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The idea of writing up a book on the hydrodynamic behavior of interacting particle systems was born after a series of lectures Claude Kipnis gave at the University of Paris 7 in the spring of 1988. At this time Claude wrote some notes in French that covered Chapters 1 and 4, parts of Chapters 2, 5 and Appendix 1 of this book. His intention was to prepare a text that was as self-contained as possible. lt would include, for instance, all tools from Markov process theory ( cf. Appendix 1, Chaps. 2 and 4) necessary to enable mathematicians and mathematical physicists with some knowledge of probability, at the Ievel of Chung (1974), to understand the techniques of the theory of hydrodynamic Iimits of interacting particle systems. In the fall of 1991 Claude invited me to complete his notes with him and transform them into a book that would present to a large audience the latest developments of the theory in a simple and accessible form. To concentrate on the main ideas and to avoid unnecessary technical difficulties, we decided to consider systems evolving in finite lattice spaces and for which the equilibrium states are product measures. To illustrate the techniques we chose two well-known particle systems, the generalized exclusion processes and the zero-range processes. We also conceived the book in such a manner that most chapters can be read independently of the others. Here are some comments that might help readers find their way.
Fysica [Mathematische ] --- Fysica [Wiskundige ] --- Lois d'échelle (Physique statistique) --- Markoff processes --- Markov [Processus de ] --- Markov models --- Markov processen --- Markov processes --- Markov-processen --- Mathematical physics --- Mathematische fysica --- Physical mathematics --- Physics -- Mathematics --- Physics [Mathematical ] --- Physique -- Mathématiques --- Physique -- Méthodes mathématiques --- Physique mathématique --- Physique théorique --- Probabiliteit--Theorie --- Probabiliteitstheorie --- Probabilities --- Probabilité [Théorie de la ] --- Probabilités --- Processus de Markov --- Scaling laws (Statistical physics) --- Schaalwetten (Statistische fysica) --- Waarschijnlijkheid--Theorie --- Waarschijnlijkheidstheorie --- Wiskundige fysica --- Hydrodynamics --- Mathematics --- Probabilities. --- Mathematical physics. --- Probability Theory and Stochastic Processes. --- Theoretical, Mathematical and Computational Physics. --- Physics --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Hydrodynamics - Mathematics
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Diffusive phenomena in statistical mechanics and in other fields arise from markovian modeling and their study requires sophisticated mathematical tools. In infinite dimensional situations, time symmetry properties can be exploited in order to make martingale approximations, along the lines of the seminal work of Kipnis and Varadhan. The present volume contains the most advanced theories on the martingale approach to central limit theorems. Using the time symmetry properties of the Markov processes, the book develops the techniques that allow us to deal with infinite dimensional models that appear in statistical mechanics and engineering (interacting particle systems, homogenization in random environments, and diffusion in turbulent flows, to mention just a few applications). The first part contains a detailed exposition of the method, and can be used as a text for graduate courses. The second concerns application to exclusion processes, in which the duality methods are fully exploited. The third part is about the homogenization of diffusions in random fields, including passive tracers in turbulent flows (including the superdiffusive behavior). There are no other books in the mathematical literature that deal with this kind of approach to the problem of the central limit theorem. Hence, this volume meets the demand for a monograph on this powerful approach, now widely used in many areas of probability and mathematical physics. The book also covers the connections with and application to hydrodynamic limits and homogenization theory, so besides probability researchers it will also be of interest to mathematical physicists and analysts.
Fluctuations (Physics). --- Markov processes. --- Martingales (Mathematics). --- Mathematics. --- Markov processes --- Fluctuations (Physics) --- Martingales (Mathematics) --- Central limit theorem --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Theory --- Mathematical Statistics --- Mathematical models --- Variations (Physics) --- Analysis, Markov --- Chains, Markov --- Markoff processes --- Markov analysis --- Markov chains --- Markov models --- Models, Markov --- Processes, Markov --- Probabilities. --- Mathematical physics. --- Probability Theory and Stochastic Processes. --- Mathematical Physics. --- Stochastic processes --- Distribution (Probability theory. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Physical mathematics --- Physics --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Distribution (Probability theory)
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Operational research. Game theory --- stochastische analyse --- kansrekening
Choose an application
Diffusive phenomena in statistical mechanics and in other fields arise from markovian modeling and their study requires sophisticated mathematical tools. In infinite dimensional situations, time symmetry properties can be exploited in order to make martingale approximations, along the lines of the seminal work of Kipnis and Varadhan. The present volume contains the most advanced theories on the martingale approach to central limit theorems. Using the time symmetry properties of the Markov processes, the book develops the techniques that allow us to deal with infinite dimensional models that appear in statistical mechanics and engineering (interacting particle systems, homogenization in random environments, and diffusion in turbulent flows, to mention just a few applications). The first part contains a detailed exposition of the method, and can be used as a text for graduate courses. The second concerns application to exclusion processes, in which the duality methods are fully exploited. The third part is about the homogenization of diffusions in random fields, including passive tracers in turbulent flows (including the superdiffusive behavior). There are no other books in the mathematical literature that deal with this kind of approach to the problem of the central limit theorem. Hence, this volume meets the demand for a monograph on this powerful approach, now widely used in many areas of probability and mathematical physics. The book also covers the connections with and application to hydrodynamic limits and homogenization theory, so besides probability researchers it will also be of interest to mathematical physicists and analysts.
Operational research. Game theory --- stochastische analyse --- kansrekening
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