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Author: Mu-fa Chen Publisher: World Scientific ISBN: 9814482900 Category : Science Languages : en Pages : 610
Book Description
This book is representative of the work of Chinese probabilists on probability theory and its applications in physics. It presents a unique treatment of general Markov jump processes: uniqueness, various types of ergodicity, Markovian couplings, reversibility, spectral gap, etc. It also deals with a typical class of non-equilibrium particle systems, including the typical Schlögl model taken from statistical physics. The constructions, ergodicity and phase transitions for this class of Markov interacting particle systems, namely, reaction-diffusion processes, are presented. In this new edition, a large part of the text has been updated and two-and-a-half chapters have been rewritten. The book is self-contained and can be used in a course on stochastic processes for graduate students.
Author: Mu-fa Chen Publisher: World Scientific ISBN: 9814482900 Category : Science Languages : en Pages : 610
Book Description
This book is representative of the work of Chinese probabilists on probability theory and its applications in physics. It presents a unique treatment of general Markov jump processes: uniqueness, various types of ergodicity, Markovian couplings, reversibility, spectral gap, etc. It also deals with a typical class of non-equilibrium particle systems, including the typical Schlögl model taken from statistical physics. The constructions, ergodicity and phase transitions for this class of Markov interacting particle systems, namely, reaction-diffusion processes, are presented. In this new edition, a large part of the text has been updated and two-and-a-half chapters have been rewritten. The book is self-contained and can be used in a course on stochastic processes for graduate students.
Author: Mufa Chen Publisher: World Scientific ISBN: 9812388117 Category : Mathematics Languages : en Pages : 610
Book Description
This book is representative of the work of Chinese probabilists on probability theory and its applications in physics. It presents a unique treatment of general Markov jump processes: uniqueness, various types of ergodicity, Markovian couplings, reversibility, spectral gap, etc. It also deals with a typical class of non-equilibrium particle systems, including the typical Schlögl model taken from statistical physics. The constructions, ergodicity and phase transitions for this class of Markov interacting particle systems, namely, reaction-diffusion processes, are presented. In this new edition, a large part of the text has been updated and two-and-a-half chapters have been rewritten. The book is self-contained and can be used in a course on stochastic processes for graduate students.
Author: Shirshendu Ganguly Publisher: ISBN: Category : Languages : en Pages : 190
Book Description
The thesis concerns asymptotic behavior of particle systems and the underlying Markov chains used to model various natural phenomena. The objective is to describe and analyze stochastic models involving spatial structure and evolution over time. Fundamental objects of interest in such systems include the equilibrium measure which the system converges to, the phenomenon of phase transition in the long term behavior and the time taken to converge to stationarity. In this thesis we present three examples highlighting the above aspects. In Chapter 2, we will discuss Competitive Erosion: a multi-particle system introduced by James Propp in 2003, as a generalization of a fundamental growth model known as Internal Diffusion Limited Aggregation. In this model, each vertex of the graph is occupied by a particle, which can be either red or blue. New red and blue particles are emitted alternately from their respective bases and perform random walk. On encountering a particle of the opposite color they remove it and occupy its position. We consider competitive erosion on discretizations of smooth planar simply connected domains. In Chapter 2 we establish positively, a conjecture of Propp regarding conformal invariance of the the model at stationarity, by showing that, with high probability the blue and the red regions are separated by an orthogonal circular arc on the disc and by a suitable hyperbolic geodesic on a general `smooth' simply connected domain. In Chapter 3, we discuss a family of conservative stochastic processes known as Activated Random Walk (ARW) which interpolates between ordinary random walk and the Stochastic Sandpile; the latter being a canonical example of Self Organized Criticality. These processes are conjectured to exhibit a sharp change in long time behavior depending on the value of certain parameters. Informally ARW is a particle system on Z with mass conservation. One starts with a mass density mu>0 of initially active particles, each of which performs a symmetric random walk at rate one and falls asleep at rate lambda. Sleepy particles become active on coming in contact with other active particles. We investigate the question of fixation/non-fixation of the process and show for small enough lambda, the critical mass density for fixation is strictly less than one. Moreover, the critical density goes to zero as lambda tends to zero. This positively answers two open questions from Dickman, Rolla, Sidoravicius (J. Stat. Phys., 2010) and Rolla, Sidoravicius (Invent. Math., 2012). In Chapter 4, we discuss a model of constrained Glauber dynamics, known as the East Process, exhibiting sharp convergence to equilibrium. The East process is a 1-D kinetically constrained interacting particle system, introduced in the physics literature in the early 90's to model liquid-glass transitions. Informally, it is a two spin (0,1) system on Z, where every site at rate one tries to randomize its spin using a fresh Bernoulli (p). However the move is suppressed unless the site to the left is in the 0 state. Thus the Glauber dynamics move is carried out only in the presence of a certain `kinetic' constraint. Spectral gap estimates of Aldous and Diaconis in 2002 imply that its mixing time on L sites has order L. Since the relaxation time is of a smaller order than the mixing time it is natural to expect a sharp convergence to equilibrium . Proving this, is the goal of this chapter, where we establish Cutoff for mixing, with an optimal window size.
Author: G.R. Grimmett Publisher: Springer Science & Business Media ISBN: 9401583269 Category : Science Languages : en Pages : 334
Book Description
This volume describes the current state of knowledge of random spatial processes, particularly those arising in physics. The emphasis is on survey articles which describe areas of current interest to probabilists and physicists working on the probability theory of phase transition. Special attention is given to topics deserving further research. The principal contributions by leading researchers concern the mathematical theory of random walk, interacting particle systems, percolation, Ising and Potts models, spin glasses, cellular automata, quantum spin systems, and metastability. The level of presentation and review is particularly suitable for postgraduate and postdoctoral workers in mathematics and physics, and for advanced specialists in the probability theory of spatial disorder and phase transition.
Author: Christian Maes Publisher: Springer ISBN: 3319677802 Category : Science Languages : en Pages : 53
Book Description
This book introduces and discusses both the fundamental aspects and the measurability of applications of time-symmetric kinetic quantities, outlining the features that constitute the non-dissipative branch of non-equilibrium physics. These specific features of non-equilibrium dynamics have largely been ignored in standard statistical mechanics texts. This introductory-level book offers novel material that does not take the traditional line of extending standard thermodynamics to the irreversible domain. It shows that although stationary dissipation is essentially equivalent with steady non-equilibrium and ubiquitous in complex phenomena, non-equilibrium is not determined solely by the time-antisymmetric sector of energy-entropy considerations. While this should not be very surprising, this book provides timely, simple reminders of the role of time-symmetric and kinetic aspects in the construction of non-equilibrium statistical mechanics.
Author: Thomas M. Liggett Publisher: Springer Science & Business Media ISBN: 3662039907 Category : Mathematics Languages : en Pages : 346
Book Description
Interactive particle systems is a branch of probability theory with close connections to mathematical physics and mathematical biology. This book takes three of the most important models in the area, and traces advances in our understanding of them since 1985. It explains and develops many of the most useful techniques in the field.
Author: R. Kubo Publisher: Springer Science & Business Media ISBN: 3642967019 Category : Science Languages : en Pages : 294
Book Description
This volume of Statistical Physics consititutes the second part of Statistical Physics (Springer Series in Solid-State Science, Vols. 30, 31) and is devoted to nonequilibrium theories of statistical mechanics. We start with an intro duction to the stochastic treatment of Brownian motion and then proceed to general problems involved in deriving a physical process from an underlying more basic process. Relaxation from nonequilibrium to equilibrium states and the response of a system to an external disturbance form the central problems of nonequilibrium statistical mechanics. These problems are treated both phenomenologically and microscopically along the lines of re cent developments. Emphasis is placed on fundamental concepts and methods rather than on applications which are too numerous to be treated exhaustively within the limited space of this volume. For information on the general aim of this book, the reader is referred to the Foreword. For further reading, the reader should consult the bibliographies, although these are not meant to be exhaustive.
Author: Mu-fa Chen
Author: Mu-fa Chen
Author: Mu Fa Chen
Author: Mu Fa Chen
Author: Mufa Chen
Author: Shirshendu Ganguly
Author: G.R. Grimmett
Author: D. Griffeath
Author: Christian Maes
Author: Thomas M. Liggett
Author: R. Kubo