File dynamics

The term file dynamics is the motion of many particles in a narrow channel.
In science: in <a href="/facts/Chemistry/SrNqkGVm">chemistry</a>, <a href="/facts/Physics/a7aH2y08">physics</a>, <a href="/facts/Mathematics/pxTouaz4">mathematics</a> and related fields, file dynamics (sometimes called, single file dynamics) is the diffusion of N (N → ∞) identical <a href="/facts/Brownian_motion/Th0g0wkr">Brownian hard spheres</a> in a quasi-one-dimensional channel of length L (L → ∞), such that the spheres do not jump one on top of the other, and the average particle's density is approximately fixed. The most famous statistical properties of this process is that the <a href="/facts/Mean_squared_displacement/85lYfXRp">mean squared displacement</a> (MSD) of a particle in the file follows, 
 
 
 
 
 M
 S
 D
 
 ≈
 
 t
 
 
 1
 2
 
 
 
 
 
 {\displaystyle \mathrm {MSD} \approx t^{\frac {1}{2}}}
 
, and its <a href="/facts/Probability_density_function/zvfybna4">probability density function</a> (PDF) is <a href="/facts/Gaussian_function/5bG8Kpp9">Gaussian</a> in position with a variance MSD.
Results in files that generalize the basic file include: 

<ul><li>In files with a density law that is not fixed, but decays as a power law with an exponent a with the distance from the origin, the particle in the origin has a MSD that scales like, 
 
 
 
 M
 S
 D
 ≈
 
 t
 
 
 
 1
 +
 a
 
 2
 
 
 
 
 
 {\displaystyle MSD\approx t^{\frac {1+a}{2}}}
 
, with a Gaussian PDF.</li>
<li>When, in addition, the particles' diffusion coefficients are distributed like a power law with exponent γ (around the origin), the MSD follows, 
 
 
 
 M
 S
 D
 ≈
 
 t
 
 
 
 1
 −
 γ
 
 
 2
 
 /
 
 (
 1
 +
 a
 )
 −
 γ
 
 
 
 
 
 
 {\displaystyle MSD\approx t^{\frac {1-\gamma }{2/(1+a)-\gamma }}}
 
, with a Gaussian PDF.</li>
<li>In anomalous files that are renewal, namely, when all particles attempt a jump together, yet, with jumping times taken from a distribution that decays as a power law with an exponent, −1 − α, the MSD scales like the MSD of the corresponding normal file, in the power of α.</li>
<li>In anomalous files of independent particles, the MSD is very slow and scales like, 
 
 
 
 M
 S
 D
 ≈
 l
 o
 
 g
 
 2
 
 
 (
 t
 )
 
 
 {\displaystyle MSD\approx log^{2}(t)}
 
. Even more exciting, the particles form clusters in such files, defining a dynamical phase transition. This depends on the anomaly power α: the percentage of particles in clusters ξ follows, 
 
 
 
 ξ
 ≈
 
 
 1
 −
 
 α
 
 3
 
 
 
 
 
 
 {\displaystyle \xi \approx {\sqrt {1-\alpha ^{3}}}}
 
.</li>
<li>Other generalizations include: when the particles can bypass each other with a constant probability upon encounter, an enhanced diffusion is seen. When the particles interact with the channel, a slower diffusion is observed. Files in embedded in two-dimensions show similar characteristics of files in one dimension.</li></ul>
Generalizations of the basic file are important since these models represent reality much more accurately than the basic file. Indeed, file dynamics are used in modeling numerous microscopic processes: the diffusion within biological and synthetic pores and porous material, the diffusion along 1D objects, such as in biological roads, the dynamics of a monomer in a polymer, etc.

File dynamics open-in-new

File dynamics