415 research outputs found
An -expansion for Small-World Networks
I construct a well-defined expansion in for diffusion
processes on small-world networks. The technique permits one to calculate the
average over disorder of moments of the Green's function, and is used to
calculate the average Green's function and fluctuations to first non-leading
order in , giving results which agree with numerics. This technique
is also applicable to other problems of diffusion in random media.Comment: 7 pages Europhysics style, 3 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
Contest based on a directed polymer in a random medium
We introduce a simple one-parameter game derived from a model describing the
properties of a directed polymer in a random medium. At his turn, each of the
two players picks a move among two alternatives in order to maximize his final
score, and minimize opponent's return. For a game of length , we find that
the probability distribution of the final score develops a traveling wave
form, , with the wave profile unusually
decaying as a double exponential for large positive and negative . In
addition, as the only parameter in the game is varied, we find a transition
where one player is able to get his maximum theoretical score. By extending
this model, we suggest that the front velocity is selected by the nonlinear
marginal stability mechanism arising in some traveling wave problems for which
the profile decays exponentially, and for which standard traveling wave theory
applies
Exact asymptotics of the freezing transition of a logarithmically correlated random energy model
We consider a logarithmically correlated random energy model, namely a model
for directed polymers on a Cayley tree, which was introduced by Derrida and
Spohn. We prove asymptotic properties of a generating function of the partition
function of the model by studying a discrete time analogy of the KPP-equation -
thus translating Bramson's work on the KPP-equation into a discrete time case.
We also discuss connections to extreme value statistics of a branching random
walk and a rescaled multiplicative cascade measure beyond the critical point
Exact Results for a Three-Body Reaction-Diffusion System
A system of particles hopping on a line, singly or as merged pairs, and
annihilating in groups of three on encounters, is solved exactly for certain
symmetrical initial conditions. The functional form of the density is nearly
identical to that found in two-body annihilation, and both systems show
non-mean-field, ~1/t**(1/2) instead of ~1/t, decrease of particle density for
large times.Comment: 10 page
Model of Cluster Growth and Phase Separation: Exact Results in One Dimension
We present exact results for a lattice model of cluster growth in 1D. The
growth mechanism involves interface hopping and pairwise annihilation
supplemented by spontaneous creation of the stable-phase, +1, regions by
overturning the unstable-phase, -1, spins with probability p. For cluster
coarsening at phase coexistence, p=0, the conventional structure-factor scaling
applies. In this limit our model falls in the class of diffusion-limited
reactions A+A->inert. The +1 cluster size grows diffusively, ~t**(1/2), and the
two-point correlation function obeys scaling. However, for p>0, i.e., for the
dynamics of formation of stable phase from unstable phase, we find that
structure-factor scaling breaks down; the length scale associated with the size
of the growing +1 clusters reflects only the short-distance properties of the
two-point correlations.Comment: 12 page
The A+B -> 0 annihilation reaction in a quenched random velocity field
Using field-theoretic renormalization group methods the long-time behaviour
of the A+B -> 0 annihilation reaction with equal initial densities n_A(0) =
n_B(0) = n_0 in a quenched random velocity field is studied. At every point (x,
y) of a d-dimensional system the velocity v is parallel or antiparallel to the
x-axis and depends on the coordinates perpendicular to the flow. Assuming that
v(y) have zero mean and short-range correlations in the y-direction we show
that the densities decay asymptotically as n(t) ~ A n_0^(1/2) t^(-(d+3)/8) for
d<3. The universal amplitude A is calculated at first order in \epsilon = 3-d.Comment: 19 pages, LaTeX using IOP-macros, 5 eps-figures. It is shown that the
amplitude of the density is universal, i.e. independent of the reaction rat
Coherent State path-integral simulation of many particle systems
The coherent state path integral formulation of certain many particle systems
allows for their non perturbative study by the techniques of lattice field
theory. In this paper we exploit this strategy by simulating the explicit
example of the diffusion controlled reaction . Our results are
consistent with some renormalization group-based predictions thus clarifying
the continuum limit of the action of the problem.Comment: 20 pages, 4 figures. Minor corrections. Acknowledgement and reference
correcte
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
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