On the evolution of Gibbs states of the lattice gradient stochastic dynamics of interacting oscillators

Grand canonical correlation functions of stochastic(Brownian) lattice linear oscillators interacting via a pair short-range potential are found in the thermodynamic limits at low activities and on a finite time interval. It is proved that their sequence is a weak solution of the BBGKY-type gradient diffision hierarchy. The initial correlation functions are Gibbsian, which corresponds to many-body positive finite-range and short-range non-positive pair interaction potentials. The utilized technique is based on an application of the Feynman–Kac formula for solutions of the Smoluchowski equation and a representation of the time-dependent correlation functions in terms of correlation functions of a Gibbs lattice oscillator path system with manybody interaction potentials.