Anomalous diffusion phenomena: A kinetic approach

Antoine Mellet[1]

  • [1] Department of Mathematics, University of Maryland College Park MD 20742 USA

Séminaire Laurent Schwartz — EDP et applications (2014-2015)

  • page 1-16
  • ISSN: 2266-0607

Abstract

top
In this talk, we review some aspects of the derivation of fractional diffusion equations from kinetic equations and in particular some applications to the description of anomalous energy transport in FPU chains. This is based on joint works with N. Ben Abdallah, L. Cesbron, S. Merino, S. Mischler, C. Mouhot and M. Puel

How to cite

top

Mellet, Antoine. "Anomalous diffusion phenomena: A kinetic approach." Séminaire Laurent Schwartz — EDP et applications (2014-2015): 1-16. <http://eudml.org/doc/275696>.

@article{Mellet2014-2015,
abstract = {In this talk, we review some aspects of the derivation of fractional diffusion equations from kinetic equations and in particular some applications to the description of anomalous energy transport in FPU chains. This is based on joint works with N. Ben Abdallah, L. Cesbron, S. Merino, S. Mischler, C. Mouhot and M. Puel},
affiliation = {Department of Mathematics, University of Maryland College Park MD 20742 USA},
author = {Mellet, Antoine},
journal = {Séminaire Laurent Schwartz — EDP et applications},
language = {eng},
pages = {1-16},
publisher = {Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Anomalous diffusion phenomena: A kinetic approach},
url = {http://eudml.org/doc/275696},
year = {2014-2015},
}

TY - JOUR
AU - Mellet, Antoine
TI - Anomalous diffusion phenomena: A kinetic approach
JO - Séminaire Laurent Schwartz — EDP et applications
PY - 2014-2015
PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
SP - 1
EP - 16
AB - In this talk, we review some aspects of the derivation of fractional diffusion equations from kinetic equations and in particular some applications to the description of anomalous energy transport in FPU chains. This is based on joint works with N. Ben Abdallah, L. Cesbron, S. Merino, S. Mischler, C. Mouhot and M. Puel
LA - eng
UR - http://eudml.org/doc/275696
ER -

References

top
  1. H. Babovsky, C. Bardos and T. Platkowski, Diffusion approximation for a Knudsen gas in a thin domain with accommodation on the boundary, Asymptotic Analysis, 3 (1991), pp. 265–289. Zbl0850.76599MR1094677
  2. C. Bardos, R. Santos and R. Sentis, Diffusion approximation and computation of the critical size, Trans. A. M. S., 284 (1984), pp. 617–649. Zbl0508.60067MR743736
  3. Giada Basile, Cédric Bernardin, and Stefano Olla. Momentum conserving model with anomalous thermal conductivity in low dimensional systems. Physical review letters, 96(20):204303, 2006. Zbl1178.82070
  4. Giada Basile, Cédric Bernardin, and Stefano Olla. Thermal conductivity for a momentum conservative model. Communications in Mathematical Physics, 287(1):67–98, 2009. Zbl1178.82070MR2480742
  5. Giada Basile, Stefano Olla, and Herbert Spohn. Energy transport in stochastically perturbed lattice dynamics. Archive for rational mechanics and analysis, 195(1):171–203, 2010. Zbl1187.82017MR2564472
  6. Naoufel Ben Abdallah, Antoine Mellet, and Marjolaine Puel. Anomalous diffusion limit for kinetic equations with degenerate collision frequency. Math. Models Methods Appl. Sci., 21(11):2249–2262, 2011. Zbl1331.76106MR2860675
  7. A. Bensoussan, J.-L. Lions, G. Papanicolaou, Boundary layers and homogenization of transport processes, Publ. RIMS Kyoto Univ., 15, 53–157 (1979). Zbl0408.60100MR533346
  8. C. Bernardin, P. Gonçalves, and M. Jara. 3/4 Fractional superdiffusion of energy in a system of harmonic oscillators perturbed by a conservative noise. ArXiv e-prints, 2014. Zbl1334.82052
  9. C. Börgers, C. Greengard, E. Thomann, The diffusion limit of free molecular flow in thin plane channels, SIAM J. Appl. Math., 52, # 4, (1992), 1057–1075. Zbl0761.76089MR1174046
  10. L. Cesbron, A. Mellet, K. Trivisa Anomalous transport of particles in Plasma physics, Applied Math. Letters, Appl. Math. Lett. 25 (2012), no. 12, 2344-2348. Zbl1248.76159MR2967841
  11. E.M. Conwell, High field electron transport in semiconductor, Solid Stat. Phys. 9 1967. 
  12. P. Debye. Vorträge über die kinetische theorie der wärme. Teubner, 1914. 
  13. P. Degond, T. Goudon, F. Poupaud, Diffusion limit for non homogeneous and non reversible processes, Indiana Univ. Math. J., 49, 1175-1198 (2000). Zbl0971.82035MR1803225
  14. I. Gentil, C. Imbert, The Lévy-Fokker-Planck equation: Φ - entropies and convergence to equilibrium, Asymptot. Anal. 59 (2008), 125-138. Zbl1170.35337MR2450356
  15. F. Golse, Anomalous diffusion limit for the Knudsen gas, Asymptotic Analysis, (1998). Zbl0974.76073MR1632712
  16. F. Golse, F. Poupaud, Limite fluide des équations de Boltzmann des semi-conducteurs pour une statistique de Fermi-Dirac, Asymptotic Analysis, 6, 135–160 (1992). Zbl0784.35084MR1193108
  17. Sabine Hittmeir and Sara Merino-Aceituno. Kinetic derivation of fractional stokes and stokes-fourier systems. http://arxiv.org/abs/1408.6400, 2014. Zbl1330.35283
  18. M. Jara, T. Komorowski, and S. Olla. Superdiffusion of energy in a chain of harmonic oscillators with noise. ArXiv e-prints, 2014. Zbl1329.82116
  19. Milton Jara, Tomasz Komorowski, and Stefano Olla. Limit theorems for additive functionals of a Markov chain. Ann. Appl. Probab., 19(6):2270–2300, 2009. Zbl1232.60018MR2588245
  20. S. Lepri, R. Livi, and A. Politi. Studies of thermal conductivity in fermipastaulam-like lattices. Chaos, 15, 2005. 
  21. Stefano Lepri, Roberto Livi, and Antonio Politi. Heat conduction in chains of nonlinear oscillators. Phys. Rev. Lett., 78, 1997. Zbl1228.82055
  22. Stefano Lepri, Roberto Livi, and Antonio Politi. On the anomalous thermal conductivity of one-dimensional lattices. Europhys. Lett., 43, 1998. 
  23. Stefano Lepri, Roberto Livi, and Antonio Politi. Thermal conduction in classical low-dimensional lattices. Physics Reports, 377(1):1–80, 2003. MR1978992
  24. Stefano Lepri, Roberto Livi, and Antonio Politi. Universality of anomalous one-dimensional heat conductivity. Physical Review, 68, 2003. Zbl1228.82055
  25. Stefano Lepri, Roberto Livi, and Antonio Politi. Studies of thermal conductivity in fermi–pasta–ulam-like lattices. Chaos: An Interdisciplinary Journal of Nonlinear Science, 15(1):015118, 2005. 
  26. Jani Lukkarinen and Herbert Spohn. Anomalous energy transport in the fpu- β chain. Communications on Pure and Applied Mathematics, 61(12):1753–1786, 2008. Zbl1214.82057MR2456185
  27. Antoine Mellet. Fractional diffusion limit for collisional kinetic equations: a moments method. Indiana Univ. Math. J., 59(4):1333–1360, 2010. Zbl1294.82032MR2815035
  28. A. Mellet, S. Merino Anomalous energy transport in FPU- β chain, Journal of Statistical Physics. Accepted. 
  29. Antoine Mellet, Stéphane Mischler, and Clément Mouhot. Fractional diffusion limit for collisional kinetic equations. Arch. Ration. Mech. Anal., 199(2):493–525, 2011. Zbl1294.82033MR2763032
  30. Stefano Olla. Energy diffusion and superdiffusion in oscillators lattice networks. In New Trends in Mathematical Physics, pages 539–547. Springer, 2009. Zbl1176.82026
  31. Rudolf Peierls. Zur kinetischen theorie der wärmeleitung in kristallen. Annalen der Physik, 395 (1929). Zbl55.0547.01
  32. D.L. Rode, Low-field electron transport in: Semiconductors and semi-metals, Vol 10 (Academic Press, New York 1975), pp. 1-52. 
  33. Herbert Spohn. Collisional invariants for the phonon boltzmann equation. Journal of statistical physics, 124(5):1131–1135, 2006. Zbl1104.82042MR2265847
  34. Herbert Spohn. The phonon boltzmann equation, properties and link to weakly anharmonic lattice dynamics. Journal of statistical physics, 124 (2006). Zbl1106.82033MR2264633
  35. Herbert Spohn. Nonlinear fluctuating hydrodynamics for anharmonic chains. Journal of Statistical Physics, 154(5):1191–1227, 2014. Zbl1291.82119MR3176405

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.