Pattern Formation in Nonlinear Chemical Reactions

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 five 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. 

Research interests: arrays of coupled chemical reactors; reaction-diffusion systems with cross diffusion; reaction-diffusion systems with density-dependent dispersal; reaction-diffusion systems with anomalous diffusion; reaction-transport systems with inertia. 

Random Fluctuations and Nonequilibrium Systems

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 forty 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.] 

Research interests: effects of spatiotemporal noise on patterns and front propagation in reaction-diffusion systems and reaction-transport systems; noise-induced transitions in population biology; fluctuations and oscillators. 

Collaborators

Daniel Campos, Grup de Física Estadística, Departament de Física, Facultat de Ciències, Universitat Autònoma de Barcelona, Spain

Stanislav I. Denisov, General and Theoretical Physics Chair, Sumy State University, Sumy State University, Ukraine

Sergei Fedotov, School of Mathematics, The University of Manchester, UK

Alexander Iomin, Department of Physics, Technion, Haifa, Israel 

Vicenç Méndez, Grup de Física Estadística, Departament de Física, Facultat de Ciències, Universitat Autònoma de Barcelona, Spain

Evgeny Zemskov, Federal Research Center for Computer Science and Control, Russian Academy of Sciences, Moscow, Russia