The following is a list of my publications. By selecting a title you get to the abstract and a link to the full text in compressed postscript (To test postscript-downloading, try this Postcard with references (13 kB)). These reprints are provided with the understanding that they will be used for limited copy personal use only. Please obtain copies from the publisher otherwise.

### Thesis: Cellular Automata for Reactive Systems

### Diffusion and Wave Propagation in Cellular Automata Models for Excitable Media

### Third Generation Cellular Automaton for Modeling Excitable Media

### Third Generation Cellular Automaton for Modeling Excitable Media

### Cellular Automaton Models for Reaction Diffusion Equations

### Fluctuation Correlations in Reaction-Diffusion Systems: Reactive Lattice Gas Automata Approach

### Spontaneous Nucleation in a Reactive Lattice Gas Automaton

### Class of Cellular Automata for Reaction-Diffusion Systems

### New Class of Cellular Automata for Reaction-Diffusion Systems Applied to the CIMA Reaction

### Reactive Cellular Automata

### Nonlinear reactions advected by a flow

### Cellular Automata for Reaction-Diffusion Systems

### Book: Simulation with Cellular Automata

### Three-dimensional Cellular Automata for Reaction-Diffusion Systems

### Coupling Microscopic and Macroscopic Cellular Automata

### JCASim - a Java system for simulating cellular automata

### Simulating reaction-diffusion cellular automata with JCASim

### Distributed Simulation Environment for Coupled Cellular Automata in Java

### The Java based cellular automata simulation system - JCASim

### Cellular Automata Approaches to Enzymatic Reaction Networks

### Translations of Cellular Automata for Efficient Simulation

### Simulation of High-Frequency Atmospheric Gas Discharges

PhD Thesis:

Two classes of cellular automata for reaction-diffusion systems are developed.
These are the * Reactive Lattice Gas Automata* and the
*Moving Average Cellular Automata*. For both classes explicit
procedures for the construction of the automata are given.
The construction can be based on given reaction mechanisms or on
partial differential equations.

The cellular automata methods are used to investigate several nonlinear reaction-diffusion systems. The reactive lattice gas automata possess intrinsic fluctuations that closely reflect fluctuations in real systems. These fluctuations are investigated in a bistable system and the correlations compared with the theoretical predictions from a Landau approach. The influence of the spatial dimension and the aspect ratio on the correlations is considered and it is demonstrated how geometrical restrictions increase the fluctuations and thereby increase the rate of transition between stable states. For investigations of macroscopic phenomena the moving average cellular automata are more efficient. They are used to simulate several models:

- The complex Ginzburg-Landau equation is used to demonstrate oscillatory spiral waves.
- The FitzHugh-Nagumo model is investigated as an example for excitable media. Excitable waves in circular and spiral form are shown, and it is demonstrated how the curvature of the wave front influences the speed of propagation. Spiral waves can be initiated either through special initial conditions or through intrinsic fluctuations. There are many analogies to the fronts in bistable media modeled with reactive lattice gas automata.
- Simulations of another reaction model (the Lengyel-Epstein model of the CIMA reaction) show Turing patterns. Here the emphasis is on the possible coexistence of different types of spatial or spatio-temporal patterns.

This work demonstrates that cellular automata are a powerful tool for modeling reaction-diffusion systems, including the effects of fluctuations.

(1) We examine general ``masks'' as discrete approximations to the diffusion equation, showing how to calculate the diffusion coefficient from the elements of the mask. (2) We combine the mask with a thresholding operation to simulate the propagation of waves in excitable media, showing that (for well-chosen masks) the waves obey a linear ``speed-curvature'' relation with slope given by the predicted diffusion coefficient. (3) We assess the utility of different masks in terms of computational efficiency and adherence to a linear speed-curvature relation.

This paper introduces a new cellular automaton model of excitable media with improved treatments of (1) diffusion and wave propagation, and (2) slow dynamics of the recovery variable. The automaton is both computationally efficient and faithful to the underlying partial differential equations.

This paper introduces a new cellular automaton model of excitable media with improved treatments of (1) diffusion and wave propagation, and (2) slow dynamics of the recovery variable. The automaton is both computationally efficient and faithful to the underlying partial differential equations.

This paper introduces a new cellular automaton model of excitable media with improved treatments of (1) diffusion and wave propagation, and (2) slow dynamics of the recovery variable. The automaton is both computationally efficient and faithful to the underlying partial differential equations.

We model reaction-diffusion systems with reactive lattice gas automata, which possess intrinsic microscopic fluctuations. We show that, within the limits of linear theory, the commonly accepted Landau equation describes correctly the measured effects of the fluctuations, as evidenced by the density autocorrelation function. We suggest that the reactive lattice gas automata constitute a powerful method for investigating reaction-diffusion systems where intrinsic fluctuations play an important role.

The influence of geometric constraints on nucleation phenomena in bistable reactions is investigated. It is demonstrated that a narrow channel with a gradual opening is an ideal geometry to facilitate nucleations. The reaction-diffusion system is simulated using reactive lattice gas models, which possess intrinsic fluctuations, as needed to initiate nucleations.

We introduce a new class of cellular automata to model reaction-diffusion systems in a quantitatively correct way. The construction of the CA from the reaction-diffusion equation relies on a moving average procedure to implement diffusion, and a probabilistic table-lookup for the reactive part. The applicability of the new CA is demonstrated using the Ginzburg-Landau equation.

We present a class of cellular automata (CAs) for modelling reaction-diffusion systems. The construction of the CA is general enough to be applicable to a large class of reaction-diffusion equations. The automata are based on a running average procedure to implement diffusion, and on a probabilistic table-lookup to implement the reaction. As an example application we present the Brandeisator (Lengyl-Epstein model for the chlorite-iodide-malonic acid reaction CIMA), which exhibits a rich set of behaviors: oscillations, hexagonal structures, stripes, and spirals. We investigate cases showing mixed states, in which different structures coexist in space: isolated spots, isolated regions of hexagons in a surrounding homogeneous region, coexistence between stripes and oscillations, and hexagons and stripes. The cellular automaton approach has the following advantages: fast simulations of large systems, easy implementation of noise in the system, and connections to other, more phenomenologically constructed CAs.

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We introduce a new class of cellular automata (CA) to model reaction-diffusion systems. The CA uses a moving average for diffusion and a probabilistic table-lookup for the reactive processes.

Flows can modify the effects of nonlinear reactions. Here we use a lattice Boltzmann method to simulate an athermal flow which advects reactant species. As an example we consider a reactive system modeled by the Brusselator which exhibits Turing structures; the system is subjected to mixing as induced by the wakes behind obstacles. The resulting effect is a modification of the reaction patterns which can be as dramatic as their disappearance.

A class of cellular automata for reaction-diffusion systems is presented. It is based on a local average for the diffusive dynamics, and closely related to finite difference schemes. The reactive dynamics is implemented as a lookup-table with probabilistic rules. The rules are derived directly and systematically from the given differential equations, using probabilistic rounding to enforce the discretization of the concentration variables. For quantitatively correct modeling, such probabilistic rules are usually necessary, but in some cases a deterministic version proves sufficient.

An introduction to cellular automata for the student or scientist of any discipline wishing to simulate real or imagined systems with spatial extent. The concept of cellular automata is described in simple terms and the design decisions for using CA in simulation are discussed in detail. Hard- and software for cellular automata is described and many examples are given, including reaction- diffusion systems, lattice gas models for fluid flow, traffic simulation and others. Each chapter concludes with some exercise problems.

**Table of contents:**

- 1 Introduction
- 1.1 Informal Definition of CA
- 1.2 Simple Example
- 1.3 Formal Definition
- 1.4 Exercises

- 2 Choices in the construction of CA
- 2.1 Lattice geometry
- 2.1.1 One dimension
- 2.1.2 Two dimensions
- 2.1.3 Three dimensions

- 2.2 Neighborhood size
- 2.3 Boundary conditions
- 2.4 Initial conditions
- 2.5 State set
- 2.6 Transition rule
- 2.6.1 Direct Specification
- 2.6.2 Small change in the rule table
- 2.6.3 Totalistic rules
- 2.6.4 Implicit specification
- 2.6.5 Multi-step rules
- 2.6.6 Probabilistic rules

- 2.7 Reversible CA
- 2.7.1 Partitioned Cellular Automata
- 2.7.2 Block Cellular Automata

- 2.8 Summary of construction choices
- 2.9 Exercises

- 2.1 Lattice geometry
- 3 Hardware and Software
- 3.1 Hardware inspired by CA: Connection Machine
- 3.2 Hardware for CA: CAM-8, CEPRA

- 3.3 Special CA Languages
- 3.4 Simulating CA in Mathematica
- 3.5 Coding alternatives
- 3.6 Exercises

- 4.1 Binary Model
- 4.2 Multi-speed Model
- 4.3 City Traffic
- 4.4 Exercises

- 5.1 Basic Model
- 5.2 Combined Model
- 5.3 Average Wetness
- 5.4 Exercises

- 6.1 Particle-Diffusion
- 6.2 Particle-Reaction
- 6.3 Moving Average for Diffusion
- 6.4 Reactive Step for Moving Average CA
- 6.4.1 Discretization operators

- 6.5 Examples of Moving Average CA
- 6.5.1 Schlögl Model
- 6.5.2 Complex Ginzburg-Landau Equation
- 6.5.3 FitzHugh-Nagumo Model, Excitable Media
- 6.5.4 Gray-Scott Model

- 6.6 Three-dimensional CA
- 6.7 Parallel Implementation
- 6.8 Summary
- 6.9 Exercises

- 7.1 Basic Lattice Gas Automata
- 7.2 Example Flow Simulation
- 7.3 Mathematical Analysis
- 7.3.1 Microscopic Description
- 7.3.2 Macroscopic Equations
- 7.3.3 Equilibrium

- 7.4 Extensions
- 7.4.1 Porous Media
- 7.4.2 Three Dimensions
- 7.4.3 Temperature
- 7.4.4 Multi-Phase Flow

- 7.5 Lattice Boltzmann Methods
- 7.6 Summary
- 7.7 Exercises

Cellular automata for reaction-diffusion systems are efficient enough to make the simulation of large three-dimensional systems feasible. The principal construction mechanisms used here are not much different from those for two-dimensional cellular automata. Diffusion is realized through a local averageing, and all the nonlinear reaction terms are collected in a table-lookup. The special issue in three dimensions is the need to increase time- and space- scales as much as possible to achieve sufficient system sizes. This can be done through the use of numerical integration schemes for constructing the lookup table, and through the use of special diffusion operators. We present examples of complex three-dimensional behaviours in an excitable reaction-diffusion system and in a model of a pattern-forming chemical reaction.

Cellular automata can model natural phenomena on many different levels of detail. Often, one specific level of detail is appropriate for the problem under investigation. But in some cases a connection between the different levels of detail is necessary. One such case is the catalytic reaction on a surface. The homogeneous crystallographic surface can reasonably well be described by mesoscopic approaches, but in the presence of defects, a microscopic simulation (where each site of the crystal lattice is individually represented) is necessary. In this paper we present a framework for coupling different types of cellular automata to achieve an efficient simulation which is nevertheless detailed enough to resolve microscopic phenomena where necessary.

The program system JCASim is a general-purpose system for simulating cellular automata in Java. It includes a stand-alone application and an applet for web presentations. The cellular automata can be specified in Java, in CDL, or using an interactive dialogue. The system supports many different lattice geometries (1-D, 2-D square, hexagonal, triangular, 3-D), neighborhoods, boundary conditions, and can display the cells using colors, text, or icons. We show several examples to demonstrate the wide applicability of the simulation system.

The program system JCASim is a general-purpose system for simulating cellular automata in Java. It includes a stand-alone application and an applet for web presentations. The cellular automata can be specified in Java, in CDL, or using an interactive dialogue. The system supports many different lattice geometries (1-D, 2-D square, hexagonal, triangular, 3-D), neighborhoods, boundary conditions, and can display the cells using colors, text, or icons. We use three kinds of cellular automata for reaction-diffusion systems to demonstrate the wide applicability of the simulation system. These are microscopic block-CA, reactive lattice gas CA, and macroscopic CA related to finite difference methods.

We describe a software environment for the coupling of different cellular automata. The system can couple CA for parallelization, can couple CA with different lattice sizes, different spatial or temporal resolutions, and even different state sets. It can also couple CA to other (possibly non-CA) simulations. The complete system, which is integrated with the CA simulation system JCASim, is written in Java and and uses remote method invocation as the communication method. The parallel efficiency is found to be satisfactory for large enough simulations.

The software system JCASim is a system for simulating cellular automata. It can simulate cellular automata (CA) with different geometries, different state sets (including state sets structured by the use of state components with different types), boundary conditions and initial conditions. The state transition function can be specified using a graphical user-interface, using Java (by programming only four methods), or in CDL, a special-purpose CA language. Cells can be represented by text, colors, or icons. JCASim also provides support for block-CA and for asynchronous models. The system is written entirely in Java to ensure portability. Here we describe the simulation system and show a number of examples, which demonstrate the ease of programming CA with JCASim. Finally, some issues surrounding execution speed are discussed.

Cellular automata simulations for enzymatic reaction networks differ from other models for reaction-diffusion systems, since enzymes and metabolites have very different properties. This paper presents a model where each lattice site can can contain at most one enzyme molecule, but many metabolite molecules. The rules are constructed to conform to the Michaelis-Menten kinetics by modeling the underlying mechanism of enzymatic conversion. Different possible approaches to rule construction are presented and analyzed, and simulations are shown for single reactions and simple enzyme networks.

Cellular automata can be described in many different ways, one of which is to use a special purpose description language. Here, the language CDL is used as the source for translations into Java or C code for computer simulations. Several coding styles are generated automatically: The state transition function can be coded as Java code, as C code with stubs for integration into a Java simulation environment, as a lookup table, or as Java code consisting of boolean functions which allow the parallel simulation of 32 or 64 cells on one processor. The coding styles are compared for several examples and it is found that the boolean function style (also called multispin-coding) realized in Java is often, but not always, significantly more efficient than even native C code.

The generation of atmospheric gas discharges is an important concern in many industrial applications. We are investigatin here a special type of high-frequency atmospheric gas discharge. The results of modelling th ebreakdown mechanism and of the develoment phase of the discharge will be presented. Furthermore we will show first numerical results.