Nonlinear Chemical Dynamics in Low-Dimensional Lattices and Fractal Sets

Nonequilibrium chemistry in confined environments: a lattice Brusselator model

In this work, we study the effect of molecular crowding on a typical example of a chemical oscillator: the Brusselator model. We adopt to this end a nonequilibrium thermodynamic description, in which the size of particles is introduced via a lattice gas model. The impenetrability and finite volume of the species are shown to affect both the reaction rates and the diffusion terms in the evolution equations for the concentrations. The corrected scheme shows a more complex dynamical behavior than its ideal counterpart, including bistability and excitability. These results help to shed light on recent experimental and computational studies in biochemistry and surface chemistry, in which it was shown that confined environments may greatly affect chemical dynamics.

Spatiotemporal oscillations and clustering in the Ziff-Gulari-Barshad model with surface reconstruction

We study the dynamics of the Ziff-Gulari-Barshad (ZGB) model on square (sq) and hexagonal-honeycomb (hex) lattices and when surface restructuring is introduced. We show that the ZGB model exhibits nonequilibrium phase transitions on the hex lattice similar to the ones already observed on the sq lattice, but the critical values of the kinetic parameters depend crucially on the substrate type. If surface reconstruction (sq<-->hex) is assumed for high lattice coverage of one of the reactive species then persistent spatiotemporal oscillations and clustering of homologous species are observed for kinetic parameter values 0.348<k1<0.393.

Spontaneous formation of dynamical patterns with fractal fronts in the cyclic lattice Lotka-Volterra model

Dynamical patterns, in the form of consecutive moving stripes or rings, are shown to develop spontaneously in the cyclic lattice Lotka-Volterra model, when realized on square lattice, at the reaction limited regime. Each stripe consists of different particles (species) and the borderlines between consecutive stripes are fractal. The interface width w between the different species scales as w(L,t) approximately L(alpha)f(t/L(z)), where L is the linear size of the interface, t is the time, and alpha and z are the static and dynamical critical exponents, respectively. The critical exponents were computed as alpha=0.49+/-0.03 and z=1.53+/-0.13 and the propagating fronts show dynamical characteristics similar to those of the Eden growth models.

Adsorption of reactive particles on a random catalytic chain: an exact solution

We study equilibrium properties of a catalytically activated annihilation A+A-->0 reaction taking place on a one-dimensional chain of length N (N--> infinity ) in which some segments (placed at random, with mean concentration p) possess special, catalytic properties. Annihilation reaction takes place as soon as any two A particles land onto two vacant sites at the extremities of the catalytic segment, or when any A particle lands onto a vacant site on a catalytic segment while the site at the other extremity of this segment is already occupied by another A particle. Noncatalytic segments are inert with respect to reaction and here two adsorbed A particles harmlessly coexist. For both "annealed" and "quenched" disorder in placement of the catalytic segments, we calculate exactly the disorder-averaged pressure per site. Explicit asymptotic formulas for the particle mean density and the compressibility are also presented.

Three-state model for cooperative desorption on a one-dimensional lattice

We develop a master equation approach to the dynamics of immobile reactants on a one-dimensional lattice, in the presence of two different species undergoing cooperative desorption. A common feature of all the schemes studied is the strong dependence of the final coverage on the initial conditions, associated with the lack of ergodicity of the invariant state. Our approach leads to full agreement with Monte Carlo simulations, both asymptotically and transiently.

Oscillatory reactive dynamics on surfaces: a lattice limit cycle model

Complex reactive dynamics on low-dimensional lattices is studied using mean-field models and Monte Carlo simulations. A lattice-compatible reactive scheme that gives rise to limit cycle behavior is constructed, involving a quadrimolecular reaction step and bimolecular adsorption and desorption steps. The resulting lattice limit cycle model is dissipative and, in the mean-field limit, exhibits sustained oscillations of the species concentrations for a wide range of parameter values. Lattice Monte Carlo simulations of the lattice limit cycle model show locally the emergence of sustained oscillations of the species concentrations. Random fluctuations of the concentrations, clustering between homologous species, and competition between the various clusters/species cause the in-phase oscillations of neighboring sites. Distant regions oscillate out of phase and spatial correlations decay exponentially with the distance. The amplitude and period of the local oscillations depend on the system parameters.

Fractal properties of the lattice Lotka-Volterra model

The lattice Lotka-Volterra (LLV) model is studied using mean-field analysis and Monte Carlo simulations. While the mean-field phase portrait consists of a center surrounded by an infinity of closed trajectories, when the process is restricted to a two-dimensional (2D) square lattice, local inhomogeneities/fluctuations appear. Spontaneous local clustering is observed on lattice and homogeneous initial distributions turn into clustered structures. Reactions take place only at the interfaces between different species and the borders adopt locally fractal structure. Intercluster surface reactions are responsible for the formation of local fluctuations of the species concentrations. The box-counting fractal dimension of the LLV dynamics on a 2D support is found to depend on the reaction constants while the upper bound of fractality determines the size of the local oscillators. Lacunarity analysis is used to determine the degree of clustering of homologous species. Besides the spontaneous clustering that takes place on a regular 2D lattice, the effects of fractal supports on the dynamics of the LLV are studied. For supports of dimensionality D(s)<2 the lattice can, for certain domains of the reaction constants, adopt a poisoned state where only one of the species survives. By appropriately selecting the fractal dimension of the substrate, it is possible to direct the system into a poisoned or oscillatory steady state at will.

Reactive dynamics on two-dimensional supports: Monte Carlo simulations and mean-field theory

Monte Carlo simulations and mean-field models are used for the study of nonequilibrium reactions taking place on the surface of a catalyst. The model represents the catalytic reduction of NO with H2 on a Pt surface. Both Monte Carlo simulations and mean-field results predict the existence of a critical surface in the parameter space where the catalyst remains active for long times. Outside this critical region the catalyst remains active for finite times only. A discrete version of the mean-field model is proposed that takes into account the discrete, two-dimensional nature of the catalyst. For homogeneous initial conditions this improved model provides better quantitative agreement with the Monte Carlo results.

Reaction-controlled cooperative desorption in a one-dimensional lattice: a dynamical approach

The spinlike dynamics of immobile reactants in a one-dimensional lattice is analyzed for two representative systems involving cooperative desorption. An exact combinatorial approach is worked out. Its failure to reproduce the results of microscopic simulations is shown to be associated with the lack of sufficiently strong ergodic properties, as a result of which the final state depends strongly on the initial conditions. A dynamical approach to the problem based on the Master equation description is subsequently developed, leading to full agreement with the microscopic simulations.