Energy Dissipation Via Coupling With a Finite Chaotic Environment

We study the flow of energy between a harmonic oscillator (HO) and an external environment consisting of N two-degrees of freedom non-linear oscillators, ranging from integrable to chaotic according to a control parameter. The coupling between the HO and the environment is bilinear in the coordinates and scales with system size with the inverse square root of N. We study the conditions for energy dissipation and thermalization as a function of N and of the dynamical regime of the non-linear oscillators. The study is classical and based on single realization of the dynamics, as opposed to ensemble averages over many realizations. We find that dissipation occurs in the chaotic regime for a fairly small N, leading to the thermalization of the HO and environment a Boltzmann distribution of energies for a well defined temperature. We develop a simple analytical treatment, based on the linear response theory, that justifies the coupling scaling and reproduces the numerical simulations when the environment is in the chaotic regime.