450 research outputs found
Particle Dynamics in a Mass-Conserving Coalescence Process
We consider a fully asymmetric one-dimensional model with mass-conserving
coalescence. Particles of unit mass enter at one edge of the chain and
coalescence while performing a biased random walk towards the other edge where
they exit. The conserved particle mass acts as a passive scalar in the reaction
process , and allows an exact mapping to a restricted ballistic
surface deposition model for which exact results exist. In particular, the
mass- mass correlation function is exactly known. These results complement
earlier exact results for the process without mass. We introduce a
comprehensive scaling theory for this process. The exact anaytical and
numerical results confirm its validity.Comment: 5 pages, 6 figure
Symmetry and species segregation in diffusion-limited pair annihilation
We consider a system of q diffusing particle species A_1,A_2,...,A_q that are
all equivalent under a symmetry operation. Pairs of particles may annihilate
according to A_i + A_j -> 0 with reaction rates k_{ij} that respect the
symmetry, and without self-annihilation (k_{ii} = 0). In spatial dimensions d >
2 mean-field theory predicts that the total particle density decays as n(t) ~
1/t, provided the system remains spatially uniform. We determine the conditions
on the matrix k under which there exists a critical segregation dimension
d_{seg} below which this uniformity condition is violated; the symmetry between
the species is then locally broken. We argue that in those cases the density
decay slows down to n(t) ~ t^{-d/d_{seg}} for 2 < d < d_{seg}. We show that
when d_{seg} exists, its value can be expressed in terms of the ratio of the
smallest to the largest eigenvalue of k. The existence of a conservation law
(as in the special two-species annihilation A + B -> 0), although sufficient
for segregation, is shown not to be a necessary condition for this phenomenon
to occur. We work out specific examples and present Monte Carlo simulations
compatible with our analytical results.Comment: latex, 19 pages, 3 eps figures include
Shift in the velocity of a front due to a cut-off
We consider the effect of a small cut-off epsilon on the velocity of a
traveling wave in one dimension. Simulations done over more than ten orders of
magnitude as well as a simple theoretical argument indicate that the effect of
the cut-off epsilon is to select a single velocity which converges when epsilon
tends to 0 to the one predicted by the marginal stability argument. For small
epsilon, the shift in velocity has the form K(log epsilon)^(-2) and our
prediction for the constant K agrees very well with the results of our
simulations. A very similar logarithmic shift appears in more complicated
situations, in particular in finite size effects of some microscopic stochastic
systems. Our theoretical approach can also be extended to give a simple way of
deriving the shift in position due to initial conditions in the
Fisher-Kolmogorov or similar equations.Comment: 12 pages, 3 figure
A phenomenological theory giving the full statistics of the position of fluctuating pulled fronts
We propose a phenomenological description for the effect of a weak noise on
the position of a front described by the Fisher-Kolmogorov-Petrovsky-Piscounov
equation or any other travelling wave equation in the same class. Our scenario
is based on four hypotheses on the relevant mechanism for the diffusion of the
front. Our parameter-free analytical predictions for the velocity of the front,
its diffusion constant and higher cumulants of its position agree with
numerical simulations.Comment: 10 pages, 3 figure
Diffusive Capture Process on Complex Networks
We study the dynamical properties of a diffusing lamb captured by a diffusing
lion on the complex networks with various sizes of . We find that the life
time and the survival probability becomes finite on scale-free networks with degree exponent
. However, for has a long-living tail on
tree-structured scale-free networks and decays exponentially on looped
scale-free networks. It suggests that the second moment of degree distribution
kn(k)n(k)\sim k^{-\sigma}\gamma<3n(k)k\approx k_{max}n(k)n(k)\sim k^2P(k)N_{tot}, which
causes the dependent behavior of and $.Comment: 9 pages, 6 figure
Asymptotic behavior of A + B --> inert for particles with a drift
We consider the asymptotic behavior of the (one dimensional) two-species
annihilation reaction A + B --> 0, where both species have a uniform drift in
the same direction and like species have a hard core exclusion. Extensive
numerical simulations show that starting with an initially random distribution
of A's and B's at equal concentration the density decays like t^{-1/3} for long
times. This process is thus in a different universality class from the cases
without drift or with drift in different directions for the different species.Comment: LaTeX, 6pp including 3 figures in LaTeX picture mod
Single-site approximation for reaction-diffusion processes
We consider the branching and annihilating random walk and with reaction rates and , respectively, and hopping rate
, and study the phase diagram in the plane. According
to standard mean-field theory, this system is in an active state for all
, and perturbative renormalization suggests that this mean-field
result is valid for ; however, nonperturbative renormalization predicts
that for all there is a phase transition line to an absorbing state in the
plane. We show here that a simple single-site
approximation reproduces with minimal effort the nonperturbative phase diagram
both qualitatively and quantitatively for all dimensions . We expect the
approach to be useful for other reaction-diffusion processes involving
absorbing state transitions.Comment: 15 pages, 2 figures, published versio
Directed Ising type dynamic preroughening transition in one dimensional interfaces
We present a realization of directed Ising (DI) type dynamic absorbing state
phase transitions in the context of one-dimensional interfaces, such as the
relaxation of a step on a vicinal surface. Under the restriction that particle
deposition and evaporation can only take place near existing kinks, the
interface relaxes into one of three steady states: rough, perfectly ordered
flat (OF) without kinks, or disordered flat (DOF) with randomly placed kinks
but in perfect up-down alternating order. A DI type dynamic preroughening
transition takes place between the OF and DOF phases. At this critical point
the asymptotic time evolution is controlled not only by the DI exponents but
also by the initial condition. Information about the correlations in the
initial state persists and changes the critical exponents.Comment: 12 pages, 10 figure
Kinetics of A+B--->0 with Driven Diffusive Motion
We study the kinetics of two-species annihilation, A+B--->0, when all
particles undergo strictly biased motion in the same direction and with an
excluded volume repulsion between same species particles. It was recently shown
that the density in this system decays as t^{-1/3}, compared to t^{-1/4}
density decay in A+B--->0 with isotropic diffusion and either with or without
the hard-core repulsion. We suggest a relatively simple explanation for this
t^{-1/3} decay based on the Burgers equation. Related properties associated
with the asymptotic distribution of reactants can also be accounted for within
this Burgers equation description.Comment: 11 pages, plain Tex, 8 figures. Hardcopy of figures available on
request from S
Exact Solutions of Anisotropic Diffusion-Limited Reactions with Coagulation and Annihilation
We report exact results for one-dimensional reaction-diffusion models A+A ->
inert, A+A -> A, and A+B -> inert, where in the latter case like particles
coagulate on encounters and move as clusters. Our study emphasized anisotropy
of hopping rates; no changes in universal properties were found, due to
anisotropy, in all three reactions. The method of solution employed mapping
onto a model of coagulating positive integer charges. The dynamical rules were
synchronous, cellular-automaton type. All the asymptotic large-time results for
particle densities were consistent, in the framework of universality, with
other model results with different dynamical rules, when available in the
literature.Comment: 28 pages in plain TeX + 2 PostScript figure
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