Stochastic self-assembly of incommensurate clusters

We examine the classic problem of homogeneous nucleation and self-assembly by deriving and analyzing a fully discrete stochastic master equation. We enumerate the highest probability steady-states, and derive exact analytical formulae for quenched and equilibrium mean cluster size distributions. Upon comparison with results obtained from the associated the mass-action Becker-D\"{o}ring (BD) equations, we find striking differences between the two corresponding equilibrium mean cluster concentrations. These differences depend primarily on the divisibility of the total available mass by the maximum allowed cluster size, and the remainder. When such mass ``incommensurability'' arises, a single remainder particle can ``emulsify'' the system by significantly broadening the equilibrium mean cluster size distribution. This discreteness-induced broadening effect is periodic in the total mass of the system but arises even when the system size is asymptotically large, provided the ratio of the total mass to the maximum cluster size is finite. 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.