Average volume of the domain visited by randomly injected spherical Brownian particles in d dimensions

In order to extend the greatly simplified Smoluchowski model for chemical reaction rates it is necessary to incorporate many-body effects. A generalization with this feature is the so-called trapping model in which random walkers move among a uniformly distributed set of traps. The solution of this model requires consideration of the distinct number of sites visited by a single n-step random walk. A recent analysis [H. Larralde et al., Phys. Rev. A 45, 1728 (1992)] has considered a generalized version of this problem by calculating the average number of distinct sites visited by N n-step random walks. A related continuum analysis is given in [A. M. Berezhkovskii, J. Stat. Phys. 76, 1089 (1994)]. We consider a slightly different version of the general problem by calculating the average volume of the Wiener sausage generated by Brownian particles generated randomly in time. The analysis shows that two types of behavior are possible: one in which there is strong overlap between the Wiener sausages of the particles, and the second in which the particles are mainly independent of one another. Either one or both of these regimes occur, depending on the dimension. © 1996 The American Physical Society.