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From Particle Systems to Partial Differential Equations II pp 273–292Cite as

Diffusive Limit for the Random Lorentz Gas

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 129))

Abstract

We review some recent results concerning the derivation of the diffusion equation and the validation of Fick’s law for the microscopic model given by the random Lorentz Gas. These results are achieved by using a linear kinetic equation as an intermediate level of description between our original mechanical system and the diffusion equation. The diffusion coefficient is given by the Green-Kubo formula associated to the generator of the stochastic process dictated by the linear Landau equation and the linear Boltzmann equation respectively, according to the weak-coupling regime and low density regime we are considering.

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References

  1. Bodineau, T., Gallagher, I., Saint-Raymond, L.: The Brownian motion as the limit of a deterministic system of hard-spheres. arxiv: 1305.3397

  2. Basile, G., Nota, A., Pulvirenti, M.: A diffusion limit for a test particle in a random distribution of scatterers. J. Stat. Phys. 155, 1087–1111 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  3. Basile, G., Nota, A., Pezzotti, F., Pulvirenti, M.: Derivation of the Fick’s law for the Lorentz model in a low density regime. Commun. Math. Phys. doi:10.1007/s00220-015-2306-z

  4. Boldrighini, C., Bunimovich, L.A., Sinai, Y.G.: On the Boltzmann equation for the Lorentz gas. J. Stat. Phys. 32, 477–501 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bunimovich, L.A., Sinai, Y.G.: Statistical properties of the Lorentz gas with periodic configuration of scatterers. Commun. Math. Phys. 78, 478–497 (1981)

    Google Scholar 

  6. Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases. Springer, Berlin (1994)

    Book  MATH  Google Scholar 

  7. Caglioti, E., Golse, F.: The Boltzmann-Grad limit of the periodic Lorentz gas in two space dimensions. C.R. Math. Acad. Sci. Paris 346, 477–482 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Caglioti, E., Golse, F.: On the Boltzmann-grad limit for the two dimensional periodic lorentz gas. J. Stat. Phys. 141, 264–317 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  9. Dürr, D., Goldstein, S., Lebowitz, J.: Asymptotic motion of a classical particle in a random potential in two dimensions: Landau model. Commun. Math. Phys. 113, 209–230 (1987)

    Article  MATH  Google Scholar 

  10. Desvillettes, L., Pulvirenti, M.: The linear Boltzmann equation for long-range forces: a derivation from particle systems. Models Methods Appl. Sci. 9, 1123–1145 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  11. Desvillettes, L., Ricci, V.: A rigorous derivation of a linear kinetic equation of Fokker-Planck type in the limit of grazing collisions. J. Stat. Phys. 104, 1173–1189 (2001)

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  13. Erdos, L., Salmhofer, M., Yau, H.-T.: Quantum diffusion of the random Schrodinger evolution in the scaling limit. Acta Math. 200(2), 211–277 (2008)

    Article  MathSciNet  Google Scholar 

  14. Gallavotti, G.: Grad-Boltzmann Limit and Lorentz’s Gas. Statistical Mechanics. A short treatise. Appendix 1.A2. Springer, Berlin (1999)

    Google Scholar 

  15. Gallagher, I., Saint-Raymond, L., Texier, B.: From Newton to Boltzmann: hard spheres and short-range potentials. In: Zurich Lectures in Advanced Mathematics (2014)

    Google Scholar 

  16. Kesten, H., Papanicolaou, G.: A limit theorem for stochastic acceleration. Commun. Math. Phys. 78, 19–63 (1981)

    Article  MathSciNet  Google Scholar 

  17. Kirkpatrick, K.: Rigorous derivation of the Landau Equation in the weak coupling limit. Commun. Pure Appl. Anal. 8, 1895–1916 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  18. Komorowski, T., Ryzhik, L.: Diffusion in a weakly random Hamiltonian flow. Commun. Math. Phys. 263, 273–323 (2006)

    Article  MathSciNet  Google Scholar 

  19. Landau, L.D., Lifshitz, E.M.: Mechanics, Course of Theoretical Physics, vol. 1. Pergamon press, Oxford (1960)

    Google Scholar 

  20. Lebowitz, J.L., Spohn, H.: Transport properties of the Lorentz Gas: Fourier’s law. J. Stat. Phys. 19, 633–654 (1978)

    Article  MathSciNet  Google Scholar 

  21. Lebowitz, J.L., Spohn, H.: Microscopic basis for Fick’s law for self-diffusion. J. Stat. Phys. 28, 539–556 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  22. Lebowitz, J.L., Spohn, H.: Steady state self-diffusion at low density. J. Stat. Phys. 29, 539–556 (1982)

    Article  Google Scholar 

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

    Google Scholar 

  24. Marklof, J., Strömbergsson, A.: The Boltzmann-Grad limit of the periodic Lorentz gas. Ann. Math. 174, 225–298 (2011)

    Article  MATH  Google Scholar 

  25. Pulvirenti, M., Saffirio, C., Simonella, S.: On the validity of the Boltzmann equation for short range potentials. Rev. Math. Phys. 26, 2 (2014)

    Article  MathSciNet  Google Scholar 

  26. Spohn, H.: The Lorentz flight process converges to a random flight process. Commun. Math. Phys. 60, 277–D0290 (1978)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

I am indebted to G. Basile, F. Pezzotti and M. Pulvirenti for their collaboration and for the illuminating discussions.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma La Sapienza, Piazzale Aldo Moro 5, 00185, Roma, Italy

    Alessia Nota

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  1. Alessia Nota
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Correspondence to Alessia Nota .

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Editors and Affiliations

  1. Centre of Mathematics, University of Minho, Braga, Portugal

    Patrícia Gonçalves

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

    Ana Jacinta Soares

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Nota, A. (2015). Diffusive Limit for the Random Lorentz Gas. In: Gonçalves, P., Soares, A. (eds) From Particle Systems to Partial Differential Equations II. Springer Proceedings in Mathematics & Statistics, vol 129. Springer, Cham. https://doi.org/10.1007/978-3-319-16637-7_10

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