Jump processes as generalized gradient flows

Peletier, Mark A.; Rossi, Riccarda; Savaré, Giuseppe; Tse, Oliver

doi:10.1007/s00526-021-02130-2

Cited by 19 publications

(8 citation statements)

References 68 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We pay attention to precisely quantifying the energy dissipation since this information is key to derive the compactness results to be used in Section 4. Denote by F n = (F n K σ ) K ∈ T ,σ ∈ E K the approximate fluxes at time step n ≥ 1, then taking inspiration in [29,30], we introduce the primal dissipation potential by setting…”

Section: Discrete Energy/dissipation Estimatesmentioning

confidence: 99%

On the square-root approximation finite volume scheme for nonlinear drift-diffusion equations

Cancès¹,

Venel²

2023

Comptes Rendus. Mathématique

View full text Add to dashboard Cite

We study a finite volume scheme for the approximation of the solution to convection diffusion equations with nonlinear convection and Robin boundary conditions. The scheme builds on the interpretation of such a continuous equation as the hydrodynamic limit of some simple exclusion jump process. We show that the scheme admits a unique discrete solution, that the natural bounds on the solution are preserved, and that it encodes the second principle of thermodynamics in the sense that some free energy is dissipated along time. The convergence of the scheme is then rigorously established thanks to compactness arguments. Numerical simulations are finally provided, highlighting the overall good behavior of the scheme.

show abstract

Section: Discrete Energy/dissipation Estimatesmentioning

confidence: 99%

On the square-root approximation finite volume scheme for nonlinear drift-diffusion equations

Cancès¹,

Venel²

2023

Comptes Rendus. Mathématique

View full text Add to dashboard Cite

show abstract

“…Our construction of an information geometry for dynamics is heavily based on the idea of using Legendre duality for the force and flux relation, proposed in the recent work of large deviations theory and the macroscopic fluctuation theorem for MJP and CRN led by A.Mieleke, R.I.A.Petterson, M.A.Peletier, D.R.M. Renger, J.Zimmer, and others 3 [75][76][77][78][79][80][81][82]. We clarified its information-geometric aspects in the context of CRN and thermodynamics in our previous work [83].…”

Section: Aim and Contributions Of This Workmentioning

confidence: 99%

“…48 Furthermore, dissipation functions have been recognized since the seminal work of Onsager [124][125][126]. However, only quadratic dissipation functions have been investigated until very recently [75][76][77][78][79][80][81][82]. This may be partly because we lack an adequate geometric language to handle the non-quadratic cases, i.e., information geometry.…”

Section: Generalized Gradient Flow and De Giorgi's Formulationmentioning

confidence: 99%

Information geometry of dynamics on graphs and hypergraphs

Kobayashi,

Loutchko,

Kamimura

et al. 2023

Info. Geo.

View full text Add to dashboard Cite

We introduce a new information-geometric structure associated with the dynamics on discrete objects such as graphs and hypergraphs. The presented setup consists of two dually flat structures built on the vertex and edge spaces, respectively. The former is the conventional duality between density and potential, e.g., the probability density and its logarithmic form induced by a convex thermodynamic function. The latter is the duality between flux and force induced by a convex and symmetric dissipation function, which drives the dynamics of the density. These two are connected topologically by the homological algebraic relation induced by the underlying discrete objects. The generalized gradient flow in this doubly dual flat structure is an extension of the gradient flows on Riemannian manifolds, which include Markov jump processes and nonlinear chemical reaction dynamics as well as the natural gradient. The information-geometric projections on this doubly dual flat structure lead to information-geometric extensions of the Helmholtz–Hodge decomposition and the Otto structure in $$L^{2}$$ L 2 -Wasserstein geometry. The structure can be extended to non-gradient nonequilibrium flows, from which we also obtain the induced dually flat structure on cycle spaces. This abstract but general framework can broaden the applicability of information geometry to various problems of linear and nonlinear dynamics.

show abstract

“…Our work, which takes a different route, not only sheds new light onto the unveiling of a gradient flow structure for (1.1), but also provides the well-posedness (in the sense of entropy solutions) of (1.1) for a general class of interaction potentials, including the Newtonian potential, which we believe to be novel. This route is inspired by deterministic particle approximations (DPA) designed for equations like (1.1) [24,27,28,34], by recent advances in the theory of generalized gradient structures [32,45], and by the asymptotic limits of such structures [43,47]. More explicitly, we introduce a DPA for the approximation of (1.1) that possesses a generalized gradient structure, and show that this approximation converges to the unique entropy solution of (1.1), which also exhibits a gradient structure.…”

Section: Introductionmentioning

confidence: 99%

On gradient flow and entropy solutions for nonlocal transport equations with nonlinear mobility

Fagioli¹,

Tse²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We prove the well-posedness of entropy solutions for a wide class of nonlocal transport equations with nonlinear mobility in one spatial dimension. The solution is obtained as the limit of approximations constructed via a deterministic system of interacting particles that exhibits a gradient flow structure. At the same time, we expose a rigorous gradient flow structure for this class of equations in terms of an Energy-Dissipation balance, which we obtain via the asymptotic convergence of functionals.

show abstract

Jump processes as generalized gradient flows

Cited by 19 publications

References 68 publications

On the square-root approximation finite volume scheme for nonlinear drift-diffusion equations

On the square-root approximation finite volume scheme for nonlinear drift-diffusion equations

Information geometry of dynamics on graphs and hypergraphs

On gradient flow and entropy solutions for nonlocal transport equations with nonlinear mobility

Contact Info

Product

Resources

About