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Discontinuity, Nonlinearity, and Complexity 9(4) (2020) 529--540 | DOI:10.5890/DNC.2020.12.006

Alexey A. Kislitsyn$^1$ , Yurii N. Orlov$^{1,2}$

$^1$ Keldysh Institute of Applied Mathematics of RAS, Miusskaya Sq., 4, Moscow 125047, Russia normalsize

$^2$ Institute of Machines Sciences named after A.A. Blagonravov of RAS, Maly Kharitonyevsky Pereulok, Moscow, 101990, Russia

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Abstract

The difference approximation of the one-dimensional Liouville equation for the sample distribution function density of the non-stationary time series estimated by the histogram is considered. The scheme with semi-group property conservation is constructed for evolution model of this sample distribution function density. We investigate the problem of appropriate Liouville equation construction for given initial and final distributions. We prove the necessary and sufficient condition of such a representation, which is a strong positivity of the initial density distribution in the inner class intervals. The determination of the corresponding Liouville velocity algorithm is constructed and its mechanical-statistical meaning is shown. The dynamical system, associated with this Liouville equation, is considered. We interpret the Liouville statistical velocity as a corresponding velocity of dynamical system, according to representation of statistical mechanics. We show, that this interpretation leads to monotonic discrete dynamical system with stationary point, corresponding to equality of initial and final distribution functions.

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