27,597 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 evolution of step meander during growth of a vicinal surface with no desorption
Step meandering due to a deterministic morphological instability on vicinal
surfaces during growth is studied. We investigate nonlinear dynamics of a step
model with asymmetric step kinetics, terrace and line diffusion, by means of a
multiscale analysis. We give the detailed derivation of the highly nonlinear
evolution equation on which a brief account has been given [Pierre-Louis et.al.
PRL(98)]. Decomposing the model into driving and relaxational contributions, we
give a profound explanation to the origin of the unusual divergent scaling of
step meander ~ 1/F^{1/2} (where F is the incoming atom flux). A careful
numerical analysis indicates that a cellular structure arises where plateaus
form, as opposed to spike-like structures reported erroneously in Ref.
[Pierre-Louis et.al. PRL(98)]. As a robust feature, the amplitude of these
cells scales as t^{1/2}, regardless of the strength of the Ehrlich-Schwoebel
effect, or the presence of line diffusion. A simple ansatz allows to describe
analytically the asymptotic regime quantitatively. We show also how
sub-dominant terms from multiscale analysis account for the loss of up-down
symmetry of the cellular structure.Comment: 23 pages, 10 figures; (Submitted to EPJ B
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