927 research outputs found
Intermittency on catalysts: three-dimensional simple symmetric exclusion
We continue our study of intermittency for the parabolic Anderson model
in a space-time random medium
, where is a positive diffusion constant, is the lattice
Laplacian on , , and is a simple symmetric exclusion
process on in Bernoulli equilibrium. This model describes the evolution
of a \emph{reactant} under the influence of a \emph{catalyst} .
In G\"artner, den Hollander and Maillard (2007) we investigated the behavior
of the annealed Lyapunov exponents, i.e., the exponential growth rates as
of the successive moments of the solution . This led to an
almost complete picture of intermittency as a function of and . In
the present paper we finish our study by focussing on the asymptotics of the
Lyaponov exponents as in the \emph{critical} dimension ,
which was left open in G\"artner, den Hollander and Maillard (2007) and which
is the most challenging. We show that, interestingly, this asymptotics is
characterized not only by a \emph{Green} term, as in , but also by a
\emph{polaron} term. The presence of the latter implies intermittency of
\emph{all} orders above a finite threshold for .Comment: 38 page
Intermittency on catalysts
The present paper provides an overview of results obtained in four recent
papers by the authors. These papers address the problem of intermittency for
the Parabolic Anderson Model in a \emph{time-dependent random medium},
describing the evolution of a ``reactant'' in the presence of a ``catalyst''.
Three examples of catalysts are considered: (1) independent simple random
walks; (2) symmetric exclusion process; (3) symmetric voter model. The focus is
on the annealed Lyapunov exponents, i.e., the exponential growth rates of the
successive moments of the reactant. It turns out that these exponents exhibit
an interesting dependence on the dimension and on the diffusion constant.Comment: 11 pages, invited paper to appear in a Festschrift in honour of
Heinrich von Weizs\"acker, on the occasion of his 60th birthday, to be
published by Cambridge University Pres
Intermittency on catalysts: Voter model
In this paper we study intermittency for the parabolic Anderson equation
with
, where is
the diffusion constant, is the discrete Laplacian,
is the coupling constant, and
is a space--time random medium.
The solution of this equation describes the evolution of a ``reactant''
under the influence of a ``catalyst'' . We focus on the case where
is the voter model with opinions 0 and 1 that are updated according to a random
walk transition kernel, starting from either the Bernoulli measure
or the equilibrium measure , where is the density of
1's. We consider the annealed Lyapunov exponents, that is, the exponential
growth rates of the successive moments of . We show that if the random walk
transition kernel has zero mean and finite variance, then these exponents are
trivial for , but display an interesting dependence on the
diffusion constant for , with qualitatively different
behavior in different dimensions. In earlier work we considered the case where
is a field of independent simple random walks in a Poisson equilibrium,
respectively, a symmetric exclusion process in a Bernoulli equilibrium, which
are both reversible dynamics. In the present work a main obstacle is the
nonreversibility of the voter model dynamics, since this precludes the
application of spectral techniques. The duality with coalescing random walks is
key to our analysis, and leads to a representation formula for the Lyapunov
exponents that allows for the application of large deviation estimates.Comment: Published in at http://dx.doi.org/10.1214/10-AOP535 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Stretched Exponential Relaxation in the Biased Random Voter Model
We study the relaxation properties of the voter model with i.i.d. random
bias. We prove under mild condions that the disorder-averaged relaxation of
this biased random voter model is faster than a stretched exponential with
exponent , where depends on the transition rates
of the non-biased voter model. Under an additional assumption, we show that the
above upper bound is optimal. The main ingredient of our proof is a result of
Donsker and Varadhan (1979).Comment: 14 pages, AMS-LaTe
Quenched Lyapunov exponent for the parabolic Anderson model in a dynamic random environment
We continue our study of the parabolic Anderson equation ¿u/¿t =k¿u+¿¿u for the space-time field u: Zd ×[0,8) ¿ R, where k ¿ [0,8) is the diffusion constant, ¿ is the discrete Laplacian, ¿ ¿ (0,8) is the coupling constant, and ¿ : Zd ×[0,8)¿R is a space-time random environment that drives the equation. The solution of this equation describes the evolution of a "reactant" u under the influence of a "catalyst" ¿, both living on Zd. In earlier work we considered three choices for ¿: independent simple random walks, the symmetric exclusion process, and the symmetric voter model, all in equilibrium at a given density. We analyzed the annealed Lyapunov exponents, i.e., the exponential growth rates of the successive moments of u w.r.t. ¿ , and showed that these exponents display an interesting dependence on the diffusion constant k, with qualitatively different behavior in different dimensions d. In the present paper we focus on the quenched Lyapunov exponent, i.e., the exponential growth rate of u conditional on ¿ . We first prove existence and derive some qualitative properties of the quenched Lyapunov exponent for a general ¿ that is stationary and ergodic w.r.t. translations in Zd and satisfies certain noisiness conditions. After that we focus on the three particular choices for ¿ mentioned above and derive some more detailed properties.We close by formulating a number of open problems
Intermittency on catalysts : symmetric exclusion
We continue our study of intermittency for the parabolic Anderson equation, i.e., the spatially discrete heat equation on the d-dimensional integer lattice with a space-time random potential. The solution of the equation describes the evolution of a "reactant" under the influence of a "catalyst". In this paper we focus on the case where the random field is an exclusion process with a symmetric random walk transition kernel, starting from Bernoulli equilibrium. We consider the annealed Lyapunov exponents, i.e., the exponential growth rates of the successive moments of the solution. We show that these exponents are trivial when the random walk is recurrent, but display an interesting dependence on the diffusion constant when the random walk is transient, with qualitatively different behavior in different dimensions. Special attention is given to the asymptotics of the exponents when the diffusion constant tends to infinity, which is controlled by moderate deviations of the random field requiring a delicate expansion argument. In Gärtner and den Hollander [10] the case of a Poisson field of independent (simple) random walks was studied. The two cases show interesting differences and similarities. Throughout the paper, a comparison of the two cases plays a crucial role
Kinetics of Radioiodinated Heptadecanoic Acid and Metabolites in the Normal and Ischaemic Canine Heart
This study was undertaken to elucidate if the myocardial elimination rate of the radioactivity after administration of radioiodinated heptadecanoic acid was related to beta-oxidation of the fatty acid or related to washout of free radioiodide. In samples of normal and ischaemic myocardium the distribution of the radioactivity over free radioiodide, heptadecanoic acid and lipids was determined. In normal myocardium the major component was free radioiodide, only a small percentage being heptadecanoic acid. In ischaemic myocardium more radiolabelled lipids were present and less free iodide when compared with normal myocardium. The percentage heptadecanoic acid was slightly increased. It is concluded that radioiodinated heptadecanoic acid behaves like the natural analogues regarding uptake and distribution. However, washout of free radioiodide determines the elimination rate as observed during a scintigraphic stud
Relaxation Height in Energy Landscapes: an Application to Multiple Metastable States
The study of systems with multiple (not necessarily degenerate) metastable
states presents subtle difficulties from the mathematical point of view related
to the variational problem that has to be solved in these cases. We introduce
the notion of relaxation height in a general energy landscape and we prove
sufficient conditions which are valid even in presence of multiple metastable
states. We show how these results can be used to approach the problem of
multiple metastable states via the use of the modern theories of metastability.
We finally apply these general results to the Blume--Capel model for a
particular choice of the parameters ensuring the existence of two multiple, and
not degenerate in energy, metastable states
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