Stefan Adams
- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199239252
- eISBN:
- 9780191716911
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199239252.003.0007
- Subject:
- Mathematics, Probability / Statistics, Analysis
Motivated by the Bose gas, this chapter introduces certain combinatorial structures. It analyses the asymptotic behaviour of empirical shape measures and of empirical path measures of N Brownian ...
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Motivated by the Bose gas, this chapter introduces certain combinatorial structures. It analyses the asymptotic behaviour of empirical shape measures and of empirical path measures of N Brownian motions with large deviations techniques. The rate functions are given as variational problems that are analysed. A symmetrized system of Brownian motions is highly correlated and has to be formulated such that standard techniques can be applied. The chapter reviews a novel spatial and a novel cycle structure approach for the symmetrized distributions of the empirical path measures. The cycle structure leads to a proof of a phase transition in the mean path measure.Less
Motivated by the Bose gas, this chapter introduces certain combinatorial structures. It analyses the asymptotic behaviour of empirical shape measures and of empirical path measures of N Brownian motions with large deviations techniques. The rate functions are given as variational problems that are analysed. A symmetrized system of Brownian motions is highly correlated and has to be formulated such that standard techniques can be applied. The chapter reviews a novel spatial and a novel cycle structure approach for the symmetrized distributions of the empirical path measures. The cycle structure leads to a proof of a phase transition in the mean path measure.
Scott Curtis
- Published in print:
- 2015
- Published Online:
- May 2016
- ISBN:
- 9780231134033
- eISBN:
- 9780231508636
- Item type:
- chapter
- Publisher:
- Columbia University Press
- DOI:
- 10.7312/columbia/9780231134033.003.0002
- Subject:
- Film, Television and Radio, Film
Chapter One examines use of film as a scientific research tool in three fields (human motion studies, physics, biology) in Germany from the 1880s to 1914. It argues that there is a close relationship ...
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Chapter One examines use of film as a scientific research tool in three fields (human motion studies, physics, biology) in Germany from the 1880s to 1914. It argues that there is a close relationship between film form and disciplinary agendas, which explains the affinity between cinema and science that Henri Bergson critiqued in Creative Evolution (1907).Less
Chapter One examines use of film as a scientific research tool in three fields (human motion studies, physics, biology) in Germany from the 1880s to 1914. It argues that there is a close relationship between film form and disciplinary agendas, which explains the affinity between cinema and science that Henri Bergson critiqued in Creative Evolution (1907).
Peter Mörters
- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199232574
- eISBN:
- 9780191716393
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199232574.003.0008
- Subject:
- Mathematics, Geometry / Topology
This chapter expounds the theory of random fractals, using tree representation as a unifying principle. Applications to the fine structure of Brownian motion are discussed.
This chapter expounds the theory of random fractals, using tree representation as a unifying principle. Applications to the fine structure of Brownian motion are discussed.
Stefan Adams and Wolfgang König
- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199239252
- eISBN:
- 9780191716911
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199239252.003.0008
- Subject:
- Mathematics, Probability / Statistics, Analysis
Bose–Einstein condensation predicts that, under certain conditions (in particular extremely low temperature), all particles will condense into one state. Some of the physical background is surveyed ...
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Bose–Einstein condensation predicts that, under certain conditions (in particular extremely low temperature), all particles will condense into one state. Some of the physical background is surveyed in this chapter. The Gross–Pitaevskii approximation for dilute systems is also discussed. Variational problems appear here naturally, as the quantum mechanical ground state is of interest. In connection with positive temperature, related probabilistic models, based on interacting Brownian motions in a trapping potential, are introduced. Again, large deviation techniques are used to determine the mean occupation measure, both for vanishing temperature and large particle number.Less
Bose–Einstein condensation predicts that, under certain conditions (in particular extremely low temperature), all particles will condense into one state. Some of the physical background is surveyed in this chapter. The Gross–Pitaevskii approximation for dilute systems is also discussed. Variational problems appear here naturally, as the quantum mechanical ground state is of interest. In connection with positive temperature, related probabilistic models, based on interacting Brownian motions in a trapping potential, are introduced. Again, large deviation techniques are used to determine the mean occupation measure, both for vanishing temperature and large particle number.
Jacques Franchi and Yves Le Jan
- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780199654109
- eISBN:
- 9780191745676
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199654109.003.0007
- Subject:
- Mathematics, Mathematical Physics
This chapter is devoted to (left and right) Brownian motions on groups of matrices, which the chapter constructs as solutions to linear stochastic differential equations. The chapter establishes in ...
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This chapter is devoted to (left and right) Brownian motions on groups of matrices, which the chapter constructs as solutions to linear stochastic differential equations. The chapter establishes in particular that the solution of such an equation lives in the subgroup associated with the Lie subalgebra generated by the coefficients of the equation. Reversed processes, Hilbert–Schmidt estimates, approximation by stochastic exponentials, Lyapunov exponents and diffusion processes are also considered. Then the chapter concentrates on important examples: the Heisenberg group, PSL(2), SO(d), PSO(1, d), the affine group A d and the Poincaré group P d +1. By means of a projection, we obtain the spherical and hyperbolic Brownian motions, and relativistic diffusion in Minkowski space.Less
This chapter is devoted to (left and right) Brownian motions on groups of matrices, which the chapter constructs as solutions to linear stochastic differential equations. The chapter establishes in particular that the solution of such an equation lives in the subgroup associated with the Lie subalgebra generated by the coefficients of the equation. Reversed processes, Hilbert–Schmidt estimates, approximation by stochastic exponentials, Lyapunov exponents and diffusion processes are also considered. Then the chapter concentrates on important examples: the Heisenberg group, PSL(2), SO(d), PSO(1, d), the affine group A d and the Poincaré group P d +1. By means of a projection, we obtain the spherical and hyperbolic Brownian motions, and relativistic diffusion in Minkowski space.
Peter Achinstein
- Published in print:
- 2001
- Published Online:
- November 2003
- ISBN:
- 9780195143898
- eISBN:
- 9780199833023
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0195143892.003.0012
- Subject:
- Philosophy, Philosophy of Science
Jean Perrin's argument for the existence of molecules on the basis of his 1908 experiments with Brownian motion is examined. Various interpretations of that argument, including hypothetico‐deductive, ...
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Jean Perrin's argument for the existence of molecules on the basis of his 1908 experiments with Brownian motion is examined. Various interpretations of that argument, including hypothetico‐deductive, Wesley Salmon's common‐cause idea, and Clark Glymour's bootstrapping claims, are examined and rejected. The argument is reconstructed as an eliminative‐causal one, and it is shown how it conforms to the requirements of potential evidence. It is also argued, against antirealist interpretations of Perrin, that Perrin himself was applying a realist argument to the existence of unobservable molecules rather than an instrumentalist one to the truth of the observational consequences of the molecular theory.Less
Jean Perrin's argument for the existence of molecules on the basis of his 1908 experiments with Brownian motion is examined. Various interpretations of that argument, including hypothetico‐deductive, Wesley Salmon's common‐cause idea, and Clark Glymour's bootstrapping claims, are examined and rejected. The argument is reconstructed as an eliminative‐causal one, and it is shown how it conforms to the requirements of potential evidence. It is also argued, against antirealist interpretations of Perrin, that Perrin himself was applying a realist argument to the existence of unobservable molecules rather than an instrumentalist one to the truth of the observational consequences of the molecular theory.
Claus Munk
- Published in print:
- 2011
- Published Online:
- September 2011
- ISBN:
- 9780199575084
- eISBN:
- 9780191728648
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199575084.003.0003
- Subject:
- Economics and Finance, Financial Economics
The price of an asset at a future point in time will typically be unknown, i.e. a random variable. In order to describe the uncertain evolution in the price of the asset over time, we need a ...
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The price of an asset at a future point in time will typically be unknown, i.e. a random variable. In order to describe the uncertain evolution in the price of the asset over time, we need a collection of random variables, namely one random variable for each point in time. Such a collection of random variables is called a stochastic process. Modern finance models therefore apply stochastic processes to represent the evolution in prices — as well as interest rates and other relevant quantities — over time. This is also the case for the dynamic interest rate models presented in this book. This chapter gives an introduction to stochastic processes and the mathematical tools needed to do calculations with stochastic processes, the so-called stochastic calculus, focusing on processes and results that will become important in later chapters.Less
The price of an asset at a future point in time will typically be unknown, i.e. a random variable. In order to describe the uncertain evolution in the price of the asset over time, we need a collection of random variables, namely one random variable for each point in time. Such a collection of random variables is called a stochastic process. Modern finance models therefore apply stochastic processes to represent the evolution in prices — as well as interest rates and other relevant quantities — over time. This is also the case for the dynamic interest rate models presented in this book. This chapter gives an introduction to stochastic processes and the mathematical tools needed to do calculations with stochastic processes, the so-called stochastic calculus, focusing on processes and results that will become important in later chapters.
Heinz-Peter Breuer and Francesco Petruccione
- Published in print:
- 2007
- Published Online:
- January 2010
- ISBN:
- 9780199213900
- eISBN:
- 9780191706349
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199213900.003.03
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter introduces the fundamentals of the description of the quantum dynamics of open systems in terms of quantum master equations, together with its most important applications. Special ...
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This chapter introduces the fundamentals of the description of the quantum dynamics of open systems in terms of quantum master equations, together with its most important applications. Special emphasis is laid on the theory of completely positive quantum dynamical semigroups, which leads to the concept of a quantum Markov process. It discusses the relaxation to equilibrium and the multi-time structure of quantum Markov processes, as well as their irreversible nature which is characterized by an appropriate entropy functional. Microscopic derivations for various quantum master equations are presented, such as the quantum optical master equation and the master equation for quantum Brownian motion. As a further application, the master equation describing continuous measurements is derived and used to study the quantum Zeno effect. The chapter also contains a treatment of non-linear, mean field quantum master equations together with applications to laser theory and super-radiance.Less
This chapter introduces the fundamentals of the description of the quantum dynamics of open systems in terms of quantum master equations, together with its most important applications. Special emphasis is laid on the theory of completely positive quantum dynamical semigroups, which leads to the concept of a quantum Markov process. It discusses the relaxation to equilibrium and the multi-time structure of quantum Markov processes, as well as their irreversible nature which is characterized by an appropriate entropy functional. Microscopic derivations for various quantum master equations are presented, such as the quantum optical master equation and the master equation for quantum Brownian motion. As a further application, the master equation describing continuous measurements is derived and used to study the quantum Zeno effect. The chapter also contains a treatment of non-linear, mean field quantum master equations together with applications to laser theory and super-radiance.
Robert M. Mazo
- Published in print:
- 2008
- Published Online:
- January 2010
- ISBN:
- 9780199556441
- eISBN:
- 9780191705625
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199556441.001.0001
- Subject:
- Physics, Condensed Matter Physics / Materials
Brownian motion is the incessant motion of small particles immersed in an ambient medium. It is due to fluctuations in the motion of the medium particles on the molecular scale. The name has been ...
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Brownian motion is the incessant motion of small particles immersed in an ambient medium. It is due to fluctuations in the motion of the medium particles on the molecular scale. The name has been carried over to other fluctuation phenomena. This book treats the physical theory of Brownian motion. The extensive mathematical theory, which treats the subject as a subfield of the general theory of random processes, is touched on but not presented in any detail. Random or stochastic process theory and statistical mechanics are the primary tools. The first eight chapters treat the stochastic theory and some applications. The next six present the statistical mechanical point of view. Then follows chapters on applications to diffusion, noise, and polymers, followed by a treatment of the motion of interacting Brownian particles. The book ends with a final chapter treating simulation, fractals, and chaos.Less
Brownian motion is the incessant motion of small particles immersed in an ambient medium. It is due to fluctuations in the motion of the medium particles on the molecular scale. The name has been carried over to other fluctuation phenomena. This book treats the physical theory of Brownian motion. The extensive mathematical theory, which treats the subject as a subfield of the general theory of random processes, is touched on but not presented in any detail. Random or stochastic process theory and statistical mechanics are the primary tools. The first eight chapters treat the stochastic theory and some applications. The next six present the statistical mechanical point of view. Then follows chapters on applications to diffusion, noise, and polymers, followed by a treatment of the motion of interacting Brownian particles. The book ends with a final chapter treating simulation, fractals, and chaos.
Claus Munk
- Published in print:
- 2013
- Published Online:
- May 2013
- ISBN:
- 9780199585496
- eISBN:
- 9780191751790
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199585496.003.0002
- Subject:
- Economics and Finance, Econometrics
Uncertainty is a key component of financial markets and thus of any asset pricing theory. This chapter provides the tools from probability theory that are being used in the asset pricing models ...
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Uncertainty is a key component of financial markets and thus of any asset pricing theory. This chapter provides the tools from probability theory that are being used in the asset pricing models covered in the remaining part of the book. The mathematical representation of uncertainty and information flow is explained. Stochastic processes are introduced with numerous examples both in discrete time and in continuous time. The important Ito’s Lemma is presented and illustrated by examples. The simultaneous handling of multiple stochastic processes is also discussed. The chapter is accessible with only little prior exposure to probability theory and continuous-time finance models.Less
Uncertainty is a key component of financial markets and thus of any asset pricing theory. This chapter provides the tools from probability theory that are being used in the asset pricing models covered in the remaining part of the book. The mathematical representation of uncertainty and information flow is explained. Stochastic processes are introduced with numerous examples both in discrete time and in continuous time. The important Ito’s Lemma is presented and illustrated by examples. The simultaneous handling of multiple stochastic processes is also discussed. The chapter is accessible with only little prior exposure to probability theory and continuous-time finance models.
Gopinath Kallianpur and P. Sundar
- Published in print:
- 2014
- Published Online:
- April 2014
- ISBN:
- 9780199657063
- eISBN:
- 9780191781759
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199657063.003.0002
- Subject:
- Mathematics, Probability / Statistics, Applied Mathematics
After defining a Brownian motion (also known as a Wiener process), a standard one-dimensional Brownian motion is constructed by the use of Haar functions. Properties of a Brownian motion such as ...
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After defining a Brownian motion (also known as a Wiener process), a standard one-dimensional Brownian motion is constructed by the use of Haar functions. Properties of a Brownian motion such as non-differentiability of almost every path, and existence of a finite quadratic variation are proved. The reflection principle and its consequences are shown.Less
After defining a Brownian motion (also known as a Wiener process), a standard one-dimensional Brownian motion is constructed by the use of Haar functions. Properties of a Brownian motion such as non-differentiability of almost every path, and existence of a finite quadratic variation are proved. The reflection principle and its consequences are shown.
Nikolai V. Brilliantov and Thorsten Pöschel
- Published in print:
- 2004
- Published Online:
- January 2010
- ISBN:
- 9780198530381
- eISBN:
- 9780191713057
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198530381.003.0015
- Subject:
- Physics, Condensed Matter Physics / Materials
This chapter analyzes Brownian motion in granular gases. Topics discusses include Boltzmann equation for the velocity distribution function of Brownian particles, Fokker–Planck equation for Brownian ...
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This chapter analyzes Brownian motion in granular gases. Topics discusses include Boltzmann equation for the velocity distribution function of Brownian particles, Fokker–Planck equation for Brownian particles, velocity distribution function for Brownian particles, and diffusion of Brownian particles.Less
This chapter analyzes Brownian motion in granular gases. Topics discusses include Boltzmann equation for the velocity distribution function of Brownian particles, Fokker–Planck equation for Brownian particles, velocity distribution function for Brownian particles, and diffusion of Brownian particles.
Nikolai V. Brilliantov and Thorsten Pöschel
- Published in print:
- 2004
- Published Online:
- January 2010
- ISBN:
- 9780198530381
- eISBN:
- 9780191713057
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198530381.001.0001
- Subject:
- Physics, Condensed Matter Physics / Materials
Kinetic Theory of Granular Gases provides an introduction to the rapidly developing theory of dissipative gas dynamics — a theory which has mainly evolved over the last decade. The book ...
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Kinetic Theory of Granular Gases provides an introduction to the rapidly developing theory of dissipative gas dynamics — a theory which has mainly evolved over the last decade. The book is aimed at readers from the advanced undergraduate level upwards and leads on to the present state of research. Throughout, special emphasis is put on a microscopically consistent description of pairwise particle collisions which leads to an impact-velocity-dependent coefficient of restitution. The description of the many-particle system, based on the Boltzmann equation, starts with the derivation of the velocity distribution function, followed by the investigation of self-diffusion and Brownian motion. Using hydrodynamical methods, transport processes and self-organized structure formation are studied. An appendix gives a brief introduction to event-driven molecular dynamics. A second appendix describes a novel mathematical technique for derivation of kinetic properties, which allows for the application of computer algebra. The text is self-contained, requiring no mathematical or physical knowledge beyond that of standard physics undergraduate level. The material is adequate for a one-semester course and contains chapter summaries as well as exercises with detailed solutions. The molecular dynamics and computer-algebra programs can be downloaded from a companion web page.Less
Kinetic Theory of Granular Gases provides an introduction to the rapidly developing theory of dissipative gas dynamics — a theory which has mainly evolved over the last decade. The book is aimed at readers from the advanced undergraduate level upwards and leads on to the present state of research. Throughout, special emphasis is put on a microscopically consistent description of pairwise particle collisions which leads to an impact-velocity-dependent coefficient of restitution. The description of the many-particle system, based on the Boltzmann equation, starts with the derivation of the velocity distribution function, followed by the investigation of self-diffusion and Brownian motion. Using hydrodynamical methods, transport processes and self-organized structure formation are studied. An appendix gives a brief introduction to event-driven molecular dynamics. A second appendix describes a novel mathematical technique for derivation of kinetic properties, which allows for the application of computer algebra. The text is self-contained, requiring no mathematical or physical knowledge beyond that of standard physics undergraduate level. The material is adequate for a one-semester course and contains chapter summaries as well as exercises with detailed solutions. The molecular dynamics and computer-algebra programs can be downloaded from a companion web page.
James Davidson
- Published in print:
- 1994
- Published Online:
- November 2003
- ISBN:
- 9780198774037
- eISBN:
- 9780191596117
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198774036.003.0027
- Subject:
- Economics and Finance, Econometrics
This chapter applies the theory of Ch. 26 to the case of the space C of continuous functions on the unit interval. It is shown how to assign probability measures to C, and Weiner measure (Brownian ...
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This chapter applies the theory of Ch. 26 to the case of the space C of continuous functions on the unit interval. It is shown how to assign probability measures to C, and Weiner measure (Brownian motion) and a number of related Gaussian cases are exhibited as examples. Weak convergence on C is discussed, and a functional CLT (a version of Donsker's theorem for martingales) is given. The multivariate generalization of the FCLT is also given.Less
This chapter applies the theory of Ch. 26 to the case of the space C of continuous functions on the unit interval. It is shown how to assign probability measures to C, and Weiner measure (Brownian motion) and a number of related Gaussian cases are exhibited as examples. Weak convergence on C is discussed, and a functional CLT (a version of Donsker's theorem for martingales) is given. The multivariate generalization of the FCLT is also given.
Jacques Franchi and Yves Le Jan
- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780199654109
- eISBN:
- 9780191745676
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199654109.001.0001
- Subject:
- Mathematics, Mathematical Physics
The idea of this book is to illustrate an interplay between distinct domains of mathematics. Firstly, this book provides an introduction to hyperbolic geometry, based on the Lorentz group PSO(1, d) ...
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The idea of this book is to illustrate an interplay between distinct domains of mathematics. Firstly, this book provides an introduction to hyperbolic geometry, based on the Lorentz group PSO(1, d) and its Iwasawa decomposition, commutation relations and Haar measure, and on the hyperbolic Laplacian. The Lorentz group plays a role in relativistic space–time analogous to rotations in Euclidean space. Hyperbolic geometry is the geometry of the unit pseudo-sphere. The boundary of hyperbolic space is defined as the set of light rays. Special attention is given to the geodesic and horocyclic flows. This book presents hyperbolic geometry via special relativity to benefit from physical intuition. Secondly, this book introduces some basic notions of stochastic analysis: the Wiener process, Itô's stochastic integral and Itô calculus. The book studies linear stochastic differential equations on groups of matrices, and diffusion processes on homogeneous spaces. Spherical and hyperbolic Brownian motions, diffusions on stable leaves, and relativistic diffusion are constructed. Thirdly, quotients of hyperbolic space under a discrete group of isometries are introduced, and form the framework in which some elements of hyperbolic dynamics are presented, especially the ergodicity of the geodesic and horocyclic flows. An analysis is given of the chaotic behaviour of the geodesic flow, using stochastic analysis methods. The main result is Sinai's central limit theorem. Some related results (including a construction of the Wiener measure) which complete the expositions of hyperbolic geometry and stochastic calculus are given in the appendices.Less
The idea of this book is to illustrate an interplay between distinct domains of mathematics. Firstly, this book provides an introduction to hyperbolic geometry, based on the Lorentz group PSO(1, d) and its Iwasawa decomposition, commutation relations and Haar measure, and on the hyperbolic Laplacian. The Lorentz group plays a role in relativistic space–time analogous to rotations in Euclidean space. Hyperbolic geometry is the geometry of the unit pseudo-sphere. The boundary of hyperbolic space is defined as the set of light rays. Special attention is given to the geodesic and horocyclic flows. This book presents hyperbolic geometry via special relativity to benefit from physical intuition. Secondly, this book introduces some basic notions of stochastic analysis: the Wiener process, Itô's stochastic integral and Itô calculus. The book studies linear stochastic differential equations on groups of matrices, and diffusion processes on homogeneous spaces. Spherical and hyperbolic Brownian motions, diffusions on stable leaves, and relativistic diffusion are constructed. Thirdly, quotients of hyperbolic space under a discrete group of isometries are introduced, and form the framework in which some elements of hyperbolic dynamics are presented, especially the ergodicity of the geodesic and horocyclic flows. An analysis is given of the chaotic behaviour of the geodesic flow, using stochastic analysis methods. The main result is Sinai's central limit theorem. Some related results (including a construction of the Wiener measure) which complete the expositions of hyperbolic geometry and stochastic calculus are given in the appendices.
Kerry E. Back
- Published in print:
- 2017
- Published Online:
- May 2017
- ISBN:
- 9780190241148
- eISBN:
- 9780190241179
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780190241148.003.0012
- Subject:
- Economics and Finance, Financial Economics
Brownian motion and concepts of the Itôs calculus are explained, including total variation, quadratic variation, Levy’s characterization of Brownian motion, the Itô integral, the difference between ...
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Brownian motion and concepts of the Itôs calculus are explained, including total variation, quadratic variation, Levy’s characterization of Brownian motion, the Itô integral, the difference between martingales and local martingales, the martingale (predictable) representation theorem , Itô’s formula (Itô’s lemma), geometric Brownian motion, covariation (joint variation) processes, the relationship between variance and expected quadratic variation, the relationship between covariance and expected covariation, and rotations of Brownian motions.Less
Brownian motion and concepts of the Itôs calculus are explained, including total variation, quadratic variation, Levy’s characterization of Brownian motion, the Itô integral, the difference between martingales and local martingales, the martingale (predictable) representation theorem , Itô’s formula (Itô’s lemma), geometric Brownian motion, covariation (joint variation) processes, the relationship between variance and expected quadratic variation, the relationship between covariance and expected covariation, and rotations of Brownian motions.
Phil Attard
- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780199662760
- eISBN:
- 9780191745287
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199662760.003.0003
- Subject:
- Physics, Condensed Matter Physics / Materials
Brownian motion is presented as a stochastic process, and as a bridge between thermodynamics and statistical mechanics. Einstein’s result for the linear growth in time of the mean square displacement ...
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Brownian motion is presented as a stochastic process, and as a bridge between thermodynamics and statistical mechanics. Einstein’s result for the linear growth in time of the mean square displacement and the consequent diffusion equation are given. Langevin’s equation and the molecular statistical basis of the fluctuation dissipation theorem are established from the relation between a stochastic process and the second entropy. The non-equilibrium probability distribution for a Brownian particle in a moving trap is derived in two complementary ways: from fluctuation theory and from fundamental statistical and physical considerations. Computer simulation results are used to illustrate the veracity of the fluctuation formulation. The time evolution of the entropy and of the probability is established in general from the second entropy. The Fokker-Planck equation and Liouville’s theorem are derived and critically analysed. A generalised non-equilibrium equipartition theorem is given.Less
Brownian motion is presented as a stochastic process, and as a bridge between thermodynamics and statistical mechanics. Einstein’s result for the linear growth in time of the mean square displacement and the consequent diffusion equation are given. Langevin’s equation and the molecular statistical basis of the fluctuation dissipation theorem are established from the relation between a stochastic process and the second entropy. The non-equilibrium probability distribution for a Brownian particle in a moving trap is derived in two complementary ways: from fluctuation theory and from fundamental statistical and physical considerations. Computer simulation results are used to illustrate the veracity of the fluctuation formulation. The time evolution of the entropy and of the probability is established in general from the second entropy. The Fokker-Planck equation and Liouville’s theorem are derived and critically analysed. A generalised non-equilibrium equipartition theorem is given.
Heinz-Peter Breuer and Francesco Petruccione
- Published in print:
- 2007
- Published Online:
- January 2010
- ISBN:
- 9780199213900
- eISBN:
- 9780191706349
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199213900.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This book treats the central physical concepts and mathematical techniques used to investigate the dynamics of open quantum systems. To provide a self-contained presentation, the text begins with a ...
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This book treats the central physical concepts and mathematical techniques used to investigate the dynamics of open quantum systems. To provide a self-contained presentation, the text begins with a survey of classical probability theory and with an introduction to the foundations of quantum mechanics, with particular emphasis on its statistical interpretation and on the formulation of generalized measurement theory through quantum operations and effects. The fundamentals of density matrix theory, quantum Markov processes, and completely positive dynamical semigroups are developed. The most important master equations used in quantum optics and condensed matter theory are derived and applied to the study of many examples. Special attention is paid to the Markovian and non-Markovian theory of environment induced decoherence, its role in the dynamical description of the measurement process, and to the experimental observation of decohering electromagnetic field states. The book includes the modern formulation of open quantum systems in terms of stochastic processes in Hilbert space. Stochastic wave function methods and Monte Carlo algorithms are designed and applied to important examples from quantum optics and atomic physics. The fundamentals of the treatment of non-Markovian quantum processes in open systems are developed on the basis of various mathematical techniques, such as projection superoperator methods and influence functional techniques. In addition, the book expounds the relativistic theory of quantum measurements and the density matrix theory of relativistic quantum electrodynamics.Less
This book treats the central physical concepts and mathematical techniques used to investigate the dynamics of open quantum systems. To provide a self-contained presentation, the text begins with a survey of classical probability theory and with an introduction to the foundations of quantum mechanics, with particular emphasis on its statistical interpretation and on the formulation of generalized measurement theory through quantum operations and effects. The fundamentals of density matrix theory, quantum Markov processes, and completely positive dynamical semigroups are developed. The most important master equations used in quantum optics and condensed matter theory are derived and applied to the study of many examples. Special attention is paid to the Markovian and non-Markovian theory of environment induced decoherence, its role in the dynamical description of the measurement process, and to the experimental observation of decohering electromagnetic field states. The book includes the modern formulation of open quantum systems in terms of stochastic processes in Hilbert space. Stochastic wave function methods and Monte Carlo algorithms are designed and applied to important examples from quantum optics and atomic physics. The fundamentals of the treatment of non-Markovian quantum processes in open systems are developed on the basis of various mathematical techniques, such as projection superoperator methods and influence functional techniques. In addition, the book expounds the relativistic theory of quantum measurements and the density matrix theory of relativistic quantum electrodynamics.
Jacques Franchi and Yves Le Jan
- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780199654109
- eISBN:
- 9780191745676
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199654109.003.0006
- Subject:
- Mathematics, Mathematical Physics
This chapter deals with the basic Itô calculus. Fundamental notions such as predictability, martingales and stopping times are introduced in the discrete case. The chapter then gives a short account ...
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This chapter deals with the basic Itô calculus. Fundamental notions such as predictability, martingales and stopping times are introduced in the discrete case. The chapter then gives a short account of the necessary background on martingales and Brownian motion, and the chapter deals finally with the basic tools of Itô's calculus: the stochastic integral and the Itô change-of-variable formula.Less
This chapter deals with the basic Itô calculus. Fundamental notions such as predictability, martingales and stopping times are introduced in the discrete case. The chapter then gives a short account of the necessary background on martingales and Brownian motion, and the chapter deals finally with the basic tools of Itô's calculus: the stochastic integral and the Itô change-of-variable formula.
Melvin Lax, Wei Cai, and Min Xu
- Published in print:
- 2006
- Published Online:
- January 2010
- ISBN:
- 9780198567769
- eISBN:
- 9780191718359
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567769.003.0003
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
Consider two physical problems describable by the same random process. The first process is the radioactive decay of a collection of nuclei. The second is the production of photoelectrons by a steady ...
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Consider two physical problems describable by the same random process. The first process is the radioactive decay of a collection of nuclei. The second is the production of photoelectrons by a steady beam of light on a photodetector. In both cases, we can let a discrete, positive, integer valued, variable n(t) represent the number of counts emitted in the time interval between 0 and t. This is the essence of the Poisson process, an example of Markovian process. Other examples of Markovian processes include the one dimensional random walk, gambler's ruin, diffusion processes and the Einstein relation, Brownian motion, Langevin theory of velocities in Brownian motion, Langevin theory of positions in Brownian motion, and chaos.Less
Consider two physical problems describable by the same random process. The first process is the radioactive decay of a collection of nuclei. The second is the production of photoelectrons by a steady beam of light on a photodetector. In both cases, we can let a discrete, positive, integer valued, variable n(t) represent the number of counts emitted in the time interval between 0 and t. This is the essence of the Poisson process, an example of Markovian process. Other examples of Markovian processes include the one dimensional random walk, gambler's ruin, diffusion processes and the Einstein relation, Brownian motion, Langevin theory of velocities in Brownian motion, Langevin theory of positions in Brownian motion, and chaos.
