Abstract
We consider a d-dimensional harmonic crystal, d ⩾ 1, and study the Cauchy problem with random initial data. The distribution μt of the solution at time t ∈ ℝ is studied. We prove the convergence of correlation functions of the measures μt to a limit for large times. The explicit formulas for the limiting correlation functions and for the energy current density (in the mean) are obtained in terms of the initial covariance. Furthermore, we prove the weak convergence of μt to a limit measure as t → ∞. We apply these results to the case when initially some infinite “parts” of the crystal have Gibbs distributions with different temperatures. In particular, we find stationary states in which there is a constant nonzero energy current flowing through the crystal. We also study the initial boundary value problem for the harmonic crystal in the half-space with zero boundary condition and obtain similar results.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C. Boldrighini, A. Pellegrinotti, and L. Triolo, “Convergence to Stationary States for Infinite Harmonic Systems,” J. Stat. Phys. 30(1), 123–155 (1983).
F. Bonetto, J. L. Lebowitz, and L. Rey-Bellet, “Fourier Law: a Challenge to Theorists,” in Mathematical Physics 2000 (A. Fokas et al. (Eds), Imperial College Press, London, 2000), pp. 128–150.
F. Bonetto, J. L. Lebowitz, and J. Lukkarinen, “Fourier’s Law for a Harmonic Crystal with Self-Consistent Stochastic Reservoirs,” J. Statist. Phys. 116(12–4), 783–813 (2004).
A. Casher and J. L. Lebowitz, “Heat Flow in Regular and Disordered Harmonic Chains,” J. Math. Phys. 12(8), 1701–1711 (1971).
I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinai, Ergodic Theory (Springer, New York, 1981).
R. L. Dobrushin and Yu. M. Sukhov, “On the Problem of the Mathematical Foundation of the Gibbs Postulate in Classical Statistical Mechanics,” in Mathematical Problems in Theoretical Physics, Lecture Notes in Physics (Springer-Verlag, Berlin, 1978), Vol. 80, pp. 325–340.
T. V. Dudnikova, A. I. Komech, and H. Spohn, “On the Convergence to Statistical Equilibrium for Harmonic Crystals,” J. Math. Phys. 44(6), 2596–2620 (2003).
T. V. Dudnikova, A. I. Komech, and N. J. Mauser, “On Two-Temperature Problem for Harmonic Crystals,” J. Statist. Phys. 114(32–4), 1035–1083 (2004).
T. V. Dudnikova and A. I. Komech, “On a Two-Temperature Problem for the Klein Gordon Equation,” Teor. Veroyattist. i Primenen. 50, 675–710 (2005) (English transl. in Theory Probab. Appl. 50(4), 5822–611 (2006)).
T. V. Dudnikova, “On the Asymptotical Normality of Statistical Solutions for Harmonic Crystals in Half-Space,” Russian J. Math. Phys. 15(4), 460–472 (2008).
T. V. Dudnikova, “On Convergence to Equilibrium for One-Dimensional Chain of Harmonic Oscillators on the Half-Line,” J. Math. Phys. 58(4), 043301 (2017).
J.-P. Eckmann, C.-A. Pillet, and L. Rey-Bellet, “nonequilibrium Statistical Mechanics of Anharmonic Chains Coupled to Two Heat Baths at Different Temperatures,” Comm. Math. Phys. 201(3), 657–697 (1999).
J.-P. Eckmann, C.-A. Pillet, and L. Rey-Bellet, “Entropy Production in Nonlinear, Thermally Driven Hamiltonian Systems,” J. Statist. Phys. 95(12–2), 305–331 (1999).
F. Fidaleo and C. Liverani, “Ergodic Properties for a Quantum Nonlinear Dynamics,” J. Statist. Phys. 97(52–6), 957–1009 (1999).
V. Jakšić and C.-A. Pillet, “Ergodic Properties of Classical Dissipative Systems. I,” Acta Math. 181(2), 245–282 (1998).
I. A. Ibragimov and Yu. V. Linnik, Independent and Stationary Sequences of Random Variables (Ed. by J.F.C. Kingman, Wolters Noordhoff, Groningen, 1971).
Y. Katznelson, An Introduction in Harmonic Analysis, 3rd edition (Cambridge University Press, 2004).
O. E. Lanford III and J. L. Lebowitz, “Time Evolution and Ergodic Properties of Harmonic Systems,” in Dynamical Systems, Theory and Applications. Lecture Notes in Physics (Springer-Verlag, Berlin, 1975), Vol. 38, pp. 144–177.
S. Lepri, R. Livi, and A. Politi, “Thermal Conduction in Classical Low-Dimensional Lattices,” Phys. Rep. 377(1), 1–80 (2003).
S. Lepri (ed.): Thermal Transport in Low Dimensions: From Statistical Physics to Nanoscale Heat Transfer (Lecture Notes in Physics, Springer, 2016), Vol. 921.
H. Nakazawa, “On the Lattice Thermal Conduction,” Supplement of the Progress of Theor. Phys. 45, 231–262 (1970).
L. Rey-Bellet and L. E. Thomas, “Exponential Convergence to nonequilibrium Stationary States in Classical Statistical Mechanics,” Comm. Math. Phys. 225(2), 305–329 (2002).
Z. Rieder, J. L. Lebowitz, and E. Lieb, “Properties of a Harmonic Crystal in a Stationary Nonequilibrium State,” J. Math. Phys. 8(5), 1073–1078 (1967).
H. Spohn and J. L. Lebowitz, “Stationary nonequilibrium States of Infinite Harmonic Systems,” Comm. Math. Phys. 54(2), 97–120 (1977).
H. Spohn, Large Scale Dynamics of Interacting Particles (Texts and Monographs in Physics, Springer-Verlag, Heidelberg, 1991).
M. I. Vishik and A. V. Fursikov, Mathematical Problems of Statistical Hydromechanics (Kluwer Academic, New York, 1988).
Acknowledgment
This work was done with the financial support from the Russian Science Foundation (Grant no. 19–71–30004).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dudnikova, T.V. Convergence to Stationary States and Energy Current for Infinite Harmonic Crystals. Russ. J. Math. Phys. 26, 428–453 (2019). https://doi.org/10.1134/S1061920819040034
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1061920819040034