Abstract
Complex macroscopic behaviour can arise in many-body systems with only very simple elements as a consequence of the combination of competition and inhomogeneity. This paper attempts to illustrate how statistical physics has driven this recognition, has contributed new insights and methodologies of wide application, influencing many fields of science, and has been stimulated in return.
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Notes
- 1.
Spin glasses were originally observed as magnetic alloys with unusual non-periodic spin ordering. They were also later recognized as having many other fascinating glassy properties.
- 2.
This restriction is not essential but represents the potentially hardest case.
- 3.
This is in contrast with traditional computer science which has been more concerned with worst instances.
- 4.
The overline indicates an average over the quenched disorder.
- 5.
There are several possible microscopic dynamics that leads to the same equilibrium/Gibbsian measure, but all such employing local dynamics lead to glassiness.
- 6.
Instead one finds a modified fluctuation-dissipation relation with the temperature normalized by the instantaneous auto-correlation.
- 7.
Sometimes one speaks of fast and slow microscopic variables but it should be emphasised that these refer to the underlying microscopic time-scales. Glassiness leads to much slower macroscopic timescales.
- 8.
i.e. uncorrelated patterns.
- 9.
The philosophy is that one gets the best price by selling when most want to buy or buying when most want to sell.
- 10.
One can make the model even more minimal by allowing each agent only one strategy \(\{\xi _{i}\}\) which (s)he either follows if its point-score is positive or acts oppositely to if the point-score is negative. This removes the random-field term and also the cusp in the tabula rasa volatility, but retains the ergodic-nonergodic transition [5].
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Acknowledgments
The author thanks the Leverhulme Trust for the award of an Emeritus Fellowship and the UK EPSRC, the EU and the ESF for support over many years during the development of the work reported here. He also thanks many colleagues throughout the world for collaborations and valuable discussions; most of their names are given in the last slide of his 2010 Blaise Pascal lecture that can be found at [9].
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Sherrington, D. (2014). Physics and Complexity: An Introduction. In: Delitala, M., Ajmone Marsan, G. (eds) Managing Complexity, Reducing Perplexity. Springer Proceedings in Mathematics & Statistics, vol 67. Springer, Cham. https://doi.org/10.1007/978-3-319-03759-2_13
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