha$ nonlinearity on the bond between sites $N/2$ and $N/2+1$ (teal hexagons), and the open-boundary Toda chain (red stars). In all three cases, the nonlinearity parameter is fixed to $\alpha = 0.25$. The initial conditions are randomly chosen at fixed energy density $E/N=0.1$, following the procedure used in Figs. \ref{['Fig:FPUT_FBC']}, \ref{['Fig:Mixed_FBC']}, and \ref{['Fig:Toda_MBC']}. For the FPUT-$\alpha$ chain, $\Lambda_1$ shows almost no dependence on $N$. In contrast, for the other two models $\Lambda_1$ approaches a finite saturation value with increasing $N$, i.e., $\log_{10}\Lambda_1$ tends to a constant as $1/N \to 0$.
• Figure 5: Rescaled Lyapunov spectra $\overline{\Lambda}(\rho) \equiv \Lambda_i/\Lambda_1$ as a function of $\rho = i/N$$(i=1,\ldots,N)$ for open-boundary Toda chains at increasing system sizes. Results are obtained from random initial conditions at fixed energy density $E/N=0.1$ for $N=32,40,48,56,64,$ and $128$. The spectra exhibit negligible dependence on $N$. The inset shows the rescaled Kolmogorov-Sinai (KS) entropy $\kappa$ as a function of $1/N$, highlighting the convergence of the $\overline{\Lambda}(\rho)$ curves with increasing system size (cf. the (c) panels of Figs. \ref{['Fig:FPUT_FBC']}, \ref{['Fig:Mixed_FBC']}, and \ref{['Fig:Toda_MBC']}).