Geometry protected probabilistic structure in many-body dynamics

Abstract

Insomuch as statistical mechanics circumvents the formidable task of addressing many-body dynamics, it remains a challenge to derive macroscopic properties from a solution to Hamiltonian equations for microscopic motion of an isolated system. Launching new attacks on this long-standing problem -- part of Hilbert's sixth problem -- is urgently important, for focus of statistical phenomena is shifting from a fictitious ensemble to an individual member, i.e. a mechanically isolated system. Here we uncover a common probabilistic structure, the concentration of measure, in Hamiltonian dynamics of two families of systems, the Fermi-Pasta-Ulam-Tsingou (FPUT) model which is finite-dimensional and (almost) ergodic, and the Gross-Pitaevskii equation (GPE) which is infinite-dimensional and suffers strong ergodicity breaking. That structure is protected by the geometry of phase space and immune to ergodicity breaking, leading to counterintuitive phenomena. Notably, an isolated FPUT behaves as a thermal ideal gas even for strong modal interaction, with the thermalization time analogous to the Ehrenfest time in quantum chaos, while an isolated GPE system, without any quantum inputs, escapes the celebrated ultraviolet catastrophe through nonlinear wave localization in the mode space, and the Rayleigh-Jeans equilibrium sets in the localization volume. Our findings may have applications in nonlinear optics and cold-atom dynamics.

Figures (3)

• Figure 1: Illustration of concentration of measure. (a) The measure of a subset $A$ undergoes a sharp transition at $\mathbb{P}(A)=1/2$, where by slightly enlarging the boundary of $A$ the full measure is attained. Thus the full measure concentrates on an extremely small region. (b) This geometric phenomenon manifests in that any Lipschitz function, no matter how complicated it is, is essentially a constant, displaying small irregular oscillations.
• Figure 2: Many-body thermal equilibrium in FPUT dynamics. We simulate long-time FPUT dynamics and perform statistical analysis for a phase trajectory. (a) Simulated and theoretical distribution are in excellent agreement, with the comparison between simulated and theoretical temperature made in the inset. (b) The simulated correlation coefficient $r(E_k, E_{k'})$(see SI SI for definition) confirms that different modes are uncorrelated, and thus the many-body equilibrium is described by the ideal-gas thermal distribution (\ref{['eq:14']}). $\varepsilon = 1$ and $\zeta = {1\over 8}$. (c) The relation (\ref{['eq:1']}) between the thermalization time and the maximal Lyapunov exponent (dashed line) is confirmed for a wide range of $\varepsilon$ and different $\zeta$ controlling the initial energy weight configuration. $\alpha$ in Eq. (\ref{['eq:1']}) is seen to depend on $\zeta$ (inset). (d) Thermalization mechanism: A projective trajectory displays self-repulsion followed by irregular motion inside a region (grey), on which the thermal distribution concentrates. Lines in black, blue and red show deviations from periodic motion in the first three periods. For simulations $n=4$, $u=1$ and $N = 1025$.
• Figure 3: GPE escapes ultraviolet catastrophe. (a) Simulations confirm the emergence of the RJ equilibrium from a single trajectory for $k\leq k_c$, and exponential localization beyond $k_c$. For each $g\ell$, two sets of $(g,\ell)$ are used and the profiles overlap perfectly. (b) Verifications of the relation (\ref{['eq:18']}) between $\xi_{\rm loc}$ and $k_c$.