ON DIFFUSION BY DISCONTINUOUS MOVEMENTS, AND ON THE TELEGRAPH EQUATION

Abstract

At time t=0 a large number of non-interacting particles start from an origin and move with a uniform velocity υ along a straight line for an interval of time τ. To begin with, half move in each direction. Thereafter, and at the end of each successive interval of time τ, each particle starts a new partial path; it still moves with speed υ, and there is a probability p that it will continue to move in the same direction as in its previous path, and a probability q (= 1−p) that the direction of its velocity will be reversed, so the directions in any two consecutive intervals are correlated with a correlation coefficient c=p−q. The partial correlations for non-consecutive intervals are zero. The difference equation is found for the fraction γ(n,ν) of the number of particles at a distance y=νυτ from the origin after a time t=nτ; it is shown how γ(n,ν) may be computed; asymptotic formulae for large n are found, both for a fixed value p/q and for a fixed value of nq/p. The limiting density distribution (and the limiting characteristic function) are found when n→∞, τ→0, with nτ=t, νυτ=y, and in the limiting operation c=1−τ/A, with A constant, so that c→1, and the speed υ is kept constant; the limiting form of the difference equation is the telegraph equation, with υ2=(LC)−1, A=L/R, where L, C, and R are the self-inductance, capacitance, and resistance per unit length; and the limiting density distribution is the solution of this equation for an instantaneous source. If p+q≠1, and there is, at the end of each interval of time τ, a non-zero probability, 1−p−q, that a particle will escape from the system, then the same limiting operation, with T/(1−p−q)=G, G constant, leads to the telegraph equation with leakage, the leakage resistance being G/C. The solution of the telegraph equation is further considered (and in particular is given in terms of Lommel's function of two variables) when one end of a long cable is held at a constant potential for t≥0.