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Derivation of the Fick’s Law for the Lorentz Model in a Low Density Regime

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We consider the Lorentz model in a slab with two mass reservoirs at the boundaries. We show that, in a low density regime, there exists a unique stationary solution for the microscopic dynamics, which converges to the stationary solution of the heat equation, namely to the linear profile of the density. In the same regime, the macroscopic current in the stationary state is given by the Fick’s law, with the diffusion coefficient determined by the Green–Kubo formula.

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  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(6), 1087–1111 (2014)

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Book  MATH  Google Scholar 

  5. 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 

  6. 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  ADS  MATH  MathSciNet  Google Scholar 

  7. Esposito, R., Pulvirenti, M.: From Particles to Fluids. Hand-Book of Mathematical Fuid Dynamics, vol. III, pp. 1–82. North-Holland, Amsterdam (2004)

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

    Article  MathSciNet  Google Scholar 

  9. Gallavotti, G.: Grad–Boltzmann limit and Lorentz’s Gas. In: Statistical Mechanics. A short treatise. Appendix 1.A2. Springer, Berlin (1999)

  10. Galves A., Kipnis C., Marchioro C., Presutti E.: Nonequilibrium measures which exhibit a temperature gradient: study of a model. Commun. Math. Phys. 81, 127–147 (1981)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  11. Kipnis C., Marchioro C., Presutti E.: Heat flow in an exactly solvable model. J. Stat. Phys. 27, 65–74 (1982)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

  15. Spohn H.: The Lorentz flight process converges to a random flight process. Commun. Math. Phys. 60, 277–290 (1978)

    Article  ADS  MATH  MathSciNet  Google Scholar 

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Correspondence to A. Nota.

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Communicated by H. Spohn

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Basile, G., Nota, A., Pezzotti, F. et al. Derivation of the Fick’s Law for the Lorentz Model in a Low Density Regime. Commun. Math. Phys. 336, 1607–1636 (2015).

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