The Role of Chaos and Resonances in Brownian Motion

Abstract

We study two points of view regarding the origin of irreversible processes. One is the “chaotic hypothesis” that says that irreversible processes are rooted in the randomness generated by chaotic dynamics. The second point of view, put forward by Prigogine’s school, is that irreversibility is rooted in non-integrable dynamics, as defined by Poincaré. Non-integrability is associated with resonances. We consider a simple model of Brownian motion, a harmonic oscillator (particle) coupled to lattice vibration modes (field). We compute numerically the “(⊄, τ) entropy”, which indicates how random are the trajectories and how close they are to Brownian trajectories. We show that (1) to obtain trajectories close to Brownian motion it is necessary to have a resonance between the particle and the lattice, which allows the transfer of information from the lattice to the particle. This resonance makes the system non-integrable in the sense of Poincaré. (2) For random initial conditions, chaos seems to play a secondary role in the Brownian motion, as the entropy is similar for both chaotic and non-chaotic dynamics. In contrast, if the initial conditions are not random, chaos plays a crucial role, as it leads to the thermalization of the lattice, which then induces the Brownian motion of the particle through resonance.