Complex Systems

A wide spectrum of scientists are showing great interest in the development of complex structures in systems far from equilibrium. Over the last decade or two, the advent of large-scale computer simulations has led to a huge increase of scientific activity in this area, which has emerged as a major subject of interdisciplinary research. The problems of interest originate in a broad range of disciplines, from mathematics, physics, chemistry and computer science to biology and economics, but they possess two common features. Firstly, they are frequently modelled as systems of a large number of objects which evolve according to simple dynamical rules, which may nevertheless result in emergent structures of great complexity at larger scales or after long times. Secondly random, or stochastic, effects are typically important: both determinism and chance are needed to describe reality. Typical questions which concern those who work in this area relate to self-organisation, the positive role of noise and fluctuations, pattern formation, self-similarity, fractals and so on.

By its very nature, it is not easy to define subdivisions of the field of complex systems. However, for the purpose of describing my research, I would like to distinguish between two different ways in which stochastic effects (also called noise) may influence the system and be analysed. The first is where the noise may be thought of as an extra aspect of the system to be modelled, and can be investigated using the techniques of the theory of stochastic processes. Of particular interest are novel mechanisms not seen in deterministic systems. The other way in which stochastic effects may be important is when the effects of noise are "creative", in the sense of evolutionary dynamics. Examples of the creative effects of noise usually stem from the production of "errors" (e.g. mutations) which may give rise to new variants and eventually lead to complex structures. This naturally finds most applications in the fields of biology, economics and social science, but is being used more and more in other areas, for example in computer science (genetic algorithms).

In the first case, the noise may be internal or external. An example of internal (or intrinsic) noise is demographic stochasticity in biological systems (noise due to the discreteness of individuals who comprise the system) and of external noise is environmental stochasticity, which has its origin in external factors not included in the model description. The nature of the model may also vary depending on the type of noise. If the model is individual-based, then the starting point might be a Markov chain or master equation (continuous time Markov chain) whereas if the noise is external, the starting point might be a phenomenological equation of the Langevin type. These factors will also govern the techniques that are used to analyse the model. If it is formulated as a master equation, a powerful technique that I frequently use is van Kampen's system size expansion. At leading order in the system size, it gives the deterministic equation(s) for the process and at next-to-leading order stochastic corrections to this. If the model is formulated as a Langevin equation, the path-integral is a useful calculational tool. The discussion on amplified stochastic cycles is mainly focused on individual based models described as master equations and analysed using the system-size expansion. The discussion on stochastic dynamics also covers systems of this type, but in addition those described as Langevin equations and analysed using path-integral methods are discussed.

The second case mentioned above relates to evolutionary dynamics. Although, in the standard evolutionary paradigm, mutations may give rise to individuals who are fitter, and so are selected, as explained in the section on neutral theories this need not be the case. Examples of neutral theories from the fields of population genetics, ecology and language evolution are described in that section. I have also been interested in applying a more standard evolutionary dynamics to construct networks, in particular food webs, where webs containing a large number of species are "grown" starting with only a very small number of species.

While I believe those working in the area of statistical mechanics have many skills to bring to modelling such systems, I also believe that close collaboration with experts in the appropriate fields is very necessary. So for instance, I have been working with Bill Croft from the Department of Linguistics, University of New Mexico in constructing a mathematical model of the evolution of language and studying it both analytically and using computer simulations. In addition, I have been involved in other ways in trying to forge links between researchers in those disciplines who have interests in the theory of complex systems. For example, I have in the past organised a course entitled "Complex Adaptive Systems" which was jointly given by members of the Departments/Schools of Biological Sciences, Computer Science, Economic Studies and Physics, and have run a "Complex Systems Discussion Group" for 10 years. I am also the principal investigator of a linked grant which is part of the NANIA collaboration involving individuals from universities in Edinburgh and Manchester. This is funded under the EPSRC "Novel computation: coping with complexity" initiative.

Interdisciplinary research of this kind is now one of the most active areas of theoretical science. I believe that the trend whereby knowledge generated by those working in statistical mechanics is applied to problems in other disciplines is one which is set to grow and will be an important feature of scientific activity in the 21st century.

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