Solutions of Master Equations and Related Random Walks on Quenched Linear Chains

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Abstract
Four separate but related contributions to the theory of quenched stochastic processes in one dimension are presented. First, Green's functions are derived for Laplace transformed master equations (here described in the language of, but not restricted to, hopping excitons) on finite chains with either periodic or free‐end boundary conditions, and with either a disruptive (substitutional impurity) or a nondisruptive quencher. We solve these problems in spectral form for short‐range quenching with arbitrary quencher location and quenching rate parameter Qo. Second, the analogous random walk situations are treated. The solution of the generating function (finite‐difference analog of the Laplace transform) equation is identical to that of the Laplace‐transformed master equation with a disruptive quencher, but not with a nondisruptive quencher. Unlike the master equation case, slowly damped oscillations of the random walk chain excitation function can exist. Other differences also exist and are discussed; these occur at short times, with large concentration of quenchers, or in the presence of uniform decay processes like fluorescence. Third, the spectral solutions of the master equations are applied to provide a closed analytical expression for the lowest moment of the chain excitation function on a particular chain in the presence of fluorescence, and to demonstrate the existence of an inflection point in the excitation function of an open chain excited and quenched at opposite ends. Fourth, we apply the coherent potential approximation to the master equation that describes a uniformly excited, randomly and nondisruptively quenched infinite chain. A closed analytic expression is obtained for the lowest moment of the excitation function in the absence of fluorescence, and is compared with previously obtained expressions for infinite chains with periodic nondisruptive quenchers or random disruptive quenchers. Finally, we discuss briefly the possibility of applying our results to the problem of metal‐ion quenching of the phosphorescence of polyriboadenylic acid.