Hopping models of charge transfer in a complex environment: Coupled memory continuous-time random walk approach Ewa Gudowska-Nowak, 1 Kinga Bochenek, 1 Agnieszka Jurlewicz, 2 and Karina Weron 3 1 Marian Smoluchowski Institute of Physics Jagellonian University, ul. Reymonta 4, 30-059 Kraków, Poland 2 Hugo Steinhaus Center for Stochastic Methods and Institute of Mathematics and Computer Science, Wroclaw University of Technology, ul. Wybrzeże Wyspiańskiego 27, 50-370 Wroclaw, Poland 3 Institute of Physics, Wroclaw University of Technology, ul. Wybrzeże Wyspiańskiego 27, 50-370 Wroclaw, Poland Received 15 September 2004; revised manuscript received 13 May 2005; published 8 December 2005 Charge transport processes in disordered complex media are accompanied by anomalously slow relaxation for which usually a broad distribution of relaxation times is adopted. To account for those properties of the environment, a standard kinetic approach in description of the system is addressed either in the framework of continuous-time random walks CTRWs or fractional diffusion. In this paper the power of the CTRW ap- proach is illustrated by use of the probabilistic formalism and limit theorems that allow one to rigorously predict the limiting distributions of the paths traversed by charges and to derive effective relaxation properties of the entire system of interest. In particular, the standard CTRW scenario is generalized to a new class of coupled memory CTRWs that effectivly can lead to the well known Havriliak-Negami response. Application of the method is discussed for nonexponential electron-transfer processes controlled by dynamics of the surround- ing medium. DOI: 10.1103/PhysRevE.72.061101 PACS numbers: 05.40-a, 82.20.Fd, 87.10.+e I. INTRODUCTION The stochastic formulation of transport phenomena in terms of a random walk process, as well as the description via the deterministic diffusion equation are two fundamental concepts in the theory of diffusion in complex systems. The best known examples are charge transport in amorphous semiconductors, rebinding kinetics in proteins, polarization fluctuations in inhomogeneous solvents, diffusion of con- taminants in complex geological formations, and diffusion of pollutants in large ecosystems. In all realms mentioned above, the complex structures, characterized by a large di- versity of elementary units and strong interaction between them, exhibit a nonpredictable or anomalous temporal evo- lution. The possibility of the dual description of the anoma- lous dynamical properties of such systems, based either on the random motion or on the differential equations for the probability density functions, has been considered in litera- ture since the late 1960s and gave rise to an extensive list of developed models 1–3. In this paper we demonstrate the power of the mathemati- cal tools underlying the concept of a continuous-time ran- dom walk CTRW by showing how the tool can be gener- alized to handle complicated situations such as diffusion- reaction schemes in complex system. The notion of the CTRW, a walk with a waiting time distribution governing the time interval between subsequent jumps of a random walker, has been introduced by Montroll and Weiss 1. The distri- bution of waiting times may stem from possible obstacles and traps that delay the particle’s motion and as a conse- quence, introduce the memory effect into the kinetics. Espe- cially fascinating in this approach was the idea of an infinite mean time between the jumps as in such a case a character- istic time scale of the process looses its common sense. This novel concept has been used by Montroll and Scher 4 to give a first explanation of experiments measuring transient electrical current in amorphous semiconductors. Since then the CTRW formalism has been successfully applied to de- scribe fully developed turbulence, transport in fractal media, intermittent chaotic systems, and relaxation phenomena. The common feature of the abovementioned applications is that they exhibit anomalous diffusion manifested by a non- Gaussian asymptotic distribution propagator, diffusion front of a distance reached at large times. At the level of the CTRW modeling, the diverging mean waiting time leads to a subdiffusive motion with the mean square displacement growing as r 2 t  t  with 0    1. When applied to the theory of Brownian motion, the CTRW scenario leads to the fractional diffusion equation 5,6 that can be treated on an equal footing with the framework used for systems with normal diffusion. Usually, in applications of the CTRW ideology, the analy- sis of the asymptotic distribution is presented within the ap- proach that is based on a formal expression for the Fourier- Laplace transform of the propagator, or otherwise, use of the fractional calculus is required 3 as a legitimate tool. Here, we present an approach to a random walk analysis which is based directly on the definition of the cumulative stochastic process. Our aim is to show that despite the extensive studies on CTRWs and their long history in physics, the powerful tool of the limit theorems 7 hidden behind the derivation of limiting distributions, has not been fully explored yet. We emphasize the possibilities of applications of that scenario in stochastic modeling of physical systems, in particular, in de- scription of the charge transport in disordered materials. The main objective is to present a clear random walk scheme leading to the nonexponential polarization relaxation ex- pressed in terms of the well-known Havriliak-Negami func- tion. Our effort is therefore directed toward bringing into light statistical conditions underlying the rigorous results and PHYSICAL REVIEW E 72, 061101 2005 1539-3755/2005/726/06110111/$23.00 ©2005 The American Physical Society 061101-1