Hidetoshi Nishimori and Gerardo Ortiz
- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780199577224
- eISBN:
- 9780191722943
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199577224.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
Phase transitions and critical phenomena have consistently been among the principal subjects of active studies in statistical physics. The simple act of transforming one state of matter or phase into ...
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Phase transitions and critical phenomena have consistently been among the principal subjects of active studies in statistical physics. The simple act of transforming one state of matter or phase into another, for instance by changing the temperature, has always captivated the curious mind. This book provides an introductory account on the theory of phase transitions and critical phenomena, a subject now recognized to be indispensable for students and researchers from many fields of physics and related disciplines. The first five chapters are very basic and quintessential, and cover standard topics such as mean-field theories, the renormalization group and scaling, universality, and statistical field theory methods. The remaining chapters develop more advanced concepts, including conformal field theory, the Kosterlitz-Thouless transition, the effects of randomness, percolation, exactly solvable models, series expansions, duality transformations, and numerical techniques. Moreover, a comprehensive series of appendices expand and clarify several issues not developed in the main text. The important role played by symmetry and topology in understanding the competition between phases and the resulting emergent collective behaviour, giving rise to rigidity and soft elementary excitations, is stressed throughout the book. Serious attempts have been directed toward a self-contained modular approach so that the reader does not have to refer to other sources for supplementary information. Accordingly, most of the concepts and calculations are described in detail, sometimes with additional/auxiliary descriptions given in appendices and exercises. The latter are presented as the topics develop with solutions found at the end of the book, thus giving the text a self-learning character.Less
Phase transitions and critical phenomena have consistently been among the principal subjects of active studies in statistical physics. The simple act of transforming one state of matter or phase into another, for instance by changing the temperature, has always captivated the curious mind. This book provides an introductory account on the theory of phase transitions and critical phenomena, a subject now recognized to be indispensable for students and researchers from many fields of physics and related disciplines. The first five chapters are very basic and quintessential, and cover standard topics such as mean-field theories, the renormalization group and scaling, universality, and statistical field theory methods. The remaining chapters develop more advanced concepts, including conformal field theory, the Kosterlitz-Thouless transition, the effects of randomness, percolation, exactly solvable models, series expansions, duality transformations, and numerical techniques. Moreover, a comprehensive series of appendices expand and clarify several issues not developed in the main text. The important role played by symmetry and topology in understanding the competition between phases and the resulting emergent collective behaviour, giving rise to rigidity and soft elementary excitations, is stressed throughout the book. Serious attempts have been directed toward a self-contained modular approach so that the reader does not have to refer to other sources for supplementary information. Accordingly, most of the concepts and calculations are described in detail, sometimes with additional/auxiliary descriptions given in appendices and exercises. The latter are presented as the topics develop with solutions found at the end of the book, thus giving the text a self-learning character.
Claus Beisbart and Stephan Hartmann (eds)
- Published in print:
- 2011
- Published Online:
- September 2011
- ISBN:
- 9780199577439
- eISBN:
- 9780191730603
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199577439.001.0001
- Subject:
- Philosophy, Philosophy of Science, Metaphysics/Epistemology
Many theories and models from physics are probabilistic. This observation raises several philosophical questions: What are probabilities in physics? Do they reflect objective chances which exist ...
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Many theories and models from physics are probabilistic. This observation raises several philosophical questions: What are probabilities in physics? Do they reflect objective chances which exist independently of the human mind? Or do they only express subjective credences and thus capture our own uncertainty about the world? Finally, which metaphysical lessons, if at all, can one draw from the largely probabilistic character of physics? The chapters collected in this volume address these questions and provide a detailed philosophical appraisal of the status of probabilities in all of physics. Particular emphasis is laid upon statistical physics and quantum mechanics. Many chapters reflect a desire to understand probabilities from physics as objective chances. These chances are characterized, e.g., as time-averages, as probabilities from a best system in the terms of David Lewis, or using the Boltzmannian typicality approach. Other chapters are sympathetic to a Bayesian view of probabilities in physics. The chapters about quantum mechanics elucidate the peculiar characteristics of quantum correlations and discuss strategies to justify the Born Rule. Finally, the chapters of this volume demonstrate how closely interpretive issues about probabilities are entangled with other foundational problems of physics such as the Reversibility Paradox, the ontology of the quantum world and the question whether the world is deterministic.Less
Many theories and models from physics are probabilistic. This observation raises several philosophical questions: What are probabilities in physics? Do they reflect objective chances which exist independently of the human mind? Or do they only express subjective credences and thus capture our own uncertainty about the world? Finally, which metaphysical lessons, if at all, can one draw from the largely probabilistic character of physics? The chapters collected in this volume address these questions and provide a detailed philosophical appraisal of the status of probabilities in all of physics. Particular emphasis is laid upon statistical physics and quantum mechanics. Many chapters reflect a desire to understand probabilities from physics as objective chances. These chances are characterized, e.g., as time-averages, as probabilities from a best system in the terms of David Lewis, or using the Boltzmannian typicality approach. Other chapters are sympathetic to a Bayesian view of probabilities in physics. The chapters about quantum mechanics elucidate the peculiar characteristics of quantum correlations and discuss strategies to justify the Born Rule. Finally, the chapters of this volume demonstrate how closely interpretive issues about probabilities are entangled with other foundational problems of physics such as the Reversibility Paradox, the ontology of the quantum world and the question whether the world is deterministic.
Jean Zinn-Justin
- Published in print:
- 2004
- Published Online:
- January 2010
- ISBN:
- 9780198566748
- eISBN:
- 9780191717994
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566748.003.0004
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics
This chapter provides a simple physical interpretation to the formal continuum limit that has led, from an integral over position variables corresponding to discrete times, to a path integral. It ...
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This chapter provides a simple physical interpretation to the formal continuum limit that has led, from an integral over position variables corresponding to discrete times, to a path integral. It shows that the integral corresponding to discrete times can be considered as the partition function of a classical statistical system in one space dimension. The continuum limit, then, corresponds to a limit where the correlation length, which characterizes the decay of correlations at large distance, diverges. This limit has some universality properties in the sense that different discretized forms lead to the same path integral. In this statistical framework, the correlation functions that have been introduced earlier appear as continuum limits of the correlation functions of classical statistical models on a one-dimensional lattice. Thus, the path integral can be used to exhibit a mathematical relation between classical statistical physics on a line and quantum statistical physics of a point-like particle at thermal equilibrium.Less
This chapter provides a simple physical interpretation to the formal continuum limit that has led, from an integral over position variables corresponding to discrete times, to a path integral. It shows that the integral corresponding to discrete times can be considered as the partition function of a classical statistical system in one space dimension. The continuum limit, then, corresponds to a limit where the correlation length, which characterizes the decay of correlations at large distance, diverges. This limit has some universality properties in the sense that different discretized forms lead to the same path integral. In this statistical framework, the correlation functions that have been introduced earlier appear as continuum limits of the correlation functions of classical statistical models on a one-dimensional lattice. Thus, the path integral can be used to exhibit a mathematical relation between classical statistical physics on a line and quantum statistical physics of a point-like particle at thermal equilibrium.
Marc Mézard and Andrea Montanari
- Published in print:
- 2009
- Published Online:
- September 2009
- ISBN:
- 9780198570837
- eISBN:
- 9780191718755
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198570837.003.0002
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter introduces the basic concepts of statistical physics. The restrictive point of view adopted here keeps to classical (non-quantum) statistical physics and treats it as a branch of ...
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This chapter introduces the basic concepts of statistical physics. The restrictive point of view adopted here keeps to classical (non-quantum) statistical physics and treats it as a branch of probability theory. The mechanism of phase transitions is described in the context of magnetic systems: ferromagnets and spin glasses.Less
This chapter introduces the basic concepts of statistical physics. The restrictive point of view adopted here keeps to classical (non-quantum) statistical physics and treats it as a branch of probability theory. The mechanism of phase transitions is described in the context of magnetic systems: ferromagnets and spin glasses.
Franck Jovanovic and Christophe Schinckus
- Published in print:
- 2017
- Published Online:
- December 2016
- ISBN:
- 9780190205034
- eISBN:
- 9780190205065
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780190205034.003.0003
- Subject:
- Economics and Finance, Financial Economics
Chapter 3 clarifies the theoretical and methodological foundations of econophysics. This crucial chapter details the contextual factors that contributed to the advent of econophysics. More precisely, ...
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Chapter 3 clarifies the theoretical and methodological foundations of econophysics. This crucial chapter details the contextual factors that contributed to the advent of econophysics. More precisely, the advances in statistical physics, the impact of the computerization of science and the new data from financial markets are presented as contextual factors that favored the development of this field. The chapter then discusses the key concepts used by econophysicists and how they contributed to a new way of using power-law distributions in physics and other sciences. This task is essential since this field is often presented in the economics literature as a strictly empirical field with no theoretical grounds. Chapter 3 refutes this claim by emphasizing the differences and the similarities with the models and methods used in financial economics.Less
Chapter 3 clarifies the theoretical and methodological foundations of econophysics. This crucial chapter details the contextual factors that contributed to the advent of econophysics. More precisely, the advances in statistical physics, the impact of the computerization of science and the new data from financial markets are presented as contextual factors that favored the development of this field. The chapter then discusses the key concepts used by econophysicists and how they contributed to a new way of using power-law distributions in physics and other sciences. This task is essential since this field is often presented in the economics literature as a strictly empirical field with no theoretical grounds. Chapter 3 refutes this claim by emphasizing the differences and the similarities with the models and methods used in financial economics.
Jochen Rau
- Published in print:
- 2017
- Published Online:
- November 2017
- ISBN:
- 9780199595068
- eISBN:
- 9780191844300
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780199595068.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
Statistical physics and thermodynamics describe the behaviour of systems on the macroscopic scale. Their methods are applicable to a wide range of phenomena: from heat engines to chemical reactions, ...
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Statistical physics and thermodynamics describe the behaviour of systems on the macroscopic scale. Their methods are applicable to a wide range of phenomena: from heat engines to chemical reactions, from the interior of stars to the melting of ice. Indeed, the laws of thermodynamics are among the most universal ones of all laws of physics. Yet this subject can prove difficult to grasp. Many view thermodynamics as merely a collection of ad hoc recipes, or are confused by unfamiliar novel concepts, such as the entropy, which have little in common with the theories to which students have got accustomed in other areas of physics. This text provides a concise yet thorough introduction to the key concepts which underlie statistical physics and thermodynamics. It begins with a review of classical probability theory and quantum theory, as well as a careful discussion of the notions of information and entropy, prior to embarking on the development of statistical physics proper. The crucial steps leading from the microscopic to the macroscopic domain are rendered transparent. In particular, the laws of thermodynamics are shown to emerge as natural consequences of the statistical framework. While the emphasis is on clarifying the basic concepts, the text also contains many applications and classroom-tested exercises, covering all major topics of a standard course on statistical physics and thermodynamics. The text is suited both for a one-semester course at the advanced undergraduate or beginning graduate level and as a self-contained tutorial guide for students in physics, chemistry, and engineering.Less
Statistical physics and thermodynamics describe the behaviour of systems on the macroscopic scale. Their methods are applicable to a wide range of phenomena: from heat engines to chemical reactions, from the interior of stars to the melting of ice. Indeed, the laws of thermodynamics are among the most universal ones of all laws of physics. Yet this subject can prove difficult to grasp. Many view thermodynamics as merely a collection of ad hoc recipes, or are confused by unfamiliar novel concepts, such as the entropy, which have little in common with the theories to which students have got accustomed in other areas of physics. This text provides a concise yet thorough introduction to the key concepts which underlie statistical physics and thermodynamics. It begins with a review of classical probability theory and quantum theory, as well as a careful discussion of the notions of information and entropy, prior to embarking on the development of statistical physics proper. The crucial steps leading from the microscopic to the macroscopic domain are rendered transparent. In particular, the laws of thermodynamics are shown to emerge as natural consequences of the statistical framework. While the emphasis is on clarifying the basic concepts, the text also contains many applications and classroom-tested exercises, covering all major topics of a standard course on statistical physics and thermodynamics. The text is suited both for a one-semester course at the advanced undergraduate or beginning graduate level and as a self-contained tutorial guide for students in physics, chemistry, and engineering.
Tom Lancaster and Stephen J. Blundell
- Published in print:
- 2014
- Published Online:
- June 2014
- ISBN:
- 9780199699322
- eISBN:
- 9780191779435
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199699322.003.0022
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
A rapid crash course in basic statistical physics is given, showing how the partition function can be used to manufacture any desired thermodynamic quantity or correlation function.
A rapid crash course in basic statistical physics is given, showing how the partition function can be used to manufacture any desired thermodynamic quantity or correlation function.
Peter Richmond, Jürgen Mimkes, and Stefan Hutzler
- Published in print:
- 2013
- Published Online:
- December 2013
- ISBN:
- 9780199674701
- eISBN:
- 9780191780066
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199674701.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
An understanding of the behaviour of financial assets and the evolution of economies has never been as important as it is today. This book looks at these complex systems from the perspective of the ...
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An understanding of the behaviour of financial assets and the evolution of economies has never been as important as it is today. This book looks at these complex systems from the perspective of the physicist. So-called ‘econophysics’ and its application to finance have made great strides in recent years. Less emphasis has been placed on the broader subject of macroeconomics, and many economics students are still taught traditional neo-classical economics. The reader is given a general primer in statistical physics, probability theory, and use of correlation functions. Much of the mathematics that is developed is frequently no longer included in undergraduate physics courses. The statistical physics of Boltzmann and Gibbs is one of the oldest disciplines within physics and it can be argued that it was first applied to ensembles of molecules as opposed to being applied to social agents only by way of historical accident. The authors argue by analogy that the theory can be applied directly to economic systems comprising assemblies of interacting agents. The necessary tools and mathematics are developed in a clear and concise manner. The body of work, now termed econophysics, is then developed. The authors show where traditional methods break down and how the probability distributions and correlation functions can be properly understood using high-frequency data. Recent work by the physics community on risk and market crashes are discussed together with new work on betting markets, as well as studies of speculative peaks that occur in housing markets.Less
An understanding of the behaviour of financial assets and the evolution of economies has never been as important as it is today. This book looks at these complex systems from the perspective of the physicist. So-called ‘econophysics’ and its application to finance have made great strides in recent years. Less emphasis has been placed on the broader subject of macroeconomics, and many economics students are still taught traditional neo-classical economics. The reader is given a general primer in statistical physics, probability theory, and use of correlation functions. Much of the mathematics that is developed is frequently no longer included in undergraduate physics courses. The statistical physics of Boltzmann and Gibbs is one of the oldest disciplines within physics and it can be argued that it was first applied to ensembles of molecules as opposed to being applied to social agents only by way of historical accident. The authors argue by analogy that the theory can be applied directly to economic systems comprising assemblies of interacting agents. The necessary tools and mathematics are developed in a clear and concise manner. The body of work, now termed econophysics, is then developed. The authors show where traditional methods break down and how the probability distributions and correlation functions can be properly understood using high-frequency data. Recent work by the physics community on risk and market crashes are discussed together with new work on betting markets, as well as studies of speculative peaks that occur in housing markets.
Jean Zinn-Justin
- Published in print:
- 2019
- Published Online:
- August 2019
- ISBN:
- 9780198787754
- eISBN:
- 9780191829840
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198787754.003.0008
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
Chapter 8 discusses effective field theory. This concept is inspired by the theory of critical phenomena in statistical physics and based on renormalization group ideas. The basic idea behind ...
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Chapter 8 discusses effective field theory. This concept is inspired by the theory of critical phenomena in statistical physics and based on renormalization group ideas. The basic idea behind effective field theory is that one starts from a microscopic model involving an infinite number of fluctuating degrees of freedom whose interactions are characterized by a microscopic scale and in which, as a result of interactions, a length that is much larger than the microscopic scale, or, equivalently, a mass much smaller than the characteristic mass scale of the initial model, is generated. The chapter illustrates this topic with examples. It also stresses that all quantum field theories as applied to particle physics or statistical physics are only effective (i.e. not fundamental) theories. Besides the problem of a phi4 type field theory with a large mass field, two more complicated examples are discussed: the Gross–Neveu and the non–linear sigma models.Less
Chapter 8 discusses effective field theory. This concept is inspired by the theory of critical phenomena in statistical physics and based on renormalization group ideas. The basic idea behind effective field theory is that one starts from a microscopic model involving an infinite number of fluctuating degrees of freedom whose interactions are characterized by a microscopic scale and in which, as a result of interactions, a length that is much larger than the microscopic scale, or, equivalently, a mass much smaller than the characteristic mass scale of the initial model, is generated. The chapter illustrates this topic with examples. It also stresses that all quantum field theories as applied to particle physics or statistical physics are only effective (i.e. not fundamental) theories. Besides the problem of a phi4 type field theory with a large mass field, two more complicated examples are discussed: the Gross–Neveu and the non–linear sigma models.
Florent Krzakala, Federico Ricci-Tersenghi, Lenka Zdeborova, Riccardo Zecchina, Eric W. Tramel, and Leticia F. Cugliandolo (eds)
- Published in print:
- 2015
- Published Online:
- March 2016
- ISBN:
- 9780198743736
- eISBN:
- 9780191803802
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198743736.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This book contains a collection of the presentations that were given in October 2013 at the Les Houches Autumn School on statistical physics, optimization, inference, and message-passing algorithms. ...
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This book contains a collection of the presentations that were given in October 2013 at the Les Houches Autumn School on statistical physics, optimization, inference, and message-passing algorithms. In the last decade, there has been increasing convergence of interest and methods between theoretical physics and fields as diverse as probability, machine learning, optimization, and inference problems. In particular, much theoretical and applied work in statistical physics and computer science has relied on the use of message-passing algorithms and their connection to the statistical physics of glasses and spin glasses. For example, both the replica and cavity methods have led to recent advances in compressed sensing, sparse estimation, and random constraint satisfaction, to name a few. This book’s detailed pedagogical lectures on statistical inference, computational complexity, the replica and cavity methods, and belief propagation are aimed particularly at PhD students, post-docs, and young researchers desiring the foundational material necessary for entering this rapidly developing field. In these lectures the reader can find detailed applications of theory to problems in community detection and clustering, signal denoising, identification of hidden cliques, error correcting codes, and constraint satisfaction.Less
This book contains a collection of the presentations that were given in October 2013 at the Les Houches Autumn School on statistical physics, optimization, inference, and message-passing algorithms. In the last decade, there has been increasing convergence of interest and methods between theoretical physics and fields as diverse as probability, machine learning, optimization, and inference problems. In particular, much theoretical and applied work in statistical physics and computer science has relied on the use of message-passing algorithms and their connection to the statistical physics of glasses and spin glasses. For example, both the replica and cavity methods have led to recent advances in compressed sensing, sparse estimation, and random constraint satisfaction, to name a few. This book’s detailed pedagogical lectures on statistical inference, computational complexity, the replica and cavity methods, and belief propagation are aimed particularly at PhD students, post-docs, and young researchers desiring the foundational material necessary for entering this rapidly developing field. In these lectures the reader can find detailed applications of theory to problems in community detection and clustering, signal denoising, identification of hidden cliques, error correcting codes, and constraint satisfaction.
Lazaros K. Gallos
- Published in print:
- 2020
- Published Online:
- July 2020
- ISBN:
- 9780190880743
- eISBN:
- 9780190880774
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780190880743.003.0018
- Subject:
- Public Health and Epidemiology, Public Health, Epidemiology
To begin understanding noncommunicable diseases in a population, researchers must understand how people are connected to each other, how they interact with each other, and if there are external ...
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To begin understanding noncommunicable diseases in a population, researchers must understand how people are connected to each other, how they interact with each other, and if there are external influences. Heterogeneity and complexity in human disease suggest new methodological and analytical ways in which the physical sciences can assist research in areas such as obesity, cardiovascular diseases, psychiatric disorders, or cancer. As it is becoming clear that human morbid states are not strictly deterministic diseases, this chapter overviews how statistical physics and nonlinear dynamics (e.g., percolation, cascades, control theory) grounded in stochastic approaches can contribute to the delineation and control of an array of complex population health outcomes.Less
To begin understanding noncommunicable diseases in a population, researchers must understand how people are connected to each other, how they interact with each other, and if there are external influences. Heterogeneity and complexity in human disease suggest new methodological and analytical ways in which the physical sciences can assist research in areas such as obesity, cardiovascular diseases, psychiatric disorders, or cancer. As it is becoming clear that human morbid states are not strictly deterministic diseases, this chapter overviews how statistical physics and nonlinear dynamics (e.g., percolation, cascades, control theory) grounded in stochastic approaches can contribute to the delineation and control of an array of complex population health outcomes.
Cole Mathis
- Published in print:
- 2020
- Published Online:
- June 2020
- ISBN:
- 9780190915650
- eISBN:
- 9780197506066
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780190915650.003.0006
- Subject:
- Biology, Bioethics
This chapter draws inspiration from statistical physics to describe a statistical category that can be termed the “living state.” References to a living state can be found throughout origin of life ...
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This chapter draws inspiration from statistical physics to describe a statistical category that can be termed the “living state.” References to a living state can be found throughout origin of life and astrobiology science. Some researchers have used the concept of the living state to explicitly place biological phenomena within the epistemological scope of statistical physics. Within this framework, biological phenomena at a given scale of organization are explained and understood by appealing to the statistical properties of the dynamics of the smaller and larger scales. This is analogous to how distinct states of matter are understood by appealing to the statistical properties of atoms, with the important distinction that statistical physicists have historically not included constraints from larger levels of organization, which are essential in determining the properties of living systems. This conception of the living state may enable astrobiologists to integrate progress from different disciplinary perspectives into a quantitative theory of life.Less
This chapter draws inspiration from statistical physics to describe a statistical category that can be termed the “living state.” References to a living state can be found throughout origin of life and astrobiology science. Some researchers have used the concept of the living state to explicitly place biological phenomena within the epistemological scope of statistical physics. Within this framework, biological phenomena at a given scale of organization are explained and understood by appealing to the statistical properties of the dynamics of the smaller and larger scales. This is analogous to how distinct states of matter are understood by appealing to the statistical properties of atoms, with the important distinction that statistical physicists have historically not included constraints from larger levels of organization, which are essential in determining the properties of living systems. This conception of the living state may enable astrobiologists to integrate progress from different disciplinary perspectives into a quantitative theory of life.
Cristopher Moore and Stephan Mertens
- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780199233212
- eISBN:
- 9780191775079
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199233212.003.0014
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
Certain formulas, such as the 3-SAT formula, undergo a phase transition from almost certain satisfiability to almost certain unsatisfiability when the number of constraints per variable reaches a ...
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Certain formulas, such as the 3-SAT formula, undergo a phase transition from almost certain satisfiability to almost certain unsatisfiability when the number of constraints per variable reaches a critical threshold. This transition is comparable to the freezing of water and also occurs in many other NP-complete problems such as graph coloring and integer partitioning. This chapter first considers some experimental results on random 3-SAT and assumes that a phase transition exists. It then explores some simple phase transitions in random graphs and shows how to compute the size of k-cores, along with the degrees at which they first appear. It also looks at random k-SAT formulas and demonstrates how to prove upper and lower bounds on the critical density of clauses. Furthermore, it describes simple search algorithms as flows through state space before concluding with a discussion of recent advances inspired by techniques in statistical physics.Less
Certain formulas, such as the 3-SAT formula, undergo a phase transition from almost certain satisfiability to almost certain unsatisfiability when the number of constraints per variable reaches a critical threshold. This transition is comparable to the freezing of water and also occurs in many other NP-complete problems such as graph coloring and integer partitioning. This chapter first considers some experimental results on random 3-SAT and assumes that a phase transition exists. It then explores some simple phase transitions in random graphs and shows how to compute the size of k-cores, along with the degrees at which they first appear. It also looks at random k-SAT formulas and demonstrates how to prove upper and lower bounds on the critical density of clauses. Furthermore, it describes simple search algorithms as flows through state space before concluding with a discussion of recent advances inspired by techniques in statistical physics.
Cristopher Moore and Stephan Mertens
- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780199233212
- eISBN:
- 9780191775079
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199233212.003.0013
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
The objects that are solutions to an NP-complete problem are difficult to count. Counting can be a subtle and complex problem even when the corresponding existence and optimisation problems are in P. ...
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The objects that are solutions to an NP-complete problem are difficult to count. Counting can be a subtle and complex problem even when the corresponding existence and optimisation problems are in P. Spanning trees and perfect matchings are simple graph-theoretic objects, and the difference between them has deep mathematical roots. A matrix's determinant is the number of spanning trees while its permanent is the number of perfect matchings. Counting is closely associated with sampling. This chapter explores how to generate random matchings, and hence count them approximately, using a Markov chain that mixes in polynomial time. It considers the special case of planar graphs, such as the square lattice, to demonstrate that the number of perfect matchings is in P. It also discusses the implications of this fact for statistical physics and looks at how to find exact solutions for many physical models in two dimensions, including the Ising model.Less
The objects that are solutions to an NP-complete problem are difficult to count. Counting can be a subtle and complex problem even when the corresponding existence and optimisation problems are in P. Spanning trees and perfect matchings are simple graph-theoretic objects, and the difference between them has deep mathematical roots. A matrix's determinant is the number of spanning trees while its permanent is the number of perfect matchings. Counting is closely associated with sampling. This chapter explores how to generate random matchings, and hence count them approximately, using a Markov chain that mixes in polynomial time. It considers the special case of planar graphs, such as the square lattice, to demonstrate that the number of perfect matchings is in P. It also discusses the implications of this fact for statistical physics and looks at how to find exact solutions for many physical models in two dimensions, including the Ising model.
James P. Sethna
- Published in print:
- 2021
- Published Online:
- April 2021
- ISBN:
- 9780198865247
- eISBN:
- 9780191897610
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198865247.003.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
Statistical mechanics explains the simple behavior of complex systems. It works by studying not a particular instance, but the typical behavior of a large collection (or ensemble) of systems, which ...
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Statistical mechanics explains the simple behavior of complex systems. It works by studying not a particular instance, but the typical behavior of a large collection (or ensemble) of systems, which is far easier to calculate. Entropy, free energies, order parameters, phases and phase transitions emerge as collective behaviors that are not manifest in the complex microscopic laws. This text will develop the statistical mechanical machinery needed to generate the new laws governing these emergent behaviors. Exercises in this chapter discuss emergence, Stirling’s formula, random matrix theory, small world networks, an NP complete problem, active matter, and topics in statistics.Less
Statistical mechanics explains the simple behavior of complex systems. It works by studying not a particular instance, but the typical behavior of a large collection (or ensemble) of systems, which is far easier to calculate. Entropy, free energies, order parameters, phases and phase transitions emerge as collective behaviors that are not manifest in the complex microscopic laws. This text will develop the statistical mechanical machinery needed to generate the new laws governing these emergent behaviors. Exercises in this chapter discuss emergence, Stirling’s formula, random matrix theory, small world networks, an NP complete problem, active matter, and topics in statistics.
Giulio Biroli
- Published in print:
- 2016
- Published Online:
- August 2016
- ISBN:
- 9780198768166
- eISBN:
- 9780191821905
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198768166.003.0003
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter provides an introduction to several important and interesting facets of out-of-equilibrium dynamics. If in the past hard condensed matter and classical statistical physics remained ...
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This chapter provides an introduction to several important and interesting facets of out-of-equilibrium dynamics. If in the past hard condensed matter and classical statistical physics remained rather separate research fields, the focus now on several kinds of out-of-equilibrium dynamics is bringing them closer together. In this chapter, the stress is upon the common concepts, with the aim of showing that there is much to gain in considering out-of-equilibrium dynamics as a research field in itself. This chapter by no means self-contained. Rather, each section is a door toward a vast research area. The most striking, useful, or important concepts and tools are presented, often in an informal and introductory way, with the hope that this will stimulate the interest of readers who will then turn to the cited specialized reviews and books to fully satisfy their curiosity and acquire more complete knowledge of the subject.Less
This chapter provides an introduction to several important and interesting facets of out-of-equilibrium dynamics. If in the past hard condensed matter and classical statistical physics remained rather separate research fields, the focus now on several kinds of out-of-equilibrium dynamics is bringing them closer together. In this chapter, the stress is upon the common concepts, with the aim of showing that there is much to gain in considering out-of-equilibrium dynamics as a research field in itself. This chapter by no means self-contained. Rather, each section is a door toward a vast research area. The most striking, useful, or important concepts and tools are presented, often in an informal and introductory way, with the hope that this will stimulate the interest of readers who will then turn to the cited specialized reviews and books to fully satisfy their curiosity and acquire more complete knowledge of the subject.
Sauro Succi
- Published in print:
- 2018
- Published Online:
- June 2018
- ISBN:
- 9780199592357
- eISBN:
- 9780191847967
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780199592357.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics, Condensed Matter Physics / Materials
Over the past near three decades, the Lattice Boltzmann method has gained a prominent role as an efficient computational method for the numerical simulation of a wide variety of complex states of ...
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Over the past near three decades, the Lattice Boltzmann method has gained a prominent role as an efficient computational method for the numerical simulation of a wide variety of complex states of flowing matter across a broad range of scales, from fully developed turbulence, to multiphase micro-flows, all the way down to nano-biofluidics and lately, even quantum-relativistic subnuclear fluids. After providing a self-contained introduction to the kinetic theory of fluids and a thorough account of its transcription to the lattice framework, this book presents a survey of the major developments which have led to the impressive growth of the Lattice Boltzmann across most walks of fluid dynamics and its interfaces with allied disciplines, such as statistical physics, material science, soft matter and biology. This includes recent developments of Lattice Boltzmann methods for non-ideal fluids, micro- and nanofluidic flows with suspended bodies of assorted nature and extensions to strong non-equilibrium flows beyond the realm of continuum fluid mechanics. In the final part, the book also presents the extension of the Lattice Boltzmann method to quantum and relativistic fluids, in an attempt to match the major surge of interest spurred by recent developments in the area of strongly interacting holographic fluids, such as quark-gluon plasmas and electron flows in graphene. It is hoped that this book may provide a source information and possibly inspiration to a broad audience of scientists dealing with the physics of classical and quantum flowing matter across many scales of motion.Less
Over the past near three decades, the Lattice Boltzmann method has gained a prominent role as an efficient computational method for the numerical simulation of a wide variety of complex states of flowing matter across a broad range of scales, from fully developed turbulence, to multiphase micro-flows, all the way down to nano-biofluidics and lately, even quantum-relativistic subnuclear fluids. After providing a self-contained introduction to the kinetic theory of fluids and a thorough account of its transcription to the lattice framework, this book presents a survey of the major developments which have led to the impressive growth of the Lattice Boltzmann across most walks of fluid dynamics and its interfaces with allied disciplines, such as statistical physics, material science, soft matter and biology. This includes recent developments of Lattice Boltzmann methods for non-ideal fluids, micro- and nanofluidic flows with suspended bodies of assorted nature and extensions to strong non-equilibrium flows beyond the realm of continuum fluid mechanics. In the final part, the book also presents the extension of the Lattice Boltzmann method to quantum and relativistic fluids, in an attempt to match the major surge of interest spurred by recent developments in the area of strongly interacting holographic fluids, such as quark-gluon plasmas and electron flows in graphene. It is hoped that this book may provide a source information and possibly inspiration to a broad audience of scientists dealing with the physics of classical and quantum flowing matter across many scales of motion.
John T. Chalker
- Published in print:
- 2017
- Published Online:
- March 2017
- ISBN:
- 9780198785781
- eISBN:
- 9780191827600
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198785781.003.0003
- Subject:
- Physics, Condensed Matter Physics / Materials
Two important markers in the history of research on spin liquids and frustrated magnetism are Anderson’s suggestion, over 40 years ago, of the resonating valence bond state as an alternative to Néel ...
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Two important markers in the history of research on spin liquids and frustrated magnetism are Anderson’s suggestion, over 40 years ago, of the resonating valence bond state as an alternative to Néel order and Ramirez’ influential review, some 20 years ago, of strongly frustrated magnets. There has been a tremendous amount of progress since then, but much remains to be done, especially in identifying experimental examples of spin liquids and understanding their properties. These lecture notes aim to provide an introduction to the field that links understanding of the classical statistical physics of these systems with approaches to their quantum mechanics.Less
Two important markers in the history of research on spin liquids and frustrated magnetism are Anderson’s suggestion, over 40 years ago, of the resonating valence bond state as an alternative to Néel order and Ramirez’ influential review, some 20 years ago, of strongly frustrated magnets. There has been a tremendous amount of progress since then, but much remains to be done, especially in identifying experimental examples of spin liquids and understanding their properties. These lecture notes aim to provide an introduction to the field that links understanding of the classical statistical physics of these systems with approaches to their quantum mechanics.
Giuseppe Mussardo
- Published in print:
- 2020
- Published Online:
- May 2020
- ISBN:
- 9780198788102
- eISBN:
- 9780191830082
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198788102.003.0005
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
Two exact combinatorial solutions of the two-dimensional Ising model are the key topics of this chapter. Although no subsequent topic depends on them, both the mathematical and the physical aspects ...
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Two exact combinatorial solutions of the two-dimensional Ising model are the key topics of this chapter. Although no subsequent topic depends on them, both the mathematical and the physical aspects of these solutions are so elegant as to deserve special attention. Chapter 5 covers how the Ising model is a pathfinder in the field of critical phenomena and has always played an important role in statistical physics, both at pedagogical and methodological levels. It describes how the Ising model is an ideal playground for several areas of pure and applied mathematics.Less
Two exact combinatorial solutions of the two-dimensional Ising model are the key topics of this chapter. Although no subsequent topic depends on them, both the mathematical and the physical aspects of these solutions are so elegant as to deserve special attention. Chapter 5 covers how the Ising model is a pathfinder in the field of critical phenomena and has always played an important role in statistical physics, both at pedagogical and methodological levels. It describes how the Ising model is an ideal playground for several areas of pure and applied mathematics.
Franck Jovanovic and Christophe Schinckus
- Published in print:
- 2017
- Published Online:
- December 2016
- ISBN:
- 9780190205034
- eISBN:
- 9780190205065
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780190205034.001.0001
- Subject:
- Economics and Finance, Financial Economics
How can we create a profitable dialogue between financial economists and econophysicists? This book moves beyond the disciplinary frontiers in order to initiate the development of a common ...
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How can we create a profitable dialogue between financial economists and econophysicists? This book moves beyond the disciplinary frontiers in order to initiate the development of a common theoretical framework that makes sense for both traditionally trained financial economists and econophysicists. Unlike other publications dedicated to econophysics, this book is written by two financial economists, and it situates econophysics in the evolution of financial economics. The major issues that concern the collaboration between the two fields are analyzed in detail. More specifically, this book explains the theoretical and methodological foundations of these two fields in an accessible vocabulary providing the first extensive analytic comparison between models and results from both fields. The book also identifies the major conceptual gatekeepers that complicate dialogue between the two communities, providing elements to overcome them. By mixing conceptual, historical, theoretical, and formal arguments, the analysis bridges the current gap between financial economists and econophysicists. This book details the recent results in econophysics that bring it closer to financial economics. So doing, it identifies what remains to be done for econophysicists to contribute significantly to financial economics. Beyond the clarification of the current situation, this book also proposes a generic model compatible with the two fields, defining minimal conditions for common models. Finally, this book provides a research agenda for a more fruitful collaboration between econophysicists and financial economists, creating new research opportunities. It thus lays the foundations for common theoretical framework and models.Less
How can we create a profitable dialogue between financial economists and econophysicists? This book moves beyond the disciplinary frontiers in order to initiate the development of a common theoretical framework that makes sense for both traditionally trained financial economists and econophysicists. Unlike other publications dedicated to econophysics, this book is written by two financial economists, and it situates econophysics in the evolution of financial economics. The major issues that concern the collaboration between the two fields are analyzed in detail. More specifically, this book explains the theoretical and methodological foundations of these two fields in an accessible vocabulary providing the first extensive analytic comparison between models and results from both fields. The book also identifies the major conceptual gatekeepers that complicate dialogue between the two communities, providing elements to overcome them. By mixing conceptual, historical, theoretical, and formal arguments, the analysis bridges the current gap between financial economists and econophysicists. This book details the recent results in econophysics that bring it closer to financial economics. So doing, it identifies what remains to be done for econophysicists to contribute significantly to financial economics. Beyond the clarification of the current situation, this book also proposes a generic model compatible with the two fields, defining minimal conditions for common models. Finally, this book provides a research agenda for a more fruitful collaboration between econophysicists and financial economists, creating new research opportunities. It thus lays the foundations for common theoretical framework and models.
