Nonlinear Langevin functionals for a driven probe

Area of Science:

• Statistical Mechanics
• Nonlinear Dynamics
• Soft Matter Physics

Background:

• Probe particles in fluids experience stochastic forces, especially under strong driving.
• These forces can exhibit complex nonlinear behaviors dependent on the driving protocol.
• Understanding these nonlinearities is crucial for modeling systems like colloids and polymers.

Purpose of the Study:

• To develop a theoretical framework for quantifying nonlinear functionals of stochastic forces acting on a probe particle.
• To connect these nonlinear functionals to equilibrium correlation functions using nonlinear response theory.
• To analyze the non-equilibrium statistics of interaction and trapping forces in driven systems.

Main Methods:

• Application of nonlinear response theory within a path integral formalism.
• Development of a Volterra series expansion for nonlinear functionals.
• Calculation of kernels in terms of connected equilibrium correlation functions.
• Simulation of a model system of nonlinearly interacting Brownian particles.

Main Results:

• The first cumulant represents the mean force, the second characterizes noise, and higher cumulants capture non-Gaussian fluctuations.
• The formalism successfully describes non-equilibrium statistics for prescribed trajectories and moving potentials.
• Simulations reveal shear-thinning (third-order response) and oscillating noise covariance (second-order response).

Conclusions:

• The Volterra series provides a powerful tool for analyzing nonlinear stochastic forces in driven systems.
• The framework links non-equilibrium phenomena to equilibrium properties through correlation functions.
• Nonlinear effects like shear-thinning and noise oscillations are observable in driven Brownian systems.