Description of the mechanisms of the emergence and dynamics of cosmological structures such as “Zel’dovich pancakes” [1] and quasi-one-dimensional threads that form the so-called “cosmic web” is currently a highly debated topic among research groups around the world. The prevailing concepts in the analysis of complex issues related to evolution scale distributions of galaxies and clusters, currently are identifying statistical patterns in sets of observational data and attempts at reasonable explanation phenomenological approximations of these regularities. At the same time, the study of the mechanism implementations of pseudo-ordering on a cosmological scale recognized by default unpromising (primarily due to the lack of a verified observational material). The generally accepted assumptions here can be considered the results of hydrodynamic and kinetic simulation in the presence of a significant number of model assumptions regarding initial data, physical assumptions regarding the influence of dark energy, dark matter (scalar fields, the presence of exotic particles, etc.). In addition, often it is difficult to assess the legitimacy of the computational aspects of the used mathematical model, including a priori specified rules passing caustic or emerging artifact–grid features.

The question of ordered relaxation of perturbations in the system along the selected direction (for filamentous large structures — along two mutually perpendicular directions) is rather nontrivial. At cosmological distances order mechanism for implementing coherence when passing from Lagrangian coordinates [1, 3] to the global description in the Euler system looks rather vague, since must take into account the existence synchronism of anisotropic damping of emerging density fluctuations (perhaps, one of the reasons for such behavior at numerical calculations can be considered a special form of averaging a self-consistent gravitational field in a system of particles; however, the choice of such averaging from a physical point of view is not entirely clear).

The task of the author in this paper is to describe the non–equilibrium dynamics of statistical ensemble of gravitationally interacting N (?1) particles, under conditions of applicability non–relativistic formalism of single-particle distribution functions, in a state close to quasi-equilibrium (which implies the presence non–stationary distributions of particles with small deviations in the norm with a sufficiently long time for changing the topology of their system). As the main physical assumption that determines validity of mathematical formalism in this paper, author assumes at this point that the kinetics of massive particles is considered in the 3-dimensional case (the transition to the case d=2 is done by changing kernels in the integral equation). In this case, there are certain restrictions on the geometric dimensions. domain R⊆R^d, in which the particle system is located: solutions Φ(x) of the nonlinear Liouville–Gelfand equation (LGE), describing the interparticle potential in the stationary version of the boundary value problem for the VPS, for the case d=3 lead to Chandrasekhar’s “infinite mass problem”, and stability a regular solution of the LGE is possible only in a simply connected compact domain. We will assume the system of particles is in  relative equilibrium state, for which actually Vlasov’s equation admits “energy substitution” for a one–particle distribution functionand the solution of the second equation of the complete VPS (i.e., LGE) refers to class “minimal”which stable near both regular and singular points of the range of parameters (“non-minimal” solutions can be considered separately as a special class of local solutions in a neighborhood singular points, in particular, under the condition that the temperature of the system is indefinite).

The author believe that a special role in the study of the structure and evolution of cosmological structures should acquire approaches based on the determined emergence of large-scale structures (coherent in the sense of having a translational invariance of substructures in the structure itself, or when comparison with similar ones existing nearby) in the cosmological background. That is “self-assembly” such structures as the result of interaction small scale disturbances creating “order out of chaos” (according to Ya.B. Zel’dovich’s “pancakes” scenario) [2] seems unlikely. The main ongoing process of structural genesis are, like classical turbulent coherent hydrodynamic structures in interpretation of O.M. Belotserkovsky [4], primary formation of a large structure with its further evolution in the form of a transition between states of relative equilibrium (which correspond to the extrema of the entropy of the system with an adiabatic change in its other thermodynamic or topological parameters); system while it can be multi-connected, just like the observed “space web”. In this paper, the author show the possibility of natural construction of models cosmological structures based on a qualitative analysis of the properties of solutions to the equations of the Vlasov–Poisson system taking into account the modification of the shape of the gravitational potential, with the inclusion of an additional term, including cosmological parameter (modified Newton potential or V. Gurzadyan’s potential [5],[6]). Set to appear multiply connected structure for the system gravitating substructures, shown the difference between the kinetic calculation and the hydrodynamic calculation for the generalized Jeans processes, using methods of bifurcation theory solutions of integral equations demonstrated the possibility of the appearance of secondary structures between nodes (“space web”).

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