486 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
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
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
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
Crossover from Rate-Equation to Diffusion-Controlled Kinetics in Two-Particle Coagulation
We develop an analytical diffusion-equation-type approximation scheme for the
one-dimensional coagulation reaction A+A->A with partial reaction probability
on particle encounters which are otherwise hard-core. The new approximation
describes the crossover from the mean-field rate-equation behavior at short
times to the universal, fluctuation-dominated behavior at large times. The
approximation becomes quantitatively accurate when the system is initially
close to the continuum behavior, i.e., for small initial density and fast
reaction. For large initial density and slow reaction, lattice effects are
nonnegligible for an extended initial time interval. In such cases our
approximation provides the correct description of the initial mean-field as
well as the asymptotic large-time, fluctuation-dominated behavior. However, the
intermediate-time crossover between the two regimes is described only
semiquantitatively.Comment: 21 pages, plain Te
Superdiffusivity of Finite-Range Asymmetric Exclusion Processes on
We consider finite-range asymmetric exclusion processes on with
non-zero drift. The diffusivity is expected to be of . We prove that in the weak (Tauberian) sense
that as . The proof employs the resolvent method to make a direct comparison with the
totally asymmetric simple exclusion process, for which the result is a
consequence of the scaling limit for the two-point function recently obtained
by Ferrari and Spohn. In the nearest neighbor case, we show further that
is monotone, and hence we can conclude that in the usual sense.Comment: Version 3. Statement of Theorem 3 is correcte
Diffusion-Limited Coalescence with Finite Reaction Rates in One Dimension
We study the diffusion-limited process in one dimension, with
finite reaction rates. We develop an approximation scheme based on the method
of Inter-Particle Distribution Functions (IPDF), which was formerly used for
the exact solution of the same process with infinite reaction rate. The
approximation becomes exact in the very early time regime (or the
reaction-controlled limit) and in the long time (diffusion-controlled)
asymptotic limit. For the intermediate time regime, we obtain a simple
interpolative behavior between these two limits. We also study the coalescence
process (with finite reaction rates) with the back reaction , and in
the presence of particle input. In each of these cases the system reaches a
non-trivial steady state with a finite concentration of particles. Theoretical
predictions for the concentration time dependence and for the IPDF are compared
to computer simulations. P. A. C. S. Numbers: 82.20.Mj 02.50.+s 05.40.+j
05.70.LnComment: 13 pages (and 4 figures), plain TeX, SISSA-94-0
Three-Species Diffusion-Limited Reaction with Continuous Density-Decay Exponents
We introduce a model of three-species two-particle diffusion-limited
reactions A+B -> A or B, B+C -> B or C, and C+A -> C or A, with three
persistence parameters (survival probabilities in reaction) of the hopping
particle. We consider isotropic and anisotropic diffusion (hopping with a
drift) in 1d. We find that the particle density decays as a power-law for
certain choices of the persistence parameter values. In the anisotropic case,
on one symmetric line in the parameter space, the decay exponent is
monotonically varying between the values close to 1/3 and 1/2. On another, less
symmetric line, the exponent is constant. For most parameter values, the
density does not follow a power-law. We also calculated various characteristic
exponents for the distance of nearest particles and domain structure. Our
results support the recently proposed possibility that 1d diffusion-limited
reactions with a drift do not fall within a limited number of distinct
universality classes.Comment: 12 pages in plain LaTeX and four Postscript files with 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
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