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Description
Complexity & Criticality (V1)This book provides a challenging and stimulating introduction to the contemporary topics of complexity and criticality, and explores their common basis of scale invariance, a central unifying theme of the book. Criticality refers to the behaviour of extended systems at a phase transition where scale invariance prevails. The many constituent microscopic parts bring about macroscopic phenomena that cannot be understood by considering a single part
This book provides a challenging and stimulating introduction to the contemporary topics of complexity and criticality, and explores their common basis of scale invariance, a central unifying theme of the book.Criticality refers to the behaviour of extended systems at a phase transition where scale invariance prevails. The many constituent microscopic parts bring about macroscopic phenomena that cannot be understood by considering a single part alone. The phenomenology of phase transitions is introduced by considering percolation, a simple model with a purely geometrical phase transition, thus enabling the reader to become intuitively familiar with concepts such as scale invariance and renormalisation. The Ising model is then introduced, which captures a thermodynamic phase transition from a disordered to an ordered system as the temperature is lowered in zero external field. By emphasising analogies between percolation and the Ising model, the reader's intuition of phase transitions is developed so that the underlying theoretical formalism may be appreciated fully. These equilibrium systems undergo a phase transition only if an external agent finely tunes certain external parameters to particular values.Besides fractals and phase transitions, there are many examples in Nature of the emergence of such complex behaviour in slowly driven non-equilibrium systems: earthquakes in seismic systems, avalanches in granular media and rainfall in the atmosphere. A class of non-equilibrium systems, not constrained by having to tune external parameters to obtain critical behaviour, is addressed in the framework of simple models, revealing that the repeated application of simple rules may spontaneously give rise to emergent complex behaviour not encoded in the rules themselves. The common basis of complexity and criticality is identified and applied to a range of non-equilibrium systems. Finally, the reader is invited to speculate whether self-organisation in non-equilibrium systems might be a unifying concept for disparate fields such as statistical mechanics, geophysics and atmospheric physics.Visit for animations for the models in the book (available for Windows and Linux), solutions to exercises, as well as a list with corrections.Shipping Notes
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★★★★★ 5
Hands free comfort!!
Size: 5 Big Kid, Color: Black
Superb comfortable, great quality and got it at such a good price. My daughter is very happy and wears these every day since a couple of months and they still look new. Best part hands free!!
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Reviewed in the United States on December 1, 2025
★★★★★ 5
good shoes
Size: 8 Toddler, Color: Black, Size: 8 Toddler, Color: Black
fit great the child like it
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Reviewed in the United States on May 21, 2026
★★★★★ 5
Toddler approved
Size: 8 Toddler, Color: Black
My kid loves these shoes. We’ve bought them multiple times. She can get them on and off easily and they stand up to her wild adventures.
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Reviewed in the United States on May 3, 2026
★★★★★ 5
So cute and so worth it!
Size: 3 Little Kid, Color: Turquoise
AMAZING for my 7 year old daughter. She can tie shoes but I no longer have to wait on her to do so. It takes 2 seconds for her to put these on and she says that they are super comfortable. We have been to Silver dollar City multiple times and she never complains about her feet hurting! They are easily washed in the washer. They hold up great. I think they are very much worth the money! Fit normal.
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Reviewed in the United States on December 8, 2025
★★★★★ 4
Our everyday shoe!
Size: 13 Little Kid, Color: Turquoise
Great shoes and good fit. Fits as expected. The sole has worn down a bit quicker than I had hoped but my kiddo wears them everyday. I would buy them again for sure!
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Reviewed in the United States on November 30, 2025