Mathematics Colloquia and Seminars

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Nucleation and molecular aggregation are important processes in numerous physical and biological systems. In many applications, these processes often take place in confined spaces, involving a finite number of particles. We examine the classic problem of homogeneous nucleation and self-assembly by deriving and analyzing a fully discrete stochastic master equation. By enumerating the highest probability steady-states, we derive exact analytical formulae for quenched and equilibrium mean cluster size distributions. Comparing results with those from mass-action models reveals striking differences between the two corresponding equilibrium mean cluster concentrations. Our findings define a new scaling regime in which results from classic mass-action theories are qualitatively inaccurate, even in the limit of large total system size. First passage times to the formation of the largest cluster will also be discussed. Finally, we investigate how transitions involving coagulation and fragmentation affect first assembly times.