Pier A. Mello and Narendra Kumar
- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198525820
- eISBN:
- 9780191712234
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525820.001.0001
- Subject:
- Physics, Condensed Matter Physics / Materials
This book presents a statistical theory of complex wave scattering and quantum transport in a class of physical systems of current interest having chaotic classical dynamics (e.g., microwave cavities ...
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This book presents a statistical theory of complex wave scattering and quantum transport in a class of physical systems of current interest having chaotic classical dynamics (e.g., microwave cavities and quantum dots) or possessing quenched randomness (e.g., disordered conductors). The emphasis here is on mesoscopic fluctuations of the sample-specific transport. The universal character of the statistical behaviour of these phenomena is revealed in a natural way through a novel maximum-entropy approach (MEA). The latter leads to the most probable distribution for the set of random matrices that describe the ensemble of disordered/chaotic samples, which are macroscopically identical but differ in microscopic details. Here, the Shannon information entropy associated with these random matrices is maximized subject to the symmetries and the constraints which are physically relevant. This non-perturbative information-theoretic approach is reminiscent of, but distinct from, the standard random-matrix theory, and indeed forms the most distinctive feature of the book.Less
This book presents a statistical theory of complex wave scattering and quantum transport in a class of physical systems of current interest having chaotic classical dynamics (e.g., microwave cavities and quantum dots) or possessing quenched randomness (e.g., disordered conductors). The emphasis here is on mesoscopic fluctuations of the sample-specific transport. The universal character of the statistical behaviour of these phenomena is revealed in a natural way through a novel maximum-entropy approach (MEA). The latter leads to the most probable distribution for the set of random matrices that describe the ensemble of disordered/chaotic samples, which are macroscopically identical but differ in microscopic details. Here, the Shannon information entropy associated with these random matrices is maximized subject to the symmetries and the constraints which are physically relevant. This non-perturbative information-theoretic approach is reminiscent of, but distinct from, the standard random-matrix theory, and indeed forms the most distinctive feature of the book.
Henning Schomerus
- Published in print:
- 2017
- Published Online:
- January 2018
- ISBN:
- 9780198797319
- eISBN:
- 9780191838774
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198797319.003.0010
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
Over the past decades, a great body of theoretical and mathematical work has been devoted to random-matrix descriptions of open quantum systems. This chapter reviews the physical origins and ...
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Over the past decades, a great body of theoretical and mathematical work has been devoted to random-matrix descriptions of open quantum systems. This chapter reviews the physical origins and mathematical structures of the underlying models, and collects key predictions which give insight into the typical system behaviour. In particular, the aim is to give an idea how the different features are interlinked. The chapter mainly focuses on elastic scattering but also includes a short detour to interacting systems, which are motivated by the overarching question of ergodicity. The first sections introduce general notions from random matrix theory, such as the 10 universality classes and ensembles of Hermitian, unitary, positive-definite, and non-Hermitian matrices. The following sections then review microscopic scattering models that form the basis for statistical descriptions, and consider signatures of random scattering in decay, dynamics, and transport. The last section touches on Anderson localization and localization in interacting systems.Less
Over the past decades, a great body of theoretical and mathematical work has been devoted to random-matrix descriptions of open quantum systems. This chapter reviews the physical origins and mathematical structures of the underlying models, and collects key predictions which give insight into the typical system behaviour. In particular, the aim is to give an idea how the different features are interlinked. The chapter mainly focuses on elastic scattering but also includes a short detour to interacting systems, which are motivated by the overarching question of ergodicity. The first sections introduce general notions from random matrix theory, such as the 10 universality classes and ensembles of Hermitian, unitary, positive-definite, and non-Hermitian matrices. The following sections then review microscopic scattering models that form the basis for statistical descriptions, and consider signatures of random scattering in decay, dynamics, and transport. The last section touches on Anderson localization and localization in interacting systems.
Jon P. Keating
- Published in print:
- 2017
- Published Online:
- January 2018
- ISBN:
- 9780198797319
- eISBN:
- 9780191838774
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198797319.003.0008
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
The aim of this chapter is to motivate and describe some recent developments concerning the applications of random matrix theory to problems in number theory. The first section provides a brief and ...
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The aim of this chapter is to motivate and describe some recent developments concerning the applications of random matrix theory to problems in number theory. The first section provides a brief and rather selective introduction to the theory of the Riemann zeta function, in particular to those parts needed to understand the connections with random matrix theory. The second section focuses on the value distribution of the zeta function on its critical line, specifically on recent progress in understanding the extreme value statistics gained through a conjectural link to log–correlated Gaussian random fields and the statistical mechanics of glasses. The third section outlines some number-theoretic problems that can be resolved in function fields using random matrix methods. In this latter case, random matrix theory provides the only route we currently have for calculating certain important arithmetic statistics rigorously and unconditionally.Less
The aim of this chapter is to motivate and describe some recent developments concerning the applications of random matrix theory to problems in number theory. The first section provides a brief and rather selective introduction to the theory of the Riemann zeta function, in particular to those parts needed to understand the connections with random matrix theory. The second section focuses on the value distribution of the zeta function on its critical line, specifically on recent progress in understanding the extreme value statistics gained through a conjectural link to log–correlated Gaussian random fields and the statistical mechanics of glasses. The third section outlines some number-theoretic problems that can be resolved in function fields using random matrix methods. In this latter case, random matrix theory provides the only route we currently have for calculating certain important arithmetic statistics rigorously and unconditionally.
Helmut Hofmann
- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780198504016
- eISBN:
- 9780191708480
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198504016.003.0006
- Subject:
- Physics, Nuclear and Plasma Physics
This chapter focuses on collective motion of isoscalar nature parameterized by shape variables. The equations of motion are derived from energy conservation as implied by self-consistency. A basic ...
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This chapter focuses on collective motion of isoscalar nature parameterized by shape variables. The equations of motion are derived from energy conservation as implied by self-consistency. A basic ingredient is the variation of the total static energy with deformation, which at finite thermal excitations has to be calculated for constant entropy. Linear response theory is exploited for the dynamics, especially for separating reactive and dissipative forces. Response functions for intrinsic, nucleonic motion are distinguished from those for collective dynamics. The origin of irreversible behavior due to the decay of simple to more complicated nucleonic configurations is described in detail. In practical applications, dressed single particle states are used in their dependence on temperature. The variation of the transport coefficients for inertia and friction with T obtained this way is confronted with that given in various other models, like in the diabatic one, in common RPA, in the random matrix model, or in the liquid drop model and for wall friction. Implications on rotational motion are discussed.Less
This chapter focuses on collective motion of isoscalar nature parameterized by shape variables. The equations of motion are derived from energy conservation as implied by self-consistency. A basic ingredient is the variation of the total static energy with deformation, which at finite thermal excitations has to be calculated for constant entropy. Linear response theory is exploited for the dynamics, especially for separating reactive and dissipative forces. Response functions for intrinsic, nucleonic motion are distinguished from those for collective dynamics. The origin of irreversible behavior due to the decay of simple to more complicated nucleonic configurations is described in detail. In practical applications, dressed single particle states are used in their dependence on temperature. The variation of the transport coefficients for inertia and friction with T obtained this way is confronted with that given in various other models, like in the diabatic one, in common RPA, in the random matrix model, or in the liquid drop model and for wall friction. Implications on rotational motion are discussed.
Pier A. Mello and Narendra Kumar
- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198525820
- eISBN:
- 9780191712234
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525820.003.0001
- Subject:
- Physics, Condensed Matter Physics / Materials
This chapter introduces the notion of complex scattering of waves, emphasizes the generality of the ideas involved, and explains its relevance to the field of nuclear physics and microwave cavities. ...
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This chapter introduces the notion of complex scattering of waves, emphasizes the generality of the ideas involved, and explains its relevance to the field of nuclear physics and microwave cavities. It then introduces the subject of coherent wave transport through mesoscopic systems, e.g., disordered conductors and chaotic cavities, with emphasis on the statistics of fluctuations observed in these systems. These fluctuations, among them the universal conductance fluctuation, arise ultimately from the complex wave interference. Various length- and time-scales defining the mesoscopic system are discussed. The idea of maximum entropy approach (MEA) is introduced as distinct from, but related to, the idea of random-matrix theory (RMT) pioneered by Wigner originally in the context of isolated resonances of complex nuclei. The contents of this chapter include complex atomic nuclei and chaotic microwave cavities; wave localization; statistical fluctuations; mesoscopic conductors: time- and length-scales, ballistic mesoscopic cavities, diffusive mesoscopic conductors, and statistical approach to mesoscopic fluctuations.Less
This chapter introduces the notion of complex scattering of waves, emphasizes the generality of the ideas involved, and explains its relevance to the field of nuclear physics and microwave cavities. It then introduces the subject of coherent wave transport through mesoscopic systems, e.g., disordered conductors and chaotic cavities, with emphasis on the statistics of fluctuations observed in these systems. These fluctuations, among them the universal conductance fluctuation, arise ultimately from the complex wave interference. Various length- and time-scales defining the mesoscopic system are discussed. The idea of maximum entropy approach (MEA) is introduced as distinct from, but related to, the idea of random-matrix theory (RMT) pioneered by Wigner originally in the context of isolated resonances of complex nuclei. The contents of this chapter include complex atomic nuclei and chaotic microwave cavities; wave localization; statistical fluctuations; mesoscopic conductors: time- and length-scales, ballistic mesoscopic cavities, diffusive mesoscopic conductors, and statistical approach to mesoscopic fluctuations.
Oriol Bohigas and Hans A. Weidenmüller
- Published in print:
- 2017
- Published Online:
- January 2018
- ISBN:
- 9780198797319
- eISBN:
- 9780191838774
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198797319.003.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
An overview of the history of random matrix theory (RMT) is provided in this chapter. Starting from its inception, the authors sketch the history of RMT until about 1990, focusing their attention on ...
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An overview of the history of random matrix theory (RMT) is provided in this chapter. Starting from its inception, the authors sketch the history of RMT until about 1990, focusing their attention on the first four decades of RMT. Later developments are partially covered. In the past 20 years RMT has experienced rapid development and has expanded into a number of areas of physics and mathematics.Less
An overview of the history of random matrix theory (RMT) is provided in this chapter. Starting from its inception, the authors sketch the history of RMT until about 1990, focusing their attention on the first four decades of RMT. Later developments are partially covered. In the past 20 years RMT has experienced rapid development and has expanded into a number of areas of physics and mathematics.
Gernot Akemann
- Published in print:
- 2017
- Published Online:
- January 2018
- ISBN:
- 9780198797319
- eISBN:
- 9780191838774
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198797319.003.0005
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter was originally presented to a mixed audience of physicists and mathematicians with some basic working knowledge of random matrix theory. The first part is devoted to the solution of the ...
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This chapter was originally presented to a mixed audience of physicists and mathematicians with some basic working knowledge of random matrix theory. The first part is devoted to the solution of the chiral Gaussian unitary ensemble in the presence of characteristic polynomials, using orthogonal polynomial techniques. This includes all eigenvalue density correlation functions, smallest eigenvalue distributions, and their microscopic limit at the origin. These quantities are relevant for the description of the Dirac operator spectrum in quantum chromodynamics with three colors in four Euclidean space-time dimensions. In the second part these two theories are related based on symmetries, and the random matrix approximation is explained. In the last part recent developments are covered, including the effect of finite chemical potential and finite space-time lattice spacing, and their corresponding orthogonal polynomials. This chapter also provides some open random matrix problems.Less
This chapter was originally presented to a mixed audience of physicists and mathematicians with some basic working knowledge of random matrix theory. The first part is devoted to the solution of the chiral Gaussian unitary ensemble in the presence of characteristic polynomials, using orthogonal polynomial techniques. This includes all eigenvalue density correlation functions, smallest eigenvalue distributions, and their microscopic limit at the origin. These quantities are relevant for the description of the Dirac operator spectrum in quantum chromodynamics with three colors in four Euclidean space-time dimensions. In the second part these two theories are related based on symmetries, and the random matrix approximation is explained. In the last part recent developments are covered, including the effect of finite chemical potential and finite space-time lattice spacing, and their corresponding orthogonal polynomials. This chapter also provides some open random matrix problems.
Helmut Hofmann
- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780198504016
- eISBN:
- 9780191708480
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198504016.003.0004
- Subject:
- Physics, Nuclear and Plasma Physics
This chapter examines nuclear properties that are related to the presence of interactions residual to the mean field. The elementary features of random matrix theory are explained, and the Wigner ...
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This chapter examines nuclear properties that are related to the presence of interactions residual to the mean field. The elementary features of random matrix theory are explained, and the Wigner form of nearest level spacings in Gaussian ensembles is derived and compared with experimental evidence. Examples from nuclear physics and other chaotic and complex systems are shown. The spreading of states into more complicated configurations is described within a schematic model. For the general case, strength functions are introduced and their mathematical structure is explained. The energy dependence of single particle widths is discussed and compared with information from experiments. Inferences are drawn on the validity of the independent-particle model. For a time-dependent description, recurrent versus irreversible behavior is studied.Less
This chapter examines nuclear properties that are related to the presence of interactions residual to the mean field. The elementary features of random matrix theory are explained, and the Wigner form of nearest level spacings in Gaussian ensembles is derived and compared with experimental evidence. Examples from nuclear physics and other chaotic and complex systems are shown. The spreading of states into more complicated configurations is described within a schematic model. For the general case, strength functions are introduced and their mathematical structure is explained. The energy dependence of single particle widths is discussed and compared with information from experiments. Inferences are drawn on the validity of the independent-particle model. For a time-dependent description, recurrent versus irreversible behavior is studied.
Aris L. Moustakas
- Published in print:
- 2017
- Published Online:
- January 2018
- ISBN:
- 9780198797319
- eISBN:
- 9780191838774
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198797319.003.0009
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
Data traffic in wireless networks has been increasing exponentially for a long time and is expected to continue this trend. The emerging data-hungry applications, such as video-on-demand and cloud ...
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Data traffic in wireless networks has been increasing exponentially for a long time and is expected to continue this trend. The emerging data-hungry applications, such as video-on-demand and cloud computing, as well as the exploding number of smart user devices demand the introduction of disruptive technologies. An analogous situation appears in the case of wireline (mostly fiber-optical) traffic, where the currently deployed infrastructure is expected to soon reach its limits, leading to the so-called capacity crunch. The aim of this chapter is to introduce the physics and mathematics community to a number of relevant problems in communications research and the types of solutions that have been used to tackle them. In the process, interested readers may be able to further acquaint themselves with research in engineering bibliography cited herein.Less
Data traffic in wireless networks has been increasing exponentially for a long time and is expected to continue this trend. The emerging data-hungry applications, such as video-on-demand and cloud computing, as well as the exploding number of smart user devices demand the introduction of disruptive technologies. An analogous situation appears in the case of wireline (mostly fiber-optical) traffic, where the currently deployed infrastructure is expected to soon reach its limits, leading to the so-called capacity crunch. The aim of this chapter is to introduce the physics and mathematics community to a number of relevant problems in communications research and the types of solutions that have been used to tackle them. In the process, interested readers may be able to further acquaint themselves with research in engineering bibliography cited herein.
Jean-Philippe Bouchaud
- Published in print:
- 2017
- Published Online:
- January 2018
- ISBN:
- 9780198797319
- eISBN:
- 9780191838774
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198797319.003.0006
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter reviews methods from random matrix theory to extract information about a large signal matrix C (for example, a correlation matrix arising in big data problems), from its noisy ...
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This chapter reviews methods from random matrix theory to extract information about a large signal matrix C (for example, a correlation matrix arising in big data problems), from its noisy observation matrix M. The chapter shows that the replica method can be used to obtain both the spectral density and the overlaps between noise-corrupted eigenvectors and the true ones, for both additive and multiplicative noise. This allows one to construct optimal rotationally invariant estimators of C based on the observation of M alone. This chapter also discusses the case of rectangular correlation matrices and the problem of random singular value decomposition.Less
This chapter reviews methods from random matrix theory to extract information about a large signal matrix C (for example, a correlation matrix arising in big data problems), from its noisy observation matrix M. The chapter shows that the replica method can be used to obtain both the spectral density and the overlaps between noise-corrupted eigenvectors and the true ones, for both additive and multiplicative noise. This allows one to construct optimal rotationally invariant estimators of C based on the observation of M alone. This chapter also discusses the case of rectangular correlation matrices and the problem of random singular value decomposition.
Grégory Schehr, Alexander Altland, Yan V. Fyodorov, Neil O'Connell, and Leticia F. Cugliandolo (eds)
- Published in print:
- 2017
- Published Online:
- January 2018
- ISBN:
- 9780198797319
- eISBN:
- 9780191838774
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198797319.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
The field of stochastic processes and random matrix theory (RMT) has been a rapidly evolving subject during the past fifteen years where the continuous development and discovery of new tools, ...
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The field of stochastic processes and random matrix theory (RMT) has been a rapidly evolving subject during the past fifteen years where the continuous development and discovery of new tools, connections, and ideas have led to an avalanche of new results. These breakthroughs have been made possible thanks, to a large extent, to the recent development of various new techniques in RMT. Matrix models have been playing an important role in theoretical physics for a long time and they are currently also a very active domain of research in mathematics. An emblematic example of these recent advances concerns the theory of growth phenomena in the Kardar–Parisi–Zhang (KPZ) universality class where the joint efforts of physicists and mathematicians during the past twenty years have unveiled the beautiful connections between this fundamental problem of statistical mechanics and the theory of random matrices, namely the fluctuations of the largest eigenvalue of certain ensemble of random matrices. These chapters not only cover this topic in detail but also present more recent developments that have emerged from these discoveries, for instance in the context of low-dimensional heat transport (on the physics side) or in the context of integrable probability (on the mathematical side).Less
The field of stochastic processes and random matrix theory (RMT) has been a rapidly evolving subject during the past fifteen years where the continuous development and discovery of new tools, connections, and ideas have led to an avalanche of new results. These breakthroughs have been made possible thanks, to a large extent, to the recent development of various new techniques in RMT. Matrix models have been playing an important role in theoretical physics for a long time and they are currently also a very active domain of research in mathematics. An emblematic example of these recent advances concerns the theory of growth phenomena in the Kardar–Parisi–Zhang (KPZ) universality class where the joint efforts of physicists and mathematicians during the past twenty years have unveiled the beautiful connections between this fundamental problem of statistical mechanics and the theory of random matrices, namely the fluctuations of the largest eigenvalue of certain ensemble of random matrices. These chapters not only cover this topic in detail but also present more recent developments that have emerged from these discoveries, for instance in the context of low-dimensional heat transport (on the physics side) or in the context of integrable probability (on the mathematical side).
Bertrand Eynard
- Published in print:
- 2017
- Published Online:
- January 2018
- ISBN:
- 9780198797319
- eISBN:
- 9780191838774
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198797319.003.0007
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter is an introduction to algebraic methods in random matrix theory (RMT). In the first section, the random matrix ensembles are introduced and it is shown that going beyond the usual Wigner ...
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This chapter is an introduction to algebraic methods in random matrix theory (RMT). In the first section, the random matrix ensembles are introduced and it is shown that going beyond the usual Wigner ensembles can be very useful, in particular by allowing eigenvalues to lie on some paths in the complex plane rather than on the real axis. As a detailed example, the Plancherel model is considered from the point of RMT. The second section is devoted to the saddle-point approximation, also called the Coulomb gas method. This leads to a system of algebraic equations, the solution of which leads to an algebraic curve called the ‘spectral curve’ which determines the large N expansion of all observables in a geometric way. Finally, the third section introduces the ‘loop equations’ (i.e., Schwinger–Dyson equations associated with matrix models), which can be solved recursively (i.e., order by order in a semi-classical expansion) by a universal recursion: the ‘topological recursion’.Less
This chapter is an introduction to algebraic methods in random matrix theory (RMT). In the first section, the random matrix ensembles are introduced and it is shown that going beyond the usual Wigner ensembles can be very useful, in particular by allowing eigenvalues to lie on some paths in the complex plane rather than on the real axis. As a detailed example, the Plancherel model is considered from the point of RMT. The second section is devoted to the saddle-point approximation, also called the Coulomb gas method. This leads to a system of algebraic equations, the solution of which leads to an algebraic curve called the ‘spectral curve’ which determines the large N expansion of all observables in a geometric way. Finally, the third section introduces the ‘loop equations’ (i.e., Schwinger–Dyson equations associated with matrix models), which can be solved recursively (i.e., order by order in a semi-classical expansion) by a universal recursion: the ‘topological recursion’.