Power-law statistics in the velocity fluctuations of Brownian particle in inhomogeneous media and driven by colored noise

TL;DR

This paper investigates the velocity fluctuations of a Brownian particle in an inhomogeneous medium driven by colored noise, revealing how environmental inhomogeneity and noise correlation affect power-law behavior and spectral properties.

Contribution

It introduces a model combining inhomogeneous environments and colored noise to analyze velocity fluctuations, highlighting effects on power-law spectra and distribution cut-offs.

Findings

Colored noise causes exponential cut-off in position distribution.

Inhomogeneity narrows the frequency range of power-law spectra.

Finite correlation time influences the spectral and distribution characteristics.

Abstract

Nonlinear stochastic differential equations generating signals with 1/f spectrum have been used so far to describe socio-economical systems. In this paper we consider the motion of a Brownian particle in an inhomogeneous environment such that the motion can be described by the equation yielding 1/f spectrum in a broad range of frequencies. The inhomogeneous environment can be a result, for example, of a linear potential affecting the Brownian particle together with the medium where steady state heat transfer is present due to the difference of temperatures at the ends of the medium. The correlation of collisions between the Brownian particle and the surrounding molecules can lead to the situation where the finite correlation time becomes important, thus we have investigated the effect of colored noise in our model. Existence of colored noise leads to the additional restriction of the diffusion and exponential cut-off of the distribution of particle positions. Narrower power law part in the distribution of the particle positions results in the narrower range of frequencies where the spectrum has power law behavior.