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Definition of Fractal Dimensions of Brownian Motion Process in the Conditions of the Fluctuating Coefficient of the Friction

Keywords:

A.N. Morozov, A.V. Skripkin


In a work the description of a Brownian motion of a particle in the viscous medium taking into account observable fluctuations of kinetic coefficient of a friction and entrainment by the Brownian particle of particles of medium surrounding it is given. In the first part of the work the classical description of a Brownian motion based on the dynamic Langevin is given. Further it is shown that the dynamic equation of movement of the Brownian particle in the presence of the fluctuating coefficient of a friction leads to the nonlinear integral equation, and fluctuations of coefficient of a friction, in the full consent with experiment, have character of flicker-noise. It is found that the account of entrainment by the Brownian particle of particles of medium instead of the differential Langevin equation leads to integral stochastic process. The received dynamic equations were used for a finding of fractal dimensions of the processes corresponding to fluctuations of a coordinate and an impulse. Comparison found fractal dimensions with a classical case at which the equation of movement of the Brownian particle represents the differential Langevin equation is given. The estimation of possibility of use the fractal dimensions of process as criterion of its degree of «non-Markovianity», and also possibilities of use of Markovian models to the processes showing non-Markovian properties is executed.
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