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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 \(t\to\infty\).
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)