Here is the continuation of the expanded map of statistical mechanics: d. Replica symmetry breaking and ultrametricity 3. Replica theory a. Replica trick and disorder averaging b. Replica symmetry and overlap distribution c. Parisi solution and hierarchical organization d. Replica field theory and renormalization group 4. Mode-coupling theory a. Memory function and self-consistent equations b. Nonlinear feedback and dynamical arrest c. Idealized glass transition and critical exponents d. Extensions and limitations of mode-coupling theory V. Computational Methods A. Monte Carlo methods 1. Metropolis algorithm a. Importance sampling and detailed balance b. Acceptance ratio and transition probabilities c. Markov chain Monte Carlo and ergodicity d. Parallel tempering and replica exchange 2. Gibbs sampling a. Conditional probability distributions b. Sequential updates and sweep schedule c. Applications in Bayesian inference and machine learning d. Collapsed Gibbs sampling and Rao-Blackwellization 3. Cluster algorithms a. Swendsen-Wang algorithm and percolation b. Wolff algorithm and embedded clusters c. Improved sampling and critical slowing down d. Generalization to other models and geometries 4. Wang-Landau sampling a. Density of states and entropic sampling b. Flat histogram method and iterative refinement c. Multicanonical ensemble and first-order transitions d. Convergence and error estimation B. Molecular dynamics 1. Equations of motion a. Newton's second law and force fields b. Hamiltonian dynamics and symplectic integrators c. Langevin dynamics and stochastic forces d. Brownian dynamics and hydrodynamic interactions 2. Integration schemes a. Verlet algorithm and velocity Verlet b. Leapfrog algorithm and time-reversibility c. Higher-order integrators and multiple time steps d. Constraint dynamics and rigid bodies 3. Force fields a. Bonded interactions and potential energy functions b. Non-bonded interactions and long-range forces c. Polarizable force fields and many-body effects d. Coarse-grained and multiscale models 4. Boundary conditions a. Periodic boundary conditions and minimum image convention b. Ewald summation and particle-mesh methods c. Spherical and ellipsoidal boundaries d. Grand canonical and osmotic ensemble C. Quantum Monte Carlo 1. Path integral Monte Carlo a. Feynman path integral and imaginary time propagator b. Polymer isomorphism and ring polymer molecular dynamics c. Bose-Einstein statistics and permutation sampling d. Fermi-Dirac statistics and sign problem 2. Variational Monte Carlo a. Trial wave function and variational principle b. Metropolis sampling and quantum expectation values c. Jastrow factors and correlation functions d. Optimization methods and energy minimization 3. Diffusion Monte Carlo a. Imaginary time Schrödinger equation and Green's function b. Importance sampling and drift-diffusion process c. Fixed-node approximation and fermion sign problem d. Projector Monte Carlo and ground state properties D. Density functional theory 1. Hohenberg-Kohn theorems a. Electron density and external potential b. Variational principle and ground state energy c. Levy constrained search and universal functional d. N-representability and v-representability 2. Kohn-Sham equations a. Non-interacting reference system and orbitals b. Kohn-Sham potential and self-consistent field c. Local density approximation and gradient corrections d. Time-dependent density functional theory and excitations 3. Exchange-correlation functionals a. Local density approximation and homogeneous electron gas b. Generalized gradient approximations and meta-GGAs c. Hybrid functionals and exact exchange d. Range-separated and double-hybrid functionals E. Renormalization group methods 1. Real-space renormalization a. Block spin transformations and decimation b. Kadanoff scaling and fixed points c. Migdal-Kadanoff approximation and bond moving d. Density matrix renormalization group and tensor networks 2. Momentum-space renormalization a. Wilsonian renormalization and effective action b. Perturbative renormalization and Feynman diagrams c. Callan-Symanzik equation and beta functions d. Renormalization group flow and critical exponents 3. Numerical renormalization group a. Logarithmic discretization and energy scales b. Iterative diagonalization and truncation c. Impurity problems and Kondo effect d. Dynamical properties and spectral functions VI. Applications A. Condensed matter physics 1. Metals and semiconductors a. Band theory and Fermi surfaces b. Electron-phonon coupling and superconductivity c. Excitons and optical properties d. Quantum Hall effect and topological insulators 2. Superconductivity a. BCS theory and Cooper pairs b. Ginzburg-Landau theory and vortices c. Josephson junctions and SQUID d. High-temperature superconductors and unconventional pairing 3. Magnetism a. Exchange interactions and spin Hamiltonians b. Ferromagnetism and Curie temperature c. Antiferromagnetism and Néel temperature d. Frustrated magnetism and spin liquids 4. Topological materials a. Berry phase and Chern numbers b. Edge states and bulk-boundary correspondence c. Majorana fermions and non-Abelian statistics d. Topological insulators and superconductors B. Chemical physics 1. Reaction kinetics a. Rate equations and reaction mechanisms b. Transition state theory and activation energy c. Kramers theory and barrier crossing d. Enzyme catalysis and Michaelis-Menten kinetics 2. Molecular dynamics simulations a. Ab initio molecular dynamics and Born-Oppenheimer approximation b. Classical force fields and empirical potentials c. Enhanced sampling methods and free energy calculations d. Quantum dynamics and nonadiabatic effects 3. Protein folding a. Levinthal's paradox and folding funnel b. Hydrophobic effect and secondary structures c. Molecular chaperones and misfolding diseases d. Folding kinetics and transition path sampling 4. Drug design a. Structure-based drug design and docking b. Ligand-based drug design and pharmacophore modeling c. Quantitative structure-activity relationships (QSAR) d. Virtual screening and high-throughput screening C. Biophysics 1. Molecular motors a. Kinesin and dynein b. Myosin and muscle contraction c. F1-ATPase and rotary motors d. Brownian ratchets and power stroke mechanisms 2. Ion channels a. Hodgkin-Huxley model and action potentials b. Ligand-gated and voltage-gated channels c. Selectivity and permeation mechanisms d. Patch-clamp techniques and single-channel recordings 3. DNA and RNA a. Double helix structure and base pairing b. Supercoiling and topological constraints c. Denaturation and melting curves d. RNA folding and secondary structures 4. Membrane physics a. Lipid bilayers and self-assembly b. Membrane elasticity and curvature c. Protein-lipid interactions and membrane domains d. Membrane fusion and exocytosis D. Astrophysics and cosmology 1. Stellar structure a. Hydrostatic equilibrium and virial theorem b. Nuclear fusion and energy generation c. Radiative transfer and opacity d. Convection and mixing length theory 2. Interstellar medium a. Molecular clouds and star formation b. HII regions and photoionization c. Dust grains and extinction d. Cosmic rays and interstellar shocks 3. Dark matter a. Rotation curves and missing mass problem b. Gravitational lensing and bullet cluster c. Weakly interacting massive particles (WIMPs) d. Axions and other dark matter candidates 4. Early universe a. Big Bang nucleosynthesis and primordial abundances b. Cosmic microwave background and anisotropies c. Inflation and quantum fluctuations d. Baryogenesis and matter-antimatter asymmetry E. Quantum information and computation 1. Quantum entanglement a. Bell states and EPR paradox b. Entanglement measures and witnesses c. Quantum teleportation and superdense coding d. Quantum key distribution and cryptography 2. Quantum algorithms a. Deutsch-Jozsa algorithm and quantum parallelism b. Grover's search algorithm and amplitude amplification c. Shor's factoring algorithm and period finding d. Quantum Fourier transform and phase estimation 3. Quantum error correction a. Decoherence and quantum noise b. Quantum error-correcting codes and stabilizers c. Fault-tolerant quantum computation and threshold theorem d. Topological quantum error correction and surface codes 4. Quantum simulation a. Quantum many-body problems and complexity b. Analog quantum simulation and trapped ions c. Digital quantum simulation and quantum circuits d. Quantum chemistry and materials science applications VII. Advanced Topics A. Field theory and critical phenomena 1. Ginzburg-Landau theory a. Order parameter and free energy functional b. Spontaneous symmetry breaking and Goldstone modes c. Fluctuations and Ginzburg criterion d. Superconductivity and superfluidity 2. Renormalization group a. Scale invariance and critical exponents b. Epsilon expansion and perturbative renormalization c. Non-perturbative renormalization group and functional RG d. Conformal bootstrap and exact results 3. Conformal field theory a. Conformal invariance and primary fields b. Virasoro algebra and central charge c. Operator product expansion and fusion rules d. Minimal models and rational CFTs 4. Topological defects a. Vortices and flux quantization b. Dislocations and Burgers vector c. Disclinations and Frank vector d. Skyrmions and topological charge B. Nonlinear dynamics and chaos 1. Bifurcations a. Saddle-node bifurcation and hysteresis b. Pitchfork bifurcation and symmetry breaking c. Hopf bifurcation and limit cycles d. Period-doubling bifurcation and Feigenbaum scenario 2. Strange attractors a. Lorenz attractor and butterfly effect b. Rössler attractor and chaotic mixing c. Hénon attractor and horseshoe map d. Multifractal attractors and generalized dimensions 3. Lyapunov exponents a. Exponential divergence and sensitive dependence b. Spectrum of Lyapunov exponents and Kaplan-Yorke dimension c. Numerical estimation and convergence properties d. Lyapunov vectors and local stability analysis 4. Fractal dimensions a. Box-counting dimension and self-similarity b. Correlation dimension and Grassberger-Procaccia algorithm c. Information dimension and multifractal spectrum d. Fractal geometry and strange sets C. Quantum many-body theory 1. Green's functions a. Single-particle Green's function and spectral function b. Two-particle Green's function and response functions c. Matsubara formalism and imaginary time d. Lehmann representation and sum rules 2. Feynman diagrams a. Perturbation theory and Wick's theorem b. Feynman rules and diagrammatic expansion c. Self-energy and Dyson equation d. Vertex functions and Bethe-Salpeter equation 3. Diagrammatic Monte Carlo a. Stochastic sampling of Feynman diagrams b. Bold diagrammatic Monte Carlo and self-consistency c. Connected diagrams and sign problem d. Skeleton diagrams and self-energy insertions 4. Tensor networks a. Matrix product states and density matrix renormalization group b. Projected entangled pair states and contraction algorithms c. Multiscale entanglement renormalization ansatz (MERA) d. Tensor network renormalization and holography D. Quantum field theory 1. Path integrals a. Feynman path integral and quantum propagator b. Euclidean path integral and statistical mechanics c. Saddle-point approximation and semiclassical limit d. Instantons and tunneling phenomena 2. Gauge theories a. Abelian gauge theory and electromagnetism b. Non-Abelian gauge theory and Yang-Mills equations c. Faddeev-Popov ghosts and BRST symmetry d. Lattice gauge theory and Wilson loops 3. Renormalization a. Regularization and ultraviolet divergences b. Renormalization group equations and running couplings c. Dimensional regularization and minimal subtraction scheme d. Anomalies and quantum corrections 4. Spontaneous symmetry breaking a. Goldstone theorem and massless bosons b. Higgs mechanism and mass generation c. Chiral symmetry breaking and pions d. Nambu-Goldstone bosons and effective field theory E. Topological phases of matter 1. Quantum Hall effect a. Integer quantum Hall effect and Landau levels b. Fractional quantum Hall effect and composite fermions c. Edge states and chiral Luttinger liquids d. Topological invariants and Chern numbers 2. Topological insulators a. Time-reversal symmetry and Z2 invariants b. Bulk-edge correspondence and helical edge states c. Spin-orbit coupling and band inversion d. Experimental realizations and transport signatures 3. Majorana fermions a. Majorana bound states and non-Abelian statistics b. Kitaev chain and p-wave superconductors c. Semiconductor-superconductor heterostructures d. Braiding operations and topological quantum computation 4. Fractional statistics a. Anyons and braid group b. Abelian anyons and fractional quantum Hall states c. Non-Abelian anyons and conformal field theory d. Topological quantum field theory and category theory VIII. Historical Perspectives and Philosophical Foundations A. Development of statistical mechanics 1. Boltzmann's work a. Kinetic theory of gases and Boltzmann equation b. H-theorem and irreversibility c. Boltzmann entropy and microscopic interpretation d. Loschmidt's paradox and reversibility objection 2. Gibbs' contributions a. Ensemble theory and statistical ensembles b. Gibbs entropy and thermodynamic limit c. Chemical potential and grand canonical ensemble d. Gibbs paradox and extensivity 3. Einstein and Bose-Einstein statistics a. Photon statistics and black-body radiation b. Bose-Einstein condensation and phase transition c. Fluctuations and Brownian motion