Statistical Dynamics of Simple Cubic Lattices. Model for the Study of Brownian Motion

The system considered is an n-dimensional cubic crystal with nearest-neighbor central and noncentral harmonic forces in which the mass M of one of the lattice particles is relatively large. It is assumed that the velocities and positions of the light particles in the system (mass m) are normally distributed, at time t=0, as in thermal equilibrium. The conditional velocity distribution for the heavy particle at time t is then a normal distribution with a time-dependent mean value. This mean value is the velocity autocorrelation function. The dispersion of the distribution is shown to be a simple function of the autocorrelation. In the limit M/m≫1 in the one- and two-dimensional lattices, the autocorrelation function is, respectively, a damped exponential and a damped oscillating exponential. These different types of statistical behavior are related to the different dynamic properties of the medium with which the heavy particle interacts.