Дымов Андрей Викторович

О стохастической модели волновой турбулентности Захарова–Львова

2020 · ARTICLE · ru

Авторы обсуждают ряд строгих результатов в стохастической модели волновой турбулентности Захарова–Львова. А именно, рассматривают уравнение Шрёдингера с (модифицированной) кубической нелинейностью и вязкостью на торе большого периода, возмущенное случайной силой, и раскладывают его решение в формальный ряд по амплитуде. Авторы показывают, что в пределе, когда амплитуда стремится к нулю, а период тора – к бесконечности, спектр энергии квадратичной срезки этого разложения сходится к решению волнового кинетического уравнения с вязкостью и внешней силой. Затем обсуждают срезки этого разложения высшего порядка.

Asymptotic estimates for singular integrals of fractions with divisors, given by products of quadratic block-forms

2020 · ARTICLE · en

The logarithmic derivative for point processes with equivalent Palm measures

2019 · ARTICLE · en

The logarithmic derivative of a point process plays a key role in the general approach, due to the third author, to constructing diffusions preserving a given point process. In this paper we explicitly compute the logarithmic derivative for determinantal processes on R with integrable kernels, a large class that includes all the classical processes of random matrix theory as well as processes associated with de Branges spaces. The argument uses the quasi-invariance of our processes established by the first author.

A functional limit theorem for the sine-process

2019 · ARTICLE · en

The main result of this paper is a functional limit theorem for the sine-process. In particular, we study the limit distribution, in the space of trajectories, for the number of particles in a growing interval. The sine-process has the Kolmogorov property and satisfies the central limit theorem, but our functional limit theorem is very different from the Donsker Invariance Principle. We show that the time integral of our process can be approximated by the sum of a linear Gaussian process and independent Gaussian fluctuations whose covariance matrix is computed explicitly. We interpret these results in terms of the Gaussian free field convergence for the random matrix models. The proof relies on a general form of the multidimensional central limit theorem under the sineprocess for linear statistics of two types: those having growing variance and those with bounded variance corresponding to observables of Sobolev regularity 1/2.

Asymptotic Behavior of a Network of Oscillators Coupled to Thermostats of Finite Energy

2018 · ARTICLE · en

We study the asymptotic behavior of a finite network of oscillators (harmonic or anharmonic) coupled to a number of deterministic Lagrangian thermostats of finite energy. In particular, we consider a chain of oscillators interacting with two thermostats situated at the boundary of the chain. Under appropriate assumptions, we prove that the vector (p, q) of moments and coordinates of the oscillators in the network satisfies (p, q)(t) → (0, q _c ) as t → ∞, where q_c is a critical point of some effective potential, so that the oscillators just stop. Moreover, we argue that the energy transport in the system stops as well without reaching thermal equilibrium. This result is in contrast to the situation when the energies of the thermostats are infinite, studied for a similar system in [14] and subsequent works, where the convergence to a nontrivial limiting regime was established. The proof is based on a method developed in [22], where it was observed that the thermostats produce some effective dissipation despite the Lagrangian nature of the system.

Nonequilibrium statistical mechanics of weakly stochastically perturbed system of oscillators

2016 · ARTICLE · en

We consider a finite region of a $d$-dimensional lattice of weakly coupled harmonic oscillators. The coupling is provided by a nearest-neighbour potential (harmonic or not) of size $\epsilon$. Each oscillator weakly interacts by force of order $\epsilon$ with its own stochastic Langevin thermostat of arbitrary positive temperature. We investigate limiting as $\epsilon\to 0$ behaviour of solutions of the system and of the local energy of oscillators on long-time intervals of order $\epsilon^{-1}$ and in a stationary regime. We show that it is governed by an effective equation which is a dissipative SDE with nondegenerate diffusion. Next we assume that the interaction potential is of size $\epsilon\lambda$, where $\lambda$ is another small parameter, independent from $\epsilon$. Solutions corresponding to this scaling describe small low temperature oscillations. We prove that in a stationary regime, under the limit $\epsilon\to 0$, the main order in $\lambda$ of the averaged Hamiltonian energy flow is proportional to the gradient of temperature. We show that the coefficient of proportionality, which we call the conductivity, admits a representation through stationary space-time correlations of the energy flow. Most of the results and convergences we obtain are uniform with respect to the number of oscillators in the system.

Nonequilibrium statistical mechanics of a solid immersed in a continuum

2016 · ARTICLE · en

In the introductory part of this survey, we briefly discuss the problems of nonequilibrium statistical physics that arise in the study of energy transport in solids as well as the results available at the moment. In the main part of the survey, we explain, compare, and generalize results obtained in our previous works. We study the dynamics and energy transport in Hamiltonian systems of particles where each particle is weakly perturbed by the interaction with its own stochastic Langevin thermostat. Such systems can be regarded as models of solids that interact weakly with a continuum.

Nonequilibrium Statistical Mechanics of Hamiltonian Rotators with Alternated Spins

2015 · ARTICLE · en

We consider a finite region of a d-dimensional lattice of nonlinear Hamiltonian rotators, where neighbouring rotators have opposite (alternated) spins and are coupled by a small potential of order $\epsilon^a,\, a\geq 1/2$. We weakly stochastically perturb the system in such a way that each rotator interacts with its own stochastic thermostat with a force of order $\epsilon$. Then we introduce action-angle variables for the system of uncoupled rotators ($\epsilon=0$) and note that the sum of actions over all nodes is conserved by the purely Hamiltonian dynamics of the system with $\epsilon>0$. We investigate the limiting (as $\epsilon \rightarrow 0$) dynamics of actions for solutions of the $\epsilon$-perturbed system on time intervals of order $\epsilon^{-1}$. It turns out that the limiting dynamics is governed by a certain stochastic equation for the vector of actions, which we call the transport equation. This equation has a completely non-Hamiltonian nature. This is a consequence of the fact that the system of rotators with alternated spins do not have resonances of the first order. The $\epsilon$-perturbed system has a unique stationary measure $\wid \mu^\epsilon$ and is mixing. Any limiting point of the family $\{\wid \mu^\epsilon\}$ of stationary measures as $\eps\ra 0$ is an invariant measure of the system of uncoupled integrable rotators. There are plenty of such measures. However, it turns out that only one of them describes the limiting dynamics of the $\epsilon$-perturbed system: we prove that a limiting point of $\{\wid\mu^\epsilon\}$ is unique, its projection to the space of actions is the unique stationary measure of the transport equation, which turns out to be mixing, and its projection to the space of angles is the normalized Lebesque measure on the torus $\mathbb{T}^N$. The results and convergences, which concern the behaviour of actions on long time intervals, are uniform in the number $N$ of rotators. Those, concerning the stationary measures, are uniform in $N$ in some natural case.

Dissipative effects in a linear Lagrangian system with infinitely many degrees of freedom

2012 · ARTICLE · en

We consider the problem of potential interaction between a finite-dimensional linear Lagrangian system and an infinite-dimensional one (a system of linear oscillators and a thermostat). We study the final dynamics of the system. Under natural assumptions, this dynamics turns out to be very simple and admits an explicit description because the thermostat produces an effective dissipation despite the energy conservation and the Lagrangian nature of the system. We use the methods of [1], where the final dynamics of the finite-dimensional subsystem is studied in the case when it has one degree of freedom and a linear potential or (under additional assumptions) polynomial potential. We consider the case of finite-dimensional subsystems with arbitrarily many degrees of freedom and a linear potential and study the final dynamics of the system of oscillators and the thermostat. The necessary assertions from [1] are given with proofs adapted to the present situation. [1] D. Treschev, “Oscillator and thermostat”, Discrete Contin. Dyn. Syst. 28:4 (2010), 1693–1712