Statistical Mechanics: Entropy, Order Parameters, and Complexity

by James P. Sethna

Overview

Statistical mechanics is our tool for deriving the laws that emerge from complex systems. Sethna's text distills the subject to be accessible to those in all realms of science and engineering — avoiding extensive use of quantum mechanics, thermodynamics, and molecular physics. Statistical mechanics explains how bacteria search for food, and how DNA replication is proof-read in biology; optimizes data compression, and explains transitions in complexity in computer science; explains the onset of chaos, and launched random matrix theory in mathematics; addresses extreme events in engineering; and models pandemics and language usage in the social sciences. Sethna's exercises introduce physicists to these triumphs and a hundred others — broadening the horizons of scholars both practicing and nascent. Flipped classrooms and remote learning can now rely on 33 pre-class exercises that test reading comprehension (Emergent vs. fundamental; Weirdness in high dimensions; Aging, entropy and DNA), and 70 in-class activities that illuminate and broaden knowledge (Card shuffling; Human correlations; Crackling noises). Science is awash in information, providing ready access to definitions, explanations, and pedagogy. Sethna's text focuses on the tools we use to create new laws, and on the fascinating simple behavior in complex systems that statistical mechanics explains.

Product Details

ISBN-13:	9780198865254
Publisher:	Oxford University Press
Publication date:	03/26/2021
Series:	Oxford Master Series in Physics
Edition description:	2nd ed.
Pages:	496
Product dimensions:	9.60(w) x 7.40(h) x 1.00(d)

About the Author

James P. Sethna, Professor of Physics, Cornell University

James P. Sethna is professor of physics at Cornell University. Sethna has used statistical mechanics to make substantive contributions in a bewildering variety of subjects — mathematics (dynamical systems and the onset of chaos), engineering (microstructure, plasticity, and fracture), statistics (information geometry, sloppy models, low-dimensional embeddings), materials science (glasses and spin glasses, liquid crystals, crackling noise, superconductivity), and popular culture (mosh pit dynamics and zombie outbreak epidemiology). He has collected cool, illustrative problems from students and colleagues over the decades, which inspired this textbook.

Table of Contents

PrefaceContentsList of figuresWhat is statistical mechanics?1.1. Quantum dice and coins1.2. Probability distributions1.3. Waiting time paradox1.4. Stirling’s formula1.5. Stirling and asymptotic series1.6. Random matrix theory1.7. Six degrees of separation1.8. Satisfactory map colorings1.9. First to fail: Weibull1.10. Emergence1.11. Emergent vs. fundamental1.12. Self-propelled particles1.13. The birthday problem1.14. Width of the height distribution1.15. Fisher information and Cram´er–Rao1.16. Distances in probability spaceRandom walks and emergent properties2.1. Random walk examples: universality and scale invariance2.2. The diffusion equation2.3. Currents and external forces2.4. Solving the diffusion equationTemperature and equilibrium3.1. The microcanonical ensemble3.2. The microcanonical ideal gas3.3. What is temperature? 3.4. Pressure and chemical potential3.5. Entropy, the ideal gas, and phase-space refinementsPhase-space dynamics and ergodicity4.1. Liouville’s theorem4.2. ErgodicityEntropy5.1. Entropy as irreversibility: engines and the heat death of the Universe5.2. Entropy as disorder5.3. Entropy as ignorance: information and memoryFree energies6.1. The canonical ensemble6.2. Uncoupled systems and canonical ensembles6.3. Grand canonical ensemble6.4. What is thermodynamics? 6.5. Mechanics: friction and fluctuations6.6. Chemical equilibrium and reaction rates6.7. Free energy density for the ideal gasQuantum statistical mechanics7.1. Mixed states and density matrices7.2. Quantum harmonic oscillator7.3. Bose and Fermi statistics7.4. Non-interacting bosons and fermions7.5. Maxwell–Boltzmann ‘quantum’ statistics7.6. Black-body radiation and Bose condensation7.7. Metals and the Fermi gasCalculation and computation8.1. The Ising model8.2. Markov chains8.3. What is a phase? Perturbation theoryOrder parameters, broken symmetry, and topology9.1. Identify the broken symmetry9.2. Define the order parameter9.3. Examine the elementary excitations9.4. Classify the topological defectsCorrelations, response, and dissipation10.1. Correlation functions: motivation10.2. Experimental probes of correlations10.3. Equal-time correlations in the ideal gas10.4. Onsager’s regression hypothesis and time correlations10.5. Susceptibility and linear response10.6. Dissipation and the imaginary part10.7. Static susceptibility10.8. The fluctuation-dissipation theorem10.9. Causality and Kramers–Kr¨onigAbrupt phase transitions11.1. Stable and metastable phases11.2. Maxwell construction11.3. Nucleation: critical droplet theory11.4. Morphology of abrupt transitionsContinuous phase transitions12.1. Universality12.2. Scale invariance12.3. Examples of critical pointsA Appendix: Fourier methodsA.1. Fourier conventionsA.2. Derivatives, convolutions, and correlationsA.3. Fourier methods and function spaceA.4. Fourier and translational symmetryReferencesIndex