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Nonequilibrium statistical mechanics of Hamiltonian rotators with alternated spins. (English) Zbl 1328.82046
The author considers a Hamiltonian system consisting of a $$d$$-dimensional lattice of $$N$$ nonlinear rotators, with neighboring rotators having opposite spin. The rotators are weakly coupled through a potential (linear or nonlinear) of size $$\varepsilon^a$$, with $$a\geq 1/2$$. Each rotator of the Hamiltonian system interacts with its own stochastic Langevin-type thermostat with a force of order $$\varepsilon$$. Introducing action-angle variables for the uncoupled Hamiltonian system (corresponding to $$\varepsilon=0$$), it can be noted that the sum of the actions is conserved by the Hamiltonian dynamics, therefore the actions play the role of the local energy. The author investigates the limiting dynamics (for $$\varepsilon \to 0$$) of the actions of the $$\varepsilon$$-perturbed system on time intervals of order $$\epsilon^{-1}$$, showing that it is governed by an autonomous system of stochastic equations, which has a completely non-Hamiltonian character. The Hamiltonian system coupled with the thermal baths is mixing and the author shows that its stationary measure converges to the product of the unique stationary measure of the autonomous system and the normalized Lebesgue measure on the $$N$$-dimensional torus. It is also proved that as $$\varepsilon \to 0$$ the convergence of the vector of the actions to a solution of the autonomous system is uniform with respect to $$N$$, while the convergence of the measures is uniform only in some natural cases.

##### MSC:
 82C70 Transport processes in time-dependent statistical mechanics 82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 34E10 Perturbations, asymptotics of solutions to ordinary differential equations
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