Characteristics of the probability function for three random-walk models of reaction–diffusion processes

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Abstract
A method is presented for calculating exactly the relative width (σ2)1/2/〈n〉, the skewness γ1, and the kurtosis γ2 characterizing the probability distribution function for three random‐walk models of diffusion‐controlled processes. For processes in which a diffusing coreactant A reacts irreversibly with a target molecule B situated at a reaction center, three models are considered. The first is the traditional one of an unbiased, n e a r e s t‐n e i g h b o rrandom walk on a d‐dimensional periodic/confining lattice with traps; the second involves the consideration of unbiased, n o n‐n e a r e s t‐n e i g h b o r (i.e., variable‐step length) walks on the same d‐dimensional lattice; and, the third deals with the case of a b i a s e d, nearest‐neighbor walk on a d‐dimensional lattice (wherein a walker experiences a potential centered at the deep trap site of the lattice). Our method, which has been described in detail elsewhere [P.A. Politowicz and J. J. Kozak, Phys. Rev. B 2 8, 5549 (1983)] is based on the use of group theoretic arguments within the framework of the theory of finite Markov processes. The approach allows the separate effects of geometry (system size N, dimensionality d, and valency ν), of the governing potential and of the medium temperature to be assessed and their respective influence on (σ2)1/2/〈n〉, γ1, and γ2 to be studied quantitatively. We determine the classes of potential functions and the regimes of temperature for which allowing variable‐length jumps or admitting a bias in the site‐to‐site trajectory of the walker produces results which are significantly different (both quantitatively and qualitatively) from those calculated assuming only unbiased, nearest‐neighbor random walks. Finally, we demonstrate that the approach provides a method for determining a continuous probability (density)distribution function consistent with the numerical data on (σ2)1/2/〈n〉, γ1, and γ2 for the processes described above. In particular we show that the first of the above reaction–diffusion models (and probability the second) can be described q u a n t i t a t i v e l y by an exponential distribution function in d=2,3 and by a Pearson type III distribution in d=1.