405 research outputs found
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
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
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
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
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
Bifurcations and chaotic dynamics in a tumour-immune-virus system
Despite mounting evidence that oncolytic viruses can be effective in treating cancer, understanding the details of the interactions between tumour cells, oncolytic viruses and immune cells that could lead to tumour control or tumour escape is still an open problem. Mathematical modelling of cancer oncolytic therapies has been used to investigate the biological mechanisms behind the observed temporal patterns of tumour growth. However, many models exhibit very complex dynamics, which renders them difficult to investigate. In this case, bifurcation diagrams could enable the visualisation of model dynamics by identifying (in the parameter space) the particular transition points between different behaviours. Here, we describe and investigate two simple mathematical models for oncolytic virus cancer therapy, with constant and immunity-dependent carrying capacity. While both models can exhibit complex dynamics, namely fixed points, periodic orbits and chaotic behaviours, only the model with immunity-dependent carrying capacity can exhibit them for biologically realistic situations, i.e., before the tumour grows too large and the experiment is terminated. Moreover, with the help of the bifurcation diagrams we uncover two unexpected behaviours in virus-tumour dynamics: (i) for short virus half-life, the tumour size seems to be too small to be detected, while for long virus half-life the tumour grows to larger sizes that can be detected; (ii) some model parameters have opposite effects on the transient and asymptotic dynamics of the tumour.Publisher PDFPeer reviewe
Growth Kinetics in Systems with Local Symmetry
The phase transition kinetics of Ising gauge models are investigated. Despite
the absence of a local order parameter, relevant topological excitations that
control the ordering kinetics can be identified. Dynamical scaling holds in the
approach to equilibrium, and the growth of typical length scale is
characteristic of a new universality class with . We suggest that the asymptotic kinetics of the 2D Ising gauge
model is dual to that of the 2D annihilating random walks, a process also known
as the diffusion-reaction .Comment: 10 pages in Tex, 2 Postscript figures appended, NSF-ITP-93-4
Reaction-diffusion with a time-dependent reaction rate: the single-species diffusion-annihilation process
We study the single-species diffusion-annihilation process with a
time-dependent reaction rate, lambda(t)=lambda_0 t^-omega. Scaling arguments
show that there is a critical value of the decay exponent omega_c(d) separating
a reaction-limited regime for omega > omega_c from a diffusion-limited regime
for omega < omega_c. The particle density displays a mean-field,
omega-dependent, decay when the process is reaction limited whereas it behaves
as for a constant reaction rate when the process is diffusion limited. These
results are confirmed by Monte Carlo simulations. They allow us to discuss the
scaling behaviour of coupled diffusion-annihilation processes in terms of
effective time-dependent reaction rates.Comment: 11 pages, 9 figures, minor correction
Slowdown for time inhomogeneous branching Brownian motion
We consider the maximal displacement of one dimensional branching Brownian
motion with (macroscopically) time varying profiles. For monotone decreasing
variances, we show that the correction from linear displacement is not
logarithmic but rather proportional to . We conjecture that this is
the worse case correction possible
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