• Lattice statistical mechanics approaches to complex systems Ian Enting
• Modelling Complex Systems George Milne
• Directed Percolation Tony Guttmann
• Statistical mechanics applied to soft matter Alf Uhlherr
• Stochastic transport models in heterogeneous porous media Mike Trefy
• Bushfire behaviour modelling in Cellular Automata Andrew Sullivan.
• Fabio Boschetti

Lattice statistical mechanics approaches to complex systems

Ian Enting,

CSIRO Atmospheric Research.

Cellular automata have been suggested as powerful models for illustrating the emergence of complex behaviour from simple rules (e.g. [1]).  Among the aims of such research are:

Much of the work in this area has concentrated on deterministic cellular automata. The generalisation to stochastic cellular automata suggests the following advantages:

• it provides a wider (and potentially more flexible) class of models;
• it provides a formalism that can interpolate between deterministic rules and thus provide a way of exploring the cellular automata rule space;
• it allows the possibility of using standard tools from statistical mechanics (renormalisation group, series expansion, statistical closures etc.) in studying cellular automata.

Dr Enting's early work in this area grew out of cellular automata modelling of growth of mixed disordered crystals in order to determine the extent to which X-ray diffraction could recover the spatial characteristics of the crystal. [2,3]. Mapping this problem onto 'standard' statistical mechanics formalisms provided new techniques for solving these models [2,4,5]. The field of stochastic cellular automata was reviewed by Rujan [6].

Ongoing research in this area builds on a long-standing collaboration with the statistical mechanics group at Melbourne University, mainly developing and applying powerful series expansion techniques [7] to investigate critical phenomena. It also reflects work on spatially complex behaviour such as the firn-ice transition that is important for interpreting concentrations of greenhouse gases recovered from bubbles in polar ice [8]. The current research is conducted as part of the CSIRO Emerging Science initiative in Complex Systems Science.

References

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Modelling Complex Systems

George J. Milne

University of Western Australia

We utilise cellular automata and process algebraic formalisms to model complex, possibly chaotic systems, constructed out of many, simple, locally connected components. This approach has been applied to urban traffic flow modelling and simulation with success being judged by benchmarking our simulations against data sampled from the physical world. Current projects in this area include bushfire, epidemic and crowd dynamics modelling and simulation. Significantly, models are created in the Circal process algebra, a process algebra originally developed to describe and analyse the behaviour of digital logic. The applicability of Circal as a powerful descriptive medium for distinct classes of concurrent systems has been demonstrated by its application to a wide variety of application domains, including fluid flow and cardiac timing models.

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Directed Percolation

A.J. Guttmann, Dept of Mathematics and Statistics, The University of Melbourne.

An ongoing project is the extension to 3 dimensions of earlier work on directed percolation in 2 dimensions.

References

1 Baxter, R.J. and Guttmann, A.J., (1988) Series expansion of the percolation probability for the directed square lattice, J. Phys. A: Math. Gen. 21,  3193.

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Statistical mechanics applied to soft matter

Alf Uhlherr CSIRO Molecular Science

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Stochastic transport models in heterogeneous porous media

Mike Trefry, CSIRO Land and Water.

Conventional theories of contaminant transport in saturated porous media have their conceptual origins in spatial-temporal random walks [1] and rely on advection-dispersion equations, similar to Fokker-Planck equations. Here the porous media properties are reflected by heterogeneities in fluid velocity fields, dispersion tensors and, for non-conservative species, reactive terms.

For certain idealised and well-behaved classes of stochastic porous media, the Central Limit Theorem tells us that the transport of conservative species obeys an asymptotic result [2], so that after large travel times the contaminant plumes tend to Gaussian spatial distributions. This "emergent" property of the transport solutions permits the construction of simplified ("macrodispersive") transport equations in which fluid velocities and dispersion tensor elements are replaced by macroscopic averages to yield the appropriate asymptotic results [3].

Recent research shows that that the macrodispersion theories are unable to represent spatial contaminant distributions in the near-source and intermediate zones, unfortunately where contamination is of most practical interest [4]. More realistic and complex porous media are even more problematic [5]. Even so, popular fractional transport theories yield tantalizing results when applied to finite-moment processes, but many basic questions remain regarding the formation statistics of porous media.

New approaches are required to help identify and classify transport symmetries and invariants in the "pre-asymptotic" domains, for representative variations in stochastic input. Presently, high-resolution numerical forward modelling is being used to generate data sets against which to test candidate theories. Perhaps cellular automata may provide a route to new understandings in stochastic transport.

References

1. Einstein, A. (1908) in Investigations on the Theory of the Brownian Movement, translation by Dover Publications in 1956 of the original manuscript.
2. Gnedenko and Kolmogorov, H. (1968) Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, Reading, Mass.
3. Dagan, G. (1989) Flow and Transport in Porous Formations, Springer-Verlag, New York.
4. Trefry, M. G., Ruan, F. P. and McLaughlin, D. (2003) Numerical simulations of pre-asymptotic transport in heterogeneous porous media: Departures from the Gaussian limit. Water Resources Research vol. 39, No. 0, XXXX, doi:10.1029/2001WR001101.
5. Ruan, F. P., Trefry, M. G. and McLaughlin, D. (2003) Pre-asymptotic transport in multiscale random fields, in preparation.

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Bushfire behaviour modelling in Cellular Automata

Andrew Sullivan

Our interest in cellular automata stems from the desire to improve the prediction of the spread of bushfires through heterogeneous fuels and to develop a platform in which the dynamics and complexity of fire behaviour can be studied.

CA is attractive because it can provide a spatial approach to modelling in an area that has traditionally been 1-dimensional and extrapolated to 2 dimensions (i.e. they predict a single point rate of forward spread and not the spread of the perimeter) ( Coleman and Sullivan 1996, Finney 1998) and has a discrete time advance. Perhaps because of this 2-dimensional spatial aspect, bushfires have been a favourite topic for many people wanting to apply their hard fought understanding of CA to real world problems.

However, previous implementations of CA (or percolation) to bushfire behaviour have been rather short on actual fire behaviour. Many attempts have not been true CA but rather raster implementation of empirical fire spread models (e.g. Green et al 1990, Guariso and Baracani 2002). CA that have used actual site state rules for modelling fire spread have been based on overly simple understanding of bushfire behaviour and their performance is questionable (e.g. Karafyllidis and Thanailakis 1997, Li and Magill 2000). Most applications of CA to bushfires have involved the study of self-organised criticality of many fires over great periods of time (e.g. Bak 1996, Malamud et al. 1998, Malamud and Turcotte 1999) which provides us with no information about the behaviour of individual bushfires.

Our current interest in CAs as applied to bushfires is of two parts. The first is the 'traditional' approach of the development of a set of CA rules that are more closely related to the behaviour of bushfires as we understand them and derived from fundamental principles. This may include multiple overlapping fuel state sites (the state change of which is determined by the behaviour of the fire in the previous time step), governance of local site rules by global variables or as some function of a range of active site states, and/or the inclusion of principles of fluid dynamics and the interaction of the atmosphere with the fire.

The second is the investigation of the possibility of using a genetic algorithm approach to develop site rules using an evolving population of rules and a fixed population of fire perimeter isopleths obtained from oblique aerial photography. At each step in evolution, the previously best performing rule from the population of rules would be tested against the entire population of fire perimeter isopleths and a measure of its performance made. To ensure that a local peak in the rule population landscape is not found, the previously most difficult fire isopleth would be tested using the entire rule population. The best rule from this evolution step would then be selected for mutation in the next step. The process would be repeated until a rule is evolved that can perform well against the entire population of isopleths.

References

Bak, P. 1996. How nature works: The Science of Self-organised Criticality. Springer-Verlag, New York.

Coleman, J.R. and Sullivan, A.L. 1996. A real-time computer application for the prediction of fire spread across the Australian landscape. Simulation 67, 230-240.

Finney, M.A. 1998. FARSITE: Fire area simulator--model development and evaluation. USDA Forest Service Research Paper RMRS-RP-4. 47 pp.

Green, D.G., Tridgell, A., and Gill, A.M. 1990. Interactive simulation of bushfires in heterogeneous fuels. Mathematical and Computer Modelling 13, 57-66.

Guariso, G., and Baracani. 2002. A simulation software of forest fires based on two-level cellular automata. In: Proceedings of the IV International Conference on Forest Fire Research 2002 and Wildland Fire Safety Summit, Luso, Portugal, 18-23 November 2002 (ed. DX Viegas).

Karafyllidis, I. and Thanailakis, A. 1997. A model predicting forest fire spread using cellular automata. Ecological Modelling 99, 87-97.

Li, X. and Magill, W. 2000. Modelling fire spread under environmental influence using a cellular automaton approach. Complexity International 8 ( Former link, no longer valid: http://www.csu.edu.au/ci/vol08/li01/)

Malamud, B.D., Morein, G., and Turcotte, D.L. 1998. Forest Fires: An example of self-organised critical behaviour. Science 281, 1840-1842.

Malamud, B.D. and Turcotte, D.L. 1999. Self-organised criticality applied to natural hazards. Natural Hazards 20, 93-116.

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