Abstract
We study the connection between a system of many independent Brownian particles on one hand and the deterministic diffusion equation on the other. For a fixed time step h > 0, a large-deviations rate functional J h characterizes the behaviour of the particle system at t = h in terms of the initial distribution at t = 0. For the diffusion equation, a single step in the time-discretized entropy-Wasserstein gradient flow is characterized by the minimization of a functional K h . We establish a new connection between these systems by proving that J h and K h are equal up to second order in h as h → 0. This result gives a microscopic explanation of the origin of the entropy-Wasserstein gradient flow formulation of the diffusion equation. Simultaneously, the limit passage presented here gives a physically natural description of the underlying particle system by describing it as an entropic gradient flow.
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References
Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in mathematics ETH Zürich. Basel: Birkhäuser, 2005
Beurling A.: An automorphism of product measures. Ann. Math. 72(1), 189–200 (1960)
Braides, A.: Gamma-Convergence for Beginners. Oxford: Oxford University Press, 2002
Berkowitz B., Scher H.: Theory of anomalous chemical transport in random fracture networks. Phys. Rev. E 57(5), 5858–5869 (1998)
Berkowitz B., Scher H., Silliman S.E.: Anomalous transport in laboratory-scale, heterogeneous porous media. Water Resour. Res. 36(1), 149–158 (2000)
Carrillo J.A., McCann R.J., Villani C.: Contractions in the 2-Wasserstein length space and thermalization of granular media. Arch. Rat. Mech. Anal. 179, 217–263 (2006)
Csiszár I.: Sanov property, generalized I-projection and a conditional limit theorem. Ann. Probab. 12(3), 768–793 (1984)
den Hollander, F.: Large Deviations. Providence, RI: Amer. Math. Soc., 2000
Dal Maso, G.: An Introduction to Γ-Convergence, Volume 8 of Progress in Nonlinear Differential Equations and Their Applications. Boston: Birkhäuser, First edition, 1993
De Masi, A., Presutti, E.: Mathematical methods for hydrodynamic limits. Lecture Notes in Mathematics, Berlin-Heidelberg-New York; Springer, 1992
Deuschel, J.D., Stroock, D.W.: Large deviations. London, New York: Academic Press, 1989
Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. Boca Raton, FL: CRC Press, 1992
Einstein A.: Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Annalen der Physik 17(4), 548–560 (1905)
Georgii, H.-O.: Gibbs Measures and Phase Transitions. Berlin: de Gruyter, 1988
Glasner K.: A diffuse-interface approach to Hele-Shaw flow. Nonlinearity 16(1), 49–66 (2003)
Giacomelli L., Otto F.: Variational formulation for the lubrication approximation of the Hele-Shaw flow. Calc. Var. Part. Diff. Eqs. 13(3), 377–403 (2001)
Gianazza U., Savaré G., Toscani G.: The Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation. Arch. Ration. Mech. Anal. 194(1), 133–220 (2009)
Huang C., Jordan R.: Variational formulations for Vlasov-Poisson-Fokker-Planck systems. Math. Meth. Appl. Sci. 23(9), 803–843 (2000)
Huang C.: A variational principle for the Kramers equation with unbounded external forces. J. Math. Anal. Appl. 250(1), 333–367 (2000)
Jordan R., Kinderlehrer D., Otto F.: Free energy and the Fokker-Planck equation. Physica D: Nonlinear Phenomena 107(2-4), 265–271 (1997)
Jordan R., Kinderlehrer D., Otto F.: The variational formulation of the Fokker-Planck Equation. SIAM J. Math. Anal. 29(1), 1–17 (1998)
Kipnis, C., Landim, C.: Scaling limits of interacting particle systems. Berlin-Heidelberg-New York: Springer Verlag, 1999
Kipnis C., Olla S.: Large deviations from the hydrodynamical limit for a system of independent Brownian particles. Stoch. and Stoch. Reps. 33(1-2), 17–25 (1990)
Léonard, C.: A large deviation approach to optimal transport. http://arXiv.org/abs/0710.1461v1 [math.PR], 2007
McCann R.J.: A convexity principle for interacting gases. Adv. Math. 128, 153–179 (1997)
Matthes, D., McCann, R.J. Savaré, G.: A family of nonlinear fourth order equations of gradient flow type. http://arXiv.org/abs/0901.0540vL [math.AP], 2009
Olver, F.W.J.: Asymptotics and Special Functions. London-New York: Academic Press, 1997
Otto F.: Lubrication approximation with prescribed nonzero contact angle. Comm. Part. Diff. Eqs. 23(11), 63–103 (1998)
Otto F.: The geometry of dissipative evolution equations: The porous medium equation. Comm. Part. Diff. Eqs. 26, 101–174 (2001)
Portegies J.W., Peletier M.A.: Well-posedness of a parabolic moving-boundary problem in the setting of Wasserstein gradient flows. Interfaces and Free Boundaries 12, 121–150 (2010)
Schrödinger, E.: Über die Umkehrung der Naturgesetze. Sitzungsber. Preuss. Akad. Wiss. Phys. Math. Kl, 144–153 (1931)
Villani, C.: Topics in Optimal Transportation. Providence, RI: Amer. Math. Soc., 2003
Villani, C.: Optimal transport: Old and new. Springer Verlag, 2009
Weeks E.R., Weitz D.A.: Subdiffusion and the cage effect studied near the colloidal glass transition. Chem. Phys. 284(1-2), 361–367 (2002)
Yeh, J.: Real Analysis: Theory of Measure and Integration. River Edge, NJ: World Scientific, 2006
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Communicated by H.-T. Yau
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Adams, S., Dirr, N., Peletier, M.A. et al. From a Large-Deviations Principle to the Wasserstein Gradient Flow: A New Micro-Macro Passage. Commun. Math. Phys. 307, 791–815 (2011). https://doi.org/10.1007/s00220-011-1328-4
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DOI: https://doi.org/10.1007/s00220-011-1328-4