: Patterns often emerge when a control parameter (like the Rayleigh number) crosses a threshold, making the uniform solution unstable to small perturbations.
For a stable homogeneous steady state to become unstable to spatial perturbations:
measures the control parameter's distance from the critical threshold. Canonical Paradigms of Pattern Formation pattern formation and dynamics in nonequilibrium systems pdf
When the pattern amplitude is no longer small—far from the instability threshold—amplitude equations are no longer valid. However, an alternative universal description, known as the , can be derived for situations where the pattern is well-formed but slowly distorted. The phase (\phi(\mathbfr, t)) describes the local position of the pattern's crests, and its dynamics are governed by a nonlinear diffusion equation. Phase dynamics provide a powerful tool for understanding phenomena such as pattern selection, defect motion, and the onset of chaos in extended systems.
When particle A affects B differently than B affects A (common in biological and social systems), new pattern-forming mechanisms arise. See recent work by Fruchart, Hanai, & Vitelli on arXiv (2021). : Patterns often emerge when a control parameter
An external source that pushes the system away from its natural, disordered state.
D_u, D_v = 0.01, 0.5 F, k = 0.035, 0.065 # FitzHugh-Nagumo parameters dt, dx = 0.1, 1.0 size = 100 However, an alternative universal description, known as the
The book opens with an inspiring "pep talk" that places nonequilibrium pattern formation in its broadest context: the Universe itself is a nonequilibrium system, having begun in a hot, dense, uniform state and subsequently evolving into the richly structured cosmos we observe today. The authors then introduce Rayleigh–Bénard convection as a paradigmatic example, using it to establish vocabulary and intuition before expanding to a wide range of other systems.