RANDOM-WALK ON A CIRCLE

We provide a theory about the random walk on a circle with radius R. T he analytical form of the probability distribution of the end to end v ector after n steps omega(theta,n) and the averaged square of end to e nd distance [r(2)] are given, Our findings are the following: when R(2 ) much greater than nl(2), [r(2)]approximate to nl(2) which restores t he result of the random walk in a 1D infinite space lattice; when R(2) much less than nl(2), [r(2)]approximate to pi(2)R(2)/3. The general e xpressions for the average end to end distance[\r\] and the kth moment [r(k)] are given.