From the diffusion equation to escape and survival probabilities

2 FROM THE DIFFUSION EQUATION TO ESCAPE AND SURVIVAL PROBABILITIES 

As previously remarked, the motion of an isolated pair of reactant radicals can be described by the diffusion equation. An isolated pair is formed at a time t0 and with a separation r0. The probability that these radicals are separated by r at a time t is p(r, t r0, f0) and this is a conditional probability (conditional upon the formation time and distance of separation). The exact location of the radical pair is unimportant only the 

The delta functions are zero unless r = r0 and t = t0, and the rate of radical pair creation is, in effect, infinitely fast. Consequently, eqn. (10) becomes 

Comparison with eqn. (A.10) of Appendix A, shows that p is the fundamental or Green s function solution of the diffusion equation, eqn. (10). 

To connect the experimental results of Chuang et al. [266] and Khudyakov et al. [267] and others with the diffusion equation (122), it is necessary to define the probability that the radical pair is extant at a time t, given that it was formed at t0 with separation r0 i.e. the survival probability, p(t r0, f0) is given by 

---