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Author: Tadahisa Funaki Publisher: Springer ISBN: 9811008493 Category : Mathematics Languages : en Pages : 147
Book Description
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 dynamics is 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.
Author: Tadahisa Funaki Publisher: Springer ISBN: 9811008493 Category : Mathematics Languages : en Pages : 147
Book Description
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 dynamics is 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.
Author: Erwin Bolthausen Publisher: Springer ISBN: 3540479449 Category : Mathematics Languages : en Pages : 469
Book Description
This volume contains lectures given at the Saint-Flour Summer School of Probability Theory during the period 8th-24th July, 1999. We thank the authors for all the hard work they accomplished. Their lectures are a work of reference in their domain. The School brought together 85 participants, 31 of whom gave a lecture concerning their research work. At the end of this volume you will find the list of participants and their papers. Finally, to facilitate research concerning previous schools we give here the number of the volume of "Lecture Notes" where they can be found: Lecture Notes in Mathematics 1975: n ° 539- 1971: n ° 307- 1973: n ° 390- 1974: n ° 480- 1979: n ° 876- 1976: n ° 598- 1977: n ° 678- 1978: n ° 774- 1980: n ° 929- 1981: n ° 976- 1982: n ° 1097- 1983: n ° 1117- 1988: n ° 1427- 1984: n ° 1180- 1985-1986 et 1987: n ° 1362- 1989: n ° 1464- 1990: n ° 1527- 1991: n ° 1541- 1992: n ° 1581- 1993: n ° 1608- 1994: n ° 1648- 1995: n ° 1690- 1996: n ° 1665- 1997: n ° 1717- 1998: n ° 1738- Lecture Notes in Statistics 1971: n ° 307- Table of Contents Part I Erwin Bolthausen: Large Deviations and Interacting Random Walks 1 On the construction of the three-dimensional polymer measure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Self-attracting random walks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3 One-dimensional pinning-depinning transitions. . . . . . . . . . . 105 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Author: Golden Krishna Publisher: New Riders ISBN: 0133890422 Category : Computers Languages : en Pages : 257
Book Description
Our love affair with the digital interface is out of control. We’ve embraced it in the boardroom, the bedroom, and the bathroom. Screens have taken over our lives. Most people spend over eight hours a day staring at a screen, and some “technological innovators” are hoping to grab even more of your eyeball time. You have screens in your pocket, in your car, on your appliances, and maybe even on your face. Average smartphone users check their phones 150 times a day, responding to the addictive buzz of Facebook or emails or Twitter. Are you sick? There’s an app for that! Need to pray? There’s an app for that! Dead? Well, there’s an app for that, too! And most apps are intentionally addictive distractions that end up taking our attention away from things like family, friends, sleep, and oncoming traffic. There’s a better way. In this book, innovator Golden Krishna challenges our world of nagging, screen-based bondage, and shows how we can build a technologically advanced world without digital interfaces. In his insightful, raw, and often hilarious criticism, Golden reveals fascinating ways to think beyond screens using three principles that lead to more meaningful innovation. Whether you’re working in technology, or just wary of a gadget-filled future, you’ll be enlighted and entertained while discovering that the best interface is no interface.
Author: Amir Dembo Publisher: Springer Science & Business Media ISBN: 9783540260691 Category : Mathematics Languages : en Pages : 300
Book Description
This volume contains two of the three lectures that were given at the 33rd Probability Summer School in Saint-Flour (July 6-23, 2003). Amir Dembo’s course is devoted to recent studies of the fractal nature of random sets, focusing on some fine properties of the sample path of random walk and Brownian motion. In particular, the cover time for Markov chains, the dimension of discrete limsup random fractals, the multi-scale truncated second moment and the Ciesielski-Taylor identities are explored. Tadahisa Funaki’s course reviews recent developments of the mathematical theory on stochastic interface models, mostly on the so-called \nabla \varphi interface model. The results are formulated as classical limit theorems in probability theory, and the text serves with good applications of basic probability techniques.
Author: Sacha Friedli Publisher: Cambridge University Press ISBN: 1316886964 Category : Mathematics Languages : en Pages : 644
Book Description
This motivating textbook gives a friendly, rigorous introduction to fundamental concepts in equilibrium statistical mechanics, covering a selection of specific models, including the Curie–Weiss and Ising models, the Gaussian free field, O(n) models, and models with Kać interactions. Using classical concepts such as Gibbs measures, pressure, free energy, and entropy, the book exposes the main features of the classical description of large systems in equilibrium, in particular the central problem of phase transitions. It treats such important topics as the Peierls argument, the Dobrushin uniqueness, Mermin–Wagner and Lee–Yang theorems, and develops from scratch such workhorses as correlation inequalities, the cluster expansion, Pirogov–Sinai Theory, and reflection positivity. Written as a self-contained course for advanced undergraduate or beginning graduate students, the detailed explanations, large collection of exercises (with solutions), and appendix of mathematical results and concepts also make it a handy reference for researchers in related areas.
Author: Clay Mathematics Institute. Summer School Publisher: American Mathematical Soc. ISBN: 0821868632 Category : Mathematics Languages : en Pages : 481
Book Description
This volume is a collection of lecture notes for six of the ten courses given in Buzios, Brazil by prominent probabilists at the 2010 Clay Mathematics Institute Summer School, ``Probability and Statistical Physics in Two and More Dimensions'' and at the XIV Brazilian School of Probability. In the past ten to fifteen years, various areas of probability theory related to statistical physics, disordered systems and combinatorics have undergone intensive development. A number of these developments deal with two-dimensional random structures at their critical points, and provide new tools and ways of coping with at least some of the limitations of Conformal Field Theory that had been so successfully developed in the theoretical physics community to understand phase transitions of two-dimensional systems. Included in this selection are detailed accounts of all three foundational courses presented at the Clay school--Schramm-Loewner Evolution and other Conformally Invariant Objects, Noise Sensitivity and Percolation, Scaling Limits of Random Trees and Planar Maps--together with contributions on Fractal and Multifractal properties of SLE and Conformal Invariance of Lattice Models. Finally, the volume concludes with extended articles based on the courses on Random Polymers and Self-Avoiding Walks given at the Brazilian School of Probability during the final week of the school. Together, these notes provide a panoramic, state-of-the-art view of probability theory areas related to statistical physics, disordered systems and combinatorics. Like the lectures themselves, they are oriented towards advanced students and postdocs, but experts should also find much of interest.
Author: Giambattista Giacomin Publisher: Springer Science & Business Media ISBN: 3642211550 Category : Language Arts & Disciplines Languages : en Pages : 140
Book Description
Understanding the effect of disorder on critical phenomena is a central issue in statistical mechanics. In probabilistic terms: what happens if we perturb a system exhibiting a phase transition by introducing a random environment? The physics community has approached this very broad question by aiming at general criteria that tell whether or not the addition of disorder changes the critical properties of a model: some of the predictions are truly striking and mathematically challenging. We approach this domain of ideas by focusing on a specific class of models, the "pinning models," for which a series of recent mathematical works has essentially put all the main predictions of the physics community on firm footing; in some cases, mathematicians have even gone beyond, settling a number of controversial issues. But the purpose of these notes, beyond treating the pinning models in full detail, is also to convey the gist, or at least the flavor, of the "overall picture," which is, in many respects, unfamiliar territory for mathematicians.
Author: P L Garrido Publisher: World Scientific ISBN: 9814554022 Category : Languages : en Pages : 390
Book Description
This book contains the invited lectures and a short account of communications at the II Granada Lectures which focused on Dynamical Systems. Key concepts such as dissipative dynamical systems, orbits, bifurcations, classical Hamiltonian chaos, KAM theorem, hyperbolic sets, time series analysis, renormalization group, quantum chaos and their applications were covered during the seminar. In addition, popular topics in computational statistical physics such as models of growth, material physics, fluids, nonequilibrium phase transitions, critical phenomena and computational astrophysics were also discussed. Written pedagogically at the graduate level, the topics were described comprehensively and supported by illustrations. This book is useful for beginners and a valuable reference for professionals in this field.
Author: Neal Noah Madras Publisher: American Mathematical Soc. ISBN: 0821829785 Category : Mathematics Languages : en Pages : 113
Book Description
Monte Carlo methods form an experimental branch of mathematics that employs simulations driven by random number generators. These methods are often used when others fail, since they are much less sensitive to the ``curse of dimensionality'', which plagues deterministic methods in problems with a large number of variables. Monte Carlo methods are used in many fields: mathematics, statistics, physics, chemistry, finance, computer science, and biology, for instance. This book is an introduction to Monte Carlo methods for anyone who would like to use these methods to study various kinds of mathematical models that arise in diverse areas of application. The book is based on lectures in a graduate course given by the author. It examines theoretical properties of Monte Carlo methods as well as practical issues concerning their computer implementation and statistical analysis. The only formal prerequisite is an undergraduate course in probability. The book is intended to be accessible to students from a wide range of scientific backgrounds. Rather than being a detailed treatise, it covers the key topics of Monte Carlo methods to the depth necessary for a researcher to design, implement, and analyze a full Monte Carlo study of a mathematical or scientific problem. The ideas are illustrated with diverse running examples. There are exercises sprinkled throughout the text. The topics covered include computer generation of random variables, techniques and examples for variance reduction of Monte Carlo estimates, Markov chain Monte Carlo, and statistical analysis of Monte Carlo output.
Author: Tadahisa Funaki
Author: Tadahisa Funaki
Author: Erwin Bolthausen
Author: Golden Krishna
Author: Amir Dembo
Author: Sacha Friedli
Author: Clay Mathematics Institute. Summer School
Author: Giambattista Giacomin
Author: P L Garrido
Author: Neal Noah Madras
Author: Stefan Adams