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arXiv:1210.2607 · Physical Review Letters

Minimal Fokker-Planck Theory for the Thermalization of Mesoscopic Subsystems

Igor Tikhonenkov, Amichay Vardi, J. R. Anglin, Doron Cohen

Published Jan 30, 2013• 38 citations
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Abstract

We explore a minimal paradigm for thermalization, consisting of two weakly coupled, low dimensional, nonintegrable subsystems. As demonstrated for Bose-Hubbard trimers, chaotic ergodicity results in a diffusive response of each subsystem, insensitive to the details of the drive exerted on it by the other. This supports the hypothesis that thermalization can be described by a Fokker-Planck equation. We also observe, however, that Levy-flight type anomalies may arise in mesoscopic systems, due to the wide range of time scales that characterize 'sticky' dynamics.

DOI
10.1103/physrevlett.110.050401

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Thermalization has been shown to occur in a number of closed quantum many-body systems, but the description of the actual thermalization dynamics is prohibitively complex. Here, we present a model-in one and two dimensions-for which we can analytically show that the evolution into thermal equilibrium is governed by a Fokker-Planck equation derived from the underlying quantum dynamics. Our approach does not rely on a formal distinction of weakly coupled bath and system degrees of freedom. The results show that transitions within narrow energy shells lead to a dynamics which is dominated by entropy and establishes detailed balance conditions that determine both the eventual equilibrium state and the nonequilibrium relaxation to it.
We consider a minimal model for quantum thermalization of coupled chaotic subsystems. The route toward ergodicity is explored as a function of the coupling strength. The results are contrasted with the predictions of standard random matrix theory and the eigenstates thermalization hypothesis. We highlight a coupling regime of disparity between the spectral statistics that indicates chaos and ergodicity measures that indicate lack of eigenstate thermalization. The analysis involves a revision of the energy-shell concept, in a way that is consistent but independent of the semiclassical perspective.
We introduce a semiclassical theory for strong localization that may arise in the context of many-body thermalization. As a minimal model for thermalization we consider a few-site Bose-Hubbard model consisting of two weakly interacting subsystems that can exchange particles. The occupation of a subsystem (x) satisfies in the classical treatment a Fokker-Planck equation with a diffusion coefficient D(x). We demonstrate that it is possible to deduce from the classical description a quantum breaktime t^{*} and, hence, the manifestations of a strong localization effect. For this purpose it is essential to take the geometry of the energy shell into account and to make a distinction between different notions of phase-space exploration.
We outline formal and physical similarities between the quantum dynamics of open systems and the mesoscopic description of classical systems affected by weak noise. The main tool of our interest is the dissipative Wigner equation, which, for suitable timescales, becomes analogous to the Fokker-Planck equation describing classical advection and diffusion. This correspondence allows, in principle, to surmise a finite resolution, other than the Planck scale, for the quantized state space of the open system, particularly meaningful when the latter underlies chaotic classical dynamics. We provide representative examples of the quantum-stochastic parallel with noisy Hopf cycles and Van der Pol-type oscillators.