The failure of isotropic continuous time random walk to model diffusion in porous media and a redeeming modification Shahar Amitai 1∗ and Raphael Blumenfeld 1,2,3 1. ESE, Imperial College London, London SW7 2AZ, UK 2. College of Science, NUDT, Changsha, Hunan, China 3. Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, UK (Dated: January 26, 2015) We test the fidelity of modelling diffusion processes of finite-size particles in porous media, sim- ulated numerically, by continuous time random walk (CTRW) of the same step size and waiting time distributions. The waiting times depend on particle size. We find that the CTRW predicts a universality-class normal-to-anomalous diffusion transition at different particle sizes. We show that the discrepancy is due to the change in connectivity (topology) of the porous media with increasing particle size. We propose a method to improve the CTRW model by adding anisotropy. This ad- justment yields good agreement with the simulated diffusion process, reinstating the CTRW, with its advantages, as an appropriate model for diffusion in confined geometries. 1. INTRODUCTION Diffusion plays a key role in a wide range of nat- ural and technological processes. A textbook mod- elling of such processes is the consideration of the diffusion of a single memory-free particle in a given medium. The nature of such a random walk is de- termined by three probability densities (PDFs): of the step size l, P l (l); of the step direction ˆ n, P n (θ,φ); and of the waiting time between steps t, P t (t). These PDFs are, in principle, position dependent, but it is standard practice to derive (or postulate) them as- suming position-independence and that P n (θ,φ) is uniform. The diffusion is then modelled as a con- tinuous time random walk (CTRW) in free space, mainly because this alleviates finite size errors due to finite samples. Specifically, the CTRW is con- structed by adding vectors of uniformly random ori- entations, whose lengths are chosen from P l , with waiting times chosen from P t . Averaging over suffi- ciently many such independent processes, the depen- dence of the mean square distance (MSD) on time is determined through MSD = DT α . In normal dif- fusion α = 1 and D is the standard diffusion coeffi- cient. When P t has a slowly decaying algebraic tail and P l does not, the random walk is sub-diffusive (α< 1) [1]. Alternatively, if P l has a slowly decay- ing algebraic tail and P t does not, the random walk is super-diffusive (α> 1), resembling a L´ evy flight [2]. Diffusion processes that have the same value of α are said to be in the same universality class [3]. Using CTRW to model diffusion in confined ge- ometries, such as porous media or different biologi- cal systems, is attractive [4–6] because it alleviates the need to simulate directly the dynamics of parti- cles within the pore space, reducing significantly the computational burden. This practice is based on the common assumption that the three aforementioned distributions alone control the random walk’s uni- versality class. The common procedure is to find first the forms of these distributions in a specific medium, using either small simulations or analytic derivation under some assumptions, and then use these to carry out a many-step CTRW in free space. It is then pre- sumed that the CTRW yields the same universality class as the diffusion in the confined geometry. The aim of this paper is to demonstrate that this is a misconception when the diffusing particles have finite sizes. We do so by comparing a simulation in an actual porous sample with a CTRW in free space, based on statistical information of the porous media. We show that the medium’s connectivity (topology) depends crucially on the particle’s size and this af- fects directly the nature and universality class of the diffusion process. We then propose a method to cor- rect for this effect, which makes it possible to still use CTRW, with its advantages, to model diffusion in confined geometries. The structure of this paper is the following. In section 2 we describe the simulated porous samples. In section 3 we give details about the diffusion pro- cess and present results for a point-like particle. In section 4 we discuss the effects of particle size on the diffusion process and present explicit results. In section 5 we describe the equivalent CTRW simula- tions and show that these yield a different universal- ity class in spite of having the same step length and waiting time distributions. We propose an explana- tion of this discrepancy. In section 6 we propose an arXiv:1501.03998v2 [cond-mat.stat-mech] 21 Jan 2015