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Statistical Mechanics: Entropy, Order Parameters, and Complexity最新文献

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Temperature and equilibrium 温度与平衡
Statistical Mechanics: Entropy, Order Parameters, and Complexity Pub Date : 2021-01-26 DOI: 10.1093/oso/9780198865247.003.0003
J. Sethna
{"title":"Temperature and equilibrium","authors":"J. Sethna","doi":"10.1093/oso/9780198865247.003.0003","DOIUrl":"https://doi.org/10.1093/oso/9780198865247.003.0003","url":null,"abstract":"Statistical mechanics explains the comprehensible behavior of microscopically complex systems by using the weird geometry of high-dimensional spaces, and by relying only on the known conserved quantity: the energy. Particle velocities and density fluctuations are determined by the geometry of spheres and cubes in dimensions with twenty three digits. Temperature, pressure, and chemical potential are defined and derived in terms of the volume of the high-dimensional energy shell, as quantified by the entropy. In particular, temperature is the inverse of the cost of buying energy from the rest of the world, and entropy is the currency being paid. Exercises discuss the weird geometry of high dimensions, how taste and smell measure chemical potentials, equilibrium fluctuations, and classic thermodynamic relations.","PeriodicalId":218123,"journal":{"name":"Statistical Mechanics: Entropy, Order Parameters, and Complexity","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130476177","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Quantum statistical mechanics 量子统计力学
Statistical Mechanics: Entropy, Order Parameters, and Complexity Pub Date : 2021-01-26 DOI: 10.1093/oso/9780198865247.003.0007
J. Sethna
{"title":"Quantum statistical mechanics","authors":"J. Sethna","doi":"10.1093/oso/9780198865247.003.0007","DOIUrl":"https://doi.org/10.1093/oso/9780198865247.003.0007","url":null,"abstract":"Quantum statistical mechanics governs metals, semiconductors, and neutron stars. Statistical mechanics spawned Planck’s invention of the quantum, and explains Bose condensation, superfluids, and superconductors. This chapter briefly describes these systems using mixed states, or more formally density matrices, and introducing the properties of bosons and fermions. We discuss in unusual detail how useful descriptions of metals and superfluids can be derived by ignoring the seemingly important interactions between their constituent electrons and atoms. Exercises explore how gregarious bosons lead to superfluids and lasers, how unsociable fermions explain transitions between white dwarfs, neutron stars, and black holes, how one calculates materials properties in semiconductors, insulators, and metals, and how statistical mechanics can explain the collapse of the quantum wavefunction during measurement.","PeriodicalId":218123,"journal":{"name":"Statistical Mechanics: Entropy, Order Parameters, and Complexity","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114659766","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 712
Correlations, response, and dissipation 相关性、响应和耗散
Statistical Mechanics: Entropy, Order Parameters, and Complexity Pub Date : 2021-01-26 DOI: 10.1093/OSO/9780198865247.003.0010
J. Sethna
{"title":"Correlations, response, and dissipation","authors":"J. Sethna","doi":"10.1093/OSO/9780198865247.003.0010","DOIUrl":"https://doi.org/10.1093/OSO/9780198865247.003.0010","url":null,"abstract":"This chapter studies how systems wiggle, and how they respond and dissipate energy when kicked. The wiggling fluctuations are described using correlation functions, the yielding and dissipation are described using susceptibilities. The intricate relations between these quantities are explored using the Onsager regression hypothesis, fluctuation--response and fluctuation--dissipation theorems, and the Kramers--Krönig relation derived from causality (the response cannot precede the kick). The powerful tools of linear response theory described here are basic tools in our exploration of materials with scattering of sound, light, X-rays, and neutrons, and have become our primary description of the behavior of materials. Exercises describe applications to noise in nanojunctions, humans on subways, magnetic spins, molecular dynamics and Ising models, liquids and magnets, materials at critical points, and fluctuations in the early Universe.","PeriodicalId":218123,"journal":{"name":"Statistical Mechanics: Entropy, Order Parameters, and Complexity","volume":"179 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133092142","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Free energies 自由能
Statistical Mechanics: Entropy, Order Parameters, and Complexity Pub Date : 2021-01-26 DOI: 10.1093/oso/9780198865247.003.0006
J. Sethna
{"title":"Free energies","authors":"J. Sethna","doi":"10.1093/oso/9780198865247.003.0006","DOIUrl":"https://doi.org/10.1093/oso/9780198865247.003.0006","url":null,"abstract":"Free energies ignore most of a system, to provide the emergent statistical ensemble describing things we care about. Free energies can ignore the external world. The cost of borrowing energy from the world is measured by the temperature, giving us the canonical ensemble and Helmholtz free energy. Similarly, borrowing particles and volume from the world gives us the grand canonical and Gibbs ensembles. Free energies can ignore unimportant internal degrees of freedom. These lead to friction and noise, and theories of chemical reactions and reaction rates. Free energies can be coarse-grained, removing short distances and times. Exercises apply free energies to molecular motors, thermodynamic relations, reaction rate theory, Zipf’s law for word frequencies, zombie outbreaks, and nucleosynthesis.","PeriodicalId":218123,"journal":{"name":"Statistical Mechanics: Entropy, Order Parameters, and Complexity","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126038433","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 7
What is statistical mechanics? 什么是统计力学?
Statistical Mechanics: Entropy, Order Parameters, and Complexity Pub Date : 2021-01-26 DOI: 10.1093/oso/9780198865247.003.0001
J. Sethna
{"title":"What is statistical mechanics?","authors":"J. Sethna","doi":"10.1093/oso/9780198865247.003.0001","DOIUrl":"https://doi.org/10.1093/oso/9780198865247.003.0001","url":null,"abstract":"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.","PeriodicalId":218123,"journal":{"name":"Statistical Mechanics: Entropy, Order Parameters, and Complexity","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130338195","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Phase-space dynamics and ergodicity 相空间动力学和遍历性
Statistical Mechanics: Entropy, Order Parameters, and Complexity Pub Date : 2021-01-26 DOI: 10.1093/oso/9780198865247.003.0004
J. Sethna
{"title":"Phase-space dynamics and ergodicity","authors":"J. Sethna","doi":"10.1093/oso/9780198865247.003.0004","DOIUrl":"https://doi.org/10.1093/oso/9780198865247.003.0004","url":null,"abstract":"This chapter provides the mathematical justification for the theory of equilibrium statistical mechanics. A Hamiltonian system which is ergodic is shown to have time-average behavior equal to the average behavior in the energy shell. Liouville’s theorem is used to justify the use of phase-space volume in taking this average. Exercises explore the breakdown of ergodicity in planetary motion and in dissipative systems, the application of Liouville’s theorem by Crooks and Jarzynski to non-equilibrium statistical mechanics, and generalizations of statistical mechanics to chaotic systems and to two-dimensional turbulence and Jupiter’s great red spot.","PeriodicalId":218123,"journal":{"name":"Statistical Mechanics: Entropy, Order Parameters, and Complexity","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130772074","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Entropy 熵
Statistical Mechanics: Entropy, Order Parameters, and Complexity Pub Date : 2021-01-26 DOI: 10.1093/oso/9780198865247.003.0005
James P. Sethna
{"title":"Entropy","authors":"James P. Sethna","doi":"10.1093/oso/9780198865247.003.0005","DOIUrl":"https://doi.org/10.1093/oso/9780198865247.003.0005","url":null,"abstract":"This chapter explores irreversibility, disorder, and ignorance as manifestations of entropy. Entropy measures irreversibility. The inevitable increase of entropy was discovered by analyzing a reversible heat engine, and implied the heat death of the Universe. Entropy measures disorder. Osmotic pressure is the entropy of ions in water; the residual entropy of glasses measures their atomic disorder. Entropy measures our ignorance about the world. Information entropy is used to compress messages and files on the Internet. Exercises span information entropy (burning information and Maxwell’s demon, reversible computation, card shuffling, Shannon entropy, entropy of DNA and aging, entropy of messy bedrooms, data compression), fundamentals of equilibration (Arnol’d cat map, proofs for and against entropy increase, phase conjugate mirrors), materials science (rubber bands and glasses), and astrophysics and cosmology (life at the end of the Universe, black hole entropy, Dyson spheres, cosmic nucleosynthesis and the arrow of time).","PeriodicalId":218123,"journal":{"name":"Statistical Mechanics: Entropy, Order Parameters, and Complexity","volume":"90 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131388572","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Continuous phase transitions 连续相变
Statistical Mechanics: Entropy, Order Parameters, and Complexity Pub Date : 2021-01-26 DOI: 10.1093/oso/9780198865247.003.0012
J. Sethna
{"title":"Continuous phase transitions","authors":"J. Sethna","doi":"10.1093/oso/9780198865247.003.0012","DOIUrl":"https://doi.org/10.1093/oso/9780198865247.003.0012","url":null,"abstract":"This chapter analyzes systems with emergent scale invariance -- fractal, self-similar behavior -- by developing the renormalization group. The renormalization group is an amazing abstraction. It describes the flow of the laws governing the system as one coarse-grains -- blurring out the short-distance or short-time details. In the huge space of possible systems (experimental and theoretical), a fixed point of the renormalization group will be the same after blurring and shrinking -- implying emergent scale invariance, a fractal self-similarity. The points which flow into the fixed point share its properties under rescaling -- implying universality, with behavior shared by theory and a wide variety of different experimental systems. The renormalization group also predicts universal power laws and universal functions, describing all behavior on long length and/or time scales. Exercises explore applications to the Ising model, the onset of lasing, superconductors, the onset of chaos, percolation, crackling noise and avalanches, earthquakes, random walks and diffusion, chemical reaction rate theory, and extreme value statistics.","PeriodicalId":218123,"journal":{"name":"Statistical Mechanics: Entropy, Order Parameters, and Complexity","volume":"94 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129075341","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Order parameters, broken symmetry, and topology 顺序参数，对称性破缺和拓扑
Statistical Mechanics: Entropy, Order Parameters, and Complexity Pub Date : 1992-04-22 DOI: 10.1093/oso/9780198865247.003.0009
J. Sethna
{"title":"Order parameters, broken symmetry, and topology","authors":"J. Sethna","doi":"10.1093/oso/9780198865247.003.0009","DOIUrl":"https://doi.org/10.1093/oso/9780198865247.003.0009","url":null,"abstract":"This chapter introduces order parameters -- the reduction of a complex system of interacting particles into a few fields that describe the local equilibrium behavior at each point in the system. It introduces an organized approach to studying a new material system -- identify the broken symmetries, define the order parameter, examine the elementary excitations, and classify the topological defects. It uses order parameters to describe crystals and liquid crystals, superfluids and magnets. It touches upon broken gauge symmetries and the Anderson/Higgs mechanism and an analogue to braiding of non-abelian quantum particles. Exercises explore sound, second sound, and Goldstone’s theorem; fingerprints and soccer balls; Landau theory and other methods for generating emergent theories from symmetries and commutation relations; topological defects in magnets, liquid crystals, and superfluids, and defect entanglement.","PeriodicalId":218123,"journal":{"name":"Statistical Mechanics: Entropy, Order Parameters, and Complexity","volume":"41 3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1992-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114287157","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 20
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