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Equilibrium Configurations of the Continuous Medium with Nonlocal Interaction. P. 123–129

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Section: Physics. Mathematics. Informatics




Tomashevsky Igor Lyudvigovich,
Institute of Mathematics, Information and Space Technologies, Northern (Arctic) Federal University named after M.V. Lomonosov (Arkhangelsk, Russia)


The model of an infinite continuous medium is considered. The interaction between spatially distant elements is characterized by the potential energy proportional to the masses of the elements and interaction potential depending on the distance between the elements. The paper investigates the state of equilibrium of such a medium with a density of material substance, positive everywhere except, perhaps, the set of points of measure zero. The geometry of the states of equilibrium is identified only by the roots of the Fourier transformation ϕ (p) of the interaction potential. Not all the states of equilibrium with a continuous distribution of material substance are stable, as they possess minimal potential energy with respect to any local redistribution of material substance. Their stability depends on the presence and position of minimum points of the Fourier transformation ϕ (p). The analysis of the states of equilibrium in the case of the Fourier transformation ϕ (p) has a single point of local minimum 0 p which is positive everywhere except maybe neighborhood of p0. Three possible cases are considered. In the first case (0 (ϕ p) < 0), if the density of material substance is positive quasi everywhere then the states of equilibrium are not stable. In the second case (0 () 0 p ϕ > ) the medium has the unique state of equilibrium, which is stable and simple. In the third case (0 (ϕ p) = 0) the medium has infinitely many states of equilibrium, which are stable and have space periodic density of material substance and the same energy. Among these states, there are simple states with constant density, as well as the states with 1D-periodical (laminal) and 2D-periodical (cylindrical) distribution of material substance. Unusual ability of the continuous medium with nonlocal interaction to create periodical crystal-like configurations of material substance can be used for a simulation and an analogue of spatial configurations and physical and mechanical properties of real polyatomic crystals.


continuous medium, nonlocal interaction, periodic functions
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