Nonlinear Langevin functionals for a driven probe

TL;DR

This paper develops a Volterra series approach using nonlinear response theory to analyze the nonlinear force cumulants on a driven probe in a fluid, revealing non-Gaussian fluctuations and nonlinear phenomena.

Contribution

It introduces a formalism expressing nonlinear force cumulants as kernels related to equilibrium correlations, applicable to various driven probe scenarios.

Findings

Nonlinear force fluctuations are characterized by higher-order cumulants.

Simulations show nonlinear effects like shear-thinning and oscillating noise covariance.

The formalism connects nonlinear response to equilibrium correlation functions.

Abstract

When a probe particle immersed in a fluid with nonlinear interactions is subject to strong driving, the cumulants of the stochastic force acting on the probe are nonlinear functionals of the driving protocol. We present a Volterra series for these nonlinear functionals, by applying nonlinear response theory in a path integral formalism, where the emerging kernels are shown to be expressed in terms of connected equilibrium correlation functions. The first cumulant is the mean force, the second cumulant characterizes the non-equilibrium force fluctuations (noise), and higher order cumulants quantify non-Gaussian fluctuations. We discuss the interpretation of this formalism in relation to Langevin dynamics. We highlight two example scenarios of this formalism: i) For a particle driven with prescribed trajectory, the formalism yields the non-equilibrium statistics of the interaction force with the fluid. ii) For a particle confined in a moving trapping potential, the formalism yields the non-equilibrium statistics of the trapping force. In simulations of a model of nonlinearly interacting Brownian particles, we find that nonlinear phenomena, such as shear-thinning and oscillating noise covariance, appear in third or second order response, respectively.