Thermalization of a Brownian particle via coupling to low-dimensional chaos.

It is shown that a paradigm of classical statistical mechanics — the thermalization of a Brownian particle — has a low-dimensional, deterministic analogue: when a heavy, slow system is coupled to fast deterministic chaos, the resultant forces drive the slow degrees of freedom toward a state of statistical equilibrium with the fast degrees. This illustrates how concepts useful in statistical mechanics may apply in situations where low-dimensional chaos exists. PACS numbers: 05.45.+b, 05.40.+j Typeset using REVTEX 1 Since the study of chaotic dynamics has clarified fundamental issues in classical statistical mechanics [1], it is worthwhile to consider the converse: when does intuition from statistical mechanics carry over to low-dimensional chaos? We all know, for instance, that a heavy particle immersed in a heat bath —a Brownian particle — is subject to both an average frictional force, and stochastic fluctuations around this average, and that the balance between these two thermalizes the particle. Now suppose the “Brownian” particle is coupled to a fast, low-dimensional, chaotic trajectory, rather than to a true heat bath. It is known that the particle then feels a dissipative force [2–4]; does the particle also (in some sense yet to be defined) “thermalize” with the chaotic trajectory? That is, does the fast chaos behave as a kind of “miniature heat reservoir”, exchanging energy with the particle in a way that brings the two into statistical equilibrium? In this paper, we pursue this question by considering the reaction forces acting on a heavy, slow system (our Brownian particle) due to its coupling to a light, fast trajectory. When the fast motion is chaotic, the forces on the particle include a conservative force, and two velocity-dependent forces, one magnetic-like, the other dissipative [4]. However (as in the case of coupling to a true thermal bath), there also exists a rapidly fluctuating, effectively stochastic force, which has not been studied in detail. We describe an approach which incorporates this force, with the others, into a unified framework. It is shown that the inclusion of this stochastic force — related to the frictional force by a fluctuation-dissipation relation [4] — causes the slow Brownian particle and the fast chaotic trajectory to evolve toward statistical equilibrium. This result provides some justification for applying statistical arguments (involving, e.g., relaxation toward equipartition of energy) to physical situations of only a few degrees of freedom. A discussion of examples — including one-body dissipation in nuclear dynamics [5], the Fermi mechanism of cosmic ray acceleration [6], and the diffusive transport of comets [7] — where such “thermal” arguments may provide insight into the physics behind more explicit calculations, will be presented in Ref. [8]. As a starting point for our discussion, we consider the framework of Ref. [4], where the position R of the slow particle parametrizes the Hamiltonian h governing the fast motion: 2 h = h(z,R), where z denotes the fast phase space coordinates. (The nature of the fast system will remain unspecified, but we take it to have a few, N ∼ 2, degrees of freedom.) This classical version of the Born-Oppenheimer framework has received considerable interest in recent years [3,4,9]. We assume that, if R were held fixed, then a fast trajectory evolving under h would ergodically and chaotically explore its energy shell (surface of constant h) in the fast phase space. This sets a fast time scale, τf , which we may take to be the Lyapunov time associated with the fast chaos. A slow time scale, τs, is set by the motion of the slow particle: it is the time required for the Hamiltonian h to change significantly. We assume τf ≪ τs; thus, the fast trajectory z(t) evolves under a slowly time-dependent Hamiltonian h. The full Hamiltonian for the combined system of slow and fast degrees is given by H(R,P, z) = P /2M + h(z,R), where P is the momentum of the slow particle, and M is its mass. (R,P, z) thus specifies a point in the full phase space of slow and fast variables. It is assumed that surfaces of constant H are bounded in the full phase space. Given this formulation, the force on the slow particle is F(t) = −∂h/∂R, evaluated along the trajectory z(t). From the point of view of the slow particle, this force fluctuates rapidly, so it is natural to separate F(t) into a slowly-changing average component, and rapid fluctuations F̃(t) around this average. In Ref. [4], Berry and Robbins introduce an approximation scheme for obtaining the net average reaction force. At leading (zeroth) order of approximation, the ergodic adiabatic invariant [2] dictates the energy of the fast system as a function of the slow coordinates, and this energy in turn serves as a potential for the slow system, giving rise to a conservative “Born-Oppenheimer” force F0. At next order, the Berry-Robbins framework yields two velocity-dependent reaction forces: deterministic friction (Fdf ) and geometric magnetism (Fgm) [10]. Geometric magnetism is a gauge force related to the geometric phase; deterministic friction (see also Ref. [3]) describes the irreversible flow of energy from the slow to the fast variables. Thus, while at leading order the fast degrees of freedom create a potential well for the slow degrees, at first order the fast motion effectively adds a magnetic field, and drains the slow system of its energy. What about the effets of the rapidly fluctuating component, F̃(t)? If the analogy with 3 ordinary Brownian motion is correct and some sort of statistical equilibration occurs, then F̃(t) ought to play a central role in the process. We now describe a framework which incorporates the effects of F̃(t) into a description of the slow particle’s evolution. In our framework we consider an ensemble of systems. Each member of the ensemble consists of a single slow particle coupled to a single fast trajectory, and represents one possible realization of the combined system of slow and fast variables. Representing this ensemble by a density φ in the full phase space, Liouville’s equation is: ∂φ ∂t + P M · ∂φ ∂R − ∂h ∂R · ∂φ ∂P + {φ, h} = 0, (1) where {·, ·} denotes the Poisson bracket with respect to the fast variables, z. Henceforth, we will ignore all information about the fast trajectory except its energy, E(t) ≡ h[z(t),R(t)] (which evolves on the slow time scale [2]). Thus, what we are really after is the evolution of W (R,P, E, t), the distribution of our ensemble in the reduced space where all fast variables other than E have been projected out. In this reduced space, F̃(t) is stochastic, which in turn suggests that W evolves diffusively. The derivation of an evolution equation for W is somewhat involved, and is sketched in the Appendix. Here we simply state the result: ∂W ∂t = − P M · ∂W ∂R + D̂ · (uW ) + ǫ 2 D̂i [