Lectures on Random Interfaces

Tadahisa Funaki (Autor)

Buch | Softcover

138 Seiten

2017 | 1st ed. 2016
Springer Verlag, Singapore
978-981-10-0848-1 (ISBN)

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Interfaces are created to separate two distinct phases in a situation in which phase coexistence occurs. This book discusses randomly fluctuating interfaces in several different settings and from several points of view: discrete/continuum, microscopic/macroscopic, and static/dynamic theories. The following four topics in particular are dealt with in the book.Assuming that the interface is represented as a height function measured from a fixed-reference discretized hyperplane, the system is governed by the Hamiltonian of gradient of the height functions. This is a kind of effective interface model called ∇φ-interface model. The scaling limits are studied for Gaussian (or non-Gaussian) random fields with a pinning effect under a situation in which the rate functional of the corresponding large deviation principle has non-unique minimizers.Young diagrams determine decreasing interfaces, and their dynamics are introduced. The large-scale behavior of such dynamicsis studied from the points of view of the hydrodynamic limit and non-equilibrium fluctuation theory. Vershik curves are derived in that limit.A sharp interface limit for the Allen–Cahn equation, that is, a reaction–diffusion equation with bistable reaction term, leads to a mean curvature flow for the interfaces. Its stochastic perturbation, sometimes called a time-dependent Ginzburg–Landau model, stochastic quantization, or dynamic P(φ)-model, is considered. Brief introductions to Brownian motions, martingales, and stochastic integrals are given in an infinite dimensional setting. The regularity property of solutions of stochastic PDEs (SPDEs) of a parabolic type with additive noises is also discussed.The Kardar–Parisi–Zhang (KPZ) equation , which describes a growing interface with fluctuation, recently has attracted much attention. This is an ill-posed SPDE and requires a renormalization. Especially its invariant measures are studied.

Erscheinungsdatum	14.01.2017
Reihe/Serie	SpringerBriefs in Probability and Mathematical Statistics
Zusatzinfo	9 Illustrations, color; 35 Illustrations, black and white; XII, 138 p. 44 illus., 9 illus. in color.
Verlagsort	Singapore
Sprache	englisch
Maße	155 x 235 mm
Themenwelt	Mathematik / Informatik ► Mathematik ► Analysis
	Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
	Naturwissenschaften ► Physik / Astronomie
Schlagworte	Dynamic Young diagrams • Introduction to stochastic partial differential equations • Kardar--Parisi--Zhang equation • Scaling limits for pinned interface model • Sharp interface limit for stochastic Allen--Cahn equations
ISBN-10	981-10-0848-5 / 9811008485
ISBN-13	978-981-10-0848-1 / 9789811008481
Zustand	Neuware