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(a) Comparison of simulated (symbols) versus analytic solutions via subordination (curves). The solid curves, largely obscured by the symbols, use the subordinator (7) for diffusion toward moving boundaries with 2H = 1.5. The dotted line assumes fixed island size of 5Ω/N 0 and 2H = 1. The particle simulations are invariant with dimensionless constant . (b) Example subordinator densities h(u, t) for stationary boundaries (thin lines) versus moving boundaries (thick lines) for β = 0.1. Each density is shown at 4 different clock times t. While similar at early time (red curves), the moving boundaries severely reduce operational time at later time (purple curves) due to increasing distance to island boundaries.
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The connection between the governing equations of chemical reaction and the underlying stochastic processes of particle collision and transformation have been developed previously along two end-member conditions: perfectly mixed and maximally diffusion-limited. The complete governing equation recognizes that in the perfectly mixed case, the particle (i.e., molecular or macro-particle) number state evolution is Markovian, but that spatial self-organization of reactants decreases the probability of reactant pairs finding themselves co-located. This decreased probability manifests itself as a subordination of the clock time: as reactant concentrations become spatially variable (unmixed), the time required for reactants to find each other increases and the random operational time that particles spend in the active reaction process is less than the clock time. For example, in the system A + B → ∅, a simple approximate calculation for the return time of a Brownian motion to a moving boundary allows a calculation of the operational time density, and the total solution is a subordination integral of the perfectly-mixed solution with a modified inverse Gaussian subordinator. The system transitions from the well-mixed solution to the asymptotic diffusion-limited solution that decays as t −d/4 in d-dimensions.
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