We introduce a general method for the systematic derivation of nonlinear reaction-diffusion equations with distributed delays. We study the interactions among different types of moving individuals (atoms, molecules, quasiparticles, biological organisms, etc). The motion of each species is described by the continuous time random walk theory, analyzed in the literature for transport problems, whereas the interactions among the species are described by a set of transformation rates, which are nonlinear functions of the local concentrations of the different types of individuals. We use the time interval between two jumps (the transition time) as an additional state variable and obtain a set of evolution equations, which are local in time. In order to make a connection with the transport models used in the literature, we make transformations which eliminate the transition time and derive a set of nonlocal equations which are nonlinear generalizations of the so-called generalized master equations. The method leads under different specified conditions to various types of nonlocal transport equations including a nonlinear generalization of fractional diffusion equations, hyperbolic reaction-diffusion equations, and delay-differential reaction-diffusion equations. Thus in the analysis of a given problem we can fit to the data the type of reaction-diffusion equation and the corresponding physical and kinetic parameters. The method is illustrated, as a test case, by the study of the neolithic transition. We introduce a set of assumptions which makes it possible to describe the transition from hunting and gathering to agriculture economics by a differential delay reaction-diffusion equation for the population density. We derive a delay evolution equation for the rate of advance of agriculture, which illustrates an application of our analysis.

This paper reviews recent developments in the field of nonlinear chemical kinetics. Five topics are dealt with: (a) new approaches to complex reaction mechanisms, stoichiometric network analysis, classification of chemical oscillators and formulation of their mechanisms by deduction from experiments, and correlation metric construction of reaction pathways from measurements; (b) thermodynamic and stochastic theory of nonequilibrium processes, the eikonal approximation, the evaluation of stochastic potentials, experimental tests of the thermodynamic and stochastic theory of relative stability, and fluctuation-dissipation relations in nonequilibrium chemical systems; (c) chemical kinetics and cellular automata and lattice gas automata; (d) theoretical approaches and experimental studies of stochastic resonance in chemical kinetics; and (e) rate processes in disordered systems, stochastic Liouville equations, stretched exponential relaxation in disordered systems, and universality classes for rate processes in systems with static or dynamic disorder.

A quantitative model is proposed for the description of the oxidation kinetics in air at room temperature of single crystalline, hydrogen-terminated, (1 0 0) silicon. The theory separates the growth kinetics of the interfacial suboxide from those of the outer stoichiometric oxide. The theory proceeds assuming that the suboxide grows along the surface at the border of oxidized-silicon clusters, while the formation of the stoichiometric oxide takes place on the top of the suboxide at a rate decaying exponentially with the oxide thickness. In these hypotheses the kinetics of suboxide formation are found to depend on the initial concentration of (defective) oxo groups, while the growth of the stoichiometric oxide is described by the Elovich equation both in the short- and long-time limits.

We consider a system made up of different physical, chemical, or biological species undergoing replication, transformation, and disappearance processes, as well as slow diffusive motion. We show that for systems with net growth the balance between kinetics and the diffusion process may lead to fast, enhanced hydrodynamic transport. Solitary waves in the system, if they exist, stabilize the enhanced transport, leading to constant transport speeds. We apply our theory to the problem of determining the original mutation position from the current geographic distribution of a given mutation. We show that our theory is in good agreement with a simulation study of the mutation problem presented in the literature. It is possible to evaluate migratory trajectories from measured data related to the current distribution of mutations in human populations. In some cases reaction-diffusion systems can generate enhanced (hydrodynamic) transport due to mechanical or electromagnetic coupling; for example, the occurrence of a reaction produces variations in density or pressure and these variations lead to convection currents (1). This type of phenomenon may occur not only in macroscopic systems but also in singlemolecule kinetics (2). In this note we report on a type of enhanced transport in reaction-diffusion systems that does not require mechanical or electromagnetic coupling. We show that under rather general conditions the growth of a species leads to enhanced transport, which may be encountered in the case of diffusing, growing populations and is independent of the detailed kinetics of the process; in particular, it may exist whether the kinetics of the process is linear or nonlinear. The analysis of the enhanced transport induced by population growth is of interest in connection with a broad range of problems in physics, chemistry, and biology, which can be described by reactiondiffusion equations.The structure of this note is the following. We suggest a deterministic reaction-diffusion model, which describes the transformation, replication, disappearance, and diffusion of a set of interacting species. We derive transport equations for the fractions of the species and show that net population growth can induce enhanced transport for these fractions, even though the motion of individual species is diffusive. We show that the enhanced transport leads to a coherent motion characterized by the same transport (hydrodynamic) velocity for all fractions, provided that the transport and rate coefficients obey a neutrality condition. Further on, we discuss the implications of possible occurrence of solitary waves during the enhanced transport. Finally, we illustrate our approach by studying the geographical spreading of mutations in human populations.

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We investigate thermal conduction described by Newton's law of cooling and by Fourier's transport equation and chemical reactions based on mass action kinetics where we detail a simple example of a reaction mechanism with one intermediate. In these cases we derive exact expressions for the entropy production rate and its differential. We show that at a stationary state the entropy production rate is an extremum if and only if the stationary state is a state of thermodynamic equilibrium. These results are exact and independent of any expansions of the entropy production rate. In the case of thermal conduction we compare our exact approach with the conventional approach based on the expansion of the entropy production rate near equilibrium. If we expand the entropy production rate in a series and keep terms up to the third order in the deviation variables and then differentiate, we find out that the entropy production rate is not an extremum at a nonequilibrium steady state. If there is a strict proportionality between fluxes and forces, then the entropy production rate is an extremum at the stationary state even if the stationary state is far away from equilibrium.

A generalized thermodynamic description of one-variable complex chemical systems is suggested on the basis of the Ross, Hunt, and Hunt (RHH) theory of nonequilibrium processes. Starting from the stationary solution of a chemical Master Equation, two complimentary, related sets of generalized state functions are introduced. The first set of functions is derived from a generalized free energy F̌X, and is used to compute the moments of stationary and non-Gaussian concentration fluctuations. Exact expressions for the cumulants of concentration are derived; a connection is made between the cumulants and the fluctuation–dissipation relations of the RHH theory. The second set of functions is derived from an excess free energy φ(x); it is used to express the conditions of existence and stability of nonequilibrium steady states. Although mathematically distinct, the formalisms based on the F̌X and φ(x) functions are physically equivalent: both lead to the same type of differential expressions and to similar global equations. A comparison is made between the RHH and Keizer’s theory of nonequilibrium processes. An appropriate choice of the integration constants occurring in Keizer’s theory is made for one-variable systems. The main differences between the two theories are: the constraints for the two theories are different; the stochastic and thermodynamic descriptions are global in RHH, whereas Keizer’s theory is local. However, both theories share some common features. Keizer’s fluctuation–dissipation relation can be recovered by using the RHH approach; it is valid even if the fluctuations are nonlinear. If the thermodynamic constraints are the same, then Keizer’s theory is a first-order approximation of RHH; this approximation corresponds to a Gaussian description of the probability of concentration fluctuations. Keizer’s theory is a good approximation of RHH in the vicinity of a stable steady state: near a steady state the thermodynamic functions of the two theories are almost identical; the chemical potential in the stationary state is of the equilibrium form in both theories. Keizer’s theory gives a very good estimate of the absolute values of the peaks of the stationary probability density of RHH theory. Away from steady states the predictions of the two theories are different; the differences do not vanish in the thermodynamic limit. The shapes of the tails of the stationary probability distributions are different; and hence the predictions concerning the relative stability are different for the two theories.

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