Attila László Nagy

Coexistence of Shocks and Rarefaction Fans: Complex Phase Diagram of a Simple Hyperbolic Particle System

This paper investigates the non-equilibrium hydrodynamic behavior of a simple totally asymmetric interacting particle system of particles, antiparticles and holes on

Dependent Double Branching Annihilating Random Walk

\mathbb {Z}

Graphs of Reactions

Z. Rigorous hydrodynamic results apply to our model with a hydrodynamic flux that is exactly calculated and shown to change convexity in some region of the model parameters. We then characterize the entropy solutions of the hydrodynamic equation with step initial condition in this scenario which include various mixtures of rarefaction fans and shock waves. We highlight how the phase diagram of the model changes by varying the model parameters.

Decomposition of Reactions

Thirty-nine detailed mechanisms for combustion of hydrogen, carbon monoxide and methanol are investigated using ReactionKinetics, a Mathematica based package published earlier. Our methods involved mainly structural and graph theoretical approaches as well as techniques which are related to the time evolution of the considered mechanisms. Our investigations support the view that the hydrogen mechanisms tend to take on a final form in these days. CO combustion mechanisms, however, showed a larger variety both in species and in reaction steps. There exist only a few mechanisms directly developed to describe methanol combustion (mechanisms developed for other purposes may contain a submechanism for methanol combustion); the big differences between them shows that the modeling community is only at the very beginning of exploring this process. Most of our results do not depend on the choice of reaction rate coefficients, the methods only use the underlying sets of reaction steps, hence they are robust and general. These investigations can be used before or in parallel with usual numerical investigations, such as pathway analysis, sensitivity analysis, parameter estimation or simulation. The package and the methods may be useful for automatic mechanism generations, testing, comparing and reduction of mechanisms as well, especially in the case of large systems.

Past, Present, and Future Programs for Reaction Kinetics

Double (or parity conserving) branching annihilating random walk, introduced in [19], is a one-dimensional non-attractive particle system in which positive and negative particles perform nearest neighbor hopping, produce two offsprings to neighboring lattice points and annihilate when they meet. Given an odd number of initial particles, positive recurrence as seen from the leftmost particle position was first proved in [2] and, subsequently in a much more general setup, in [16]. These results assume that jump rates of the various moves do not depend on the configuration of the particles not involved in these moves. The present article deals with the case when the jump rates are affected by the locations of several particles in the system. Motivation for such models comes from non-attractive interacting particle systems with particle conservation. Under suitable assumptions we establish the existence of the process, and prove that the one-particle state is positive recurrent. We achieve this by arguments similar to those appeared in [16]. We also extend our results to some cases of long range jumps, when branching can also occur to non-neighboring sites. We outline and discuss several particular examples of models where our results apply.

Approximations of the Models

This chapter focuses on the formal analysis of the most relevant model describing the time evolution of concentrations of reactions. This model is deterministic; both the time and the state space are continuous. The corresponding mathematical object is a system of ordinary differential equations, called the induced kinetic differential equation of the reaction in question. This model can be constructed in different ways and has different forms; we are going to study these forms in the present chapter. Special emphasis is laid on those reactions which are endowed with mass action type kinetics: In these cases the induced kinetic differential equation is a special polynomial differential equation. The right-hand side of the induced kinetic differential equation of reactions can be factorized in many different ways, and these factorizations facilitate the analyses of the stationary states (see Chap. 7) and the dynamic behavior (see Chap. 8), as well; furthermore, they help solve some of the inverse problems of reaction kinetics (see Chap. 11). A remark with far-reaching consequences is that although the induced kinetic differential equation is a nonlinear differential equation, it has many linear features. At the end of the chapter, we introduce and discuss models with diffusion, i.e. space inhomogeneities, and those in which the change and interplay with chemistry of a further physical parameter, temperature, is also taken into consideration.

Time-Dependent Behavior of the Concentrations

There are a series of graphs which are able to represent chemical reactions so as to lead to consequences on their (static or dynamic) behavior. We start with the Feinberg–Horn–Jackson graph, the one which may be the most well known for the chemists. Strictly connected to this concept is reversibility and weak reversibility and also the concept of central importance: deficiency. The Volpert graph is almost as well known, especially for those involved in metabolism research. Indexing of the Volpert graph has relevant consequences on the time evolution of the concentrations. Relationships between the two graphs are studied. Finally, the species–complex–linkage class graph and the complex graph are finally introduced. Further graphs (such as the influence diagram, the graph of atomic fluxes, the state space of the stochastic model) even more closely related to time-dependent behavior can and will only be introduced in later chapters.

ReactionKinetics—A Mathematica package with applications

The first step in chemical kinetic research often is the determination of an overall reaction reflecting the summarized stoichiometry of a reaction. However, such an overall reaction is the result of a set of elementary steps. Finding out the constituent steps is usually the goal to be reached by the joint effort of the chemist and the mathematician. Here the mathematical tools useful in this area have been collected and presented.