Computational physics

Numerical analysis · Simulation Data analysis · Visualization

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A computer simulation, a computer model or a computational model is a computer program, or network of computers, that attempts to simulate an abstract model of a particular system. Computational physics is the study and implementation of numerical Algorithms in order to solve problems in Physics for which a quantitative theory already exists Numerical analysis is the study of Algorithms for the problems of continuous mathematics (as distinguished from Discrete mathematics) Data analysis is the process of looking at and summarizing Data with the intent to extract useful Information and develop conclusions See also Visualization and Information graphics Visualization is any technique for creating Images Diagrams or Computer programs (also software programs, or just programs) are instructions for a Computer. Simulation is the imitation of some real thing state of affairs or process Scientific modelling is the process of generating abstract, conceptual, Graphical and or mathematical models. Computer simulations have become a useful part of mathematical modelling of many natural systems in physics (computational physics), chemistry and biology, human systems in economics, psychology, and social science and in the process of engineering new technology, to gain insight into the operation of those systems, or to observe their behavior. Note The term model has a different meaning in Model theory, a branch of Mathematical logic. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Computational physics is the study and implementation of numerical Algorithms in order to solve problems in Physics for which a quantitative theory already exists Chemistry (from Egyptian kēme (chem meaning "earth") is the Science concerned with the composition structure and properties Foundations of modern biology There are five unifying principles Economics is the social science that studies the production distribution, and consumption of goods and services. Psychology (from Greek grc ψῡχή psȳkhē, "breath life soul" and grc -λογία -logia) is an Academic and The social sciences comprise academic disciplines concerned with the study of the social life of human groups and individuals including Anthropology, Communication studies Engineering is the Discipline and Profession of applying technical and scientific Knowledge and Technology is a broad concept that deals with a Species ' usage and knowledge of Tools and Crafts and how it affects a species' ability to control and adapt ^[1]

Computer simulations vary from computer programs that run a few minutes, to network-based groups of computers running for hours, to ongoing simulations that run for days. The scale of events being simulated by computer simulations has far exceeded anything possible (or perhaps even imaginable) using the traditional paper-and-pencil mathematical modeling: over 10 years ago, a desert-battle simulation, of one force invading another, involved the modeling of 66,239 tanks, trucks and other vehicles on simulated terrain around Kuwait, using multiple supercomputers in the DoD High Performance Computer Modernization Program; ^[2] a 1-billion-atom model of material deformation (2002); a 2. Note The term model has a different meaning in Model theory, a branch of Mathematical logic. The State of Kuwait ( دولة الكويت IPA [dawlatt̪ alkuwajt̪]) is a sovereign Arab Emirate on the coast of the Persian Gulf, enclosed The United States Department of Defense ( DOD or DoD) is the federal department charged with coordinating and supervising all agencies and functions of the government 64-million-atom model of the complex maker of protein in all organisms, a ribosome, in 2005;^[3] and the Blue Brain project at EPFL (Switzerland), began in May 2005, to create the first computer simulation of the entire human brain, right down to the molecular level. Ribosomes ( from ribo nucleic acid and "Greek soma ( meaning body") are complexes of RNA and Protein that Blue Brain is a project begun in May 2005 to create a Computer simulation of the brain of mammals including the human Brain, down to the molecular level ^[4]

Simulation versus modelling

Traditionally, the formal modelling, or modeling, of systems has been via a mathematical model, which attempts to find analytical solutions to problems which enables the prediction of the behaviour of the system from a set of parameters and initial conditions. Note The term model has a different meaning in Model theory, a branch of Mathematical logic.

While computer simulations might use some algorithms from purely mathematical models, computers can combine simulations with reality of actual events, such as generating input responses, to simulate test subjects who are no longer present. Whereas the missing test subjects are being modeled/simulated, the system they use could be the actual equipment, revealing performance limits or defects in long-term use by the simulated users.

Note that the term computer simulation is broader than computer modelling, which implies that all aspects are being modelled in the computer representation. However, computer simulation also includes generating inputs from simulated users to run actual computer software or equipment, with only part of the system being modelled: an example would be flight simulators which can run machines as well as actual flight software. A flight simulator is a system that tries to copy or simulate, the experience of flying an aircraft

Computer simulations are used in many fields, including science, technology, entertainment, and business planning and scheduling. Science (from the Latin scientia, meaning " Knowledge " or "knowing" is the effort to discover, and increase human understanding Technology is a broad concept that deals with a Species ' usage and knowledge of Tools and Crafts and how it affects a species' ability to control and adapt See also Entertainment (disambiguation and The Entertainer (disambiguation Entertainment is an activity designed to give people A business (also called firm or an enterprise) is a legally recognized organizational entity designed to provide goods and/or services to

History

Computer simulation was developed hand-in-hand with the rapid growth of the computer, following its first large-scale deployment during the Manhattan Project in World War II to model the process of nuclear detonation. The World War II Manhattan Project developed the first Nuclear weapon (atomic bomb World War II, or the Second World War, (often abbreviated WWII) was a global military conflict which involved a majority of the world's nations, including A nuclear weapon is an explosive device that derives its destructive force from Nuclear reactions either fission or a combination of fission and fusion. It was a simulation of 12 hard spheres using a Monte Carlo algorithm. Hard spheres are widely used as model particles in the statistical mechanical theory of fluids and solids Monte Carlo methods are a class of Computational Algorithms that rely on repeated Random sampling to compute their results Computer simulation is often used as an adjunct to, or substitution for, modeling systems for which simple closed form analytic solutions are not possible. There are many different types of computer simulation; the common feature they all share is the attempt to generate a sample of representative scenarios for a model in which a complete enumeration of all possible states of the model would be prohibitive or impossible. Computer models were initially used as a supplement for other arguments, but their use later became rather widespread.

Data preparation

The data input/output for the simulation can be either through formatted textfiles or a pre- and postprocessor. A text file (sometimes spelled "textfile" is a kind of Computer file that is structured as a sequence of lines. In Computer aided engineering (CAE a preprocessor is a program which provides a Graphical user interface (GUI to define physical properties

Types

Computer models can be classified according to several criteria including:

• Stochastic or deterministic (and as a special case of deterministic, chaotic) - see External links below for examples of stochastic vs. A stochastic process, or sometimes random process, is the counterpart to a deterministic process (or Deterministic system) in Probability theory. In Computer science, a deterministic algorithm is an Algorithm which in informal terms behaves predictably deterministic simulations
• Steady-state or dynamic
• Continuous or discrete (and as an important special case of discrete, discrete event or DE models)
• Local or distributed. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output Discrete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally discrete in the sense of not supporting or requiring the In discrete -event Simulation, the operation of a System is represented as a chronological sequence of events. Distributed computing deals with Hardware and Software Systems containing more than one processing element or Storage element concurrent

For example:

• Steady-state models use equations defining the relationships between elements of the modelled system and attempt to find a state in which the system is in equilibrium. Such models are often used in simulating physical systems, as a simpler modelling case before dynamic simulation is attempted.
• Dynamic simulations model changes in a system in response to (usually changing) input signals.
• Stochastic models use random number generators to model chance or random events; they are also called Monte Carlo simulations. A stochastic process, or sometimes random process, is the counterpart to a deterministic process (or Deterministic system) in Probability theory. A random number generator (often abbreviated as RNG is a computational or physical device designed to generate a sequence of Numbers or symbols that lack any Monte Carlo methods are a class of Computational Algorithms that rely on repeated Random sampling to compute their results
• A discrete event simulation (DES) manages events in time. In discrete -event Simulation, the operation of a System is represented as a chronological sequence of events. Most computer, logic-test and fault-tree simulations are of this type. In this type of simulation, the simulator maintains a queue of events sorted by the simulated time they should occur. The simulator reads the queue and triggers new events as each event is processed. It is not important to execute the simulation in real time. It's often more important to be able to access the data produced by the simulation, to discover logic defects in the design, or the sequence of events.
• A continuous dynamic simulation performs numerical solution of differential-algebraic equations or differential equations (either partial or ordinary). In Mathematics, differential algebraic equations (DAEs are a general form of Differential equation, given in implicit form A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i In Mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one Independent variable, and one or more of its Periodically, the simulation program solves all the equations, and uses the numbers to change the state and output of the simulation. Applications include flight simulators, construction and management simulation games, chemical process modeling, and simulations of electrical circuits. Construction and management simulation games (or CMS s are a type of Simulation game which task players to build expand or manage fictional communities or projects Chemical process modeling is a Computer modeling technique used in Chemical engineering process design. An electrical network is an interconnection of Electrical elements such as Resistors Inductors Capacitors Transmission lines Voltage Originally, these kinds of simulations were actually implemented on analog computers, where the differential equations could be represented directly by various electrical components such as op-amps. An analog computer (spelt analogue in British English is a form of Computer that uses continuous physical phenomena such as electrical mechanical An operational amplifier, often called an op-amp, is a DC - coupled high- Gain electronic voltage amplifier with differential By the late 1980s, however, most "analog" simulations were run on conventional digital computers that emulate the behavior of an analog computer. The 1980s was the decade spanning from January 1 1980 to December 31 1989. A computer is a Machine that manipulates data according to a list of instructions. An emulator duplicates (provides an emulation of the functions of one System using a different system so that the second system behaves like (and appears to
• A special type of discrete simulation which does not rely on a model with an underlying equation, but can nonetheless be represented formally, is agent-based simulation. In agent-based simulation, the individual entities (such as molecules, cells, trees or consumers) in the model are represented directly (rather than by their density or concentration) and possess an internal state and set of behaviors or rules which determine how the agent's state is updated from one time-step to the next.
• distributed models run on a network of interconnected computers, possibly through the Internet. Distributed computing deals with Hardware and Software Systems containing more than one processing element or Storage element concurrent The Internet is a global system of interconnected Computer networks Simulations dispersed across multiple host computers like this are often referred to as "distributed simulations". There are several standards for distributed simulation, including Aggregate Level Simulation Protocol (ALSP), Distributed Interactive Simulation (DIS), the High Level Architecture (simulation) (HLA) and the Test and Training Enabling Architecture (TENA). The Aggregate Level Simulation Protocol (ALSP is a protocol and supporting software that enables simulations to interoperate with one another Distributed Interactive Simulation ( DIS) is an open standard for conducting real-time platform-level Wargaming across multiple host computers and is used worldwide The High Level Architecture ( HLA) is a general purpose architecture for distributed Computer simulation Systems.

CGI computer simulation

Formerly, the output data from a computer simulation was sometimes presented in a table, or a matrix, showing how data was affected by numerous changes in the simulation parameters. The use of the matrix format was related to traditional use of the matrix concept in mathematical models; however, psychologists and others noted that humans could quickly perceive trends by looking at graphs or even moving-images or motion-pictures generated from the data, as displayed by computer-generated-imagery (CGI) animation. Note The term model has a different meaning in Model theory, a branch of Mathematical logic. Computer animation Computer-generated imagery (also known as CGI) is the application of the field of Computer graphics or more specifically 3D computer graphics Although observers couldn't necessarily read out numbers, or spout math formulas, from observing a moving weather chart, they might be able to predict events (and "see that rain was headed their way"), much faster than scanning tables of rain-cloud coordinates. In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each point Such intense graphical displays, which transcended the world of numbers and formulae, sometimes also led to output that lacked a coordinate grid or omitted timestamps, as if straying too far from numeric data displays. Today, weather forecasting models tend to balance the view of moving rain/snow clouds against a map that uses numeric coordinates and numeric timestamps of events. Weather forecasting is the application of science and technology to predict the state of the atmosphere for a future time and a given location

Similarly, CGI computer simulations of CAT scans can simulate how a tumor might shrink or change, during an extended period of medical treatment, presenting the passage of time as a spinning view of the visible human head, as the tumor changes. Computed tomography (CT is a Medical imaging method employing Tomography. A brain tumor is any intracranial Tumor created by abnormal and uncontrolled cell division, normally either in the Brain itself ( Neurons

Other applications of CGI computer simulations are being developed to graphically display large amounts of data, in motion, as changes occur during a simulation run.

Computer simulation in science

Generic examples of types of computer simulations in science, which are derived from an underlying mathematical description:

• a numerical simulation of differential equations which cannot be solved analytically, theories which involve continuous systems such as phenomena in physical cosmology, fluid dynamics (e. A computer simulation, a computer model or a computational model is a Computer program, or network of computers that attempts to simulate an A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the Physical cosmology, as a branch of Astronomy, is the study of the large-scale structure of the Universe and is concerned with fundamental questions about its Fluid dynamics is the sub-discipline of Fluid mechanics dealing with fluid flow: Fluids ( Liquids and Gases in motion g. climate models, roadway noise models, roadway air dispersion models), continuum mechanics and chemical kinetics fall into this category. This article is about the theories and mathematics of climate modeling Roadway noise is the collective sound energy emanating from Motor vehicles In the USA it contributes more to environmental noise exposure than any other noise source and is Roadway air dispersion modeling is the study of Air pollutant transport from a roadway or other linear emitter Continuum mechanics is a branch of Mechanics that deals with the analysis of the Kinematics and mechanical behavior of materials modeled as a continuum e Chemical kinetics, also known as reaction kinetics is the study of rates of chemical processes
• a stochastic simulation, typically used for discrete systems where events occur probabilistically, and which cannot be described directly with differential equations (this is a discrete simulation in the above sense). Stochastic (from the Greek "Στόχος" for "aim" or "guess" means Random. Probability is the likelihood or chance that something is the case or will happen Phenomena in this category include genetic drift, biochemical or gene regulatory networks with small numbers of molecules. In Population genetics, genetic drift is the accumulation of random events that change the makeup of a gene pool slightly but often compound over time Biochemistry is the study of the chemical processes in living Organisms It deals with the Structure and function of cellular components such as A Gene regulatory network (also called a GRN or genetic regulatory network) is a collection of DNA segments in a cell which interact with each (see also: Monte Carlo method). Monte Carlo methods are a class of Computational Algorithms that rely on repeated Random sampling to compute their results

Specific examples of computer simulations follow:

• statistical simulations based upon an agglomeration of a large number of input profiles, such as the forecasting of equilibrium temperature of receiving waters, allowing the gamut of meteorological data to be input for a specific locale. Temperature is a physical property of a system that underlies the common notions of hot and cold something that is hotter generally has the greater temperature Meteorology (from Greek grc μετέωρος metéōros, "high in the sky" and grc -λογία -logia) is the Interdisciplinary This technique was developed for thermal pollution forecasting . Thermal pollution is a Temperature change in natural bodies of water caused by human influence

• agent based simulation has been used effectively in ecology, where it is often called individual based modeling and has been used in situations for which individual variability in the agents cannot be neglected, such as population dynamics of salmon and trout (most purely mathematical models assume all trout behave identically). Ecology (from Greek grc οἶκος oikos, "house(hold" and grc -λογία -logia) is the scientific study of Population dynamics is the study of marginal and long-term changes in the numbers individual weights and age composition of individuals in one or several Populations and Salmon is the common name for several species of Fish of the family Salmonidae. Trout is the common name given to a number of Species of Freshwater Fish belonging to the Salmonidae family

• time stepped dynamic model. In hydrology there are several such hydrology transport models such as the SWMM and DSSAM Models developed by the U.S. Environmental Protection Agency for river water quality forecasting. An hydrological transport model is a Mathematical model used to simulate river or Stream flow and calculate water quality parameters The United States Environmental Protection Agency Storm Water Management Model ( SWMM) is a dynamic Rainfall - runoff - subsurface runoff The DSSAM Model (Dynamic Stream Simulation and AssessmentModel is a Computer simulation developed for the Truckee River to analyze water quality

• computer simulations have also been used to formally model theories of human cognition and performance, e. g. ACT-R

• computer simulation using molecular modelling for drug discovery

• Computational fluid dynamics simulations are used to simulate the behaviour of flowing air, water and other fluids. ACT-R (pronounced act-ARE: Adaptive Control of Thought--Rational is a Cognitive architecture mainly developed by John Robert Anderson at Carnegie Molecular modelling is a collective term that refers to theoretical methods and computational techniques to model or mimic the behaviour of Molecules The techniques In Medicine, Biotechnology and Pharmacology, drug discovery is the process by which drugs are discovered and/or designed Computational fluid dynamics (CFD is one of the branches of Fluid mechanics that uses Numerical methods and algorithms to solve and analyze problems that involve There are one-, two- and three- dimensional models used. A one dimensional model might simulate the effects of water hammer in a pipe. Water hammer (or more generally fluid hammer) is a Pressure surge or wave resulting when a fluid in motion is forced to stop or change direction suddenly (Momentum A two-dimensional model might be used to simulate the drag forces on the cross-section of an aeroplane wing. A three-dimensional simulation might estimate the heating and cooling requirements of a large building.

• An understanding of statistical thermodynamic molecular theory is fundamental to the appreciation of molecular solutions. Development of the Potential Distribution Theorem (PDT) allows one to simplify this complex subject to down-to-earth presentations of molecular theory.

Notable, and sometimes controversial, computer simulations used in science include: Donella Meadows' World3 used in the Limits to Growth, James Lovelock's Daisyworld and Thomas Ray's Tierra. Donella "Dana" Meadows ( March 13, 1941 Elgin Illinois, USA - February 20, 2001, New Hampshire) was a pioneering The World3 model was a Computer simulation of interactions between population industrial growth food production and limits in the Ecosystems of the Earth Limits to Growth is a 1972 book modeling the consequences of a rapidly growing World population and finite resource supplies commissioned by the Dr James Ephraim Lovelock, CH, CBE, FRS (born 26 July 1919) is an independent scientist author researcher environmentalist and Daisyworld, a Computer simulation, is a hypothetical world Orbiting a Sun whose Temperature is slowly increasing in the simulation Tierra is a Computer simulation developed by Ecologist Thomas S

Simulation environments for physics and engineering

Graphical environments to design simulations have been developed. Special care was taken to handle events (situations in which the simulation equations are not valid and have to be changed). The open project Open Source Physics was started to develop reusable libraries for simulations in Java, together with Easy Java Simulations, a complete graphical environment that generates code based on these libraries. Open Source Physics, or OSP is a project sponsored by the National Science Foundation and Davidson College, whose mission is to spread the use of open source code Open Source Physics, or OSP is a project sponsored by the National Science Foundation and Davidson College, whose mission is to spread the use of open source code

Computer simulation in practical contexts

Computer simulations are used in a wide variety of practical contexts, such as:

• analysis of air pollutant dispersion using atmospheric dispersion modeling
• design of complex systems such as aircraft and also logistics systems. Air pollution is the human introduction into the atmosphere of Chemicals Particulate matter, or Biological materials that cause harm or discomfort Atmospheric dispersion modeling is the mathematical simulation of how air pollutants disperse in the ambient atmosphere. Logistics is the management of the flow of Goods, Information and other resources including Energy and people between the point of origin and the point
• design of Noise barriers to effect roadway noise mitigation
• flight simulators to train pilots
• weather forecasting
• behavior of structures (such as buildings and industrial parts) under stress and other conditions
• design of industrial processes, such as chemical processing plants
• Strategic Management and Organizational Studies
• Reservoir simulation for the petroleum engineering to model the subsurface reservoir
• Process Engineering Simulation tools. A noise barrier (also called a soundwall, sound berm, sound barrier, or acoustical barrier) is an exterior structure designed to protect Noise mitigation is a set of strategies to reduce Noise pollution. A flight simulator is a system that tries to copy or simulate, the experience of flying an aircraft Weather forecasting is the application of science and technology to predict the state of the atmosphere for a future time and a given location Strategic management is the art science and craft of formulating implementing and evaluating cross-functional decisions that will enable an organization to achieve its objectives Reservoir simulation is an area of Reservoir engineering in which computer models are used to predict the flow of fluids (typically oil water and gas through Porous
• Robot simulators for the design of robots and robot control algorithms
• Traffic engineering to plan or redesign parts of the street network from single junctions over cities to a national highway network, see for example VISSIM. A robotics suite is a visual environment for robot control and Simulation. VisSim is a visual block diagram language for modeling simulating and analyzing dynamic systems

The reliability and the trust people put in computer simulations depends on the validity of the simulation model, therefore verification and validation are of crucial importance in the development of computer simulations. The term validity (also called logical truth, analytic truth, or necessary truth) as it occurs in Logic refers generally to a property of Scientific modelling is the process of generating abstract, conceptual, Graphical and or mathematical models. Another important aspect of computer simulations is that of reproducibility of the results, meaning that a simulation model should not provide a different answer for each execution. Although this might seem obvious, this is a special point of attention in stochastic simulations, where random numbers should actually be semi-random numbers. An exception to reproducibility are human in the loop simulations such as flight simulations and computer games. A personal computer Game (also known as a computer game or simply PC game) is a Video game played on a Personal computer, rather Here a human is part of the simulation and thus influences the outcome in a way that is hard if not impossible to reproduce exactly.

Computer graphics can be used to display the results of a computer simulation. Computer graphics are Graphics created by Computers and more generally the Representation and Manipulation of Pictorial Data Animations can be used to experience a simulation in real-time e. The bouncing ball animation (below consists of these 6 frames g. in training simulations. In some cases animations may also be useful in faster than real-time or even slower than real-time modes. For example, faster than real-time animations can be useful in visualizing the buildup of queues in the simulation of humans evacuating a building. Furthermore, simulation results are often aggregated into static images using various ways of scientific visualization. See also Visualization and Information graphics Visualization is any technique for creating Images Diagrams or

In debugging, simulating a program execution under test (rather than executing natively) can detect far more errors than the hardware itself can detect and, at the same time, log useful debugging information such as instruction trace, memory alterations and instruction counts. This technique can also detect buffer overflow and similar "hard to detect" errors as well as produce performance information and tuning data. In Computer security and programming, a buffer overflow, or buffer overrun, is an anomalous condition where a process attempts to

Pitfalls

Although sometimes ignored in computer simulations, it is very important to perform sensitivity analysis to ensure that the accuracy of the results are properly understood. Sensitivity analysis (SA is the study of how the variation (uncertainty in the output of a Mathematical model can be apportioned qualitatively or quantitatively to different For example, the probabilistic risk analysis of factors determining the success of an oilfield exploration program involves combining samples from a variety of statistical distributions using the Monte Carlo method. Monte Carlo methods are a class of Computational Algorithms that rely on repeated Random sampling to compute their results If, for instance, one of the key parameters (i. e. the net ratio of oil-bearing strata) is known to only one significant figure, then the result of the simulation might not be more precise than one significant figure, although it might (misleadingly) be presented as having four significant figures.

See also

• ACT-R
• Articulatory synthesis
• Artificial life
• CAVE
• Computer-aided design
• Computer simulation and organizational studies
• Dry Lab
• Earth Simulator
• Emulator
• Experiment in silico
• List of computer simulation software
• Mathematical model
• Molecular dynamics
• Simcyp Simulator
• Simulated reality
• Social simulation

External links

Organizations

• EUROSIM - Federation of European Simulation Societies
• Institute for Simulation and Training, University of Central Florida
• Simulation Interoperability Standards Organization
• The Society for Modeling and Simulation International (Formerly the Society of Computer Simulation)
• United States Defense Modeling and Simulation Office
• The System Dynamics Society
• The Computational Modelling Group at Cambridge University's Department of Chemical Engineering
• Liophant Simulation
• United Simulation Team - Genoa University

Education

• Simulation-An Enabling Technology in Software Engineering
• Sabanci University School of Languages Podcasts: Computer Simulation by Prof. David M. Goldsman
• IMTEK Mathematica Supplement (IMS) (some Mathematica-specific tutorials here)
• The Creative Learning Exchange
• McLeod Institute of Simulation Science

Examples

• A portfolio of free public simulations from the University of Florida
• Integrated Land Use, Transportation, Environment, (ILUTE) Modeling System
• Nanorobotics Simulation - Computational Nanomechatronics Lab. at Center for Automation in Nanobiotech (CAN)
• Online traffic simulation
• Shakemovie Caltech's Online Seismic Event Simulation
• Global Politics Simulation
• Industrial & Educational Examples of Modelling & Simulation
• Matlab SUrrogate MOdeling Toolbox - SUMO Toolbox - Matlab code for Surrogate Simulation Models

References

• R. Frigg and S. Hartmann, Models in Science. Entry in the Stanford Encyclopedia of Philosophy.
• S. Hartmann, The World as a Process: Simulations in the Natural and Social Sciences, in: R. Hegselmann et al. (eds. ), Modelling and Simulation in the Social Sciences from the Philosophy of Science Point of View, Theory and Decision Library. Dordrecht: Kluwer 1996, 77-100.
• P. Humphreys, Extending Ourselves: Computational Science, Empiricism, and Scientific Method. Oxford: Oxford University Press, 2004.