1. The probability of a particle escaping a given cell so that it can it can move either to the cell’s right or left can be calculated as: Where  is defined as: Our analysis to calculate the diffusion coefficient D starts by using the expression for the dispersion in one dimension: where, is a recursive relation of the periodically flashing infinite potentials where  is the jumping frequency. Combining these two equations, D can be expressed in terms of A : Expressing A in terms of the escaping probabilities R and L, Replacing A in the equation for the diffusion coefficient to describe the diffusion coefficient in terms of the escaping probabilities In order to study the current of Brownian motion in this environment, the average particle current is Using the escaping probabilities for R and L, the first moment of the distribution is calculated The current can now be calculated as: Diffusion plays an essential role in a wide variety of physical and chemical phenomena. In our study, we focus on cellular media which consist of consecutive finite cells separated by permeable walls. This medium has been identified with multiple materials ranging from biological tissues to soap suds. It has been observed that transport properties, such as diffusion, are highly sensitive to structural changes in the medium. Introducing a bias, such as randomly distributed barriers along a simple geometrical representation of a medium, affects a particle’s motion and its eventual diffusion across the medium. A formulation based on a microscopic model and a diffusion relaxation condition is used to derive an equation for the diffusion coefficient as a function of the concentration of barriers, the lifetime of such barriers, and the strength of a constant external field. A general analytic expression to calculate the diffusion coefficient with randomly distributed barriers on a one dimensional lattice is proposed; using Monte Carlo simulations, we are attempting find a significant correlation in the diffusion coefficients calculated via simulation and those calculated via the proposed analytical equation. This representation of the model consists of n+1 sites. Cell k is surrounded by two other cells of varying sizes. p and 1-p are the microscopic jumping probabilities for a particle to move to its nearest neighbor. R and L are the probabilities of escaping the cell. Combined Effect of Randomly Distributed Barriers and External Fields on the Diffusion Coefficient of a Single Particle Mario M. Apodaca 1, Juan M. R. Parrondo 2, and Hernan L. Martinez 1 1 Department of Chemistry, California State University, Dominguez Hills, Carson, CA 2 Grupo Interdisciplinar de Sistemas Complejos (GISC) and Departamento de Física Atómica, Molecular y Nuclear, Universidad Complutense, Madrid, Spain Understanding anomalous diffusion is of key importance in describing a variety of relevant phenomena in biology, chemistry, and physics. We gratefully acknowledge support from the National Institutes of Health through Grants Numbers: NIH MBRS RISE R25 GM62252 NIH NIGMS/MBRS SCORE S06 GM08156 Abstract Motivation Objective The purpose of this work is to study the diffusion coefficient of a single particle moving along a geometrical representation of cellular media while under the influence of a constant external field. We use a statistical microscopic model of a biased random walk to study this phenomenon. Diffusion: Theory versus Simulation Calculate the diffusion coefficient of a single particle in one dimension with randomly distributed barriers that are placed at random locations along the lattice when reappearing after time t. Analytical Expression Schematic Representation of the Model Monte Carlo Simulations We simulate a single particle on a one-dimensional discrete lattice with an infinite series of consecutive cells. The particle can jump to either of its nearest neighbors provided a barrier does not obstruct its path. Barriers are randomly distributed. Trajectories consist of 3 x 10 4 units of time. A total of 10 5 trajectories for each selection c, , p are explored. Periodic boundary conditions are used to simulate an infinite lattice. Preliminary Results We have found an excellent match between our computer simulations and our analytical calculations both qualitatively and quantitatively. Small discrepancies can be observed for low values of the product of the variables ct. We believe this is due to the fact the assumption of a compact random walk within a given cage during the time the gates are closed is not met. Future Research Acknowledgements Current: Theory versus Simulation