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Daniel T. Gillespie

Abstract

In an attempt to resolve the confusion over the “stochastic completeness” of the stochastic coalescence equation, an analysis is made of an idealized cloud consisting of a number of large “drops” falling through very many small, equal-size “droplets” with a constant drop-droplet collection kernel. It is shown how three superficially equivalent physical interpretations of the drop-droplet collection kernel lead to three quite different models for the growth of the drops. The implications of and relationships between the models are drawn out in detail. The stochastic completeness controversy is apparently a consequence of a failure to distinguish clearly between two of these three conceptual approaches.

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Daniel T. Gillespie

Abstract

A computer-oriented Monte Carlo algorithm for simulating the stochastic coalescence process in a warm cloud is presented. The computational procedures used in the algorithm are rigorously derived from the fundamental premise of the stochastic model. The algorithm is more general, more accurate and more efficient than previously proposed Monte Carlo procedures; in addition, it is more exact than computations based on the stochastic coalescence equation, since it takes proper account of correlations that are artificially ignored in the derivation of that equation. The mechanics of the algorithm, its advantages and its limitations are discussed in general terms; applications to specific coalescence kernels are deferred to future publication.

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Daniel T. Gillespie

Abstract

The stochastic coalescence model for droplet growth in warm clouds is analyzed, with a view to clarifying the theoretical foundations and significance of the well-known stochastic coalescence equation. It is suggested that the analysis of the model is most logically carried out in terms of a function P (n, m; t which is defined as the probability that the number of cloud droplets consisting of m molecules at time t will be n. A time-evolution equation for P (n, m; t is derived, and under certain stated assumptions it is deduced that: 1) the mean value of P (n, m; t with respect to n satisfies the stochastic coalescence equation; and 2) regardless of the initial conditions, the graph of P (n, m; t vs n will approach the Poisson shape as t →∞ to with an estimable “relaxation tirne.” The implications of these results for the stochastic fluctuations in the number of cloud droplets are examined. It is found that a distinction must be made between fluctuations in droplet concentration arising from the assumed stochasticity of the coalescence process, and fluctuations in droplet concentration arising from the hypothesis that the droplets are randomly positioned in the cloud; the former fluctuations are normally very much smaller than the latter.

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