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Toward the Fourier law for a weakly interacting anharmonic crystal
Authors:
Carlangelo Liverani and Stefano Olla
Journal:
J. Amer. Math. Soc. 25 (2012), 555-583
MSC (2010):
Primary 82C70, 60F17, 80A20
Posted:
December 1, 2011
MathSciNet review:
2869027
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Additional Information
Abstract: For a system of weakly interacting anharmonic oscillators, perturbed by an energy-preserving stochastic dynamics, we prove an autonomous (stochastic) evolution for the energies at large time scale (with respect to the coupling parameter). It turns out that this macroscopic evolution is given by the so-called conservative (nongradient) Ginzburg-Landau system of stochastic differential equations. The proof exploits hypocoercivity and hypoellipticity properties of the uncoupled dynamics.
- 1.
Cédric
Bernardin and Stefano
Olla, Fourier’s law for a microscopic model of heat
conduction, J. Stat. Phys. 121 (2005), no. 3-4,
271–289. MR 2185330
(2006j:82058), http://dx.doi.org/10.1007/s10955-005-7578-9
- 2.
F.
Bonetto, J.
L. Lebowitz, and L.
Rey-Bellet, Fourier’s law: a challenge to theorists,
Mathematical physics 2000, Imp. Coll. Press, London, 2000,
pp. 128–150. MR 1773043
(2001g:82067), http://dx.doi.org/10.1142/9781848160224_0008
- 3.
Sandra
Cerrai and Philippe
Clément, Well-posedness of the martingale problem for some
degenerate diffusion processes occurring in dynamics of populations,
Bull. Sci. Math. 128 (2004), no. 5, 355–389. MR 2066345
(2005h:60249), http://dx.doi.org/10.1016/j.bulsci.2004.03.004
- 4.
Dmitry Dolgopyat, Carlangelo Liverani, Energy transfer in a fast-slow Hamiltonian system, Communications in Mathematical Physics, 308 (2011), N. 1, 201-225.
- 5.
M.
I. Freĭdlin, Fluctuations in dynamical systems with
averaging, Dokl. Akad. Nauk SSSR 226 (1976),
no. 2, 273–276 (Russian). MR 0410796
(53 #14539)
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M.
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D. Wentzell, Random perturbations of dynamical systems, 2nd
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Translated from the 1979 Russian original by Joseph Szücs. MR 1652127
(99h:60128)
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Lars
Hörmander, Hypoelliptic second order differential
equations, Acta Math. 119 (1967), 147–171. MR 0222474
(36 #5526)
- 8.
Yuri
Kifer, Some recent advances in averaging, Modern dynamical
systems and applications, Cambridge Univ. Press, Cambridge, 2004,
pp. 385–403. MR 2093312
(2005h:37067)
- 9.
T. Komorowski, C. Landim, S. Olla, Fluctuations in Markov Processes, Springer, to appear (2012).
- 10.
S. Olla, M. Sasada, Macroscopic energy diffusion for a chain of anharmonic oscillators, arXiv 2011 vol. math-ph, http://arxiv.org/abs/1109.5297v1.
- 11.
S.
Olla, S.
R. S. Varadhan, and H.-T.
Yau, Hydrodynamical limit for a Hamiltonian system with weak
noise, Comm. Math. Phys. 155 (1993), no. 3,
523–560. MR 1231642
(94k:60158)
- 12.
Z. Rieder, J. L. Lebowitz, and E. Lieb, Properties of a Harmonic Crystal in a Stationary Nonequilibrium State, J. Math. Phys. 8 (1967), 1073.
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Sunder
Sethuraman, S.
R. S. Varadhan, and Horng-Tzer
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exclusion processes, Comm. Pure Appl. Math. 53
(2000), no. 8, 972–1006. MR 1755948
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S.
R. S. Varadhan, Nonlinear diffusion limit for a system with nearest
neighbor interactions. II, diffusions on fractals (Sanda/Kyoto, 1990)
Pitman Res. Notes Math. Ser., vol. 283, Longman Sci. Tech., Harlow,
1993, pp. 75–128. MR 1354152
(97a:60144)
- 15.
Cédric
Villani, Hypocoercivity, Mem. Amer. Math. Soc. 202 (2009), no. 950, iv+141. MR 2562709
(2011e:35381), http://dx.doi.org/10.1090/S0065-9266-09-00567-5
- 1.
- C. Bernardin, S. Olla, Fourier's law for a microscopic model of heat conduction, Journal of Statistical Physics, 118 (2005), nos. 3/4, 271-289. MR 2185330 (2006j:82058)
- 2.
- F. Bonetto, J.L. Lebowitz, L. Rey-Bellet, Fourier's law: A challenge to theorists, Mathematical Physics 2000, Imperial College Press, London, 2000, pp.128-150. MR 1773043 (2001g:82067)
- 3.
- S. Cerrai, Ph. Clément, Well-posedness of the martingale problem for some degenerate diffusion processes occurring in dynamics of populations, Bull. Sci. Math. 128 (2004) 355-389. MR 2066345 (2005h:60249)
- 4.
- Dmitry Dolgopyat, Carlangelo Liverani, Energy transfer in a fast-slow Hamiltonian system, Communications in Mathematical Physics, 308 (2011), N. 1, 201-225.
- 5.
- M. I. Freidlin, Fluctuations in dynamical systems with averaging (Russian)Dokl. Akad. Nauk SSSR 226 (1976), no. 2, 273-276. MR 0410796 (53:14539)
- 6.
- M.I. Freidlin, A.D. Wentzell, Random Perturbations of Dynamical Systems, 2nd edn., Springer, Heidelberg, 1998. MR 1652127 (99h:60128)
- 7.
- L. Hormander, Hypoelliptic second order differential equations. Acta Math. 119 (1967), 147-171. MR 0222474 (36:5526)
- 8.
- Y. Kifer, Some recent advances in averaging. In: Modern Dynamical Systems and Applications, pp. 385-403. Cambridge University Press, Cambridge, 2004. MR 2093312 (2005h:37067)
- 9.
- T. Komorowski, C. Landim, S. Olla, Fluctuations in Markov Processes, Springer, to appear (2012).
- 10.
- S. Olla, M. Sasada, Macroscopic energy diffusion for a chain of anharmonic oscillators, arXiv 2011 vol. math-ph, http://arxiv.org/abs/1109.5297v1.
- 11.
- S. Olla, S. Varadhan, H. Yau, Hydrodynamical limit for a Hamiltonian system with weak noise, Commun. Math. Phys. 155 (1993), 523-560. MR 1231642 (94k:60158)
- 12.
- Z. Rieder, J. L. Lebowitz, and E. Lieb, Properties of a Harmonic Crystal in a Stationary Nonequilibrium State, J. Math. Phys. 8 (1967), 1073.
- 13.
- S. Sethuraman, S.R.S. Varadhan, H.T. Yau, Diffusive limit of a tagged particle in asymmetric simple exclusion processes, Comm. Pure Appl. Math. 53 (2000), no. 8, 972-1006. MR 1755948 (2001k:60146)
- 14.
- S.R.S. Varadhan, Nonlinear diffusion limit for a system with nearest neighbor interactions. II, Asymptotic problems in probability theory: stochastic models and diffusions on fractals (Sanda/Kyoto, 1990), 75-128, Pitman Res. Notes Math. Ser., 283, Longman Sci. Tech., Harlow, 1993. MR 1354152 (97a:60144)
- 15.
- Cedric Villani, Hypocoercivity. Mem. Amer. Math. Soc. 202 (2009), no. 950, 141 pp. MR 2562709 (2011e:35381)
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Additional Information
Carlangelo Liverani
Affiliation:
Dipartimento di Matematica, II Università di Roma (Tor Vergata), Via della Ricerca Scientifica, 00133 Roma, Italy
Email:
liverani@mat.uniroma2.it
Stefano Olla
Affiliation:
CEREMADE, UMR CNRS 7534, Université Paris-Dauphine, 75775 Paris-Cedex 16, France and INRIA - Université Paris Est, CERMICS, Projet MICMAC, Ecole des Ponts ParisTech, 6 & 8 Av. Pascal, 77455 Marne-la-Vallée Cedex 2, France
Email:
olla@ceremade.dauphine.fr
DOI:
http://dx.doi.org/10.1090/S0894-0347-2011-00724-8
PII:
S 0894-0347(2011)00724-8
Keywords:
Weak coupling,
scaling limits,
hypoellipticity,
hypocoercivity,
Ginzburg-Landau dynamics,
heat equation
Received by editor(s):
January 18, 2011
Received by editor(s) in revised form:
November 6, 2011
Posted:
December 1, 2011
Additional Notes:
This paper has been partially supported by the European Advanced Grant Macroscopic Laws and Dynamical Systems (MALADY) (ERC AdG 246953), by Agence Nationale de la Recherche, under grant ANR-07-BLAN-2-184264 (LHMSHE), and by MIUR under the grant PRIN 2007B3RBEY
Article copyright:
© Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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