Hostname: page-component-7d8f8d645b-p72pn Total loading time: 0 Render date: 2023-05-29T21:30:44.194Z Has data issue: false Feature Flags: { "useRatesEcommerce": true } hasContentIssue false

Coarse graining of a Fokker–Planck equation with excluded volume effects preserving the gradient flow structure

Published online by Cambridge University Press:  22 September 2020

Department of Applied Mathematics and Theoretical Physics, University of Cambridge, CambridgeCB3 0WA, UK,
Department Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstrasse 11, 91058Erlangen, Germany,
Mathematical Institute, University of Oxford, OxfordOX2 6GG, UK, email:
Rights & Permissions[Opens in a new window]


HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The propagation of gradient flow structures from microscopic to macroscopic models is a topic of high current interest. In this paper, we discuss this propagation in a model for the diffusion of particles interacting via hard-core exclusion or short-range repulsive potentials. We formulate the microscopic model as a high-dimensional gradient flow in the Wasserstein metric for an appropriate free-energy functional. Then we use the JKO approach to identify the asymptotics of the metric and the free-energy functional beyond the lowest order for single particle densities in the limit of small particle volumes by matched asymptotic expansions. While we use a propagation of chaos assumption at far distances, we consider correlations at small distance in the expansion. In this way, we obtain a clear picture of the emergence of a macroscopic gradient structure incorporating corrections in the free-energy functional due to the volume exclusion.

Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (, which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
© The Author(s), 2020. Published by Cambridge University Press


Adams, S., Dirr, N., Peletier, M. & Zimmer, J. (2013) Large deviations and gradient flows. Phil. Trans. R. Soc. A 371, 20120341–815.CrossRefGoogle ScholarPubMed
Adams, S., Dirr, N., Peletier, M. A. & Zimmer, J. (2011) From a large-deviations principle to the Wasserstein Gradient flow: a new micro-macro passage. Commun. Math. Phys. 307, 791815.CrossRefGoogle Scholar
Ambrosio, L., Gigli, N. & Savaré, G. (2005) Gradient flows in metric spaces and in the space of probability measures. In: Lectures in Mathematics, ETH Zürich, Basel: Birkhäuser Verlag.Google Scholar
Arnrich, S., Mielke, A., Peletier, M. A., Savaré, G. & Veneroni, M. (2012) Passing to the limit in a Wasserstein gradient flow: from diffusion to reaction. Calculus Variations Part. Diff. Equ. 44, 419454.CrossRefGoogle Scholar
Barker, J. A. & Henderson, D. (1967) Perturbation theory and equation of State for Fluids. II. A successful theory of liquids. J. Chem. Phys. 47, 47144721.CrossRefGoogle Scholar
Beirlant, J., Dudewicz, E., Györfi, L. & van der Meulen, E. C. (1997) Nonparametric entropy estimation: an overview. Int. J. Math. Stat. Sci. 6, 1739.Google Scholar
Benamou, J.-D. & Brenier, Y. (2000) A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numerische Mathematik 84, 375393.CrossRefGoogle Scholar
Berman, R. J. & Önnheim, M. (2019) Propagation of Chaos for a class of first order models with singular mean field interactions. SIAM J. Math. Anal. 51, 159196.CrossRefGoogle Scholar
Bodnar, M. & Velázquez, J. J. L. (2005) Derivation of macroscopic equations for individual cell-based models: a formal approach. Math. Meth. Appl. Sci. 28, 17571779.CrossRefGoogle Scholar
Bolley, F., Cañizo, J. A. & Carrillo, J. A. (2011) Stochastic mean-field limit: non-Lipschitz forces and swarming. Math. Models Methods Appl. Sci. 21, 21792210.CrossRefGoogle Scholar
Brenier, Y. (2003) Extended Monge-Kantorovich theory. In: Optimal Transportation and Applications. Lecture Notes in Mathematics (Fondazione C.I.M.E., Firenze), Vol. 1813, Berlin, Heidelberg: Springer.CrossRefGoogle Scholar
Bresch, D., Jabin, P.-E. & Wang, Z. (2019) On mean-field limits and quantitative estimates with a large class of singular kernels: application to the Patlak-Keller-Segel model. C. R. Math. Acad. Sci. Paris 357, 708720.CrossRefGoogle Scholar
Brezzi, F. & Fortin, M. (2012) Mixed and Hybrid Finite Element Methods. New York: Springer.Google Scholar
Bruna, M., Burger, M., Ranetbauer, H. & Wolfram, M.-T. (2017) Cross-diffusion systems with excluded-volume effects and asymptotic gradient flow structures. J. Nonlinear Sci. 27, 687719.CrossRefGoogle Scholar
Bruna, M. & Chapman, S. J. (2012) Diffusion of multiple species with excluded-volume effects. J. Chem. Phys. 137, 204116.CrossRefGoogle ScholarPubMed
Bruna, M. & Chapman, S. J. (2012) Excluded-volume effects in the diffusion of hard spheres. Phys. Rev. E 85, 011103.CrossRefGoogle ScholarPubMed
Bruna, M., Chapman, S. J. & Robinson, M. (2017) Diffusion of particles with short-range interactions. SIAM J. Appl. Math. 77, 22942316.CrossRefGoogle Scholar
Burger, M. (2017) Transport metrics for Vlasov hierarchies. In: Benamou Jean-David, Ehrlacher Virginie, Matthes Daniel: Applications of Optimal Transportation in the Natural Sciences. Oberwolfach Rep. 14 (2017), 339416.Google Scholar
Burger, M., Capasso, V. & Morale, D. (2007) On an aggregation model with long and short range interactions. Nonlinear Anal. Real World Appl. 8, 939958.CrossRefGoogle Scholar
Carrillo, J. A., Chertock, A. & Huang, Y. (2015) A finite-volume method for nonlinear nonlocal equations with a gradient flow structure. Commun. Comput. Phys. 17, 233258.CrossRefGoogle Scholar
Carrillo, J. A., Delgadino, M. G. & Pavliotis, G. A. (2020) A proof of the mean-field limit for λ-convex potentials by Γ-convergence. J. Funct. Anal. 279, 108734.CrossRefGoogle Scholar
Carrillo, J. A., McCann, R. J. & Villani, C. (2003) Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates. Rev. Mat. Iberoamericana 19, 9711018.CrossRefGoogle Scholar
Fathi, M. & Simon, M. (2016) The gradient flow approach to hydrodynamic limits for the simple exclusion process. In: From Particle Systems to Partial Differential Equations III. Cham: Springer, pp. 167184.CrossRefGoogle Scholar
Henderson, D. (2010) Rowlinson’s concept of an effective hard sphere diameter. J. Chem. Eng. Data 55, 45074508.CrossRefGoogle ScholarPubMed
Jabin, P.-E. & Wang, Z. (2017) Mean field limit for stochastic particle systems. In: Active Particles, Vol. 1. Cham: Birkhäuser, pp. 379402.CrossRefGoogle Scholar
Jabin, P.-E. & Wang, Z. (2018) Quantitative estimates of propagation of chaos for stochastic systems with W -1∞ kernels. Invent. Math. 214, 523591.CrossRefGoogle Scholar
Jordan, R., Kinderlehrer, D. & Otto, F. (1998) The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29, 117.CrossRefGoogle Scholar
Kaiser, M., Jack, R. L. & Zimmer, J. (2019) A variational structure for interacting particle systems and their hydrodynamic scaling limits. Commun. Math. Sci. 17, 739780.CrossRefGoogle Scholar
Lasry, J.-M. & Lions, P.-L. (2007) Mean field games. Japanese J. Math. 2, 229260.CrossRefGoogle Scholar
Mielke, A. (2016) On evolutionary Γ-convergence for gradient systems. In: Muntean, A., Rademacher, J., Zagaris, A., ed., Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity. Cham: Springer, pp. 187249.CrossRefGoogle Scholar
Mielke, A., Montefusco, A. & Peletier, M. A. (2020) Exploring families of energy-dissipation landscapes via tilting–three types of EDP convergence. arXiv preprint arXiv:2001.01455.Google Scholar
Mielke, A. & Stephan, A. (2020) Coarse-graining via EDP-convergence for linear fast-slow reaction systems. Math. Mod. Meth. Appl. S., in press.CrossRefGoogle Scholar
Oelschlager, K. (1984) A martingale approach to the law of large numbers for weakly interacting stochastic processes. Ann. Probab. 12, 458479.CrossRefGoogle Scholar
Oelschläger, K. (1985) A law of large numbers for moderately interacting diffusion processes. Probab. Theory Rel. 69, 279322.Google Scholar
Otto, F. (2001) The geometry of dissipative evolution equations: the porous medium equation. Comm. Part. Diff. Equ. 26, 101174.CrossRefGoogle Scholar
Robinson, M. & Bruna, M. (2017) Particle-based and meshless methods with Aboria. SoftwareX, 6 IS, pp. 172178.CrossRefGoogle Scholar
Sandier, E. & Serfaty, S. (2004) Gamma-convergence of gradient flows with applications to Ginzburg-Landau. Commun. Pure Appl. Math. 57, 16271672.CrossRefGoogle Scholar
Santambrogio, F. (2015) Optimal Transport for Applied Mathematicians, Vol. 55. New York: Birkäuser, NY, pp. 5863.CrossRefGoogle Scholar
Serfaty, S. (2011) Gamma-convergence of gradient flows on Hilbert and metric spaces and applications. Discrete Contin. Dyn. Syst. 31, 14271451.CrossRefGoogle Scholar
Sznitman, A.-S. (1991) Topics in propagation of chaos. In: École d’Été de Probabilités de Saint-Flour XIX—1989, Vol. 1464, Lecture Notes in Mathematics. Berlin: Springer, pp. 165251.CrossRefGoogle Scholar
van Dyke, M. (1964) Perturbation Methods in Fluid Mechanics , In: Applied Mathematics and Mechanics, Vol. 8. New York, London: Academic Press.Google Scholar
Villani, C. (2003) Topics in optimal transportation. In: Graduate Studies in Mathematics, Vol. 58. Providence, RI: American Mathematical Society.CrossRefGoogle Scholar