Sinossi
There are many ways of introducing the concept of probability in classical, deterministic physics. This volume is concerned with one approach, known as 'the method of arbitrary functions', which was first considered by Poincare. Essentially, the method proceeds by associating some uncertainty to our knowledge of both the initial conditions and the values of the physical constants that characterize the evolution of a physical system. By modeling this uncertainty by a probability density distribution, it is then possible to analyze how the state of the system evolves through time. This approach may be applied to a wide variety of classical problems and the author considers here examples as diverse as bouncing balls, simple and coupled harmonic oscillators, integrable systems (such as spinning tops), planetary motion, and billiards. An important aspect of this account is to study the speed of convergence for solutions in order to determine the practical relevance of the method of arbitrary functions for specific examples. Consequently, both new results on convergence, and tractable upper bounds are derived and applied.
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Contenuti
1 Introduction.- 1.1 The Simple Harmonic Oscillator.- 1.2 Philosophical Interpretations.- 1.3 Coupled Harmonic Oscillators.- 1.4 Mathematical Results.- 1.5 Calculating Rates of Convergence.- 1.6 Hopf’s Approach.- 1.7 Physical and Statistical Independence.- 1.8 Statistical Regularity of a Dynamical System.- 1.9 More Applications.- 2 Preliminaries.- 2.1 Basic Notation.- 2.2 Weak-star Convergence.- 2.3 Variation Distance.- 2.4 Sup Distance.- 2.5 Some Concepts from Number Theory.- 3 One Dimensional Case.- 3.1 Mathematical Results.- 3.1.1 Weak-star Convergence.- 3.1.2 Bounds on the Rate of Convergence.- 3.1.3 Exact Rates of Convergence.- 3.1.4 Fastest Rate of Convergence.- 3.2 Applications.- 3.2.1 A Bouncing Ball.- 3.2.2 Coin Tossing.- 3.2.3 Throwing a Dart at a Wall.- 3.2.4 Poincaré’s Roulette Argument.- 3.2.5 Poincaré’s Law of Small Planets.- 3.2.6 An Example from the Dynamical Systems Literature.- 4 Higher Dimensions.- 4.1 Mathematical Results.- 4.1.1 Weak-star Convergence.- 4.1.2 Bounds on the Rate of Convergence.- 4.1.3 Exact Rates of Convergence.- 4.2 Applications.- 4.2.1 Lagrange’s Top and Integrable Systems.- 4.2.2 Coupled Harmonic Oscillators.- 4.2.3 Billiards.- 4.2.4 Gas Molecules in a Room.- 4.2.5 Random Number Generators.- 4.2.6 Repeated Observations.- 5 Hopf’s Approach.- 5.1 Force as a Function of Only Velocity: One Dimensional case.- 5.2 Force as a Function of Only Velocity: Higher Dimensions.- 5.3 The Force also Depends on the Position.- 5.4 Statistical Regularity of a Dynamical System.- 5.5 Physical and Statistical Independence.- 5.6 The Method of Arbitrary Functions and Ergodic Theory.- 5.7 Partial Statistical Regularity.- 6 Non Diagonal Case.- 6.1 Mathematical Results.- 6.1.1 Convergence in the Variation Distance.- 6.1.2 Weak-star Convergence.- 6.1.3 Rates of Convergence.- 6.2 Linear Differential Equations.- 6.3 Automorphisms of the n-dimensional Torus.- References.
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Taschenbuch. Condizione: Neu. This item is printed on demand - it takes 3-4 days longer - Neuware -There are many ways of introducing the concept of probability in classical, i. e, deter ministic, physics. This work is concerned with one approach, known as 'the method of arbitrary funetionJ. ' It was put forward by Poincare in 1896 and developed by Hopf in the 1930's. The idea is the following. There is always some uncertainty in our knowledge of both the initial conditions and the values of the physical constants that characterize the evolution of a physical system. A probability density may be used to describe this uncertainty. For many physical systems, dependence on the initial density washes away with time. Inthese cases, the system's position eventually converges to the same random variable, no matter what density is used to describe initial uncertainty. Hopf's results for the method of arbitrary functions are derived and extended in a unified fashion in these lecture notes. They include his work on dissipative systems subject to weak frictional forces. Most prominent among the problems he considers is his carnival wheel example, which is the first case where a probability distribution cannot be guessed from symmetry or other plausibility considerations, but has to be derived combining the actual physics with the method of arbitrary functions. Examples due to other authors, such as Poincare's law of small planets, Borel's billiards problem and Keller's coin tossing analysis are also studied using this framework. Finally, many new applications are presented. 168 pp. Englisch. Codice articolo 9780387977409
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Taschenbuch. Condizione: Neu. This item is printed on demand - Print on Demand Titel. Neuware -There are many ways of introducing the concept of probability in classical, i. e, deter ministic, physics. This work is concerned with one approach, known as 'the method of arbitrary funetionJ. ' It was put forward by Poincare in 1896 and developed by Hopf in the 1930's. The idea is the following. There is always some uncertainty in our knowledge of both the initial conditions and the values of the physical constants that characterize the evolution of a physical system. A probability density may be used to describe this uncertainty. For many physical systems, dependence on the initial density washes away with time. Inthese cases, the system's position eventually converges to the same random variable, no matter what density is used to describe initial uncertainty. Hopf's results for the method of arbitrary functions are derived and extended in a unified fashion in these lecture notes. They include his work on dissipative systems subject to weak frictional forces. Most prominent among the problems he considers is his carnival wheel example, which is the first case where a probability distribution cannot be guessed from symmetry or other plausibility considerations, but has to be derived combining the actual physics with the method of arbitrary functions. Examples due to other authors, such as Poincare's law of small planets, Borel's billiards problem and Keller's coin tossing analysis are also studied using this framework. Finally, many new applications are presented.Springer-Verlag KG, Sachsenplatz 4-6, 1201 Wien 168 pp. Englisch. Codice articolo 9780387977409
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Taschenbuch. Condizione: Neu. Druck auf Anfrage Neuware - Printed after ordering - There are many ways of introducing the concept of probability in classical, i. e, deter ministic, physics. This work is concerned with one approach, known as 'the method of arbitrary funetionJ. ' It was put forward by Poincare in 1896 and developed by Hopf in the 1930's. The idea is the following. There is always some uncertainty in our knowledge of both the initial conditions and the values of the physical constants that characterize the evolution of a physical system. A probability density may be used to describe this uncertainty. For many physical systems, dependence on the initial density washes away with time. Inthese cases, the system's position eventually converges to the same random variable, no matter what density is used to describe initial uncertainty. Hopf's results for the method of arbitrary functions are derived and extended in a unified fashion in these lecture notes. They include his work on dissipative systems subject to weak frictional forces. Most prominent among the problems he considers is his carnival wheel example, which is the first case where a probability distribution cannot be guessed from symmetry or other plausibility considerations, but has to be derived combining the actual physics with the method of arbitrary functions. Examples due to other authors, such as Poincare's law of small planets, Borel's billiards problem and Keller's coin tossing analysis are also studied using this framework. Finally, many new applications are presented. Codice articolo 9780387977409
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Condizione: Gut. Zustand: Gut | Sprache: Englisch | Produktart: Bücher | There are many ways of introducing the concept of probability in classical, i. e, deter ministic, physics. This work is concerned with one approach, known as "the method of arbitrary funetionJ. " It was put forward by Poincare in 1896 and developed by Hopf in the 1930's. The idea is the following. There is always some uncertainty in our knowledge of both the initial conditions and the values of the physical constants that characterize the evolution of a physical system. A probability density may be used to describe this uncertainty. For many physical systems, dependence on the initial density washes away with time. Inthese cases, the system's position eventually converges to the same random variable, no matter what density is used to describe initial uncertainty. Hopf's results for the method of arbitrary functions are derived and extended in a unified fashion in these lecture notes. They include his work on dissipative systems subject to weak frictional forces. Most prominent among the problems he considers is his carnival wheel example, which is the first case where a probability distribution cannot be guessed from symmetry or other plausibility considerations, but has to be derived combining the actual physics with the method of arbitrary functions. Examples due to other authors, such as Poincare's law of small planets, Borel's billiards problem and Keller's coin tossing analysis are also studied using this framework. Finally, many new applications are presented. Codice articolo 1532351/203
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