SKU: 23948880777

Complexity & Criticality (V1)

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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.
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SKU: 23948880777

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Jasmine Unger
Fort Morgan, US
★★★★★ 5
Great slip lead
Color: Blue, Size: Large
Love the double handle for walks
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Reviewed in the United States on April 6, 2026
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Papa
Boise, US
★★★★★ 4
Second handle too high up on leash
Color: Black, Size: Large
Slip lead length provides your reactive dog with too much range to lunge. The second handle is great but it is too high up on the leash. If the slip lead length was shorter and the second handle was lower, this product would be awesome.
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Reviewed in the United States on December 3, 2025
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Jill Crowell
Fort Morgan, US
★★★★★ 5
Great lead alternative!
Color: Blue, Size: Large
This works great for an alternative to just a slip lead. I work in a kennel and it makes it a lot easier to grip for the dogs that pull a lot.
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Reviewed in the United States on January 28, 2026
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nicole ernst
Pawtucket, US
★★★★★ 5
Great training tool
Color: Red, Size: Large
This leash works as a great training tool. Currently training, my dog leash manners, and it is working well.
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Reviewed in the United States on March 29, 2026
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Nancy
Battle Creek, US
★★★★★ 3
Won’t stay in place
Color: Blue, Size: Large
While the leash was nicely made, the lead would not stay in place. The slide to hold the “O” ring in place kept sliding there for the leash would loosen. The padded handle was nice for sure
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Reviewed in the United States on November 1, 2025

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