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Weak limits of probability distributions in systems with nonstationary perturbations. (Russian, English) Zbl 1083.37507
Dokl. Akad. Nauk, Ross. Akad. Nauk 389, No. 5, 605-607 (2003); translation in Dokl. Math. 67, No. 2, 283-285 (2003).
A system of differential equations $$\dot x = \omega,$$ $$\dot \omega = f(t),$$ is considered where $$x = (x_1, x_2, \dots, x_n\pmod{2\pi})$$ are angular coordinates of the $$n$$-dimensional torus, $$\omega = (\omega_1, \omega_2, \dots, \omega_n) \in \mathbb R^n,$$ $$f$$ is a prescribed vector function. Let $$\Gamma$$ be the phase space of the system and $$\rho$$ a non-negative measurable function on $$\Gamma$$ with $$\int_\Gamma \rho(x,\omega)\,dx\,d\omega=1$$. The time evolution of the probability measure with density $$\rho$$ determines a family of probability measures with densities $$\rho_t,\,t\geq0$$, where $$\rho_0=\rho$$. The main result of the paper is that, under some integrability conditions on $$f$$, we have $$\lim_{t\to\pm\infty}\int_\Gamma\rho_t(x,\omega)\varphi(x)\,dx\,d\omega= \int_{\mathbb T^n}\varphi(x)\,dx$$, which means that the projection of the measure with density $$\rho_t$$ to the configuration space $$\mathbb T^n$$ converges to the normalized Lebesgue measure on $$\mathbb T^n$$ as $$t\to\infty$$.
##### MSC:
 37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems 34E10 Perturbations, asymptotics of solutions to ordinary differential equations 37A60 Dynamical aspects of statistical mechanics 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 70H05 Hamilton’s equations 82C05 Classical dynamic and nonequilibrium statistical mechanics (general)