Correlation function of the tangent vectors

Next let us consider the time correlation function of the tangent vector 

An important conclusion is derived from eqn (6.43) by a geometrical interpretation of (u(s, t) u(s, 0)). First note that if u(s) and (s ) are the tangent vectors of a Gaussian chain of step length a, the integral 

This is equal to unity at time r = 0. As time passes, the original tube segment s becomes empty of the primitive chain. If this happens u(s, 0) becomes totally uncorrelated to u(s, t) and the contribution from such case to ip(js, t) is zero. Therefore s, t) represents the probability that the original tube segment s is remaining at time t. Indeed from eqns (6.43) and (6.45), it follows that 

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