Application of integral transforms to a description of the Brownian motion by a non-Markovian random process

Abstract

The one-dimensional Brownian motion and the Brownian motion of a spherical particle in an infinite medium are described by the conventional methods and integral transforms considering the entrainment of surrounding particles of the medium by the Brownian particle. It is demonstrated that fluctuations of the Brownian particle velocity represent a non-Markovian random process. A harmonic oscillator in a viscous medium is also considered within the framework of the examined model. It is demonstrated that for rheological models, random dynamic processes are also non-Markovian in character.Application of the theory of Markovian processes to a description of the Brownian motion in actual physical media is approximate, because it ignores the special features of the interaction of the Brownian particles with particles of the medium [1, 2]. We note also that the processes proceeding in physical and technical systems are often non-Markovian ones [3][4][5]. In particular, the flicker noise observed in various physical processes can serve as an example of a non-Markovian process [6]. Various physical fluctuations of the kinetic coefficients observed experimentally (for example, fluctuations of the electrical conductivity) have the spectral density characteristic typical of the flicker noise. The flicker noise is the main type of noise limiting the sensitivity of electronic devices at low frequencies [7]. Actual radio engineering signals with amplitude and phase modulation by a combination of deterministic and random processes also belong to the non-Markovian process [5].We note also that when a Markovian random process acts on a dynamic system, its response represents a non-Markovian random process. The sum of two Markovian processes is a non-Markovian process. The processes observed after integration of a Markovian process or finding of a sliding average of the process with independent values are also non-Markovian ones [4]. In particular, the coordinate of the Brownian particle calculated as an integral of its velocity is not always described by the model of a random Markovian process. The Wiener approximation is correct for the Brownian particles only for sufficiently long time intervals much longer than the particle relaxation time.The above-indicated reasons demonstrate that we practically always use non-Markovian random processes to describe and to analyze actual physical processes and technical devices, and models of the Markovian process can be considered only as a first approximation. BROWNIAN MOTION AS A MARKOVIAN PROCESSTo describe the Brownian motion, the approach based on the application of the stochastic Langevin equation is conventionally used [1,8]. This approach allows the researchers to take advantage of the well-developed theory of stochastic differential systems [9, 10] which determines all necessary statistical characteristics of the velocity fluctuations of the Brownian particle motion.

Description of evaporation of a spherical liquid drop by a non-Markovian random process based on integral stochastic equations

“…Thus, in [6] it was demonstrated that consideration of entrainment by the Brownian particle of the surrounding particles calls for the application of integral equations and the theory of non-Markovian processes.…”

Propagation of heat in the space around a cylindrical surface as a non-Markovian random process

“…We note that the indicated approach to description of Brownian motion, the evaporation of a liquid droplet in the atmosphere, and the motion of a particle in a medium with a fluctuating friction factor makes it possible to substantially refine results obtained with classical methods [8,9].…”