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. 
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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