Temperature and equilibrium

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.

Order parameters, broken symmetry, and topology

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.

Free energies

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.

Phase-space dynamics and ergodicity

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.

Correlations, response, and dissipation

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.

Random walks and emergent properties

Random walks and diffusion provide our most basic example of how new laws emerge in statistical mechanics. Random walks have fractal properties, with fluctuations on all scales. The microscopic nature of the random steps is unimportant to the emergent fractal behavior. The diffusion equation is the law which emerges when many random walks are combined; it describes the evolution of any locally conserved density at macroscopic distances and times. The chapter introduces powerful Fourier and Green’s function methods for solving for the resulting behavior. Exercises explore this emergence, with applications to perfume, polymers, stock markets, the Sun, bacteria, and flocking of animals.

What is statistical mechanics?

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.

Abrupt phase transitions

This chapter studies abrupt phase transitions, familiar from boiling water, raindrops, snowflake formation, and frost. At these transitions, the properties change abruptly -- the ice cube and the water in which it floats are at the same temperature and pressure, but have quite different densities and rigidities. The chapter studies the coexistence between two phases by matching their free energies, and discover the Maxwell equal-area construction. It examines the barriers to raindrop formation and discovers nucleation and critical droplet theory. It examines how the transition proceeds after nucleation, and discovers coarsening (familiar from the segregation of oil and water in well-shaken salad dressing), dendrites (snowflakes and frost patterns), and martensitic structures (important in steel). Exercises explore nucleation of dislocations, of cracks, and of droplets in the Ising model; complex free energies and nucleation rates; dendrites in surface growth and snowflakes; and martensites, minimizing sequences, and origami.

Continuous phase transitions

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.

Entropy

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).