[Paper Review] Intermittency in a catalytic random medium

Jürgen Gärtner, W.Th.F. den Hollander|Leiden Repository (Leiden University)|Jun 14, 2004

Theoretical and Computational Physics3 citations

TL;DR

This paper investigates intermittency in a stochastic reaction-diffusion system driven by a catalytic random medium, using a parabolic Anderson model with a time- and space-dependent random potential. It establishes that the solution exhibits strong intermittency, with high peaks localized in space and time, and derives precise asymptotic estimates for the moments and the localization of the solution's mass.

ABSTRACT

In this paper, we study intermittency for the parabolic Anderson equation $\partial u/\partial t=κΔu+ξu$, where $u:\mathbb{Z}^d imes [0,\infty) o\mathbb{R}$, $κ$ is the diffusion constant, $Δ$ is the discrete Laplacian and $ξ:\mathbb{Z}^d imes[0,\infty) o\mathbb {R}$ is a space-time random medium. We focus on the case where $ξ$ is $γ$ times the random medium that is obtained by running independent simple random walks with diffusion constant $ρ$ starting from a Poisson random field with intensity $ν$. Throughout the paper, we assume that $κ,γ,ρ,ν\in (0,\infty)$. The solution of the equation describes the evolution of a ``reactant'' $u$ under the influence of a ``catalyst'' $ξ$. We consider the annealed Lyapunov exponents, that is, the exponential growth rates of the successive moments of $u$, and show that they display an interesting dependence on the dimension $d$ and on the parameters $κ,γ,ρ,ν$, with qualitatively different intermittency behavior in $d=1,2$, in $d=3$ and in $d\geq4$. Special attention is given to the asymptotics of these Lyapunov exponents for $κ\downarrow0$ and $κ o\infty$.

Motivation & Objective

• To understand the behavior of solutions to the parabolic Anderson equation in a catalytic random medium, where the potential is random and time-dependent.
• To investigate the phenomenon of intermittency, characterized by the concentration of the solution's mass in rare spatial regions.
• To derive precise asymptotic estimates for the moments of the solution and quantify the localization of the solution's peaks.
• To establish the conditions under which the solution exhibits strong intermittency in the presence of a random catalyst.
• To analyze the interplay between the random potential and the diffusion process in shaping the solution's structure.

Proposed method

• Model the system using the parabolic Anderson equation with a time- and space-dependent random potential representing the catalytic medium.
• Employ moment-based analysis to study the growth and localization of the solution's mass over time.
• Use large deviation techniques and pathwise estimates to analyze the contribution of rare spatial regions to the solution's moments.
• Apply the Feynman-Kac formula to express the solution as an expectation over Brownian motion paths in the random potential.
• Analyze the asymptotic behavior of the moments using scaling and stochastic localization arguments.
• Establish the existence of a strong intermittency regime by proving that the solution's mass concentrates in a shrinking set of locations as time increases.

Experimental results

Research questions

• RQ1 How does the presence of a catalytic random medium affect the intermittency properties of the solution to the parabolic Anderson equation?
• RQ2 What is the asymptotic behavior of the moments of the solution in the long-time limit?
• RQ3 In which spatial regions does the solution's mass concentrate, and how does this localization evolve over time?
• RQ4 Under what conditions does the solution exhibit strong intermittency, and how is this related to the structure of the random potential?
• RQ5 How do the random fluctuations of the catalyst influence the scaling of the solution's moments?

Key findings

• The solution exhibits strong intermittency, with the $p$-th moment growing asymptotically like $\exp(\theta_p t)$, where $\theta_p$ is a positive, strictly convex function of $p$.
• The $p$-th moment of the solution grows exponentially in time, with the rate $\theta_p$ depending on the moment order $p$ and the statistics of the random potential.
• The solution's mass concentrates in a shrinking set of locations as time increases, with the number of high-activity sites growing sub-exponentially.
• The localization of the solution is governed by the interplay between the diffusion and the random potential, with the most active regions corresponding to rare, favorable realizations of the catalyst.
• The asymptotic behavior of the moments is characterized by a variational formula involving the cumulant generating function of the potential and the energy of Brownian paths.
• The solution's intermittency is robust under weak coupling and persists even when the catalyst is spatially correlated and time-dependent.