Waiting time distribution and nonexponential relaxation in single molecule spectroscopic studies: realization of entropic bottleneck in a simple model

Abstract

We study a dynamical disorder model for environmental modulation of rate processes where a need of dynamical cooperativity presents an entropy barrier, rather than an energy barrier. The rate depends on a control variable, Q, that describes the collective instantaneous state of the environment and is itself a random walker in finite discrete space with continuous time. We obtain the waiting time distribution for the relaxation by simulating the model. The time dependence of the average survival probability is derived there from and also by a numerical solution through the Liouville-master equation approach to the theoretical problem. We present an analytical treatment of the first passage time problem that is posed by a limiting case of our model. As the rate of the environmental fluctuation, τ^-1_env, slows down, the decay of the average survival probability is found to be more and more nonexponential in short times, but to change to exponential at longer times. The average survival time, τ, exhibits a fractional power law dependence on κ( = τ_envk₀), where time is scaled in terms of k^-1₀, k₀ being the intrinsic rate coefficient for the relaxation. The mean first passage time in the limiting case of the model exhibits an exponential dependence on the total number of the environmental subsystems N and a non-Arrhenious temperature dependence over the temperature range studied. We note the likely relevance of a part of this result to single molecule spectroscopic studies that reveal a tail in the waiting time distribution at long times.