354 research outputs found
Fractal dimension of transport coefficients in a deterministic dynamical system
In many low-dimensional dynamical systems transport coefficients are very
irregular, perhaps even fractal functions of control parameters. To analyse
this phenomenon we study a dynamical system defined by a piece-wise linear map
and investigate the dependence of transport coefficients on the slope of the
map. We present analytical arguments, supported by numerical calculations,
showing that both the Minkowski-Bouligand and Hausdorff fractal dimension of
the graphs of these functions is 1 with a logarithmic correction, and find that
the exponent controlling this correction is bounded from above by 1 or
2, depending on some detailed properties of the system. Using numerical
techniques we show local self-similarity of the graphs. The local
self-similarity scaling transformations turn out to depend (irregularly) on the
values of the system control parameters.Comment: 17 pages, 6 figures; ver.2: 18 pages, 7 figures (added section 5.2,
corrected typos, etc.
Asymptotic expansion for reversible A + B <-> C reaction-diffusion process
We study long-time properties of reversible reaction-diffusion systems of
type A + B C by means of perturbation expansion in powers of 1/t (inverse
of time). For the case of equal diffusion coefficients we present exact
formulas for the asymptotic forms of reactant concentrations and a complete,
recursive expression for an arbitrary term of the expansions. Taking an
appropriate limit we show that by studying reversible reactions one can obtain
"singular" solutions typical of irreversible reactions.Comment: 6 pages, no figures, to appear in PR
Asymptotics of Reaction-Diffusion Fronts with One Static and One Diffusing Reactant
The long-time behavior of a reaction-diffusion front between one static (e.g.
porous solid) reactant A and one initially separated diffusing reactant B is
analyzed for the mean-field reaction-rate density R(\rho_A,\rho_B) =
k\rho_A^m\rho_B^n. A uniformly valid asymptotic approximation is constructed
from matched self-similar solutions in a reaction front (of width w \sim
t^\alpha where R \sim t^\beta enters the dominant balance) and a diffusion
layer (of width W \sim t^{1/2} where R is negligible). The limiting solution
exists if and only if m, n \geq 1, in which case the scaling exponents are
uniquely given by \alpha = (m-1)/2(m+1) and \beta = m/(m+1). In the diffusion
layer, the common ad hoc approximation of neglecting reactions is given
mathematical justification, and the exact transient decay of the reaction rate
is derived. The physical effects of higher-order kinetics (m, n > 1), such as
the broadening of the reaction front and the slowing of transients, are also
discussed.Comment: final version, new title & combustion reference
Diffusion-controlled annihilation with initially separated reactants: The death of an particle island in the particle sea
We consider the diffusion-controlled annihilation dynamics with
equal species diffusivities in the system where an island of particles is
surrounded by the uniform sea of particles . We show that once the initial
number of particles in the island is large enough, then at any system's
dimensionality the death of the majority of particles occurs in the {\it
universal scaling regime} within which of the particles die at
the island expansion stage and the remaining at the stage of its
subsequent contraction. In the quasistatic approximation the scaling of the
reaction zone has been obtained for the cases of mean-field ()
and fluctuation () dynamics of the front.Comment: 4 RevTex pages, 1 PNG figure and 1 EPS figur
Reaction Front in an A+B -> C Reaction-Subdiffusion Process
We study the reaction front for the process A+B -> C in which the reagents
move subdiffusively. Our theoretical description is based on a fractional
reaction-subdiffusion equation in which both the motion and the reaction terms
are affected by the subdiffusive character of the process. We design numerical
simulations to check our theoretical results, describing the simulations in
some detail because the rules necessarily differ in important respects from
those used in diffusive processes. Comparisons between theory and simulations
are on the whole favorable, with the most difficult quantities to capture being
those that involve very small numbers of particles. In particular, we analyze
the total number of product particles, the width of the depletion zone, the
production profile of product and its width, as well as the reactant
concentrations at the center of the reaction zone, all as a function of time.
We also analyze the shape of the product profile as a function of time, in
particular its unusual behavior at the center of the reaction zone
Localization-delocalization transition of a reaction-diffusion front near a semipermeable wall
The A+B --> C reaction-diffusion process is studied in a system where the
reagents are separated by a semipermeable wall. We use reaction-diffusion
equations to describe the process and to derive a scaling description for the
long-time behavior of the reaction front. Furthermore, we show that a critical
localization-delocalization transition takes place as a control parameter which
depends on the initial densities and on the diffusion constants is varied. The
transition is between a reaction front of finite width that is localized at the
wall and a front which is detached and moves away from the wall. At the
critical point, the reaction front remains at the wall but its width diverges
with time [as t^(1/6) in mean-field approximation].Comment: 7 pages, PS fil
Formation of Liesegang patterns: A spinodal decomposition scenario
Spinodal decomposition in the presence of a moving particle source is
proposed as a mechanism for the formation of Liesegang bands. This mechanism
yields a sequence of band positions x_n that obeys the spacing law
x_n~Q(1+p)^n. The dependence of the parameters p and Q on the initial
concentration of the reagents is determined and we find that the functional
form of p is in agreement with the experimentally observed Matalon-Packter law.Comment: RevTex, 4 pages, 4 eps figure
Hopping motion of lattice gases through nonsymmetric potentials under strong bias conditions
The hopping motion of lattice gases through potentials without
mirror-reflection symmetry is investigated under various bias conditions. The
model of 2 particles on a ring with 4 sites is solved explicitly; the resulting
current in a sawtooth potential is discussed. The current of lattice gases in
extended systems consisting of periodic repetitions of segments with sawtooth
potentials is studied for different concentrations and values of the bias.
Rectification effects are observed, similar to the single-particle case. A
mean-field approximation for the current in the case of strong bias acting
against the highest barriers in the system is made and compared with numerical
simulations. The particle-vacancy symmetry of the model is discussed.Comment: 8 pages (incl. 6 eps figures); RevTeX 3.
Band Formation during Gaseous Diffusion in Aerogels
We study experimentally how gaseous HCl and NH_3 diffuse from opposite sides
of and react in silica aerogel rods with porosity of 92 % and average pore size
of about 50 nm. The reaction leads to solid NH_4Cl, which is deposited in thin
sheet-like structures. We present a numerical study of the phenomenon. Due to
the difference in boundary conditions between this system and those usually
studied, we find the sheet-like structures in the aerogel to differ
significantly from older studies. The influence of random nucleation centers
and inhomogeneities in the aerogel is studied numerically.Comment: 7 pages RevTex and 8 figures. Figs. 4-8 in Postscript, Figs. 1-3 on
request from author
Granular Matter: a wonderful world of clusters in far-from-equilibrium systems
In this paper, we recall various features of non equilibrium granular
systems. Clusters with specific properties are found depending on the packing
density, going from loose (a granular gas) to sintered (though brittle)
polycrystalline materials. The phase space available can be quite different.
Unexpected features, with respect to standard or expected ones in classical
fluids or solids, are observed, - like slow relaxation processes or anomalous
electrical and thermoelectrical transport property dependences. The cases of
various pile structures and the interplay between classical phase transitions
and self-organized criticality for avalanches are also outlined.Comment: 7 figures, 37 refs., to be published in Physica
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