A perturbative theory for Brownian vortexes

TL;DR

This paper develops a perturbative theoretical framework for understanding Brownian vortexes, stochastic systems driven by non-conservative forces, explaining experimental phenomena observed in colloidal particles in optical traps.

Contribution

It introduces a perturbative expansion of the Fokker-Planck equation to model Brownian vortexes, capturing their complex behaviors and responses to temperature changes.

Findings

First-order solution modifies Boltzmann relation

Explains vortex direction and topology changes

Matches experimental observations in optical tweezers

Abstract

Brownian vortexes are stochastic machines that use static non-conservative force fields to bias random thermal fluctuations into steadily circulating currents. The archetype for this class of systems is a colloidal sphere in an optical tweezer. Trapped near the focus of a strongly converging beam of light, the particle is displaced by random thermal kicks into the nonconservative part of the optical force field arising from radiation pressure, which then biases its diffusion. Assuming the particle remains localized within the trap, its time-averaged trajectory traces out a toroidal vortex. Unlike trivial Brownian vortexes, such as the biased Brownian pendulum, which circulate preferentially in the direction of the bias, the general Brownian vortex can change direction and even topology in response to temperature changes. Here we introduce a theory based on a perturbative expansion of the Fokker-Planck equation for weak non-conservative driving. The first-order solution takes the form of a modified Boltzmann relation and accounts for the rich phenomenology observed in experiments on micrometer-scale colloidal spheres in optical tweezers.