Minimal Fokker-Planck Theory for the Thermalization of Mesoscopic Subsystems

Igor Tikhonenkov, Amichay Vardi, James R. Anglin, and Doron Cohen

Phys. Rev. Lett. 110, 050401 – Published 30 January 2013

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.

Received 10 October 2012

DOI:https://doi.org/10.1103/PhysRevLett.110.050401

Authors & Affiliations

Igor Tikhonenkov¹, Amichay Vardi¹, James R. Anglin², and Doron Cohen³

¹Department of Chemistry, Ben Gurion University of the Negev, Beer Sheva 84105, Israel
²OPTIMAS Research Center and Fachbereich Physik, Technische Universität Kaiserslautern, D-67653 Kaiserslautern, Germany
³Department of Physics, Ben Gurion University of the Negev, Beer Sheva 84105, Israel

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Vol. 110, Iss. 5 — 1 February 2013

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Images

Figure 1
The energy spectrum of the unperturbed BHH. In panel (a), the scaled eigenenergies $ε_{n}$ of $H_{0}$ are plotted vs the scaled interaction parameter, $u$ , for $N = 35$ particles. The level spacing statistics is characterized by the Brody parameter ( $0 < q < 1$ ), which is displayed in panel (b) for a system with $N = 120$ particles. In the energy range where the motion is chaotic, $q \sim 1$ . Square symbols indicate the preparations that were used for the simulations in Fig. 2.Reuse & Permissions
Figure 2
The quantum probability distribution $p_{n} (t)$ for representative simulations is imaged as a function of time (right). The short-time energy-spreading profile is determined by the perturbation matrix $| W_{n, n_{0}} |^{2}$ (left). The number of particles is $N = 50$ . The upper set is for $u = 50$ and the lower is for $u = 5$ . The strength of the driving is $K_{d} / K_{0} = 0.1$ . For the time axis, we use dimensionless units, $τ = (E_{\max } - E_{0}) t / ℏ$ , and the scaled driving frequency in both simulations is $Ω \approx 0.03$ . The image of the initial level is vertically zoomed and it has the energy $ε \approx 0.5$ . The boundaries of the chaotic sea in the lower image are indicated by the horizontal dashed lines.Reuse & Permissions
Figure 3
The scaled variance as a function of the scaled time in the classical (left) and in the quantum (right) simulations. The value of the scaled interaction parameter is $u = 5$ (upper panels) and $u = 50$ (lower panels). Note the different scale of the vertical axis for the quantum vs classical simulations in panels (c) and (d). Clearly quantum-to-classical correspondence fails in the quasi-integrable regime. The values of $K_{d} / K_{0}$ are 0.025 (blue), 0.05 (green), 0.075 (red), and 0.1 (cyan). The driving frequency is as in Fig. 2.Reuse & Permissions
Figure 4
(a) The evolving spreading profile $ρ (E)$ , referring to the simulation that has been imaged in the lower panel of Fig. 2. The solid lines (calculation) and the associated symbols (simulation) are for $τ = 149$ (narrower), and $τ = 299$ (wider), and $τ = 448$ (widest). The lines are based on the numerical solution of Eq. (1). (b) The density of states $g (E)$ (dotted line) and the diffusion coefficient $D (E)$ (solid line) were deduced from the diagonalization of the BHH and Eq. (5), see Ref. 4 for technical details.Reuse & Permissions
Figure 5
On the left, the scaled energy $ε (t)$ as a function of time is plotted for a few representative trajectories. The right panel displays the average value and the dispersion of $ε$ within the time interval $0 < τ < 40000$ , for the representative trajectories (symbols), as well as for many other trajectories (points). Low dispersion values reflect the finite probability to encounter sticky motion. The parameters are the same as in the lower panel of Fig. 2, with initial points that have the energy $ε = 0.3$ .Reuse & Permissions

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