4,273 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
Nonlinear diffusion from Einstein's master equation
We generalize Einstein's master equation for random walk processes by
considering that the probability for a particle at position to make a jump
of length lattice sites, is a functional of the particle
distribution function . By multiscale expansion, we obtain a
generalized advection-diffusion equation. We show that the power law (with ) follows from the requirement
that the generalized equation admits of scaling solutions (). The solutions have a -exponential form
and are found to be in agreement with the results of Monte-Carlo simulations,
so providing a microscopic basis validating the nonlinear diffusion equation.
Although its hydrodynamic limit is equivalent to the phenomenological porous
media equation, there are extra terms which, in general, cannot be neglected as
evidenced by the Monte-Carlo computations.}Comment: 7 pages incl. 3 fig
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
Nonextensive diffusion as nonlinear response
The porous media equation has been proposed as a phenomenological
``non-extensive'' generalization of classical diffusion. Here, we show that a
very similar equation can be derived, in a systematic manner, for a classical
fluid by assuming nonlinear response, i.e. that the diffusive flux depends on
gradients of a power of the concentration. The present equation distinguishes
from the porous media equation in that it describes \emph{% generalized
classical} diffusion, i.e. with scaling, but with a generalized
Einstein relation, and with power-law probability distributions typical of
nonextensive statistical mechanics
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
Heavy Quark Diffusion from the Lattice
We study the diffusion of heavy quarks in the Quark Gluon Plasma using the
Langevin equations of motion and estimate the contribution of the transport
peak to the Euclidean current-current correlator. We show that the Euclidean
correlator is remarkably insensitive to the heavy quark diffusion coefficient
and give a simple physical interpretation of this result using the free
streaming Boltzmann equation. However if the diffusion coefficient is smaller
than , as favored by RHIC phenomenology, the transport
contribution should be visible in the Euclidean correlator. We outline a
procedure to isolate this contribution.Comment: 24 pages, 5 figure
Parental views from rural Cambodia on disability causation and change
Purpose. This study explored the beliefs of Cambodian parents of children with cerebral palsy regarding disability causation and their perceptions of the effectiveness of interventions in bringing about change in their child. Results. Beliefs around disability causation were mixed, with equal numbers of participants attributing their child's disability to biomedical causes as to traditional causes incorporating elements of Theravada Buddhism, animism and Brahmanism. While many participants had initially sought traditional interventions for their child, few found them to be effective and most had subsequently utilised medical and rehabilitation services. Parents whose children were moderately or severely impaired perceived both traditional interventions and rehabilitation to be less effective than parents of children with mild impairments. Participants generally judged the effectiveness of interventions based on functional changes in their child. Conclusions. The complexity of Khmer belief systems was reflected in the range of participants' perceptions of the cause of their child's disability, yet beliefs around disability causation did not appear to have determined their care-seeking behaviour or their perceptions of effectiveness of interventions
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