Classical dissipation and asymptotic equilibrium via interaction with chaotic systems

Abstract

We study the energy flow between a one dimensional oscillator and a chaotic system with two degrees of freedom in the weak coupling limit. The oscillator's observables are averaged over an initially microcanonical ensemble of trajectories of the chaotic system, which plays the role of an environment for the oscillator.We show numerically that the oscillator's average energy exhibits irreversible dynamics and 'thermal' equilibrium at long times. We use linear response theory to describe the dynamics at short times and we derive a condition for the absorption or dissipation of energy by the oscillator from the chaotic system. The equilibrium properties at long times, including the average equilibrium energies and the energy distributions, are explained with the help of statistical arguments. We also check that the concept of temperature defined in terms of the 'volume entropy' agrees very well with these energy distributions.

Energy dissipation via coupling with a finite chaotic environment

“…Similar results were obtained in [9] with N = 1 but performing ensemble averages over several thousands of trajectories. The results displayed in Fig.…”

“…The main result of this paper is that a small chaotic environment can indeed behave as an effective infinite reservoir, promoting energy flow and equilibration with smaller systems in a single realization of the dynamics. This is to be contrasted with previous results where a single chaotic system plays the role of the environment and thermodynamical behavior is achieved via ensemble averaging over many realizations with random initial conditions [9]. For the same values of parameters the results with N = 100 are roughly equivalent to averaging over 30 000 initial conditions.…”

“…In these cases, and in particular for N = 1, exponential dissipation can be observed only in an ensemble averaged treatment of the energy [9].…”

“…is the response function obtained by Bonanca [9] for the case of a single QS coupled with the oscillator. Substituting (17) in (16) and (15), we obtain…”

“…Second order corrections lead to a force proportional to the velocity whose symmetric part corresponds to friction and whose antisymmetric part acts as a geometric magnetism [2,3]. Since the publication of these pioneering works, a large number of papers have explored the possibility of dissipation caused by small environments, both classically [7][8][9][10][11] and quantum mechanically [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28], where decoherence is the main focus.…”