Random Walks on Lattices. III. Calculation of First-Passage Times with Application to Exciton Trapping on Photosynthetic Units

Abstract

The following statistical problem arises in the theory of exciton trapping in photosynthetic units: Given an infinite periodic lattice of unit cells, each containing N points of which (N − 1) are chlorophyll molecules and one is a trap; if an exciton is produced with equal probability at any nontrapping point, how many steps on the average are required before the exciton reaches a trapping center for the first time? It is shown that, when steps can be taken to near-neighbor lattice points only, as N → ∞, our required number of steps is 〈n〉∼{N2/6, linear chain,π−1NlogN, square lattice,1.5164N, single cubic lattice.The correction terms for medium and relatively small N are obtained for a number of lattices.