The Encounter Probability for Random Walkers in a Confined Space
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The Encounter Probability for Random Walkers in a Confined Space James P. Lavine Display and Components, Image Sensor Solutions Eastman Kodak Company 1999 Lake Avenue Rochester, NY 14650-2008, U.S.A. ABSTRACT Particles diffusing in a confined space should encounter one another with a probability that depends on the size and dimension of the space. The present work uses pairs of random walkers on a lattice to investigate the encounter probability in one, two, and three spatial dimensions. There is an initial rapid decay of the survival-time distribution that is followed by an exponential decay in time. The characteristic time for this latter decay is strongly dependent on the model space size and scales as a power law in the size. The exponent of the power law depends on the number of spatial dimensions. For a fixed L, the exponential tail of the survival-time distribution has a similar slope when the initial separation of the two walkers is varied. The spacing between the exponential decay curves scales with the initial separation in 1-D, but not in 2-D or 3-D. In addition, the mapping of two random walkers to an equivalent single walker is explored. 1. INTRODUCTION Particles diffusing in a confined space should encounter one another with a probability that depends on the size and dimension of the space. This encounter probability is useful in predicting reaction rates in molecular systems [1], in calculating collision rates of excitations [2], and in considerations of dark current in the pixels of solid-state image sensors [3]. The latter involves metal complexes such as copper pairs, an iron-boron complex, or a gold and iron complex in silicon. The dark current generation rate is different for single atoms and for complexes [4]. Treatments of encounter probabilities in the literature appear to go back to Noyes [1], who bases his approach on the classic review paper of Chandrasekhar [5]. Noyes uses the solution of the diffusion equation in three dimensions for independent particles. Textbooks [6] favor the diffusion equation with a relative diffusion coefficient, which is the sum of the diffusion coefficients for the diffusing particles. One equivalent diffusing particle is considered, and the encounter occurs when the diffusing particle reaches a trap or sink. The present work uses pairs of random walkers on a lattice to investigate the encounter probability in one, two, and three spatial dimensions. Pairs are used to prepare for future work with spatially dependent diffusion coefficients and correlation studies. The second section describes the numerical method, and the third section contains the computational results. The fourth section discusses the results with a single, equivalent walker. The final section has the conclusions. 2. NUMERICAL METHODS
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A random walk approach is used on a lattice with sites that are one unit spacing apart in each active dimension. The model space runs from –L to L in each dimension that is active. At each time step, both random walkers take a step. I
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