Bogolyubov generating functional method in statistical mechanics and the analog of the transformation to collective variables

Bogolyubov generating functional method in statistical physics and “collective”-variables transform within grand canonical ensemble

“…The result presented above can be obtained by means of formal calculations based on generalized functions and operator theories [4,8]. Indeed, as → 0, one obtains…”

“…To calculate the functional L (l) (f ), f ∈ S(R 3 ; R), corresponding to the long-range part V (l) of the full potential energy V : Φ → Φ, we use the analog of the Bogolyubov-Zubarev [5] "collective"-variables transform within the grand canonical ensemble suggested earlier in [4,6,7]. Namely, let L (l)…”

“…. , x n ), n ∈ Z + , are suitable "correlation" functions [1,4,6], and σ[n] denotes a partition of the set {1, 2, . .…”

“…Based on expressions (1.8) and (1.9), we can now formulate the following proposition: [1,3,4] the Bogolyubov-type functional equation…”

The order parameter and susceptibility of the 3D Ising-like system in an external field near the phase transition point

“…Thus, the problem needs to be treated very carefully in this case. From this point of view the Yukhnovskii's method of a phase transition description [5] used in this work is quite consistent with Bogolyubov's(Jr.) ideas of using the canonical collective variable transformation approach to the corresponding Bogolyubov's functional equation [6][7][8] for the correlation functions of a simple magnet system Hamiltonian instead of that for the standard Ising model. The related functional equation splitting, compatible with the Bogolyubov's principle of correlations weakening, proves to be equivalent to the suitable mean-field approximation of higher order, giving rise to a closed solution in the thermodynamical limit.…”