925 research outputs found
Generalized diffusion equation
Modern analyses of diffusion processes have proposed nonlinear versions of
the Fokker-Planck equation to account for non-classical diffusion. These
nonlinear equations are usually constructed on a phenomenological basis. Here
we introduce a nonlinear transformation by defining the -generating function
which, when applied to the intermediate scattering function of classical
statistical mechanics, yields, in a mathematically systematic derivation, a
generalized form of the advection-diffusion equation in Fourier space. Its
solutions are discussed and suggest that the -generating function approach
should be a useful tool to generalize classical diffusive transport
formulations.Comment: 5 pages with 3 figure
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
Propagation-Dispersion Equation
A {\em propagation-dispersion equation} is derived for the first passage
distribution function of a particle moving on a substrate with time delays. The
equation is obtained as the continuous limit of the {\em first visit equation},
an exact microscopic finite difference equation describing the motion of a
particle on a lattice whose sites operate as {\em time-delayers}. The
propagation-dispersion equation should be contrasted with the
advection-diffusion equation (or the classical Fokker-Planck equation) as it
describes a dispersion process in {\em time} (instead of diffusion in space)
with a drift expressed by a propagation speed with non-zero bounded values. The
{\em temporal dispersion} coefficient is shown to exhibit a form analogous to
Taylor's dispersivity. Physical systems where the propagation-dispersion
equation applies are discussed.Comment: 12 pages+ 5 figures, revised and extended versio
A microscopic approach to nonlinear Reaction-Diffusion: the case of morphogen gradient formation
We develop a microscopic theory for reaction-difusion (R-D) processes based
on a generalization of Einstein's master equation with a reactive term and we
show how the mean field formulation leads to a generalized R-D equation with
non-classical solutions. For the -th order annihilation reaction
, we obtain a nonlinear reaction-diffusion equation
for which we discuss scaling and non-scaling formulations. We find steady
states with either solutions exhibiting long range power law behavior (for
) showing the relative dominance of sub-diffusion over reaction
effects in constrained systems, or conversely solutions (for )
with finite support of the concentration distribution describing situations
where diffusion is slow and extinction is fast. Theoretical results are
compared with experimental data for morphogen gradient formation.Comment: Article, 10 pages, 5 figure
International Conference: Alternative Histories of Electronic Music, 2016
Video proceedings of the first international conference on ‘Alternative Histories of Electronic Music’ (AHEM2016), held at the Science Museum, London, 14-16 April 2016. The conference programme booklet (PDF), and video recordings of the opening address and closing remarks (MP4), may be downloaded here. Video recordings of the individual talks and keynote addresses delivered at the conference may be accessed via the video proceedings index, also downloadable here.
This conference was held as part of the AHRC-funded project ‘Hugh Davies: Electronic Music Innovator’, led by Dr James Mooney (University of Leeds) in partnership with Dr Tim Boon (Science Museum).
Project reference: AH/M005216/1
Molecular theory of anomalous diffusion - Application to Fluorescence Correlation Spectroscopy
The nonlinear theory of anomalous diffusion is based on particle interactions
giving an explicit microscopic description of diffusive processes leading to
sub-, normal, or super-diffusion as a result competitive effects between
attractive and repulsive interactions. We present the explicit analytical
solution to the nonlinear diffusion equation which we then use to compute the
correlation function which is experimentally measured by correlation
spectroscopy. The theoretical results are applicable in particular to the
analysis of fluorescence correlation spectroscopy of marked molecules in
biological systems. More specifically we consider the case of fluorescently
labeled lipids and we find that the nonlinear correlation spectrum reproduces
very well the experimental data indicating sub-diffusive molecular motion of
lipid molecules in the cell membrane.Comment: 13 pages including one figure; Submitted to the Journal of
Statistical Physics. arXiv admin note: text overlap with arXiv:1302.557
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