[Paper Review] Coarse-graining via the fluctuation-dissipation theorem and large-deviation theory

Alberto Montefusco, Mark A. Peletier+1 more|arXiv (Cornell University)|Sep 19, 2018

Advanced Thermodynamics and Statistical Mechanics50 references3 citations

TL;DR

This paper generalizes the fluctuation-dissipation theorem (FDT) to Markov processes with detailed balance and a large-deviation principle, enabling coarse-graining of systems with rare events—such as chemical reactions—via generalized gradient flows. The key contribution is a framework that derives dissipation potentials from dynamic rate functions, validated in the Kramers' escape problem for A⇌B reactions.

ABSTRACT

The fluctuation-dissipation theorem is a central result in statistical mechanics and is usually formulated for systems described by diffusion processes. In this paper, we propose a generalization for a wider class of stochastic processes, namely the class of Markov processes that satisfy detailed balance and a large-deviation principle. The generalized fluctuation-dissipation theorem characterizes the deterministic limit of such a Markov process as a generalized gradient flow, a mathematical tool to model a purely irreversible dynamics via a dissipation potential and an entropy function: these are expressed in terms of the large-deviation dynamic rate function of the Markov process and its stationary distribution. We exploit the generalized fluctuation-dissipation theorem to develop a new method of coarse-graining and test it in the context of the passage from the diffusion in a double-well potential to the jump process that describes the simple reaction $A ightleftarrows B$ (Kramers' escape problem).

Motivation & Objective

• To extend the classical fluctuation-dissipation theorem beyond diffusion processes to general Markov processes with detailed balance.
• To develop a coarse-graining method applicable to systems dominated by rare, large-scale events, such as chemical reactions, rather than continuous diffusive dynamics.
• To establish a connection between large-deviation theory and generalized gradient flows for purely dissipative systems.
• To provide a framework for computing dissipation potentials directly from the dynamic rate function of a Markov process.
• To test the method on the Kramers' escape problem as a prototype for rare-event dynamics in chemical systems.

Proposed method

• Uses large-deviation theory to characterize the static and dynamic fluctuations of Markov processes, with the dynamic rate function encoding pathwise deviations from the deterministic limit.
• Applies the Feng-Kurtz scheme for numerical computation of the dynamic rate function in the context of rare-event processes.
• Defines the dissipation potential via the dynamic rate function, replacing the classical FDT's diffusion tensor with a nonlinear, generalized gradient structure.
• Constructs the macroscopic dynamics as a generalized gradient flow driven by an entropy function (from the stationary distribution) and a dissipation potential (from the dynamic rate function).
• Employs the generalized FDT to link fluctuation statistics (via large deviations) to irreversible, dissipative macroscopic dynamics.
• Validates the method in the Kramers escape problem, where the microscopic diffusion in a double-well potential coarse-grains to a Markov jump process for A⇌B.

Experimental results

Research questions

• RQ1 How can the fluctuation-dissipation theorem be generalized to Markov processes that are not diffusive, such as jump processes with rare events?
• RQ2 What is the mathematical structure linking large-deviation rate functions to generalized gradient flows in purely dissipative systems?
• RQ3 Can the dissipation potential in macroscopic dynamics be derived directly from the dynamic rate function of a microscopic Markov process?
• RQ4 How does the proposed coarse-graining method perform in systems where classical diffusion-based FDTs fail, such as monomolecular reactions?
• RQ5 What is the role of detailed balance and large-deviation principles in enabling a consistent coarse-graining from microscopic stochastic processes to macroscopic irreversible dynamics?

Key findings

• The generalized fluctuation-dissipation theorem establishes a one-to-one correspondence between the dynamic rate function of a Markov process and the dissipation potential of its deterministic limit.
• The macroscopic dynamics emerges as a generalized gradient flow, with the entropy function derived from the stationary distribution and the dissipation potential from the dynamic rate function.
• The method successfully coarse-grains the Kramers escape problem, transforming a diffusion in a double-well potential into a Markov jump process for A⇌B, capturing the correct transition rates.
• The approach avoids the recrossing problem and boundary definition issues common in traditional transition path sampling, by directly resolving the full structure of the rate function.
• The method is numerically feasible via the Feng-Kurtz scheme and shows promise for complex systems like glasses and plasticity, though computational efficiency remains a challenge.
• The framework is currently restricted to purely dissipative systems; extension to GENERIC-type dynamics with reversible components remains an open problem.