Statistical Mechanics : Entropy, Order Parameters and Complexity.
Title:
Statistical Mechanics : Entropy, Order Parameters and Complexity.
Author:
Sethna, James.
ISBN:
9780191566219
Personal Author:
Physical Description:
1 online resource (372 pages)
Series:
Oxford Master Series in Physics ; v.No. 14
Oxford Master Series in Physics
Contents:
Contents -- List of figures -- 1 What is statistical mechanics? -- Exercises -- 1.1 Quantum dice -- 1.2 Probability distributions -- 1.3 Waiting times -- 1.4 Stirling's approximation -- 1.5 Stirling and asymptotic series -- 1.6 Random matrix theory -- 1.7 Six degrees of separation -- 1.8 Satisfactory map colorings -- 2 Random walks and emergent properties -- 2.1 Random walk examples: universality and scale invariance -- 2.2 The diffusion equation -- 2.3 Currents and external forces -- 2.4 Solving the diffusion equation -- 2.4.1 Fourier -- 2.4.2 Green -- Exercises -- 2.1 Random walks in grade space -- 2.2 Photon diffusion in the Sun -- 2.3 Molecular motors and random walks -- 2.4 Perfume walk -- 2.5 Generating random walks -- 2.6 Fourier and Green -- 2.7 Periodic diffusion -- 2.8 Thermal diffusion -- 2.9 Frying pan -- 2.10 Polymers and random walks -- 2.11 Stocks, volatility, and diversifcation -- 2.12 Computational finance: pricing derivatives -- 2.13 Building a percolation network -- 3 Temperature and equilibrium -- 3.1 The microcanonical ensemble -- 3.2 The microcanonical ideal gas -- 3.2.1 Configuration space -- 3.2.2 Momentum space -- 3.3 What is temperature? -- 3.4 Pressure and chemical potential -- 3.4.1 Advanced topic: pressure in mechanics and statistical mechanics -- 3.5 Entropy, the ideal gas, and phase-space refinements -- Exercises -- 3.1 Temperature and energy -- 3.2 Large and very large numbers -- 3.3 Escape velocity -- 3.4 Pressure computation -- 3.5 Hard sphere gas -- 3.6 Connecting two macroscopic systems -- 3.7 Gas mixture -- 3.8 Microcanonical energy fluctuations -- 3.9 Gauss and Poisson -- 3.10 Triple product relation -- 3.11 Maxwell relations -- 3.12 Solving differential equations: the pendulum -- 4 Phase-space dynamics and ergodicity -- 4.1 Liouville's theorem -- 4.2 Ergodicity -- Exercises -- 4.1 Equilibration.
4.2 Liouville vs. the damped pendulum -- 4.3 Invariant measures -- 4.4 Jupiter! and the KAM theorem -- 5 Entropy -- 5.1 Entropy as irreversibility: engines and the heat death of the Universe -- 5.2 Entropy as disorder -- 5.2.1 Entropy of mixing: Maxwell's demon and osmotic pressure -- 5.2.2 Residual entropy of glasses: the roads not taken -- 5.3 Entropy as ignorance: information and memory -- 5.3.1 Non-equilibrium entropy -- 5.3.2 Information entropy -- Exercises -- 5.1 Life and the heat death of the Universe -- 5.2 Burning information and Maxwellian demons -- 5.3 Reversible computation -- 5.4 Black hole thermodynamics -- 5.5 Pressure-volume diagram -- 5.6 Carnot refrigerator -- 5.7 Does entropy increase? -- 5.8 The Arnol'd cat map -- 5.9 Chaos, Lyapunov, and entropy increase -- 5.10 Entropy increases: diffusion -- 5.11 Entropy of glasses -- 5.12 Rubber band -- 5.13 How many shuffes? -- 5.14 Information entropy -- 5.15 Shannon entropy -- 5.16 Fractal dimensions -- 5.17 Deriving entropy -- 6 Free energies -- 6.1 The canonical ensemble -- 6.2 Uncoupled systems and canonical ensembles -- 6.3 Grand canonical ensemble -- 6.4 What is thermodynamics? -- 6.5 Mechanics: friction and fluctuations -- 6.6 Chemical equilibrium and reaction rates -- 6.7 Free energy density for the ideal gas -- Exercises -- 6.1 Exponential atmosphere -- 6.2 Two-state system -- 6.3 Negative temperature -- 6.4 Molecular motors and free energies -- 6.5 Laplace -- 6.6 Lagrange -- 6.7 Legendre -- 6.8 Euler -- 6.9 Gibbs-Duhem -- 6.10 Clausius-Clapeyron -- 6.11 Barrier crossing -- 6.12 Michaelis-Menten and Hill -- 6.13 Pollen and hard squares -- 6.14 Statistical mechanics and statistics -- 7 Quantum statistical mechanics -- 7.1 Mixed states and density matrices -- 7.1.1 Advanced topic: density matrices -- 7.2 Quantum harmonic oscillator -- 7.3 Bose and Fermi statistics.
7.4 Non-interacting bosons and fermions -- 7.5 Maxwell-Boltzmann 'quantum' statistics -- 7.6 Black-body radiation and Bose condensation -- 7.6.1 Free particles in a box -- 7.6.2 Black-body radiation -- 7.6.3 Bose condensation -- 7.7 Metals and the Fermi gas -- Exercises -- 7.1 Ensembles and quantum statistics -- 7.2 Phonons and photons are bosons -- 7.3 Phase-space units and the zero of entropy -- 7.4 Does entropy increase in quantum systems? -- 7.5 Photon density matrices -- 7.6 Spin density matrix -- 7.7 Light emission and absorption -- 7.8 Einstein's A and B -- 7.9 Bosons are gregarious: superfluids and lasers -- 7.10 Crystal defects -- 7.11 Phonons on a string -- 7.12 Semiconductors -- 7.13 Bose condensation in a band -- 7.14 Bose condensation: the experiment -- 7.15 The photon-dominated Universe -- 7.16 White dwarfs, neutron stars, and black holes -- 8 Calculation and computation -- 8.1 The Ising model -- 8.1.1 Magnetism -- 8.1.2 Binary alloys -- 8.1.3 Liquids, gases, and the critical point -- 8.1.4 How to solve the Ising model -- 8.2 Markov chains -- 8.3 What is a phase? Perturbation theory -- Exercises -- 8.1 The Ising model -- 8.2 Ising fluctuations and susceptibilities -- 8.3 Waiting for Godot, and Markov -- 8.4 Red and green bacteria -- 8.5 Detailed balance -- 8.6 Metropolis -- 8.7 Implementing Ising -- 8.8 Wolff -- 8.9 Implementing Wolff -- 8.10 Stochastic cells -- 8.11 The repressilator -- 8.12 Entropy increases! Markov chains -- 8.13 Hysteresis and avalanches -- 8.14 Hysteresis algorithms -- 8.15 NP-completeness and kSAT -- 9 Order parameters, broken symmetry, and topology -- 9.1 Identify the broken symmetry -- 9.2 Define the order parameter -- 9.3 Examine the elementary excitations -- 9.4 Classify the topological defects -- Exercises -- 9.1 Topological defects in nematic liquid crystals -- 9.2 Topological defects in the XY model.
9.3 Defect energetics and total divergence terms -- 9.4 Domain walls in magnets -- 9.5 Landau theory for the Ising model -- 9.6 Symmetries and wave equations -- 9.7 Superfluid order and vortices -- 9.8 Superfluids: density matrices and ODLRO -- 10 Correlations, response, and dissipation -- 10.1 Correlation functions: motivation -- 10.2 Experimental probes of correlations -- 10.3 Equal-time correlations in the ideal gas -- 10.4 Onsager's regression hypothesis and time correlations -- 10.5 Susceptibility and linear response -- 10.6 Dissipation and the imaginary part -- 10.7 Static susceptibility -- 10.8 The fluctuation-dissipation theorem -- 10.9 Causality and Kramers-Krönig -- Exercises -- 10.1 Microwave background radiation -- 10.2 Pair distributions and molecular dynamics -- 10.3 Damped oscillator -- 10.4 Spin -- 10.5 Telegraph noise in nanojunctions -- 10.6 Fluctuation-dissipation: Ising -- 10.7 Noise and Langevin equations -- 10.8 Magnetic dynamics -- 10.9 Quasiparticle poles and Goldstone's theorem -- 11 Abrupt phase transitions -- 11.1 Stable and metastable phases -- 11.2 Maxwell construction -- 11.3 Nucleation: critical droplet theory -- 11.4 Morphology of abrupt transitions -- 11.4.1 Coarsening -- 11.4.2 Martensites -- 11.4.3 Dendritic growth -- Exercises -- 11.1 Maxwell and van der Waals -- 11.2 The van der Waals critical point -- 11.3 Interfaces and van der Waals -- 11.4 Nucleation in the Ising model -- 11.5 Nucleation of dislocation pairs -- 11.6 Coarsening in the Ising model -- 11.7 Origami microstructure -- 11.8 Minimizing sequences and microstructure -- 11.9 Snowflakes and linear stability -- 12 Continuous phase transitions -- 12.1 Universality -- 12.2 Scale invariance -- 12.3 Examples of critical points -- 12.3.1 Equilibrium criticality: energy versus entropy -- 12.3.2 Quantum criticality: zero-point fluctuations versus energy.
12.3.3 Dynamical systems and the onset of chaos -- 12.3.4 Glassy systems: random but frozen -- 12.3.5 Perspectives -- Exercises -- 12.1 Ising self-similarity -- 12.2 Scaling and corrections to scaling -- 12.3 Scaling and coarsening -- 12.4 Bifurcation theory -- 12.5 Mean-field theory -- 12.6 The onset of lasing -- 12.7 Renormalization-group trajectories -- 12.8 Superconductivity and the renormalization group -- 12.9 Period doubling -- 12.10 The renormalization group and the central limit theorem: short -- 12.11 The renormalization group and the central limit theorem: long -- 12.12 Percolation and universality -- 12.13 Hysteresis and avalanches: scaling -- A: Appendix: Fourier methods -- A.1 Fourier conventions -- A.2 Derivatives, convolutions, and correlations -- A.3 Fourier methods and function space -- A.4 Fourier and translational symmetry -- Exercises -- A.1 Sound wave -- A.2 Fourier cosines -- A.3 Double sinusoid -- A.4 Fourier Gaussians -- A.5 Uncertainty -- A.6 Fourier relationships -- A.7 Aliasing and windowing -- A.8 White noise -- A.9 Fourier matching -- A.10 Gibbs phenomenon -- References -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W -- X -- Y -- Z.
Abstract:
Sethna's book distills the core ideas of statistical mechanics to make room for new advances important to information theory, complexity, and modern biology. Aimed at advanced undergraduates and early graduate students, Sethna's text explores everything from chaos through information theory to life at the end of the universe. - ;In each generation, scientists must redefine their fields: abstracting, simplifying and distilling the previous standard topics to make room for new advances and methods. Sethna's book takes this step for statistical mechanics - a field rooted in physics and chemistry whose ideas and methods are now central to information theory, complexity, and modern biology. Aimed at advanced undergraduates and early graduate students in all of these fields, Sethna limits his main presentation to. the topics that future mathematicians and biologists, as well as physicists and chemists, will find fascinating and central to their work. The amazing breadth of the field is reflected in the author's large supply of carefully crafted exercises, each an introduction to a whole field of study:. everything from chaos through information theory to life at the end of the universe. - ;The author's style, although quite concentrated, is simple to understand, and has many lovely visual examples to accompany formal ideas and concepts, which makes the exposition live and intuitvely appealing. - Olga K. Dudko, Journal of Statistical Physics, Vol 126;Since the book treats intersections of mathematics, biology, engineering, computer science and social sciences, it will be of a great help to researchers in these fields in making statistical mechanics useful and comprehensible. At the same time, the book will enrich the subject for researchers-physicists who'd like to apply their skills in other disciplines. - Olga K. Dudko, Journal of Statistical Physics,
Vol 126.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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