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June 2018, 11(3): 615-645. doi: 10.3934/krm.2018026

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  March 2018

Fund Project: 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

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.

Citation: Tomasz Komorowski, Stefano Olla, Marielle Simon. Macroscopic evolution of mechanical and thermal energy in a harmonic chain with random flip of velocities. Kinetic & Related Models, 2018, 11 (3) : 615-645. doi: 10.3934/krm.2018026
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. JaraT. 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. LukkarinenM. 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.

show all references

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. JaraT. 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. LukkarinenM. 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.

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