arXiv subjects

T. V. Dudnikova

Publications and source records attributed to T. V. Dudnikova.

22 records · Page 2Linked to original sources

arXiv2005

On Convergence to Equilibrium Distribution, II. The Wave Equation in Odd Dimensions, with Mixing

The paper considers the wave equation, with constant or variable coefficients in $\R^n$, with odd $n\geq 3$. We study the asymptotics of the distribution $μ_t$ of the random solution at time $t\in\R$ as $t\to\infty$. It is assumed that the initial measure $μ_0$ has zero mean, translation-invariant covariance matrices, and finite expected energy density. We also assume that $μ_0$ satisfies a Rosenblatt- or Ibragimov-Linnik-type space mixing condition. The main result is the convergence of $μ_t$ to a Gaussian measure $μ_\infty$ as $t\to\infty$, which gives a Central Limit Theorem (CLT) for the wave equation. The proof for the case of constant coefficients is based on an analysis of long-time asymptotics of the solution in the Fourier representation and Bernstein's `room-corridor' argument. The case of variable coefficients is treated by using a version of the scattering theory for infinite energy solutions, based on Vainberg's results on local energy decay.

math-ph↗

arXiv2005

Local stationarity for lattice dynamics in the harmonic approximation

We consider the lattice dynamics in the harmonic approximation for We consider the lattice dynamics in the harmonic approximation for a simple hypercubic lattice with arbitrary unit cell. The initial data are random according to a probability measure which enforces slow spatial variation on the linear scale $ε^{-1}$. We establish two time regimes. For times of order $ε^{-γ}$, $0<γ<1$, locally the measure converges to a Gaussian measure which is space-time stationary with a covariance inherited from the initial (in general, non-Gaussian) measure. For times of order $ε^{-1}$ this local space covariance changes in time and is governed by a semiclassical transport equation.

math-ph↗

arXiv2002

On two-temperature problem for harmonic crystals

We consider the dynamics of a harmonic crystal in $d$ dimensions with $n$ components,$d,n \ge 1$. The initial date is a random function with finite mean density of the energy which also satisfies a Rosenblatt- or Ibragimov-Linnik-type mixing condition. The random function converges to different space-homogeneous processes as $x_d\to\pm\infty$, with the distributions $μ_\pm$. We study the distribution $μ_t$ of the solution at time $t\in\R$. The main result is the convergence of $μ_t$ to a Gaussian translation-invariant measure as $t\to\infty$. The proof is based on the long time asymptotics of the Green function and on Bernstein's `room-corridor' argument. The application to the case of the Gibbs measures $μ_\pm=g_\pm$ with two different temperatures $T_{\pm}$ is given. Limiting mean energy current density is $- (0,...,0,C(T_+ - T_-))$ with some positive constant $C>0$ what corresponds to Second Law.

math-ph↗

arXiv2002

On the convergence to statistical equilibrium for harmonic crystals

We consider the dynamics of a harmonic crystal in $d$ dimensions with $n$ components, $d,n$ arbitrary, $d,n\ge 1$, and study the distribution $μ_t$ of the solution at time $t\in\R$. The initial measure $μ_0$ has a translation-invariant correlation matrix, zero mean, and finite mean energy density. It also satisfies a Rosenblatt- resp. Ibragimov-Linnik type mixing condition. The main result is the convergence of $μ_t$ to a Gaussian measure as $t\to\infty$. The proof is based on the long time asymptotics of the Green's function and on Bernstein's ``room-corridors'' method.

math-ph↗