Geodesic
On the evolution of Gibbs states of the lattice gradient stochastic dynamics of interacting oscillators
Teoriâ slučajnyh processov, Tome 15 (2009) no. 1, pp. 61-82.

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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.
Keywords: Lattice gradient stochastic dynamics, Gibbs state, grand canonical correlation functions. 
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W. I. Skrypnik. On the evolution of Gibbs states of the lattice gradient stochastic dynamics of interacting oscillators. Teoriâ slučajnyh processov, Tome 15 (2009) no. 1, pp. 61-82. http://geodesic.mathdoc.fr/item/THSP_2009_15_1_a6/

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