[Paper Review] Variational structures beyond gradient flows: a macroscopic fluctuation-theory perspective

Robert I. A. Patterson, D. R. Michiel Renger+1 more|arXiv (Cornell University)|Mar 26, 2021

Advanced Thermodynamics and Statistical Mechanics4 citations

TL;DR

This paper develops a generalized variational structure for macroscopic evolution equations beyond gradient flows by incorporating fluxes into the large-deviation framework, enabling a decomposition into dissipative and non-dissipative components even without detailed balance. The key contribution is a unified abstract theory that generalizes gradient flows, FIR inequalities, and macroscopic fluctuation theory to non-quadratic rate functions, validated across particle systems like jump processes, zero-range processes, and chemical reaction networks in complex balance.

ABSTRACT

Macroscopic equations arising out of stochastic particle systems in detailed balance (called dissipative systems or gradient flows) have a natural variational structure, which can be derived from the large-deviation rate functional for the density of the particle system. While large deviations can be studied in considerable generality, these variational structures are often restricted to systems in detailed balance. Using insights from macroscopic fluctuation theory, in this work we aim to generalise this variational connection beyond dissipative systems by augmenting densities with fluxes, which encode non-dissipative effects. Our main contribution is an abstract framework, which for a given flux-density cost and a quasipotential, provides a decomposition into dissipative and non-dissipative components and a generalised orthogonality relation between them. We then apply this abstract theory to various stochastic particle systems -- independent copies of jump processes, zero-range processes, chemical-reaction networks in complex balance and lattice-gas models.

Motivation & Objective

• To extend variational structures beyond gradient flows in systems without detailed balance, where classical large-deviation-based variational formulations fail.
• To unify and generalize existing frameworks such as FIR inequalities and macroscopic fluctuation theory into a single abstract action functional.
• To derive a decomposition of the rate functional into dissipative and non-dissipative components using fluxes and quasipotentials.
• To establish a generalized orthogonality relation between dissipative and non-dissipative parts in the absence of detailed balance.
• To validate the theory on concrete stochastic particle systems, including independent jump processes, zero-range processes, and complex-balanced chemical reaction networks.

Proposed method

• Formulate an abstract action functional $(\rho,j) \mapsto \int_0^T \mathcal{L}(\rho(t),j(t))\,dt$ representing the large-deviation rate of particle densities and fluxes.
• Introduce a quasipotential $\mathcal{V}(\rho)$ as the large-deviation rate of the invariant measure, enabling the definition of a dual flux $j^* = \nabla \mathcal{V}(\rho)$.
• Decompose the Lagrangian $\mathcal{L}(\rho,j)$ into a dissipative part $\mathcal{D}(\rho,j)$ and a non-dissipative part $\mathcal{N}(\rho,j)$ via a generalized orthogonality condition.
• Use time-reversal symmetry and flux-flux duality to separate symmetric (dissipative) and antisymmetric (non-dissipative) contributions in the rate functional.
• Apply the abstract framework to specific systems: independent particles on a graph, zero-range processes, complex-balanced reaction networks, and lattice-gas models.
• Derive new variational formulations for macroscopic equations that generalize classical gradient-flow structures by including non-equilibrium, non-dissipative effects.

Experimental results

Research questions

• RQ1 Can a variational structure be constructed for macroscopic equations derived from stochastic particle systems without detailed balance?
• RQ2 How can non-dissipative (non-gradient) effects be systematically incorporated into large-deviation-based variational formulations?
• RQ3 Is there a general decomposition of the large-deviation rate functional into dissipative and non-dissipative components that generalizes both gradient flows and FIR inequalities?
• RQ4 Can the abstract theory be applied to non-quadratic rate functions in macroscopic fluctuation theory?
• RQ5 Under what conditions does the quasipotential correspond to a solution of the Hamilton-Jacobi equation in the absence of detailed balance?

Key findings

• The abstract theory provides a general decomposition of the Lagrangian $\mathcal{L}(\rho,j)$ into dissipative and non-dissipative components, even without detailed balance.
• For complex-balanced chemical reaction networks, the quasipotential $\mathcal{V}(\rho) = \sum_{x} s(\rho_x \mid \pi_x)$ satisfies the Hamilton-Jacobi equation if and only if complex balance holds with respect to the stationary measure $\pi$.
• The theory generalizes the classical connection between large deviations and gradient flows to non-equilibrium systems by including fluxes as independent variables.
• The framework extends FIR inequalities to a full variational structure, fully characterizing the macroscopic dynamics rather than just bounding free-energy differences.
• In systems like independent particles on a graph and zero-range processes, the theory yields new variational formulations that recover known macroscopic equations as minimizers of the augmented action functional.
• The abstract orthogonality condition between dissipative and non-dissipative components generalizes the Hilbert-space decomposition of quadratic rate functions to non-quadratic settings.