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 Teor. Veroyatnost. i Primenen., 2005, Volume 50, Issue 4, Pages 675–710 (Mi tvp125)

On a two-temperature problem for Klein–Gordon equation

T. V. Dudnikovaa, A. I. Komechb

a Electrostal' Polytechnic Institute
b M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We consider the Klein–Gordon equation in $\mathbf{R}^n$, $n\geq 2$, with constant or variable coefficients. The initial datum is a random function with a finite mean density of the energy and satisfies a Rosenblatt- or Ibragimov–Linnik-type mixing condition. We also assume that the random function is close to different space-homogeneous processes as $x_n\to\pm\infty$, with the distributions $\mu_\pm$. We study the distribution $\mu_t$ of the random solution at time $t\in\mathbf{R}$. The main result is the convergence of $\mu_t$ to a Gaussian translation-invariant measure as $t\to\infty$ that means the central limit theorem for the Klein–Gordon equation. The proof is based on the Bernstein “room-corridor” method and oscillatory integral estimates. The application to the case of the Gibbs measures $\mu_\pm=g_\pm$ with two different temperatures $T_{\pm}$ is given. It is proved that limit mean energy current density formally is $-\infty\cdot(0,…,0,T_+-T_-)$ for the Gibbs measures, and it is finite and equals $-C(0,…,0,T_+-T_-)$ with some positive constant $C>0$ for the smoothed solution. This corresponds to the second law of thermodynamics.

Keywords: Klein–Gordon equation, Cauchy problem, random initial data, mixing condition, Fourier transform, weak convergence of measures, Gaussian measures, covariance functions and matrices, characteristic functional.

DOI: https://doi.org/10.4213/tvp125

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English version:
Theory of Probability and its Applications, 2006, 50:4, 582–611

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Revised: 09.05.2005

Citation: T. V. Dudnikova, A. I. Komech, “On a two-temperature problem for Klein–Gordon equation”, Teor. Veroyatnost. i Primenen., 50:4 (2005), 675–710; Theory Probab. Appl., 50:4 (2006), 582–611

Citation in format AMSBIB
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This publication is cited in the following articles:
1. Dudnikova T.V., Komech A.I., “On the convergence to a statistical equilibrium in the crystal coupled to a scalar field”, Russ. J. Math. Phys., 12:3 (2005), 301–325
2. T. V. Dudnikova, “Convergence to equilibrium of the wave equation in $\mathbb R^n$ with odd $n\geqslant3$”, Russian Math. Surveys, 61:1 (2006), 168–170
3. Dudnikova T.V., “On ergodic properties for harmonic crystals”, Russ. J. Math. Phys., 13:2 (2006), 123–130
4. Dudnikova T.V., “On the asymptotical normality of statistical solutions for harmonic crystals in half-space”, Russ. J. Math. Phys., 15:4 (2008), 460–472
5. Dudnikova T.V., “Convergence to equilibrium distribution. The Klein–Gordon equation coupled to a particle”, Russ. J. Math. Phys., 17:1 (2010), 77–95
6. Dudnikova T.V., “Lattice dynamics in the half-space. Energy transport equation”, J. Math. Phys., 51:8 (2010), 083301, 25 pp.
7. T. V. Dudnikova, “Deriving hydrodynamic equations for lattice systems”, Theoret. and Math. Phys., 169:3 (2011), 1668–1682
8. T. V. Dudnikova, “On the Asymptotic Normality of a Harmonic Crystal Coupled to a Wave Field”, Math. Notes, 99:6 (2016), 942–945
9. Dudnikova T.V., “On the Asymptotical Normality of Statistical Solutions For Wave Equations Coupled to a Particle”, Russ. J. Math. Phys., 24:2 (2017), 172–194
10. T. V. Dudnikova, “O neravnovesnykh sostoyaniyakh kristallicheskoi reshetki”, Preprinty IPM im. M. V. Keldysha, 2018, 015, 26 pp.
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