Lectures on Random Interfaces

 USD 39.99

 USD 39.99
Descripción de editorial
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 fixedreference discretized hyperplane, the system is governed by the Hamiltonian of gradient of the height functions. This is a kind of effective interface model called ∇φinterface model. The scaling limits are studied for Gaussian (or nonGaussian) random fields with a pinning effect under a situation in which the rate functional of the corresponding large deviation principle has nonunique minimizers.Young diagrams determine decreasing interfaces, and their dynamics are introduced. The largescale behavior of such dynamics is studied from the points of view of the hydrodynamic limit and nonequilibrium 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 timedependent 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 illposed SPDE and requires a renormalization. Especially its invariant measures are studied.
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