14,855 research outputs found
Microscopic dynamics underlying the anomalous diffusion
The time dependent Tsallis statistical distribution describing anomalous
diffusion is usually obtained in the literature as the solution of a non-linear
Fokker-Planck (FP) equation [A.R. Plastino and A. Plastino, Physica A, 222, 347
(1995)]. The scope of the present paper is twofold. Firstly we show that this
distribution can be obtained also as solution of the non-linear porous media
equation. Secondly we prove that the time dependent Tsallis distribution can be
obtained also as solution of a linear FP equation [G. Kaniadakis and P.
Quarati, Physica A, 237, 229 (1997)] with coefficients depending on the
velocity, that describes a generalized Brownian motion. This linear FP equation
is shown to arise from a microscopic dynamics governed by a standard Langevin
equation in presence of multiplicative noise.Comment: 4 pag. - no figures. To appear on Phys. Rev. E 62, September 200
Microscopic dynamics underlying the anomalous diffusion
The time dependent Tsallis statistical distribution describing anomalous
diffusion is usually obtained in the literature as the solution of a non-linear
Fokker-Planck (FP) equation [A.R. Plastino and A. Plastino, Physica A, 222, 347
(1995)]. The scope of the present paper is twofold. Firstly we show that this
distribution can be obtained also as solution of the non-linear porous media
equation. Secondly we prove that the time dependent Tsallis distribution can be
obtained also as solution of a linear FP equation [G. Kaniadakis and P.
Quarati, Physica A, 237, 229 (1997)] with coefficients depending on the
velocity, that describes a generalized Brownian motion. This linear FP equation
is shown to arise from a microscopic dynamics governed by a standard Langevin
equation in presence of multiplicative noise.Comment: 4 pag. - no figures. To appear on Phys. Rev. E 62, September 200
Molecular theory of anomalous diffusion
We present a Master Equation formulation based on a Markovian random walk
model that exhibits sub-diffusion, classical diffusion and super-diffusion as a
function of a single parameter. The non-classical diffusive behavior is
generated by allowing for interactions between a population of walkers. At the
macroscopic level, this gives rise to a nonlinear Fokker-Planck equation. The
diffusive behavior is reflected not only in the mean-squared displacement
( with ) but also in the existence
of self-similar scaling solutions of the Fokker-Planck equation. We give a
physical interpretation of sub- and super-diffusion in terms of the attractive
and repulsive interactions between the diffusing particles and we discuss
analytically the limiting values of the exponent . Simulations based on
the Master Equation are shown to be in agreement with the analytical solutions
of the nonlinear Fokker-Planck equation in all three diffusion regimes.Comment: Published text with additional comment
Probing microscopic origins of confined subdiffusion by first-passage observables
Subdiffusive motion of tracer particles in complex crowded environments, such
as biological cells, has been shown to be widepsread. This deviation from
brownian motion is usually characterized by a sublinear time dependence of the
mean square displacement (MSD). However, subdiffusive behavior can stem from
different microscopic scenarios, which can not be identified solely by the MSD
data. In this paper we present a theoretical framework which permits to
calculate analytically first-passage observables (mean first-passage times,
splitting probabilities and occupation times distributions) in disordered media
in any dimensions. This analysis is applied to two representative microscopic
models of subdiffusion: continuous-time random walks with heavy tailed waiting
times, and diffusion on fractals. Our results show that first-passage
observables provide tools to unambiguously discriminate between the two
possible microscopic scenarios of subdiffusion. Moreover we suggest experiments
based on first-passage observables which could help in determining the origin
of subdiffusion in complex media such as living cells, and discuss the
implications of anomalous transport to reaction kinetics in cells.Comment: 21 pages, 3 figures. Submitted versio
Short note on the emergence of fractional kinetics
In the present Short Note an idea is proposed to explain the emergence and
the observation of processes in complex media that are driven by fractional
non-Markovian master equations. Particle trajectories are assumed to be solely
Markovian and described by the Continuous Time Random Walk model. But, as a
consequence of the complexity of the medium, each trajectory is supposed to
scale in time according to a particular random timescale. The link from this
framework to microscopic dynamics is discussed and the distribution of
timescales is computed. In particular, when a stationary distribution is
considered, the timescale distribution is uniquely determined as a function
related to the fundamental solution of the space-time fractional diffusion
equation. In contrast, when the non-stationary case is considered, the
timescale distribution is no longer unique. Two distributions are here
computed: one related to the M-Wright/Mainardi function, which is Green's
function of the time-fractional diffusion equation, and another related to the
Mittag-Leffler function, which is the solution of the fractional-relaxation
equation
Anomalous transport in the crowded world of biological cells
A ubiquitous observation in cell biology is that diffusion of macromolecules
and organelles is anomalous, and a description simply based on the conventional
diffusion equation with diffusion constants measured in dilute solution fails.
This is commonly attributed to macromolecular crowding in the interior of cells
and in cellular membranes, summarising their densely packed and heterogeneous
structures. The most familiar phenomenon is a power-law increase of the MSD,
but there are other manifestations like strongly reduced and time-dependent
diffusion coefficients, persistent correlations, non-gaussian distributions of
the displacements, heterogeneous diffusion, and immobile particles. After a
general introduction to the statistical description of slow, anomalous
transport, we summarise some widely used theoretical models: gaussian models
like FBM and Langevin equations for visco-elastic media, the CTRW model, and
the Lorentz model describing obstructed transport in a heterogeneous
environment. Emphasis is put on the spatio-temporal properties of the transport
in terms of 2-point correlation functions, dynamic scaling behaviour, and how
the models are distinguished by their propagators even for identical MSDs.
Then, we review the theory underlying common experimental techniques in the
presence of anomalous transport: single-particle tracking, FCS, and FRAP. We
report on the large body of recent experimental evidence for anomalous
transport in crowded biological media: in cyto- and nucleoplasm as well as in
cellular membranes, complemented by in vitro experiments where model systems
mimic physiological crowding conditions. Finally, computer simulations play an
important role in testing the theoretical models and corroborating the
experimental findings. The review is completed by a synthesis of the
theoretical and experimental progress identifying open questions for future
investigation.Comment: review article, to appear in Rep. Prog. Phy
Inferring diffusion in single live cells at the single molecule level
The movement of molecules inside living cells is a fundamental feature of
biological processes. The ability to both observe and analyse the details of
molecular diffusion in vivo at the single molecule and single cell level can
add significant insight into understanding molecular architectures of diffusing
molecules and the nanoscale environment in which the molecules diffuse. The
tool of choice for monitoring dynamic molecular localization in live cells is
fluorescence microscopy, especially so combining total internal reflection
fluorescence (TIRF) with the use of fluorescent protein (FP) reporters in
offering exceptional imaging contrast for dynamic processes in the cell
membrane under relatively physiological conditions compared to competing single
molecule techniques. There exist several different complex modes of diffusion,
and discriminating these from each other is challenging at the molecular level
due to underlying stochastic behaviour. Analysis is traditionally performed
using mean square displacements of tracked particles, however, this generally
requires more data points than is typical for single FP tracks due to
photophysical instability. Presented here is a novel approach allowing robust
Bayesian ranking of diffusion processes (BARD) to discriminate multiple complex
modes probabilistically. It is a computational approach which biologists can
use to understand single molecule features in live cells.Comment: combined ms (1-37 pages, 8 figures) and SI (38-55, 3 figures
Anomalous transport resolved in space and time by fluorescence correlation spectroscopy
A ubiquitous observation in crowded cell membranes is that molecular
transport does not follow Fickian diffusion but exhibits subdiffusion. The
microscopic origin of such a behaviour is not understood and highly debated.
Here we discuss the spatio-temporal dynamics for two models of subdiffusion:
fractional Brownian motion and hindered motion due to immobile obstacles. We
show that the different microscopic mechanisms can be distinguished using
fluorescence correlation spectroscopy (FCS) by systematic variation of the
confocal detection area. We provide a theoretical framework for space-resolved
FCS by generalising FCS theory beyond the common assumption of spatially
Gaussian transport. We derive a master formula for the FCS autocorrelation
function, from which it is evident that the beam waist of an FCS experiment is
a similarly important parameter as the wavenumber of scattering experiments.
These results lead to scaling properties of the FCS correlation for both
models, which are tested by in silico experiments. Further, our scaling
prediction is compatible with the FCS half-value times reported by Wawrezinieck
et al. [Biophys. J. 89, 4029 (2005)] for in vivo experiments on a transmembrane
protein.Comment: accepted for publication in Soft Matte
Anomalous diffusion and Tsallis statistics in an optical lattice
We point out a connection between anomalous quantum transport in an optical
lattice and Tsallis' generalized thermostatistics. Specifically, we show that
the momentum equation for the semiclassical Wigner function that describes
atomic motion in the optical potential, belongs to a class of transport
equations recently studied by Borland [PLA 245, 67 (1998)]. The important
property of these ordinary linear Fokker--Planck equations is that their
stationary solutions are exactly given by Tsallis distributions. Dissipative
optical lattices are therefore new systems in which Tsallis statistics can be
experimentally studied.Comment: 4 pages, 1 figur
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