PRACTICAL IMPLEMENTATION OF THE IDEA OF A REDUCED DESCRIPTION OF DENSITY FIELD FLUCTUATIONS USING A TWO-LEVEL STATISTICAL METHOD
UDC 531.19; 538.911
Key words: two-level statistical method, variational method, mean force potential, heterogeneous system, nanoparticle, fluctuating density field, effective Landau Hamiltonian, elementary density fluctuations, short description in fluctuation theory.
For citation: Narkevich I. I., Farafontova E. V. Practical implementation of the idea of a reduced description of density field fluctuations using a two-level statistical method. Proceedings of BSTU, issue 3, Physics and Mathematics. Informatics, 2022, no. 2 (260), pp. 49–54 (In Russian). DOI: https://doi.org/10.52065/2520-6141-2022-260-2-49-54.
Abstract
Inside of a two-level statistical method, a statistical expression for a grand thermodynamic potential of an inhomogeneous system is obtained. With its help, a method was developed for numerical variational calculation of the density profiles of a medium in the vicinity of the boundary of spherical crystalline nanoparticles, which are in equilibrium with a gaseous medium. As a result, a correlation was established between the structural parameters of a heterogeneous system and the thermodynamic characteristics of crystalline nanoparticles, taking into account the spatial relaxation of the lattice at their boundary. The two-level statistical approach used in this purpose for description of the inhomogeneous systems properties is a symbiosis of the Bogolyubov – Born – Green – Kirkwood – Yvon (BBGKI) correlative functions method, the Rott conditional correlative functions method and the thermodynamic density functionals method. Exactly the joint use of these three methods has made it possible to effectively solve the two main problems of modern statistical physics. This includes the nesessity of linking (break) the chains of integro-differential equations for correlative functions with the simultaneous solution of the question of how to normalize these functions, taking into account the inhomogeneities of the density field in singleand multi-component condensed systems, i.e., in crystalline and liquid systems.
The results obtained in the statistical description of inhomogeneous media created the prerequisites for the practical implementation of the previously formulated idea of a reduced description of density field fluctuations in equilibrium systems. The proposed statistical approach in the theory of fluctuations is an alternative of the phenomenological theories known from the literature, which use the effective Landau Hamiltonian.
In this article, inside a two-level statistical method, it is proposed to represent the system fluctuating density field under study as a system of elementary density fluctuations that arise against the background of a homogeneous medium with an average density. Elementary fluctuations interacting with this medium and with each other form a statistical subsystem of quasi-particles. Their interactions are proposed to be described using the corresponding effective potentials, the expressions for which are obtained using the statistical expression for a large thermodynamic potential, which is a functional with respect to the fluctuating density field.
References
- Narkevich I. I. Molekulyarno-statisticheskaya teoriya neodnorodnykh kondensirovannykh sred. Dissertatsiya doktora fiziko-matematicheskikh nauk [Molecular-statistical theory of the non-homogeneous condensed matter. Dissertation DSc (Physics and Mathematics)]. St. Petersburg, 1993. 223 p. (In Russian).
- Narkevich I. I. Dvukhurovnevyy statisticheskiy metod opisaniya neodnorodnykh sistem. Simbioz metodov korrelyativnykh funktsiy i termodinamicheskikh funktsionalov plotnosti [Two-level statistical method for describing heterogeneous systems. Symbiosis of methods of correlative functions and thermodynamic functionals of density]. Norderstedt, LAP LAMBERT Academic Publishing RU, 2019. 114 p. (In Russian).
- Oden J. Konechnyye elementy v nelineynoy mekhanike sploshnykh sred [Finite elements in nonlinear continuum mechanics]. Moscow, Mir Publ., 1976. 464 p. (In Russian).
- Narkevich I. Statistical theory of nonuniform systems and reduced description in the density fluctuation theory. Physica, 1982, no 112 A, pp. 167–192.
- Narkevich I. I. Abbreviated description of inhomogeneous systems based on conditional spatial density correlation functions. Izvestiya AN BSSR [News of the Academy of Sciences of the BSSR], series of physical and athematical sciences, 1980, no 5, pp 107–112 (In Russian).
- Landau L. D., Lifshits E. M. Statisticheskaya fizika. [Statistical physics: in 2 parts], Moscow, Nauka Publ., 1987, part 1. 586 p.
- Narkevich I. I., Farafontova E. V. Two-level statistical description of structure of homogeneous macroscopic system and spherical crystalline nanoparticles. Nanoscience and Technology: International Journal, 2019, no. 10 (4), pp. 365–376.
