APPLICATION OF THE STOCHASTIC HARMONIC OSCILLATOR MODEL IN TEACHING THE SECTION "FLUCTUATIONS" IN A TECHNICAL UNIVERSITY

Abstract

Nanotechnologies represent one of the priorities of the progress in all fields of the modern science and technology. The thermal fluctuations play a significant role for the nanosized objects. However, in modern textbooks and educational standards, studying thermal fluctuations have not received enough attention. The purpose of the present paper is to fill in this lacune partially. The presentation of the material is focused on the stochastic harmonic oscillator model, i.e. on the vibrational system experiencing the action of the dissipative and stochastic (random) forces. In the paper, we present several examples of specific modern physical experiments and phenomena requiring the above model for their description. The Chandrasekhar method for constructing the probability density depending simultaneously upon two fluctuating variables (being for our problem the generalized coordinate and velocity) is enunciated. This method is not widely known. We also enunciate the Lagrange method of constants variation; this method forms the basis for obtaining the solution of the differential equation describing the harmonic oscillator underwent stochastic forces. Using this method, we obtain the specific formulas designed to describe the time dependence of the variance of the generalized oscillator coordinate, the variance of its generalized velocity, and the correlator of these two as well as the mean values of the generalized coordinate and velocity. These formulas are valid for the case of the damped oscillations when the oscillator eigen frequency exceeds the damping coefficient. The resulting formulas are analyzed for the limiting cases. In particular, we show that at small values of time the variance of the stochastic harmonic oscillator velocity evolves with time in the same way as the variance of the coordinate for free Brownian particle.