Correlations in Single File Diffusion: Open and Closed Systems

2014 ◽  
Vol 09 (04) ◽  
pp. 367-379 ◽  
Author(s):  
Ashwani Kr. Tripathi ◽  
Deepak Kumar
Keyword(s):  
Diffusion Constant ◽  
Mean Square Displacement ◽  
Hard Core ◽  
Constant Density ◽  
Special Issue ◽  
Tagged Particle ◽  
Single File ◽  
First Case ◽  
Single File Diffusion ◽  
Core Particles

We present a discussion of positional and velocity correlations of particles in single-file diffusion, based on some earlier work. We consider two physical situations: (a) An open system of N hard-core particles on an infinite line. (b) A large system with a fixed density of hard-core particles at an arbitrary temperature. In the first case (a), moments and correlations show unusual behavior. The average displacement of a particle is nonzero and grows as t1/2. Furthermore it depends on the position of the particle. Particles on the right of center are pushed right and those on the left are pushed left. The mean-square displacement also depends on the position. The diffusion constant is small for particles around the center but grows rapidly toward edges. Certain correlations in particle displacement grow with separation. For the second case (b) we give exact results for velocity-velocity auto-correlator of a tagged particle and establish that with time this correlator becomes negative and approaches zero as a power-law t-3/2 at long times. The mobility of the tagged particle is shown to decrease rapidly with density as has been observed in experiments. [Formula: see text] Special Issue Comments: This article presents mathematical results on the dynamics in expanding files, and constant density files. This article is connected to the Special Issue articles about advanced statistical properties in single file dynamics29 and files with force and advanced formulations.30

Single-File System with Absorbing Boundary: Tracer Dynamics and First-Passage Properties

2015 ◽  
Vol 10 (02) ◽  
pp. 85-96
Author(s):  
Artem Ryabov
Keyword(s):  
Diffusion Coefficient ◽  
Hard Core ◽  
Tracer Diffusion ◽  
Constant Density ◽  
Special Issue ◽  
Absorbing Boundary ◽  
Exact Probability ◽  
Tagged Particle ◽  
Single File ◽  
Number Of Particles

In this paper, we review the tagged particle dynamics in a semi-infinite system with an absorbing boundary. The emphasis is on an interplay between the hard-core interparticle interaction and the absorption process. The exact probability density function for the position of a tagged particle is derived by means of probabilistic arguments. First, the initially homogeneous system with constant density of particles is studied. In this setting, the dynamics of the tracer conditioned on nonabsorption becomes subdiffusive, the generalized diffusion coefficient being different from that reported for the system without absorbing boundary. Second, the case when the initial number of particles is finite is discussed. In this case, in the long time limit the tracer diffusion is normal and the hard-core interaction manifests itself through the renormalization of the tracer diffusion coefficient. The Gaussian distribution derived for infinite single-file systems is, in the present semi-infinite setting, replaced by the Rayleigh distribution. [Formula: see text]Special Issue Comments: This article presents results on the dynamics of a tagged particle in open systems, where the number of particles is not conserved in time. This article is related to the Special Issue articles about advanced statistical properties in single file dynamics,1 the calculation of correlations,2 files with force3 and the zig-zag patterns in files.4

Longitudinal and Transverse Single File Diffusion in Quasi-1D Systems

2014 ◽  
Vol 09 (04) ◽  
pp. 333-348 ◽  
Author(s):  
Christophe Coste ◽  
Jean-Baptiste Delfau ◽  
Michel Saint Jean
Keyword(s):  
Normal Modes ◽  
Mean Square Displacement ◽  
Transverse Mode ◽  
Pitchfork Bifurcation ◽  
Thermal Bath ◽  
Special Issue ◽  
Translational Invariance ◽  
Single File ◽  
A Chain ◽  
Single File Diffusion

We review our recent results on Single File Diffusion (SFD) of a chain of particles that cannot cross each other, in a thermal bath, with long ranged interactions, and arbitrary damping. We exhibit new behaviors specifically associated to small systems and to small damping. The fluctuation dynamics is explained by the decomposition of the particles' motion in the normal modes of the chain. For longitudinal fluctuations, we emphasize the relevance of the soft mode linked to the translational invariance of the system to the long time SFD behavior. We show that close to the zigzag threshold, the transverse fluctuations also exhibit the SFD behavior, characterized by a mean square displacement that increases as the square root of time. This cannot be explained by the single file ordering, and the SFD behavior results from the strong correlation of the transverse displacements of neighbouring particles near the bifurcation. Extending our analytical modelization, we demonstrate the existence of this subdiffusive regime near the zigzag transition, in the thermodynamic limit. The zigzag transition is a supercritical pitchfork bifurcation, and we show that the transverse SFD behavior is closely linked to the vanishing of the frequency of the zigzag transverse mode at the bifurcation threshold. [Formula: see text] Special Issue Comments: This article presents mathematical results on the dynamics in files with longitudinal movements. This article is connected to the Special Issue articles about advanced statistical properties in single file dynamics,28 expanding files,63 and files with force and advanced formulations.29

Single File Dynamics Advances with a Focus on Biophysical Relevance

2014 ◽  
Vol 09 (04) ◽  
pp. 307-331 ◽  
Author(s):  
Ophir Flomenbom
Keyword(s):  
Power Law ◽  
Mean Square Displacement ◽  
Constant Density ◽  
Special Issue ◽  
Dynamical Phase Transition ◽  
Gaussian Probability Density Function ◽  
Dynamical Phase ◽  
Single File ◽  
Biophysical Processes ◽  
Probability Density Function Pdf

In this review (appearing in the Special Issue on single file dynamics in biophysics and related extensions), three recently treated variants in file dynamics are presented: files with density that is not fixed, files with heterogeneous particles, and files with slow particles. The results in these files include:• In files with a density law that is not fixed, but decays as a power law with an exponent a the distance from the origin, the particle in the origin mean square displacement (MSD) scales like MSD ~ t[1+a]/2, with a Gaussian probability density function (PDF). This extends the scaling, MSD ~ t1/2, seen in a constant density file.• When, in addition, the particles' diffusion coefficients are distributed like a power law with an exponent γ (around the origin), the MSD follows MSD ~ t[1-γ]/[2/(1+a) - γ], with a Gaussian PDF.• In anomalous files that are renewal, namely, when all particles attempt a jump together, yet, with jump times taken from a PDF that decays as a power law with an exponent -1 - ε, ψ(t) ~ t-1-ε, the MSD scales like the MSD of the corresponding normal file, in the power ε.• In anomalous files of independent particles, the MSD is very slow and scales like MSD ~ log2(t). Even more exciting, the particles form clusters, defining a dynamical phase transition: depending on the anomaly power ε, the percentage of particles in clusters ξ follows [Formula: see text], yet when ε > 1, fluidity rather than clusters is seen.We talk about utilizing these results while focusing on biophysical processes and applications: dynamics in channels, membranes, biosensors, etc.[Formula: see text] Special Issue Comments: In this article, results about recently suggested variants in single file dynamics appear: heterogeneous files and slow files, yet also, the relevance with biophysical processes. It is related to the Special Issue articles about expansions in files,61files with force,62and the zig zag occurrences in files.63

scholarly journals Insights from Single-File Diffusion into Cooperativity in Higher Dimensions

2016 ◽  
Vol 11 (01) ◽  
pp. 9-38 ◽  
Author(s):  
Takeshi Ooshida ◽  
Susumu Goto ◽  
Takeshi Matsumoto ◽  
Michio Otsuki
Keyword(s):  
Large Scale ◽  
Mean Square Displacement ◽  
Colloidal Suspensions ◽  
Higher Dimensions ◽  
Colloidal Systems ◽  
One Dimensional ◽  
Single File ◽  
The Mean ◽  
The One ◽  
Single File Diffusion

Diffusion in colloidal suspensions can be very slow due to the cage effect, which confines each particle within a short radius on one hand, and involves large-scale cooperative motions on the other. In search of insight into this cooperativity, here the authors develop a formalism to calculate the displacement correlation in colloidal systems, mainly in the two-dimensional (2D) case. To clarify the idea for it, studies are reviewed on cooperativity among the particles in the one-dimensional (1D) case, i.e. the single-file diffusion (SFD). As an improvement over the celebrated formula by Alexander and Pincus on the mean-square displacement (MSD) in SFD, it is shown that the displacement correlation in SFD can be calculated from Lagrangian correlation of the particle interval in the one-dimensional case, and also that the formula can be extended to higher dimensions. The improved formula becomes exact for large systems. By combining the formula with a nonlinear theory for correlation, a correction to the asymptotic law for the MSD in SFD is obtained. In the 2D case, the linear theory gives description of vortical cooperative motion.

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