The Wiener trajectories approach

3 The Wiener trajectories approach An original formalism for the treatment of many-particle effects in the A -f B B reaction was developed in a series of papers by Berezhkovskii, Machnovskii and Suris [54-59]. It is based on the so-called Wiener trajectories and related the Wiener sausages concept (the spatial region visited by a spherical Brownian particle during its random walks) [55, 60, 61]. It was shown that the convential survival probability for a walker among traps, which could be presented in a form [47] 

Since the particle will not die during the time t if it spends all this time inside a cavity free from traps, we introduce a d-dimensional sphere of an arbitrary radius R, surrounding the starting point of a particle A and write down evident inequality 

Up to now, no restriction have been imposed on the radius of the sphere R. Now we shall optimise our estimation choosing R — Rt so that the right-hand side of the inequality (5.2.39) is maximal at a certain instant time t. Substituting the value Rt obtained in this way, into equation (5.2.39) one obtains the probability of particle survival  

This estimate is of a special interest since it predicts a higher probability of particle survival than the conventional expression having the form (J 3) 

A comparison of this equation with equations (5.2.40) shows that the fluc-tuational decrease of particle disappearance rate takes place both for Db = 0 and Db Y if the trap diffusion takes place slowly enough, 

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