Advanced Search
Article Contents
Article Contents

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

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 Full Text(HTML) Related Papers Cited by
  • 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.

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


    \begin{equation} \\ \end{equation}
  • 加载中
  • [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. GantmakherThe 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. KelleyGeneral 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.
  • 加载中

Article Metrics

HTML views(273) PDF downloads(166) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint