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We consider a finite region of a d-dimensional lattice of nonlinearHamiltonian rotators, where neighbouring rotators have opposite spins and arecoupled by a small potential of order $\varepsilon^a,\, a\geq1/2$. We weaklystochastically perturb the system in such a way that each rotator interactswith its own stochastic Langevin-type thermostat with a force of order$\varepsilon$. Then we introduce the action-angle variables for the system ofuncoupled rotators ($\varepsilon=0$) and note that the sum of actions over allnodes is conserved by the purely Hamiltonian dynamics of the system with$\varepsilon>0$. We investigate the limiting (as $\varepsilon \rightarrow 0$)dynamics of actions for solutions of the $\varepsilon$-perturbed system on timeintervals of order $\varepsilon^{-1}$. It turns out that the limiting dynamicsis governed by a certain autonomous (stochastic) equation for the vector ofactions. This equation has a completely non-Hamiltonian nature.  The $\varepsilon$-perturbed system has a unique stationary measure$\widetilde \mu^\varepsilon$ and is mixing. Any limiting point of the family$\{\widetilde \mu^\varepsilon\}$ of stationary measures as$\varepsilon\rightarrow 0$ is an invariant measure of the system of uncoupledintegrable rotators. There are plenty of such measures. However, it turns outthat only one of them describes the limiting dynamics of the$\varepsilon$-perturbed system: we prove that a limiting point of$\{\widetilde\mu^\varepsilon\}$ is unique, its projection to the space ofactions is the unique stationary measure of the autonomous equation above,which turns out to be mixing, and its projection to the space of angles is thenormalized Lebesque measure on the torus $\mathbb{T}^N$.  Most of results and convergences we obtain are uniform in the number $N$ ofrotators.

Statistical mechanics of nonequilibrium systems of rotators with alternated spins