- Clustering in anomalous files of independent particles (2010-2011)

Highlights of [1]:

A previously un-described variant of the famous process of file dynamics (the diffusion of classical hard spheres in a quasi one dimensional channel) made of anomalous, independent, particles (whose jumping times are taken independently from, ψ_{α}(t)~t^{-1-α}, 0<α<1), exhibits a unique and exciting behavior: formation of clusters. We characterize this phenomenon mathematically, enabling a precise control through α, and prove that it’s stable through many numerical tests. We think that the clustering in this system may explain the mysterious phenomenon of rafts in biological membranes and can be used in regulating biological channels.

[1] O. Flomenbom

Clustering in anomalous files of independent particles

EPL**94**, 58001 (2011).

- New results in single file dynamics (2008-2010)

- How enzyme works? (2003-2005)

The dynamics of a tagged particle in a 1D system of N hard sphere was studied in a series of papers [1], [2], [3].
Firstly, we’ve studied a file with an initial particles’ density,

r=r_{0}l^{-a}, 0≤a≤1,

where l is a macroscopic length scale. We have derived the following formula for the mean square displacement (MSD) [1]:

$<\; r2>\; ~$r_{0} ^{(a-1)}
< r ^{2} _{free} >^{(1+a)/2}. (1)

Here < r^{2} > _{free} is the scaling law for a free particle,
so our result holds for any underlying dynamics.
This formula shows explicitly how the particles' density affects the tagged particle's diffusion,
thus enables to enhance very accurately its propagation while adjusting the power *a*.
We have also found that for a normal diffusion file,

the probability density function (PDF) of the tagged particle is Gaussian in position for any value of a.

Now, equation (1) holds when all particles are the same. For a heterogeneous file with a distribution of diffusion coefficients,

W(D)~D^{- g}, 0≤g≤1,

the MSD scales as [2],

< r^{2} > ~ t^{ m }, m = [1-g ] /
[2/(1+a)g ]. (2)

The PDF is still a Gaussian in position.

In [3], we look on hetereogeneous-renewal-anomalous files. A renewal anomalous file is a file in which the effective waiting time PDF for individual jumps for a sphere is of the form:

y(t)~t^{-1- a }, 0<a≤1,

for large t, and all the particles attemp to jump at the same time. There are several schemes that exhibit this scaling for y(t), yet each one of them has different dynamical properties. In a renewal-anomalous file, we can use the results for the MSD in normal dynamics with the substitution,

t→t^{ α }. (3)

We also find that in an anomalous file of independent particles, the dynamics are much slower than in its renewal counterpart.

[1] O. Flomenbom, A. Taloni

On single file and less dense processes

Europhys. Lett.**83**, 20004-p1-p6 (2008).

[2] O. Flomenbom

Dynamics of heterogeneous hard spheres in a file

Phys. Rev. E**82**, 031126 (2010).

[3] O. Flomenbom

Renewal-anomalous-heterogeneous files

Phys. Lett. A**374**, 4331 (2010).

r=r

where l is a macroscopic length scale. We have derived the following formula for the mean square displacement (MSD) [1]:

$<\; r2>\; ~$r

Here < r

the probability density function (PDF) of the tagged particle is Gaussian in position for any value of a.

Now, equation (1) holds when all particles are the same. For a heterogeneous file with a distribution of diffusion coefficients,

W(D)~D

the MSD scales as [2],

< r

The PDF is still a Gaussian in position.

In [3], we look on hetereogeneous-renewal-anomalous files. A renewal anomalous file is a file in which the effective waiting time PDF for individual jumps for a sphere is of the form:

y(t)~t

for large t, and all the particles attemp to jump at the same time. There are several schemes that exhibit this scaling for y(t), yet each one of them has different dynamical properties. In a renewal-anomalous file, we can use the results for the MSD in normal dynamics with the substitution,

t→t

We also find that in an anomalous file of independent particles, the dynamics are much slower than in its renewal counterpart.

[1] O. Flomenbom, A. Taloni

On single file and less dense processes

Europhys. Lett.

[2] O. Flomenbom

Dynamics of heterogeneous hard spheres in a file

Phys. Rev. E

[3] O. Flomenbom

Renewal-anomalous-heterogeneous files

Phys. Lett. A

To fully utilize RD forms in the analysis of the data, we developed a toolbox that builds a RD form from a two-state trajectory. See Software descriptions for details.