Nonlinear nonequilibrium system

Gas-solid fluidization is a typical nonlinear nonequilibrium system with multiscale structure. In particular, the mesoscale structure in terms of bubbles or clusters, which can be characterized by nonequilibrium features in terms of bimodal velocity distribution, energy nonequipartition, and correlated density fluctuations, is the critical factor. 

When a nonlinear system ewolwes under far-from-equilibrium conditions in the vicinity of a bifurcation point, a purely deterministic description often proved to be incomplete. The fluctuations of the dynamical variables can play an essential role and obstruct the observation of a transition expected by a deterministic analysis. In the framework of the deterministic approach, the stability of the different states according to the values of the control parameters is studied through a mathematical analysis of the velocity field. In particular, the theory of normal forms leads to the determination of the various kinds of attractors [l,2]. As far as we are concerned with the stochastic approach, the rrLa te.n. equation, has been widely used to analyze bifurcations of homogeneous or spatially ordered steady states or of limit cycles [3,4]. Our aim in the present contribution is to insist on the generality of the method to analyze various kinds of bifurcations in nonlinear nonequilibrium systems. The general procedure proposed to obtain a local description of the probability, which allows us to determine the system s attractors, turns out to display marked analogies with the theory of normal forms. 

Recently there has been an increasing interest in self-oscillatory phenomena and also in formation of spatio-temporal structure, accompanied by the rapid development of theory concerning dynamics of such systems under nonlinear, nonequilibrium conditions. The discovery of model chemical reactions to produce self-oscillations and spatio-temporal structures has accelerated the studies on nonlinear dynamics in chemistry. The Belousov-Zhabotinskii(B-Z) reaction is the most famous among such types of oscillatory chemical reactions, and has been studied most frequently during the past couple of decades [1,2]. The B-Z reaction has attracted much interest from scientists with various discipline, because in this reaction, the rhythmic change between oxidation and reduction states can be easily observed in a test tube. As the reproducibility of the amplitude, period and some other experimental measures is rather high under a found condition, the mechanism of the B-Z reaction has been almost fully understood until now. The most important step in the induction of oscillations is the existence of auto-catalytic process in the reaction network. 

The fundamental question in transport theory is Can one describe processes in nonequilibrium systems with the help of (local) thermodynamic functions of state (thermodynamic variables) This question can only be checked experimentally. On an atomic level, statistical mechanics is the appropriate theory. Since the entropy, 5, is the characteristic function for the formulation of equilibria (in a closed system), the deviation, SS, from the equilibrium value, S0, is the function which we need to use for the description of non-equilibria. Since we are interested in processes (i.e., changes in a system over time), the entropy production rate a = SS is the relevant function in irreversible thermodynamics. Irreversible processes involve linear reactions (rates 55) as well as nonlinear ones. We will be mainly concerned with processes that occur near equilibrium and so we can linearize the kinetic equations. The early development of this theory was mainly due to the Norwegian Lars Onsager. Let us regard the entropy S(a,/3,. ..) as a function of the (extensive) state variables a,/ ,. .. .which are either constant (fi,.. .) or can be controlled and measured (a). In terms of the entropy production rate, we have (9a/0f=a)... 

Ray Kapral came to Toronto from the United States in 1969. His research interests center on theories of rate processes both in systems close to equilibrium, where the goal is the development of a microscopic theory of condensed phase reaction rates,89 and in systems far from chemical equilibrium, where descriptions of the complex spatial and temporal reactive dynamics that these systems exhibit have been developed.90 He and his collaborators have carried out research on the dynamics of phase transitions and critical phenomena, the dynamics of colloidal suspensions, the kinetic theory of chemical reactions in liquids, nonequilibrium statistical mechanics of liquids and mode coupling theory, mechanisms for the onset of chaos in nonlinear dynamical systems, the stochastic theory of chemical rate processes, studies of pattern formation in chemically reacting systems, and the development of molecular dynamics simulation methods for activated chemical rate processes. His recent research activities center on the theory of quantum and classical rate processes in the condensed phase91 and in clusters, and studies of chemical waves and patterns in reacting systems at both the macroscopic and mesoscopic levels. 

Far from thermodynamic equiHbrium we find nonfinear interdependence of thermodynamic fluxes and forces. In this case, the Onsager reciprocal relations are generally not satisfied, and the formafism developed in Chapter 2 is not fuUy applicable for analysis of the state of open systems. Analysis of systems that are far from thermodynamic equilibrium is the subject of nonlinear nonequilibrium thermodynamics. 

Computer experiments after FPU have indeed been made to examine under what conditions, how the system breaks the ergodicity, and how long nonequilibrium states persist. In particular, nonlinear lattice systems, first investigated by FPU to model the vibrational oscillations around an equilibrium point of solids, have been used as a canonical model to explore such issues [2]. 

Alan Turing s paper entitled The Chemical Basis of Morphogenesis [440] ranks without doubt among the most important papers of the last century. In that seminal work Turing laid the foundation for the theory of chemical pattern formation. Turing showed that diffusion can have nontrivial effects in nonequilibrium systems. The interplay of diffusion with nonlinear kinetics can destabilize the uniform steady state of reaction-diffusion systems and generate stable, stationary concentration patterns. To quote from the abstract,... 

Kawezynski, L., Model of reaction diffusion system as generator of old Hebrew Alphabet, in Selforganization in Nonequilibrium Systems, Proceedings of International Conference in Nonlinear Science, Belgrade, Serbia and Montenegro, September, pp. 89-91, Belgrade, 2004. 

With experimental quantification of mesoscale structures, statistical analysis based on nonequilibrium distribution may require novel mathematical skills to unravel the complex dependence of stress-strain relation on the mass exchange between phases. The structure-dependent energy analysis may help elucidate the dependence between energy dissipation and structural parameters, and how to relate such structural-dependent analysis and the nonlinear nonequilibrium thermodynamics remains a challenge, especially for the scale-dependent granular flow or fluidization systems. 

Graham,R. Statistical Theory of Instabilities in Stationary Nonequilibrium Systems with Applications to Lasers and Nonlinear Optics (Vol. 66) Haake.F. Statistical Treatment of Open Systems by (jeneralized Master Equations (Vol. 66) Schwabl.F., Thirring.W. Quantum Theory of Laser Radiation (Vol. 36)... 

Fig. 2. Amplitude of spatial correlation function (a) and range (b) in a nonlinear chemical system, plotted against the nonequilibrium constraints...

Among various aspects of nonequilibrium phenomena the problem of transient behaviour is one of the most interesting ones. In particular, the relaxation from the unstable state was the subject of many studies Cl]. Recent experiments C23 show that stochastic effects may be very important in explaining irreproducibility of results for some nonlinear chemical systems. 

In recent years, problems related to the stability and the nonlinear dynamics of nonequilibrium systems invaded a great number of fields ranging from abstract mathematics to biology. One of the most striking aspects of this development is that subjects reputed to be "classical" and "well-established" like chemistry, turned out to give rise to a rich variety of phenomena leading to multiple steady states and hysteresis, oscillatory behavior in time, spatial patterns, or propagating wave fronts. 

The hierarchy that characterizes life is maintained through a multiplicity of catabolic and anabolic reactions, governed by a complex set of mechanisms that control the rate and timing of these processes. Thus, while this book will focus primarily on intermediary metabolism and its control mechanisms, in this chapter we will consider metabolic control from the viewpoint of nonlinear, nonequilibrium thermodynamics, which deals with the issue of how molecular events can be coupled and amplified so that they are expressed on a macroscopic level. The concepts herein expressed may be difficult to grasp at first reading. With perseverance, the beauty of the concepts will become apparent and their great importance for biological systems will become evident. 

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