Relations between the Friction Constant and the Force Correlation Integral in Brownian Movement Theory A Comparison of the Microscopic and Macroscopic Theories

Abstract

Kirkwood's expression for the friction constant is compared with the expression implicit in Brownian movement theory. Relations between the friction constant β and the fluctuating force F(t) are obtained in the case of a white spectrum and of a general spectrum. The former relation resembles Kirkwood's expression and depends on the correlation function of F(t) being sufficiently sharply peaked to permit the integral to reach a plateau value in a time τ1 such that βτ1 1. Since Kirkwood's expression involves the total force X(t) = F(t) - βu, the correlation integral of this force is calculated in the macroscopic theory of Brownian movement. The infinite integral ∫∞0 <X(t)X(t + τ)>tdτ is found to be zero, but the incomplete integral ∫τ10 <X(t)X(t + τ)>tdτ is found to be equal to the incomplete correlation integral of the fluctuating force ∫τ10 <F(t)F(t + τ)>tdτ provided that the upper limit of integration τ1 is large enough for the correlation function <F(t)F(t+τ)>t to have decayed to zero, but is sufficiently small to ensure that βτ1 1. It follows that there is correspondence between the microscopic and macroscopic theories provided this condition holds.