Perturbative theory for Brownian vortexes

Abstract

Brownian vortexes are stochastic machines that use static non-conservative force fields to bias random thermal fluctuations into steadily circulating currents [1,2]. The archetype for this class of systems is a colloidal sphere in an optical tweezer [1,3,4]. 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 [5], which then biases its diffusion [1,3]. 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.

Role of nonconservative scattering forces and damping on Brownian particles in optical traps

“…Because of the presence of these scattering forces, even static optical traps represent non-equilibrium systems as the particles trapped within them are driven by the non-conservative forces and thus are not described by a Gibbs-Boltzmann probability distribution. For instance, it has been demonstrated [6][7][8][9] both theoretically and experimentally that scattering forces induce non-equilibrium currents, so called Brownian vortices, for trapped overdamped Brownian particles. At atmospheric pressures and near room temperatures, the underlying dynamics of a trapped particle can be taken to be overdamped Brownian.…”

“…First a general theory for the probability distribution of the perturbed steady state due to a small non-conservative force is developed for general damped Brownian dynamics. The modification of the steady state probability distribution function for the full phase space in the optical model of [9] is found and the resulting marginal probability distributions for the particle position and velocity is computed. We demonstrate that the Brownian vortices found in the current of the particle position persists and remains geometrically the same as that found in [9].…”

“…The modification of the steady state probability distribution function for the full phase space in the optical model of [9] is found and the resulting marginal probability distributions for the particle position and velocity is computed. We demonstrate that the Brownian vortices found in the current of the particle position persists and remains geometrically the same as that found in [9]. However the amplitude of the vortices depends on the damping parameters, we will show that this damping parameter can be tuned in order to optimize the non-equilibrium current.…”

“…For the optical trap model of [9] we also compute the spectral density of the particle position for general damping. Our result recovers the computation of [10] for overdamped Brownian motion and generalizes it to all damping regimes.…”

Calibration of optical tweezers for force microscopy

“…When these requirements are not fulfilled, the scalar and vector potential may be used to describe the nonconservative force field. This was presented in Moyses et al [2015] for the discussion of Brownian vortices, and Irrera et al [2016] and Toe et al [2016] which discusses optically trapped nanowires.…”