Fluctuation effects on chemical wave fronts

Renormalization of stochastic differential equations with multiplicative noise using effective potential methods

We present a method to renormalize stochastic differential equations subjected to multiplicative noise. The method is based on the widely used concept of effective potential in high-energy physics and has already been successfully applied to the renormalization of stochastic differential equations subjected to additive noise. We derive a general formula for the one-loop effective potential of a single ordinary stochastic differential equation (with arbitrary interaction terms) subjected to multiplicative Gaussian noise (provided the noise satisfies a certain normalization condition). To illustrate the usefulness (and limitations) of the method, we use the effective potential to renormalize a toy chemical model based on a simplified Gray-Scott reaction. In particular, we use it to compute the scale dependence of the toy model's parameters (in perturbation theory) when subjected to a Gaussian power-law noise with short time correlations.

Stochastic simulation of reaction-diffusion systems: A fluctuating-hydrodynamics approach

We develop numerical methods for stochastic reaction-diffusion systems based on approaches used for fluctuatinghydrodynamics (FHD). For hydrodynamicsystems, the FHD formulation is formally described by stochastic partial differential equations (SPDEs). In the reaction-diffusion systems we consider, our model becomes similar to the reaction-diffusion master equation (RDME) description when our SPDEs are spatially discretized and reactions are modeled as a source term having Poissonfluctuations. However, unlike the RDME, which becomes prohibitively expensive for an increasing number of molecules, our FHD-based description naturally extends from the regime where fluctuations are strong, i.e., each mesoscopic cell has few (reactive) molecules, to regimes with moderate or weak fluctuations, and ultimately to the deterministic limit. By treating diffusion implicitly, we avoid the severe restriction on time step size that limits all methods based on explicit treatments of diffusion and construct numerical methods that are more efficient than RDME methods, without compromising accuracy. Guided by an analysis of the accuracy of the distribution of steady-state fluctuations for the linearized reaction-diffusion model, we construct several two-stage (predictor-corrector) schemes, where diffusion is treated using a stochastic Crank-Nicolson method, and reactions are handled by the stochastic simulation algorithm of Gillespie or a weakly second-order tau leaping method. We find that an implicit midpoint tau leaping scheme attains second-order weak accuracy in the linearized setting and gives an accurate and stable structure factor for a time step size of an order of magnitude larger than the hopping time scale of diffusing molecules. We study the numerical accuracy of our methods for the Schlögl reaction-diffusion model both in and out of thermodynamic equilibrium. We demonstrate and quantify the importance of thermodynamicfluctuations to the formation of a two-dimensional Turing-like pattern and examine the effect of fluctuations on three-dimensional chemical front propagation. By comparing stochastic simulations to deterministic reaction-diffusion simulations, we show that fluctuations accelerate pattern formation in spatially homogeneous systems and lead to a qualitatively different disordered pattern behind a traveling wave.

Front waves in the early RNA world: The Schlögl model and the logistic growth model

Front wave solutions of nonlinear reaction-diffusion models describing the spatio-temporal growth of RNA populations in the early RNA world are discussed. A two-variable model for RNA enzymes and enzyme complex molecules as well as single-variable models obtained via adiabatic elimination of the complex molecules are considered. In both models, the focus is on enzyme diffusion in one spatial dimension, assuming that the diffusion of complex molecules can be neglected. It is shown that one of the single-variable models corresponds to a Schlögl model of front propagation. In general, for the single-variable models it is found that front speed corresponds to the minimal speed of traveling fronts. In contrast, the two-variable model exhibits even slower front propagation. Front propagation might be an important factor in competitive evolutionary processes in the early RNA world.

Spatiotemporal clustering and temporal order in the excitable BZ reaction

The prototype experimental example of "spontaneous" pattern formation in an unstirred chemical medium is the oscillatory Belousov-Zhabotinsky (BZ) reaction: target patterns of outward-moving concentric rings are readily observed when the reaction is run in a thin layer in a Petri dish. In many experimental runs, new target centers appeared to form closer to pre-existing target centers than expected in a randomized model. Here we describe a simple direct test for the presence of temporal order in the spatiotemporal dynamics of target nucleation, and apply this test to detect significant temporal order in target formation in the ferroin-catalyzed BZ reaction. We also describe how mixing heterogeneity can generate temporal order, even in the absence of heterogeneous physical nucleating centers.

Exactly soluble noisy traveling-wave equation appearing in the problem of directed polymers in a random medium

We calculate exactly the velocity and diffusion constant of a microscopic stochastic model of N evolving particles which can be described by a noisy traveling-wave equation with a noise of order N(-1/2). Our model can be viewed as the infinite range limit of a directed polymer in random medium with N sites in the transverse direction. Despite some peculiarities of the traveling-wave equations in the absence of noise, our exact solution allows us to test the validity of a simple cutoff approximation and to show that, in the weak noise limit, the position of the front can be completely described by the effect of the noise on the first particle.

Directed particle diffusion under "burnt bridges" conditions

We study random walks on a one-dimensional lattice that contains weak connections, so-called "bridges." Each time the walker crosses the bridge from the left or attempts to cross it from the right, the bridge may be destroyed with probability p; this restricts the particle's motion and directs it. Our model, which incorporates asymmetric aspects in an otherwise symmetric hopping mechanism, is very akin to "Brownian ratchets" and to front propagation in autocatalytic A+B-->2A reactions. The analysis of the model and Monte Carlo simulations show that for large p the velocity of the directed motion is extremely sensitive to the distribution of bridges, whereas for small p the velocity can be understood based on a mean-field analysis. The single-particle model advanced by us here allows an almost quantitative understanding of the front's position in the A+B-->2A many-particle reaction.

Macroscopic effects of the perturbation of the particle velocity distribution in a trigger wave

Departure from the equilibrium particle velocity distribution induced by a chemical reaction is studied in an inhomogeneous chemical bistable system in which a trigger wave can propagate. These nonequilibrium effects influence the speed and shape of the trigger wave propagating between the two stable stationary states. In contrast to the Fisher front, the discretization of the variables and the internal fluctuations do not sensitively modify the macroscopic properties of the trigger wave. Analytical results deduced from the Boltzmann equation agree well with microscopic simulations using Bird's method. Both approaches lead to large relative corrections to the front speed, in particular for parameter values close to that corresponding to a stationary interface between the two stable states.

Front propagation in one-dimensional autocatalytic reactions: the breakdown of the classical picture at small particle concentrations

The autocatalytic scheme A+B-->2A in a discrete particle system is studied in one dimension via Monte Carlo simulations. We find considerable differences in the results for the front velocities and front forms compared to the classical, continuous picture, which is only valid in the limit of very small reaction probabilities p. Interestingly, we also obtain front propagation velocities fairly below the classical minimal velocity.