My research concerns basic aspects of nonlinear chemical dynamics, in particular pattern formation in far-from-equilibrium systems. Spatiotemporal patterns are a ubiquitous feature of natural systems. Examples range from spiral galaxies to cloud streets to banded patterns in rocks to excitation waves across the heart to the quintessential example of pattern formation, embryogenesis. The key element of pattern-forming mechanisms is often the same: the coupling of local nonlinear transformation processes with transport processes. The simplest systems that display such a coupling are chemically reacting and diffusing systems. The self-organization of patterns in reaction-diffusion systems requires nonequilibrium conditions and nonlinear kinetics. Many reactions in natural or industrial systems are indeed governed by nonlinear kinetics, for example, substrate inhibition (enzymatic processes) and autocatalysis (auto-oxidation, chain branching, self-heating). Reactions that produce temporal self-organization, such as periodic oscillations or deterministic chaos, have been widely studied during the last four decades. Besides the famous Belousov-Zhabotinsky reaction, many other reactions have been found to display temporal structures.
Spatial patterns in chemical reaction-diffusion systems have received particular attention since Turing's pioneering work [A. M. Turing, "The Chemical Basis of Morphogenesis," Phil. Trans. R. Soc. Lond. B 237, 37 – 72 (1952)]. Turing showed that the coupling of diffusion with nonlinear kinetics can destabilize the homogeneous steady state in nonequilibrium systems and generate stable, stationary concentration patterns. In addition to these Turing patterns, reaction-diffusion systems are capable of exhibiting a large variety of spatial, temporal, and spatiotemporal patterns. The formation and dynamics of such patterns in chemical systems continues to pose fundamental and challenging problems. While complex spatial and temporal behavior is encountered in many disciplines, chemical systems are especially advantageous for the study of complex behavior in space and time, since the underlying coupling of reactions and diffusion can be rigorously characterized. Reaction-diffusion models also provide a general theoretical framework for describing pattern formation in a variety of fields, such as biology, ecology, physics, and materials science.
My current research projects in this area address pattern formation in arrays of coupled chemical reactors, i.e., spatially discrete reaction-diffusion systems, and pattern formation in reaction-transport systems where dispersal does not follow simple diffusive motion:
(i) Most experimental and theoretical studies of coupled chemical reactors have considered the effect of coupling on chemical oscillators. Only a few studies have investigated the effects of coupling steady-state units, and little is known about stationary patterns in such systems. I am studying the influence of network topology on pattern formation in diffusively-coupled well-stirred reactors. A particular focus are diffusion-induced instabilities, i.e., Turing instabilities.
Diffusive coupling between well-stirred reactors (point reactors) requires physical proximity. Complex network topologies, e.g., small-world networks which require some long-range connections, are impractical to implement with arrays of diffusively coupled point reactors. Both workhorses of experimental nonlinear chemical dynamics, the Belousov-Zhabotinsky (BZ) reaction and the chlorine dioxide-iodine-malonic acid (CDIMA) reaction, are photosensitive or exist in photosensitive variants. Therefore, photochemical feedback coupling represents a very attractive method to realize arrays of reactors with complex network topologies. I am studying the pattern-forming capabilities of such arrays, their stability properties, and the effects of the network topology on pattern selection.
The research on networks of coupled reactors is conducted in collaboration with Professor Peter K. Moore, Department of Mathematics, Southern Methodist University.
(ii) Brownian motion, or the diffusion equation, is commonly used to model the spatial spread of molecules and organisms, but it has well-known unphysical features. Brownian particles lack inertia; their direction of motion in successive time intervals is uncorrelated. This implies that particles move with infinite velocity. There is some probability, though exponentially small, that a dispersing particle will travel an infinite distance from its current position in a small but nonzero amount of time. The lack of inertia also implies that the motion of the dispersing particles is unpredictable even on the smallest scales. The persistent random walk is the simplest generalization of the ordinary random walk (Brownian motion) that takes into account inertia or persistence effects and remedies these unphysical features. Reaction random walks are a natural generalization of reaction-diffusion systems; they replace Brownian motion by a persistent random walk to account for inertial effects in the dispersal process. My research addresses the effect of inertia on various pattern forming mechanisms in nonlinear reaction systems, and my main effort is directed towards understanding the spatiotemporal behavior of reaction random walks.
(iii) In situations where persistence is negligible, the medium may give rise to other deviations from standard diffusive behavior. Anomalous diffusion, where the mean square displacement of a particle does not grow linearly with time, i.e., ‹x(t)2› ~ ta, is ubiquitous, though not due to any single universal cause. If 0‹a‹1, the process is subdiffusive; if a›1, it is superdiffusive. Superdiffusion is encountered, for example, in turbulent fluids, layered velocity fields, or rotating flows, and subdiffusion in porous systems, amorphous semiconductors, or disordered materials. Anomalous diffusion is often caused by memory effects and Lévy-type statistics. Specifically, superdiffusion is observed for random walks with long-tail jump-length distributions, and subdiffusion for long-tail waiting-time distributions. The latter type of distributions can be caused by "traps" that have an infinite mean waiting-time. My research addresses the derivation of proper evolution equations for nonlinear chemical systems with anomalous diffusion and the investigation of their spatiotemporal dynamics. A particular focus is the effect of anomalous diffusion on Turing instabilities. Together with postdoctoral research associate Aniruddha Yadav, I have derived a generalized reaction-diffusion equation for transport with memory. We have shown that the non-Markovian nature of the transport results in a non-trivial combination of reactions and spatial dispersal. We have applied our generalized reaction-diffusion equation to subdiffusive transport and derived the conditions for a Turing instability.
Other topics of our research in this area concern the effects of the microscopic details of the transport process on front propagation and critical patch size.
The research on reaction-transport systems with memory is conducted in collaboration with Professor Vicenç Méndez, Grup de Física Estadística, Departament de Física, Facultat de Ciències, Universitat Autònoma de Barcelona, Spain, Professor Sergei Fedotov, School of Mathematics, The University of Manchester, UK, and Professor Peter K. Moore, Department of Mathematics, Southern Methodist University.
A second area of my research concerns basic aspects of random fluctuations in nonlinear, nonequilibrium systems. Random fluctuations exist in all natural systems, though they are often treated as a mere nuisance that can be safely ignored. Fluctuations that arise from the discrete nature of chemical and physical systems at the microscopic level are known as internal fluctuations. They are unavoidable and cannot be eliminated in experiments. Their influence on the behavior of the system can be safely neglected for macroscopic systems under most circumstances, since the amplitude of internal fluctuations decreases as the system size increases. Nonequilibrium systems are open systems; they are coupled to their surroundings. Random changes in the latter are a second source of random fluctuations, known as external noise. The amplitude of external noise is independent of the system size, and we showed about thirty years ago that there are many situations where such fluctuations cannot be neglected. On the contrary, external noise can drastically change the behavior of a nonequilibrium system and can actually have an ordering influence. [For a review, see, W. Horsthemke and R. Lefever: "Noise-Induced Transitions. Theory and Applications in Physics, Chemistry, and Biology." Springer Verlag, Berlin, 1984.]
My current research projects in this area address the effect of random inhomogeneities on the transport of particles, the origins of anomalous diffusion, and the effect of spatiotemporal noise on patterns in reaction-diffusion systems and reaction-transport systems. This research is carried out in collaboration with Professor Stanislav I. Denisov, Department of Mechanics and Mathematics, Sumy State University, Ukraine.