arXiv subjects

Yu. M. Suhov

Publications and source records attributed to Yu. M. Suhov.

3 recordsLinked to original sources

arXiv2005

On Convergence to Equilibrium Distribution, I. The Klein - Gordon Equation with Mixing

Consider the Klein-Gordon equation (KGE) in $\R^n$, $n\ge 2$, with constant or variable coefficients. We study the distribution $μ_t$ of the random solution at time $t\in\R$. We assume that the initial probability measure $μ_0$ has zero mean, a translation-invariant covariance, and a finite mean energy density. We also asume that $μ_0$ satisfies a Rosenblatt- or Ibragimov-Linnik-type mixing condition. The main result is the convergence of $μ_t$ to a Gaussian probability measure as $t\to\infty$ which gives a Central Limit Theorem for the KGE. 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 an `averaged' version of the scattering theory for infinite energy solutions, based on Vainberg's results on local energy decay.

math-ph↗

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↗

arXiv2004

Gibbs Measures For SOS Models On a Cayley Tree

We consider a nearest-neighbor SOS model, spin values $0,1,..., m$, $m\geq 2$, on a Cayley tree of order $k$ . We mainly assume that $m=2$ and study translation-invariant (TI) and `splitting' (S) Gibbs measures (GMs). For $m=2$, in the anti-ferromagnetic (AFM) case, a symmetric TISGM is unique for all temperatures. In the ferromagnetic (FM) case, for $m=2$, the number of symmetric TISGMs varies with the temperature: here we identify a critical inverse temperature, $β^1_{\rm{cr}}$ ($=T_{\rm{cr}}^{\rm{STISG}}$) $\in (0,\infty)$ such that $\forall$ $0\leq β\leqβ^1_{\rm{cr}}$, there exists a unique symmetric TISGM $μ^*$ and $\forall$ $β>β^1_{\rm{cr}}$ there are exactly three symmetric TISGMs : $μ^*_+$, $μ^*_{\rm m}$ and $μ^*_-$ For $β>β^1_{\rm{cr}}$ we also construct a continuum of distinct, symmertric SGMs which are non-TI. Our second result gives complete description of the set of periodic Gibbs measures for the SOS model on a Cayley tree. We show that (i) for an FM SOS model, for any normal subgroup of finite index, each periodic SGM is in fact TI. Further, (ii) for an AFM SOS model, for any normal subgroup of finite index, each periodic SGM is either TI or has period two (i.e., is a chess-board SGM).

math.PR↗