235 research outputs found
Transfer across Random versus Deterministic Fractal Interfaces
A numerical study of the transfer across random fractal surfaces shows that
their responses are very close to the response of deterministic model
geometries with the same fractal dimension. The simulations of several
interfaces with prefractal geometries show that, within very good
approximation, the flux depends only on a few characteristic features of the
interface geometry: the lower and higher cut-offs and the fractal dimension.
Although the active zones are different for different geometries, the electrode
reponses are very nearly the same. In that sense, the fractal dimension is the
essential "universal" exponent which determines the net transfer.Comment: 4 pages, 6 figure
Explicit Construction of the Brownian Self-Transport Operator
Applying the technique of characteristic functions developped for
one-dimensional regular surfaces (curves) with compact support, we obtain the
distribution of hitting probabilities for a wide class of finite membranes on
square lattice. Then we generalize it to multi-dimensional finite membranes on
hypercubic lattice. Basing on these distributions, we explicitly construct the
Brownian self-transport operator which governs the Laplacian transfer. In order
to verify the accuracy of the distribution of hitting probabilities, numerical
analysis is carried out for some particular membranes.Comment: 30 pages, 9 figures, 1 tabl
Percolation-dependent Reaction Rates in the Etching of Disordered Solids
A prototype statistical model for the etching of a random solid is
investigated in order to assess the influence of disorder and temperature on
the dissolution kinetics. At low temperature, the kinetics is dominated by
percolation phenomena, and the percolation threshold determines the global
reaction time. At high temperature, the fluctuations of the reaction rate are
Gaussian, whereas at low temperature they exhibit a power law tail due to
chemical avalanches. This is an example where microscopic disorder directly
induces non-classical chemical kinetics.Comment: Revtex, 4 pages, 5 figure
Transition from Knudsen to molecular diffusion in activity of absorbing irregular interfaces
We investigate through molecular dynamics the transition from Knudsen to
molecular diffusion transport towards 2d absorbing interfaces with irregular
geometry. Our results indicate that the length of the active zone decreases
continuously with density from the Knudsen to the molecular diffusion regime.
In the limit where molecular diffusion dominates, we find that this length
approaches a constant value of the order of the system size, in agreement with
theoretical predictions for Laplacian transport in irregular geometries.
Finally, we show that all these features can be qualitatively described in
terms of a simple random-walk model of the diffusion process.Comment: 4 pages, 4 figure
Surface Hardening and Self-Organized Fractality Through Etching of Random Solids
When a finite volume of etching solution is in contact with a disordered
solid, complex dynamics of the solid-solution interface develop. If the etchant
is consumed in the chemical reaction, the dynamics stop spontaneously on a
self-similar fractal surface. As only the weakest sites are corroded, the solid
surface gets progressively harder and harder. At the same time it becomes
rougher and rougher uncovering the critical spatial correlations typical of
percolation. From this, the chemical process reveals the latent percolation
criticality hidden in any random system. Recently, a simple minimal model has
been introduced by Sapoval et al. to describe this phenomenon. Through analytic
and numerical study, we obtain a detailed description of the process. The time
evolution of the solution corroding power and of the distribution of resistance
of surface sites is studied in detail. This study explains the progressive
hardening of the solid surface. Finally, this dynamical model appears to belong
to the universality class of Gra dient Percolation.Comment: 14 pages, 15 figures (1457 Kb
Optimal branching asymmetry of hydrodynamic pulsatile trees
Most of the studies on optimal transport are done for steady state regime
conditions. Yet, there exists numerous examples in living systems where supply
tree networks have to deliver products in a limited time due to the pulsatile
character of the flow. This is the case for mammals respiration for which air
has to reach the gas exchange units before the start of expiration. We report
here that introducing a systematic branching asymmetry allows to reduce the
average delivery time of the products. It simultaneously increases its
robustness against the unevitable variability of sizes related to
morphogenesis. We then apply this approach to the human tracheobronchial tree.
We show that in this case all extremities are supplied with fresh air, provided
that the asymmetry is smaller than a critical threshold which happens to fit
with the asymmetry measured in the human lung. This could indicate that the
structure is adjusted at the maximum asymmetry level that allows to feed all
terminal units with fresh air.Comment: 4 pages, 4 figure
Localized low-frequency Neumann modes in 2d-systems with rough boundaries
We compute the relative localization volumes of the vibrational eigenmodes in
two-dimensional systems with a regular body but irregular boundaries under
Dirichlet and under Neumann boundary conditions. We find that localized states
are rare under Dirichlet boundary conditions but very common in the Neumann
case. In order to explain this difference, we utilize the fact that under
Neumann conditions the integral of the amplitudes, carried out over the whole
system area is zero. We discuss, how this condition leads to many localized
states in the low-frequency regime and show by numerical simulations, how the
number of the localized states and their localization volumes vary with the
boundary roughness.Comment: 7 pages, 4 figure
Self-stabilised fractality of sea-coasts through damped erosion
Erosion of rocky coasts spontaneously creates irregular seashores. But the
geometrical irregularity, in turn, damps the sea-waves, decreasing the average
wave amplitude. There may then exist a mutual self-stabilisation of the waves
amplitude together with the irregular morphology of the coast. A simple model
of such stabilisation is studied. It leads, through a complex dynamics of the
earth-sea interface, to the appearance of a stationary fractal seacoast with
dimension close to 4/3. Fractal geometry plays here the role of a morphological
attractor directly related to percolation geometry.Comment: 4 pages, 5 figure
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