Constraints and Relations in Driven and Active Diffusions

Abstract

In near-equilibrium thermodynamic systems, the fluctuation-dissipation theorem characterizes the responses of arbitrary observables due to small perturbations in terms of experimentally measurable equilibrium correlation functions. However, in far-from-equilibrium systems, the fluctuation-dissipation theorem is no longer valid and the approach to re-establishing the connection between the responses and correlations for nonequilibrium steady states requires detailed knowledge of the system’s microscopic dynamics, which is usually prohibitively difficult to obtain.

This dissertation proposes a new perspective on studying nonequilibrium static responses in any system that can be modeled as a diffusion process with periodic boundary conditions. For one-dimensional diffusion processes I analyze the static response to perturbations of nonequilibrium steady states and demonstrate that an arbitrary perturbation can be broken up into a combination of three specific classes of perturbations that can be fruitfully addressed individually. For each class I derive a simple formula that quantitatively characterizes the response in terms of the strength of nonequilibrium driving valid arbitrarily far from equilibrium. Among the three classes of perturbations, I show that the perturbation in mobility has an important physical meaning of violation of the fluctuation-dissipation theorem. This motivates us to generalize the study in mobility perturbation for higher dimensions. I present the general Fourier expansion approach that can be used for studying this problem and the challenges for a general model. Then I show for several special cases some analytical or numerical bounds for mobility perturbation responses. Finally, for active Brownian particles, an application of studying the response in hydrodynamics gives the nonequilibrium Green-Kubo relations. This nonequilibrium Green-Kubo relation is numerically verified using molecular dynamics simulations and data analysis.