Macroscopic evolution of mechanical and thermal energy in a harmonic chain with random flip of velocities

Tomasz Komorowski; Stefano Olla; Marielle Simon

doi:10.3934/krm.2018026

Article Contents

2018, Volume 11, Issue 3: 615-645. Doi: 10.3934/krm.2018026

This issue Previous Article Propagation of stretched exponential moments for the Kac equation and Boltzmann equation with Maxwell molecules Next Article The rigorous derivation of the Linear Landau equation from a particle system in a weak-coupling limit

Macroscopic evolution of mechanical and thermal energy in a harmonic chain with random flip of velocities

1.
Institute of Mathematics, Polish Academy Of Sciences, Warsaw, Poland
2.
Université Paris-Dauphine, PSL Research University, CNRS UMR 7534, CEREMADE, 75016 Paris, France
3.
Inria, Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé, 59000 Lille, France

Received: March 2017

Revised: September 2017

Published online: March 5, 2018

T.K. acknowledges the support of the Polish National Science Center grant DEC-2012/07/B/ST1/03320. The work of M.S. was supported by the ANR-14-CE25-0011 project (EDNHS) of the French National Research Agency (ANR), and by the Labex CEMPI (ANR-11-LABX-0007-01). S.O. has been partially supported by the ANR-15-CE40-0020-01 grant LSD. 6.

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

We consider an unpinned chain of harmonic oscillators with periodic boundary conditions, whose dynamics is perturbed by a random flip of the sign of the velocities. The dynamics conserves the total volume (or elongation) and the total energy of the system. We prove that in a diffusive space-time scaling limit the profiles corresponding to the two conserved quantities converge to the solution of a diffusive system of differential equations. While the elongation follows a simple autonomous linear diffusive equation, the evolution of the energy depends on the gradient of the square of the elongation.

Keywords:

Mathematics Subject Classification: 60K35, 74A25, 82C22.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	G. Basile, C. Bernardin, M. Jara, T. Komorowski and S. Olla, Thermal conductivity in harmonic lattices with random collisions, Thermal Transport in Low Dimensions, Lecture Notes in Physics, Springer, 921 (2016), 215-237.
[2]	C. Bernardin and S. Olla, Fourier law and fluctuations for a microscopic model of heat conduction, J. Stat. Phys., 121 (2005), 271-289. doi: 10.1007/s10955-005-7578-9.
[3]	C. Bernardin and S. Olla, Transport properties of a chain of anharmonic oscillators with random flip of velocities, J. Stat. Phys., 145 (2011), 1224-1255. doi: 10.1007/s10955-011-0385-6.
[4]	F. R. Gantmakher, The Theory of Matrices, Hirsch Chelsea Publishing Co., New York, 1959.
[5]	M. Jara, T. Komorowski and S. Olla, Superdiffusion of energy in a system of harmonic oscillators with noise, Commun. Math. Phys., 339 (2015), 407-453. doi: 10.1007/s00220-015-2417-6.
[6]	J. L. Kelley, General Topology, Springer-Verlag, New York-Berlin, 1975.
[7]	T. Komorowski and S. Olla, Ballistic and superdiffusive scales in macroscopic evolution of a chain of oscillators, Nonlinearity, 29 (2016), 962-999. doi: 10.1088/0951-7715/29/3/962.
[8]	T. Komorowski and S. Olla, Diffusive propagation of energy in a non-acoustic chain, Arch. Ration. Mech. Anal., 223 (2017), 95-139. doi: 10.1007/s00205-016-1032-9.
[9]	J. Lukkarinen, Thermalization in harmonic particle chains with velocity flips, J. Stat. Phys., 155 (2014), 1143-1177. doi: 10.1007/s10955-014-0930-1.
[10]	J. Lukkarinen, M. Marcozzi and A. Nota, Harmonic chain with velocity flips: Thermalization and kinetic theory, J. Stat. Phys., 165 (2016), 809-844. doi: 10.1007/s10955-016-1647-0.
[11]	J. Lukkarinen and H. Spohn, Kinetic limit for wave propagation in a random medium, Arch. Ration. Mech. Anal., 183 (2006), 93-162.
[12]	M. Simon, Hydrodynamic limit for the velocity-flip model, Stoch. Proc. and Appl., 123 (2013), 3623-3662. doi: 10.1016/j.spa.2013.05.005.
[13]	H. T. Yau, Relative entropy and hydrodynamics of Ginzburg-Landau models, Lett. Math. Phys., 22 (1991), 63-80. doi: 10.1007/BF00400379.