Skip to main content
SpringerLink
Log in
Book cover

Meeting on Particle Systems and PDE's

PSPDE 2016: From Particle Systems to Partial Differential Equations pp 1–12Cite as

Linear Boltzmann Equations: A Gradient Flow Formulation

  • Conference paper
  • First Online:

  • 453 Accesses

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 258))

Abstract

I present some results obtained together with D. Benedetto and L. Bertini on a gradient flow formulation of linear kinetic equations, in terms of an entropy dissipation inequality. The setting includes the current as a dynamical variable. As an application I discuss the diffusive limit of linear Boltzmann equations and show that the rescaled entropy inequality asymptotically provides the corresponding inequality for heat equation.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Adams, S., Dirr, N., Peletier, M.A., Zimmer, J.: From a large-deviations principle to the Wasserstein gradient flow: a new micro-macro passage. Commun. Math. Phys. 307(3), 791–815 (2011)

    Article  MathSciNet  Google Scholar 

  2. Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Spaces of Probability Measures. Lectures in Mathematics. ETH Zurich, Birkhuser (2005)

    MATH  Google Scholar 

  3. Bardos, C., Santos, R., Sentis, R.: Diffusion approximation and computation of the critical size. Trans. Am. Math. Soc. 284, 617–649 (1984)

    Article  MathSciNet  Google Scholar 

  4. Basile, G., Olla, S., Spohn, H.: Energy transport in stochastically perturbed lattice dynamics. Arch. Ration. Mech. Anal. 195(1), 171–203 (2010)

    Article  MathSciNet  Google Scholar 

  5. Basile, G., Benedetto, D., Bertini, L.: A gradient flow approach to linear Boltzmann equation. Preprint on arXiv:1707.09204 (2017)

  6. Benamou, J.D., Brenier, Y.: A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 84, 375–393 (2000)

    Article  MathSciNet  Google Scholar 

  7. Bensoussan, A., Lions, J.L., Papanicolaou, G.: Boundary Layers and Homogenization of Transport Processes, vol. 15, pp. 53–157. RIMS Kyoto University (1979)

    Google Scholar 

  8. Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Large deviations of the empirical current in interacting particle systems. Theory Probab. Appl. 51(1), 2–27 (2007)

    Article  MathSciNet  Google Scholar 

  9. Bertini, L., Faggionato, A., Gabrielli, D.: Flows, currents, and cycles for Markov chains: large deviation asymptotics. Stoch. Process. Appl. 125(7), 2786–2819 (2015)

    Article  MathSciNet  Google Scholar 

  10. Bodineau, T., Gallagher, I., Saint-Raymond, L.: The Brownian motion as the limit of a deterministic system of hard-spheres. Invent. Math. 203(2), 493–553 (2016)

    Article  MathSciNet  Google Scholar 

  11. Dawson, D.A., Gärtner, J.: Large deviations from the McKean-Vlasov limit for weakly interacting diffusions. Stochastics 20(4), 247–308 (1987)

    Article  MathSciNet  Google Scholar 

  12. De Giorgi, E., Marino, A., Tosques, M.: Problemi di evoluzione in spazi metrici e curve di massima pendenza. Atti Acc. Lincei Rend. Cl. Sci. Fis. Mat. Natur. Serie 8 68(3), 180–187 (1980)

    Google Scholar 

  13. Donsker, M.D., Varadhan, S.R.S.: Asymptotic evaluation of certain Markov process expectations for large time. I. II. Comm. Pure Appl. Math. 28, 1–47, ibid. 28, 279–301 (1975)

    Google Scholar 

  14. Erbar, M.: Gradient flows of the entropy for jump processes. Ann. Inst. H. Poincar Probab. Statist. 50(3), 920–945 (2014)

    Article  MathSciNet  Google Scholar 

  15. Erbar, M.: A gradient flow approach to the Boltzmann equation. Preprint on arXiv:1603.00540 (2017)

  16. Esposito, R., Pulvirenti, M.: From particles to fluids. In: Hand-Book of Mathematical Fluid Dynamics III, pp. 1–82. North-Holland, Amsterdam (2004)

    Google Scholar 

  17. Gallavotti, G.: Grad Boltzmann limit and Lorentzs Gas. In: Statistical Mechanics. A Short Treatise. Appendix 1.A2. Springer, Berlin (1999)

    Chapter  Google Scholar 

  18. Gigli, N.: On the Heat flow on metric measure spaces: existence, uniqueness and stability. Calc. Var. Part. Diff. Eq. 39, 101–120 (2010)

    Article  MathSciNet  Google Scholar 

  19. Jordan, R., Kinderlehrer, D., Otto, F.: The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29, 1–17 (1998)

    Article  MathSciNet  Google Scholar 

  20. Kipnis, C., Olla, S.: Large deviations from the hydrodynamical limit for a system of independent Brownian particles. Stoch. Stoch. Rep. 33(1–2), 17–25 (1990)

    Article  MathSciNet  Google Scholar 

  21. Larsen, E., Keller, J.B.: Asymptotic solution of neutron transport problems for small mean free paths. J. Math. Phys. 15, 75–81 (1974)

    Article  MathSciNet  Google Scholar 

  22. Lorentz, H.A.: The motion of electrons in metallic bodies. Proc. Acad. Amst. 7, 438–453 (1905)

    Google Scholar 

  23. Maas, J.: Gradient flows of the entropy for finite Markov chains. J. Funct. Anal. 261, 2250–2292 (2011)

    Article  MathSciNet  Google Scholar 

  24. Mielke, A.: Geodesic convexity of the relative entropy in reversible Markov chains. Calc. Var. Partial Differ. Equ. 48, 1–31 (2013)

    Article  MathSciNet  Google Scholar 

  25. Mielke, A., Peletier, M.A., Renger, D.R.M.: On the relation between gradient flows and the large-deviation principle, with applications to Markov chains and diffusion. Potential Anal. 41(4), 1293–1327 (2014)

    Article  MathSciNet  Google Scholar 

  26. Otto, F.: The geometry of dissipative evolution equations: the porous medium equation

    Google Scholar 

  27. Sandier, E., Serfaty, S.: Gamma-convergence of gradient flows with applications to Ginzburg-Landau. Commun. Pure Appl. Math. 57(12), 1627–1672 (2004)

    Article  MathSciNet  Google Scholar 

  28. Spohn, H.: Kinetic equations from Hamiltonian dynamics: Markovian limits. Rev. Mod. Phys. 52(3), 569–615 (1980)

    Article  MathSciNet  Google Scholar 

  29. van Beijeren, H., Lanford III, O.E., Lebowitz, J.L., Spohn, H.: Equilibrium time correlation functions in the low-density limit. J. Stat. Phys. 22(2), 237–257 (1980)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma “La Sapienza”, P.le Aldo Moro 2, 00185, Roma, Italy

    Giada Basile

Authors
  1. Giada Basile
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Giada Basile .

Editor information

Editors and Affiliations

  1. Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, University of Lisbon, Lisbon, Portugal

    Patrícia Gonçalves

  2. Center of Mathematics, University of Minho, Braga, Portugal

    Ana Jacinta Soares

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Basile, G. (2018). Linear Boltzmann Equations: A Gradient Flow Formulation. In: Gonçalves, P., Soares, A. (eds) From Particle Systems to Partial Differential Equations . PSPDE 2016. Springer Proceedings in Mathematics & Statistics, vol 258. Springer, Cham. https://doi.org/10.1007/978-3-319-99689-9_4

Download citation

Publish with us

Policies and ethics

Search

Navigation