Long-range order in nonequilibrium systems of interacting Brownian linear oscillators (Q1871894)

There are considered the interacting one-dimensional Brownian oscillators \(q_\Lambda(t)= (q_x(t), x\in\Lambda)\), \(t\geq 0\), described by the stochastic differential equations \[ dq_x(t)= -\beta\partial_x U_g(q_\Lambda(t)) dt+ dW_x(t),\qquad x\in\Lambda,\;t>0, \] where \(\beta> 0\), \(\Lambda\) is a hypercube in \(\mathbb{Z}^d\), \(d> 1\), \(W_x\) are independent standard Brownian motions, \(\partial_x= {\partial\over\partial q_x}\), \[ U_g(q_\Lambda)= \sum_{x\in\Lambda} (\eta g^{-n} q^{2n}_x- q^2_x)+ \sum_{\langle x,y\rangle\in\Lambda} (q_x- q_y)^2, \] \(g,\eta> 0\), \(n\in\mathbb{Z}^+\), \(n> 1\), \(\langle x,y\rangle\) means that \(x\) and \(y\) are nearest neighbours, assuming that the initial distributions are Gibbsian. Conditions are found under which the thermodynamic limits exist and for sufficiently large \(g\) the ferromagnetic long-range order occurs for the spins \(s_x(t):= \text{sign }q_x(t)\).