Stochastic equation for reptation dynamics

1 Stochastic equation for reptation dynamics Although the above probabilistic description is quite useful in understanding the essence of reptation dynamics, it becomes progressively more difficult to proceed with the calculation for other types of time correlation function. For example, it is not easy to calculate the mean square displacement of a primitive chain segment (R(s, t)-R(s, 0)) ) by this method. In this section we shall describe a convenient method for calculating general time correlation functions. 

First we derive a simple mathematical equation for reptation dynamics. Let A (0 be the distance that the primitive chain moves in a time interval between t and t + Ar, then 

Equation (6.22) is not correct for all 5 if s + A (r) is not between 0 and L, R(s, t + At) should be on the newly created part of the tube (see Fig. 6.6). Writing down this condition in a formal mathematical equation is cumbersome. However, such an expression is not needed in practice since the condition can usually be accounted for by the fact that the distribution of the tangent vectors at the chain ends ii(0, r) and (L, t) are independent of the previous conformation of the primitive chain since their correlation time is infinitesimal, t 

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