256 research outputs found
Diffusion and Correlations in Lattice Gas Automata
We present an analysis of diffusion in terms of the spontaneous density
fluctuations in a non-thermal two-species fluid modeled by a lattice gas
automaton. The power spectrum of the density correlation function is computed
with statistical mechanical methods, analytically in the hydrodynamic limit,
and numerically from a Boltzmann expression for shorter time and space scales.
In particular we define an observable -- the weighted difference of the species
densities -- whose fluctuation correlations yield the diffusive mode
independently of the other modes so that the corresponding power spectrum
provides a measure of diffusion dynamics solely. Automaton simulations are
performed to obtain measurements of the spectral density over the complete
range of wavelengths (from the microscopic scale to the macroscopic scale of
the automaton universe). Comparison of the theoretical results with the
numerical experiments data yields the following results: (i) the spectral
functions of the lattice gas fluctuations are in accordance with those of a
classical `non-thermal' fluid; (ii) the Landau-Placzek theory, obtained as the
hydrodynamic limit of the Boltzmann theory, describes the spectra correctly in
the long wavelength limit; (iii) at shorter wavelengths and at moderate
densities the complete Boltzmann theory provides good agreement with the
simulation data. These results offer convincing validation of lattice gas
automata as a microscopic approach to diffusion phenomena in fluid systems.Comment: 9 pages (revtex source), 12 Postscript figure
Dynamical systems theory for music dynamics
We show that, when music pieces are cast in the form of time series of pitch
variations, the concepts and tools of dynamical systems theory can be applied
to the analysis of {\it temporal dynamics} in music. (i) Phase space portraits
are constructed from the time series wherefrom the dimensionality is evaluated
as a measure of the {\pit global} dynamics of each piece. (ii) Spectral
analysis of the time series yields power spectra () close to
{\pit red noise} () in the low frequency range. (iii) We define an
information entropy which provides a measure of the {\pit local} dynamics in
the musical piece; the entropy can be interpreted as an evaluation of the
degree of {\it complexity} in the music, but there is no evidence of an
analytical relation between local and global dynamics. These findings are based
on computations performed on eighty sequences sampled in the music literature
from the 18th to the 20th century.Comment: To appear in CHAOS. Figures and Tables (not included) can be obtained
from [email protected]
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
A New Class of Cellular Automata for Reaction-Diffusion Systems
We introduce a new class of cellular automata to model reaction-diffusion
systems in a quantitatively correct way. The construction of the CA from the
reaction-diffusion equation relies on a moving average procedure to implement
diffusion, and a probabilistic table-lookup for the reactive part. The
applicability of the new CA is demonstrated using the Ginzburg-Landau equation.Comment: 4 pages, RevTeX 3.0 , 3 Figures 214972 bytes tar, compressed,
uuencode
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
Lattice gas automaton approach to "Turbulent Diffusion"
A periodic Kolmogorov type flow is implemented in a lattice gas automaton.
For given aspect ratios of the automaton universe and within a range of
Reynolds number values, the averaged flow evolves towards a stationary
two-dimensional type flow. We show the analogy between the streamlines of
the flow in the automaton and the phase plane trajectories of a dynamical
system. In practice flows are commonly studied by seeding the fluid with
suspended particles which play the role of passive tracers. Since an actual
flow is time-dependent and has fluctuations, the tracers exhibit interesting
intrinsic dynamics. When tracers are implemented in the automaton and their
trajectories are followed, we find that the tracers displacements obey a
diffusion law, with ``super-diffusion'' in the direction orthogonal to the
direction of the initial forcing.Comment: 7 revtex4 pages including 3 figure
Viscous fingering in miscible, immiscible and reactive fluids
With the Lattice Boltzmann method (using the BGK approximation) we
investigate the dynamics of Hele-Shaw flow under conditions corresponding to
various experimental systems. We discuss the onset of the instability
(dispersion relation), the static properties (characterization of the
interface) and the dynamic properties (growth of the mixing zone) of simulated
Hele-Shaw systems. We examine the role of reactive processes (between the two
fluids) and we show that they have a sharpening effect on the interface similar
to the effect of surface tension.Comment: 6 pages with 2 figure, to be published in J.Mod.Phys
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