[Paper Review] Entropy dissipation of Fokker-Planck equations on graphs

Shui-Nee Chow, Wuchen Li+1 more|arXiv (Cornell University)|Jan 17, 2017

Geometric Analysis and Curvature Flows4 citations

TL;DR

This paper establishes the exponential convergence of solutions to nonlinear Fokker-Planck equations on finite graphs, formulated as gradient flows in the discrete Wasserstein space. It derives an explicit analytic bound for the entropy dissipation rate, whose continuous analog is linked to Yano’s formula in Riemannian geometry, extending classical results to discrete settings with applications in probability, PDEs, and statistical physics.

ABSTRACT

We study the nonlinear Fokker-Planck equation on graphs, which is the gradient flow in the space of probability measures supported on the nodes with respect to the discrete Wasserstein metric. The energy functional driving the gradient flow consists of a Boltzmann entropy, a linear potential and a quadratic interaction energy. We show that the solution converges to the Gibbs measures exponentially fast with a rate that can be given analytically. The continuous analog of this asymptotic rate is related to the Yano's formula.

Motivation & Objective

• To study the long-time behavior of nonlinear Fokker-Planck equations on finite graphs as gradient flows in the discrete 2-Wasserstein metric.
• To establish exponential convergence of solutions to Gibbs measures under a free energy functional combining Boltzmann entropy, linear potential, and quadratic interaction energy.
• To derive an explicit analytic formula for the exponential convergence rate in the discrete setting.
• To connect the asymptotic convergence rate to the continuous Yano’s formula from Riemannian geometry, thereby unifying discrete and continuous entropy dissipation theories.

Proposed method

• Formulates the Fokker-Planck equation on graphs as a gradient flow in the space of probability measures with respect to the discrete 2-Wasserstein metric.
• Defines the discrete Wasserstein metric using skew-symmetric vector fields, flux functions, and a discrete inner product with a logarithmic mean or arithmetic mean for flux regularization.
• Constructs the free energy functional as the sum of Boltzmann entropy, linear potential, and quadratic interaction energy, ensuring convexity and uniqueness of equilibrium under suitable conditions.
• Applies integration by parts and geodesic analysis on the discrete probability simplex to compute first and second-order variations of the free energy along paths.
• Derives the Hessian of the free energy in the discrete Wasserstein geometry, leading to a quadratic form that bounds the convergence rate.
• Establishes a continuous analog of the discrete convergence rate, showing its equivalence to Yano’s formula in Riemannian geometry via curvature and second derivative terms.

Experimental results

Research questions

• RQ1 What is the rate of exponential convergence of solutions to the nonlinear Fokker-Planck equation on finite graphs?
• RQ2 How does the discrete Wasserstein gradient flow structure on graphs relate to the continuous case in terms of entropy dissipation?
• RQ3 Can the convergence rate be bounded analytically in the discrete setting, and how does it compare to known continuous formulas?
• RQ4 What is the geometric interpretation of the convergence rate in terms of curvature and Hessian structure on graphs?
• RQ5 How does the discrete Hessian of the free energy relate to the continuous Yano’s formula in Riemannian geometry?

Key findings

• The solution to the nonlinear Fokker-Planck equation on graphs converges to the Gibbs measure exponentially fast, with a rate that can be computed analytically.
• The convergence rate is bounded by a quadratic form involving the Hessian of the free energy in the discrete Wasserstein metric, which is derived via geodesic variation and integration by parts.
• The continuous analog of the discrete convergence rate matches the Yano’s formula, linking the discrete gradient flow to Riemannian curvature in the limit.
• The Hessian of the free energy along geodesics is expressed as a double integral over the graph involving second derivatives of the energy functional and divergence terms.
• For the linear entropy case, the Hessian reduces to the $L^2$-norm of the divergence of the velocity field, which matches the Ricci curvature and Hessian trace terms in Yano’s formula.
• The framework generalizes the continuous gradient flow theory to discrete graphs, providing a rigorous foundation for numerical schemes and mean-field models in complex networks.