Classical Chaos as an Environment for Dissipation

Abstract

We study an isolated three-dimensional system composed of a slow one-freedom system of interest interacting with a soft-chaotic environment. We present numerical evidence that the averaged motion of the slow system is dissipative, though not exactly governed by the Langevin equation. An analytical reasoning is proposed to explain the results, circumventing the unsufficiency of the adiabatic hypothesis.

Classical dissipation and asymptotic equilibrium via interaction with chaotic systems

“…Notice that the quantity E o (t) is very different from E o ( z , p z ), which is the energy of the average trajectory, considered in [3].…”

“…Low dimensional chaotic systems can, under appropriate circumstances, play the role of thermodynamical heat baths [1,2,3,4,5,6,7]. If a slow system with few degrees of freedom is weakly coupled to a fast chaotic system, the slow system's average trajectory can dissipate energy into the chaotic one at short times.…”

“…Jarzynski [4] showed later that coupling to a low-dimensional chaotic motion can also lead to a 'thermalization' of the system of interest, very much like the thermalization of a Brownian particle interacting with a large thermal bath. Finally, Carvalho and de Aguiar [3] established a connection between de formalism developed by Caldeira and Leggett [9] for describing dissipation via coupling with a thermal bath and via coupling with a chaotic system.…”

“…The present work has some similarities with that of ref. [3], which also considered the model of an oscillator coupled to a chaotic bath to study dissipation at short times. Here we study both the short and long time limits, showing that the coupling with the chaotic system may indeed lead to an equilibration very much like that caused by the coupling with a large thermal reservoir.…”

Energy transfer dynamics and thermalization of two oscillators interacting via chaos

“…The spectral function, which is related to the distribution of frequencies of the normal modes, can be chosen to model several types of thermal baths [3][4][5]. Other representations of the environment have also been explored, from spin systems [6] (or two-level atoms) to chaotic systems [7][8][9][10][11]. The latter is particularly important to model coupling to small external systems where the chaotic nature of the trajectories compensates for the small number of degrees of freedom in the decay of correlation functions.…”