[Paper Review] Front propagation in an exclusion one-dimensional reactive dynamics

Milton Andrés Jara Ramírez, Gregorio R. Moreno Flores+1 more|ArXiv.org|Mar 6, 2007

Stochastic processes and statistical mechanics10 references3 citations

TL;DR

This paper studies a one-dimensional exclusion process where moving X particles activate static Y particles upon contact, modeling a pulled front in reactive dynamics. Using regeneration times and coupling techniques, it establishes a law of large numbers and functional central limit theorem for the front position and activated particle count, confirming ballistic motion with Gaussian fluctuations observed in prior simulations.

ABSTRACT

We consider an exclusion process representing a reactive dynamics of a pulled front on the integer lattice, describing the dynamics of first class $X$ particles moving as a simple symmetric exclusion process, and static second class $Y$ particles. When an $X$ particle jumps to a site with a $Y$ particle, their position is intechanged and the $Y$ particle becomes an $X$ one. Initially, there is an arbitrary configuration of $X$ particles at sites $..., -1,0$, and $Y$ particles only at sites $1,2,...$, with a product Bernoulli law of parameter $ρ,0

Motivation & Objective

• To rigorously analyze front propagation in a one-dimensional exclusion process with reactive dynamics where X particles activate static Y particles.
• To establish the existence of a deterministic front velocity and normal fluctuations, confirming Monte Carlo simulations in similar models.
• To prove convergence of the particle configuration as seen from the front to a unique invariant measure.
• To develop a novel regeneration time framework tailored to this exclusion process with activation.

Proposed method

• Introduces a labeled particle-hole process with regeneration times defined by the first return of labeled particles to the front after each regeneration event.
• Uses a stopping time $ D_n $ to detect when labeled particles from prior regeneration cycles reach the front, enabling the definition of regeneration epochs $ \kappa_n $.
• Applies the strong Markov property and translation invariance to prove regeneration structure, ensuring i.i.d. increments after regeneration.
• Defines the state space $ \Omega' $ and the Skorohod space of paths to analyze the process $ (r_t, \eta(t)) $, the front and particle configuration.
• Employs coupling arguments and conditional independence via $ \mathcal{G}_1 $ to prove the regeneration property.
• Uses the infinitesimal generator $ Lf $ to describe the jump dynamics of X particles and activation of Y particles at the front.

Experimental results

Research questions

• RQ1 Does the front position $ r_t $ grow ballistically, i.e., does $ r_t / t $ converge almost surely to a positive constant $ v $?
• RQ2 Do the fluctuations of the front position satisfy a functional central limit theorem with non-degenerate variance?
• RQ3 Is the empirical measure of X particles as seen from the front invariant and unique in the long-time limit?
• RQ4 Can a regeneration structure be constructed in this exclusion process with activation, despite the lack of i.i.d. increments?
• RQ5 How does the number of activated Y particles $ p_t $ scale with time, and does it converge to a deterministic rate $ w $?

Key findings

• The front position $ r_t $ satisfies a strong law of large numbers: $ \lim_{t \to \infty} r_t / t = v $ almost surely for some $ v > 0 $, independent of the initial configuration.
• The rescaled front position $ \epsilon^{-1/2}(r_{\epsilon^{-1}t} - v\epsilon^{-1}t) $ converges weakly to a Brownian motion with non-degenerate variance.
• The number of activated Y particles $ p_t $ satisfies $ \lim_{t \to \infty} p_t / t = w $ almost surely for some $ w > 0 $, independent of initial conditions.
• The process $ (r_t, \eta(t)) $ as seen from the front converges in law to a unique invariant measure.
• The regeneration times $ \kappa_n $ are finite almost surely, and the process regenerates at these times with i.i.d. increments.
• The regeneration structure is proven via a coupling argument showing that the future evolution after $ \kappa_1 $ is distributed as the original process started from $ \delta_0 $, given $ D_0 = \infty $.