Diffusion of Tagged Particle in an Exclusion Process E. Barkai Department of Physics, Bar Ilan University, Ramat-Gan 52900, Israel R. Silbey Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA We study the diffusion of tagged hard core interacting particles under the influence of an external force field. Using the Jepsen line we map this many particle problem onto a single particle one. We obtain general equations for the distribution and the mean square displacement 〈 ( x$_{T}$ ) $^{2}$〉 of the tagged center particle valid for rather general external force fields and initial conditions. A wide range of physical behaviors emerge which are very different than the classical single file sub-diffusion 〈 ( x$_{T}$ ) $^{2}$〉 ∼ t 1 / 2 found for uniformly distributed particles in an infinite space and in the absence of force fields. For symmetric initial conditions and potential fields we find 〈 ( x$_{T}$ ) $^{2}$〉 = R (1 −R ) 2 N $_{r}$2 where 2 N is the (large) number of particles in the system, R is a single particle reflection coefficient obtained from the single particle Green function and initial conditions, and r its derivative. We show that this equation is related to the mathematical theory of order statistics and it can be used to find 〈 ( x$_{T}$ ) $^{2}$〉 even when the motion between collision events is not Brownian (e.g. it might be ballistic, or anomalous diffusion). As an example we derive the Percus relation for non Gaussian diffusion. PACS numbers: 05.60.k, 02.50.Ey, 05.40.Jc, 05.70.Ln Systems of particles governed by stochastic dynamics and exclusion interactions have been studied for a long time [1–3]. One aspect of this problem is the motion of a tagged particle sometimes called the tracer particle [4–7]. The diffusion of a tagged particle, in a one dimensional system of Brownian particles, interacting via hard core interaction is a model for the motion of a single molecule in a crowded one dimensional environment such as a bi- ological pore or channel [8–11], and in experimentally studied physical systems such as zeolites [12], confined colloid particles [13–15] and charged spheres in circular channels [16]. Since particles do not pass each other such diffusion processes are called single file diffusion. Confinement of a tagged Brownian particle, due to its interaction with other Brownian particles, leads to a slow-down of the diffusion of the tagged particle 〈 ( x$_{T}$ ) $^{2}$〉 ∝ t 1 / 2 instead of normal diffusion 〈 ( x$_{T}$ ) $^{2}$〉 ∝ t when the particles in the system are uniformly dis- tributed [4, 6]. Such many body problems can be treated using methods which exploit the relation between the dy- namics of the interacting tagged particle and the motion of a particle free of interactions [5, 6, 17–20]. In recent years at-least six new directions of research emerged. (i) The effect of an external field acting on all the particles [21] or on the tagged particle only [22, 23] has attracted attention since pores induce entropic barriers [10] and are generally inhomogeneous. In this category the ex- amples of single file motion in a periodic potential [24] or in a box [25] were considered in detail. (ii) Initial conditions may have a profound impact on diffusion of the tagged particle. For example if particles start as a narrow Gaussian packet the diffusion of the tagged cen- ter particle increases linearly in time 〈 ( x$_{T}$ ) $^{2}$〉 ∝ t [26] (see details below). Power law initial conditions induce Here we provide a very general theory of single file dif- fusion of the center tagged particle, valid in the presence or the absence of an external potential field V ( x ), as well as for thermal and non-thermal initial conditions. Our main results reproduce previously obtained formulas and many new ones, by mapping the many particle problem onto a single particle model. Our method, explained in Sec. II, exploits the theoretical concept of the Jepsen line [5], is limited to hard core point particles, but, as we show in Sec. V, is not limited to Brownian particles. 〈 ( x$_{T}$ ) $^{2}$〉 ∝ t ξ and ξ is neither 1 nor 1 / 2 [27] (see further details in text). When an external field is acting on the system, it is important to consider non uniform initial conditions where the density of particles is determined naturally from Boltzmann’s distribution. (iii) In some investigations the underlying motion is not normal diffu- sion, instead the particles may be non-Brownian, and fol- lowing anomalous kinetics [27, 28]. (iv) Usually hard core interactions are considered, although the hard problem of more general interactions has been recently treated in [23, 29, 30]. Screened hydrodynamic interactions which seem important at short times were investigated in [31] both theoretically and experimentally. Granular single file diffusion with inelastic collisions shows the typical t 1 / 2 sub-diffusion [32, 33]. (v) Interacting particles in systems with quenched disorder is yet another challenge. The motion of a tagged particle was recently treated in the context of single file diffusion in the Sinai model [34]. (vi) Finally, if the motion of the tagged particle is not normal Brownian motion, what stochastic theory replaces the usual Brownian-Langevin framework? In this direction interesting connections to fractional cal- culus emerged [23, 35, 36]. Roughly speaking and under certain conditions half order time derivatives ( d 1 / $^{2}$/dt1 / $^{2}$) enter in the Langevin equation [23] and fractional Brow- nian noise replaces white noise. I. INTRODUCTION arXiv:1001.0732v1 [cond-mat.stat-mech] 5 Jan 2010