Thermalization in high-dimensional systems: the (weak) role of chaos

Abstract

In their seminal work, Fermi, Pasta, Ulam and Tsingou explored the connection between statistical mechanics and dynamical properties, such as chaos and ergodicity. Even today, seventy years later, the topic is not fully understood: while most results of statistical mechanics require the ergodic hypothesis to be rigorously proved, there are many indications that these predictions, both in and out of equilibrium, hold even in the absence of a rigorous form of ergodicity. Motivated by the above considerations, in this work we reconsider the point of view that the relevant ingredients for the validity of statistical mechanics are the large number of degrees of freedom and the choice of extensive observables, while the details of the dynamics do not play an essential role. This is the idea behind Khinchin's famous proof of the typicality of macroscopic observables at equilibrium. We extend this perspective to the context of non equilibrium, by investigating the thermalization properties of both harmonic (integrable) and nonharmonic (chaotic) oscillator chains initially prepared in out-of-equilibrium conditions. In integrable systems, thermalization occurs, or not, depending on the observable. In the chaotic regime, instead, thermalization is reached by any observable, although the relaxation timescale might be larger than the observation time.

Figures (6)

• Figure 1: $X(t)$ vs. $t$ for $N=1024$ and $X(0)= X_{eq}+10 \sigma_{eq}$ in a chaotic piston (a), and in a non chaotic piston (b). Red lines represent $X(t)$ for a single realization; black lines refer to the ensemble average $\langle X(t) \rangle$. Figure reproduced from cerino2016role.
• Figure 2: Kullback–Leibler divergence between the empirical distribution of the momenta and the Maxwell–Boltzmann, as a function of time. Panel a) shows the behavior of $K[P_e(p,t)| P_{MB}(p)]$ for different values of $N$. The system is initialized in a far-from-equilibrium state where only a small fraction of the modes is excited. The convergence to equilibrium takes place on a time scale that is proportional to $N$. Panels b), c) and d) show the detail of the empirical distribution $P_e(p,t)$ at different times, for the $N=10^4$ case. Figure reproduced from Baldovin2023.
• Figure 3: Solid lines: evolution of the on-site energy of the half-chain $E_\text{half}^\text{os}$ and of its kinetic $K_\text{half}^\text{os}$ and potential $U_\text{half}^\text{os}$ contributions over time, for varying $N$. The initial conditions are set according to \ref{['eqn:initialConditionLinearCase']} with $T=0.5/N$ and $k_R=2$. Dotted lines: theoretical prediction for the long-time average according to \ref{['eqn:theoPredictionHalfChain']} and \ref{['eqn:half_energy']}. The observables approach thermalization at their expected microcanonical value for $N\gg 1$.
• Figure 4: Time average of the observable $E^{\text{os}}_j$ defined in Eq. \ref{['eqn:energyOS']}, as a function of time $t$, averaged on all sites $j$. Averages are estimated as sample means over integer times. Panel (a) focuses on short time scales, panel (b) on the long-time behaviour. In both panels, blue triangles represent the unperturbed harmonic dynamics (energy $E_0$), while green circles stand for the perturbed non-harmonic one. The harmonic dynamics with corrected total energy $E=E_0+\Delta$ is represented by red squares. Horizontal lines indicate the corresponding analytical value of $\left\langle E_k^{\text{os}}\right\rangle_\beta$, provided by Eq. \ref{['eq:eos_beta']}, with color code as before. The two times $t_1$ and $t_2$ highlighted in panel (b) are better explored in Fig. \ref{['fig:modes']}. Parameters: $\beta=0.1$, $E_0/N=1$, $N=256$.
• Figure 5: Time-averaged distribution of the harmonic energy among the normal modes. Both the harmonic case with corrected energy [panels (a) and (c)] and the nonlinearly perturbed evolution [panels (b) and (d)] are considered. We focus on the two times $t_1=2^{19}$ [panels (a-b)] and $t_2=2^{28}$ [panels (c-d)] highlighted in Fig. \ref{['fig:observable']}. Parameters as in Fig. \ref{['fig:observable']}.