[Paper Review] Structures of nonequilibrium fluctuations: dissipation and activity

Bram Wynants|arXiv (Cornell University)|Nov 18, 2010

Advanced Thermodynamics and Statistical Mechanics52 references3 citations

TL;DR

This paper develops a unified framework for nonequilibrium fluctuations by decomposing response functions into entropy flux and 'traffic'—a measure of system activity—using large deviation theory. It derives explicit rate functions for occupation and current fluctuations in terms of these quantities, recovering equilibrium principles in the close-to-equilibrium limit.

ABSTRACT

We discuss research done in two important areas of nonequilibrium statistical mechanics: fluctuation dissipation relations and dynamical fluctuations. In equilibrium systems the fluctuation-dissipation theorem gives a simple relation between the response of observables to a perturation and correlation functions in the unperturbed system. Our contribution here is an investigation of the form of the response function for systems out of equilibrium. Furthermore, we use the theory of large deviations to examine dynamical fluctuations in systems out of equilibrium. In dynamical fluctuation theory we consider two kinds of observables: occupations (describing the fraction of time the system spends in each configuration) and currents (describing the changes of configuration the system makes). We explain how to compute the rate functions of the large deviations, and what the physical quantities are that govern their form.

Motivation & Objective

• To understand how fluctuation-dissipation relations break down in nonequilibrium systems and to identify the structural components governing response.
• To characterize dynamical fluctuations in terms of occupation and current statistics using large deviation theory.
• To identify entropy flux and 'traffic' as the fundamental thermodynamic quantities governing nonequilibrium response and fluctuations.
• To recover known equilibrium principles, such as minimum entropy production, in the close-to-equilibrium regime.
• To provide a general formalism applicable to Markov jump processes and diffusions, linking microscopic dynamics to macroscopic fluctuation behavior.

Proposed method

• Uses the path-probability measure and Radon-Nikodym derivative to define the action and irreversibility of trajectories.
• Introduces 'excess entropy flux' and 'excess traffic' as path-wise quantities derived from time-reversed dynamics.
• Applies large deviation theory to compute the rate function for joint fluctuations of occupation and current, expressing it in terms of entropy and traffic.
• Derives the response function as the sum of two correlation functions: one linked to entropy exchange (dissipation), the other to traffic (activity).
• Establishes that the rate function for occupations depends solely on traffic, and traffic acts as a thermodynamic potential for currents.
• Uses scaling limits of Markov jump processes to derive overdamped diffusion dynamics, recovering the Fokker-Planck equation with a forcing term.

Experimental results

Research questions

• RQ1 How can the response function in nonequilibrium systems be decomposed into physically meaningful components?
• RQ2 What is the role of entropy flux and 'traffic' in governing the statistics of dynamical fluctuations?
• RQ3 How do the rate functions for occupation and current fluctuations depend on thermodynamic quantities like entropy and traffic?
• RQ4 Can the minimum entropy production principle be derived from the large deviation formalism in the close-to-equilibrium regime?
• RQ5 How do Markov jump processes scale to diffusive dynamics, and what is the role of local detailed balance in this limit?

Key findings

• The response function in nonequilibrium systems decomposes into two correlation functions: one related to entropy flux (dissipation) and one to traffic (activity).
• The rate function for joint fluctuations of occupation and current is explicitly expressed in terms of entropy flux and traffic, providing a complete statistical description.
• The rate function for occupations alone depends exclusively on traffic, indicating traffic's fundamental role in fluctuation statistics.
• Traffic is shown to act as a thermodynamic potential for currents, analogous to free energy in equilibrium systems.
• In the close-to-equilibrium regime, the formalism recovers the minimum entropy production principle as a variational principle.
• The scaling limit of Markov jump processes on a lattice yields a diffusion process with a drift term that includes a forcing field, consistent with the Fokker-Planck equation and local detailed balance.