Gigantic map of statistical mechanics Statistical Mechanics I. Fundamentals A. Probability theory 1. Random variables 2. Probability distributions 3. Expectation values 4. Moments 5. Central limit theorem B. Thermodynamics 1. Laws of thermodynamics a. Zeroth law b. First law c. Second law d. Third law 2. Thermodynamic potentials a. Internal energy b. Enthalpy c. Helmholtz free energy d. Gibbs free energy 3. Equations of state 4. Phase transitions C. Classical mechanics 1. Hamiltonian mechanics 2. Lagrangian mechanics 3. Poisson brackets 4. Liouville's theorem D. Quantum mechanics 1. Postulates of quantum mechanics 2. Schrödinger equation 3. Heisenberg uncertainty principle 4. Operators and observables 5. Density matrix formalism II. Ensembles A. Microcanonical ensemble 1. Definition and properties 2. Entropy and temperature 3. Applications B. Canonical ensemble 1. Definition and properties 2. Partition function 3. Thermodynamic quantities 4. Applications C. Grand canonical ensemble 1. Definition and properties 2. Grand partition function 3. Chemical potential 4. Applications D. Other ensembles 1. Isothermal-isobaric ensemble 2. Isobaric-isoenthalpic ensemble 3. Generalized ensembles III. Statistical Mechanics of Interacting Systems A. Ideal gas 1. Classical ideal gas 2. Quantum ideal gas a. Fermi-Dirac statistics b. Bose-Einstein statistics B. Real gases 1. Van der Waals equation 2. Virial expansion C. Liquids 1. Radial distribution function 2. Integral equations a. Ornstein-Zernike equation b. Percus-Yevick equation 3. Perturbation theory D. Solids 1. Lattice dynamics 2. Phonons 3. Specific heat 4. Thermal expansion E. Magnetic systems 1. Ising model 2. Heisenberg model 3. Mean-field theory 4. Spin waves F. Quantum fluids 1. Superfluidity 2. Bose-Einstein condensation 3. Fermi liquids G. Polymers and soft matter 1. Polymer chain models 2. Flory-Huggins theory 3. Reptation theory 4. Colloidal suspensions IV. Non-equilibrium Statistical Mechanics A. Linear response theory 1. Fluctuation-dissipation theorem 2. Green-Kubo relations B. Transport phenomena 1. Diffusion 2. Viscosity 3. Thermal conductivity 4. Electrical conductivity C. Stochastic processes 1. Markov processes 2. Master equation 3. Fokker-Planck equation 4. Langevin equation D. Irreversible thermodynamics 1. Onsager reciprocal relations 2. Prigogine's minimum entropy production principle E. Glassy systems and spin glasses 1. Structural glasses 2. Spin glasses 3. Replica theory 4. Mode-coupling theory V. Computational Methods A. Monte Carlo methods 1. Metropolis algorithm 2. Gibbs sampling 3. Cluster algorithms 4. Wang-Landau sampling B. Molecular dynamics 1. Equations of motion 2. Integration schemes 3. Force fields 4. Boundary conditions C. Quantum Monte Carlo 1. Path integral Monte Carlo 2. Variational Monte Carlo 3. Diffusion Monte Carlo D. Density functional theory 1. Hohenberg-Kohn theorems 2. Kohn-Sham equations 3. Exchange-correlation functionals E. Renormalization group methods 1. Real-space renormalization 2. Momentum-space renormalization 3. Numerical renormalization group VI. Applications A. Condensed matter physics 1. Metals and semiconductors 2. Superconductivity 3. Magnetism 4. Topological materials B. Chemical physics 1. Reaction kinetics 2. Molecular dynamics simulations 3. Protein folding 4. Drug design C. Biophysics 1. Molecular motors 2. Ion channels 3. DNA and RNA 4. Membrane physics D. Astrophysics and cosmology 1. Stellar structure 2. Interstellar medium 3. Dark matter 4. Early universe E. Quantum information and computation 1. Quantum entanglement 2. Quantum algorithms 3. Quantum error correction 4. Quantum simulation VII. Advanced Topics A. Field theory and critical phenomena 1. Ginzburg-Landau theory 2. Renormalization group 3. Conformal field theory 4. Topological defects B. Nonlinear dynamics and chaos 1. Bifurcations 2. Strange attractors 3. Lyapunov exponents 4. Fractal dimensions C. Quantum many-body theory 1. Green's functions 2. Feynman diagrams 3. Diagrammatic Monte Carlo 4. Tensor networks D. Quantum field theory 1. Path integrals 2. Gauge theories 3. Renormalization 4. Spontaneous symmetry breaking E. Topological phases of matter 1. Quantum Hall effect 2. Topological insulators 3. Majorana fermions 4. Fractional statistics VIII. Historical Perspectives and Philosophical Foundations A. Development of statistical mechanics 1. Boltzmann's work 2. Gibbs' contributions 3. Einstein and Bose-Einstein statistics 4. Fermi-Dirac statistics B. Interpretations of probability 1. Classical probability 2. Frequentist interpretation 3. Bayesian interpretation 4. Quantum probability C. Foundations of thermodynamics 1. Carnot's work 2. Clausius and entropy 3. Kelvin and the absolute temperature scale 4. Gibbs and the thermodynamic potentials D. Philosophical issues 1. Reductionism and emergent properties 2. Arrow of time and irreversibility 3. Maxwell's demon and information theory 4. Quantum measurement problem IX. Current Research Frontiers A. Quantum thermodynamics 1. Quantum heat engines 2. Quantum fluctuation theorems 3. Quantum coherence and correlations 4. Quantum thermal machines B. Active matter and biological systems 1. Flocking and swarming 2. Active Brownian particles 3. Cytoskeletal dynamics 4. Collective behavior in living systems C. Machine learning and data-driven approaches 1. Neural networks for potential energy surfaces 2. Generative models for molecular design 3. Reinforcement learning for materials discovery 4. Compressed sensing and sparse sampling D. Quantum simulation and quantum computing 1. Cold atom systems 2. Superconducting qubits 3. Trapped ions 4. Quantum algorithms for statistical mechanics E. Non-equilibrium quantum dynamics 1. Quantum quenches 2. Floquet engineering 3. Many-body localization 4. Quantum thermalization and eigenstate thermalization hypothesis X. Interdisciplinary Connections A. Information theory and computation 1. Landauer's principle 2. Algorithmic complexity and Kolmogorov complexity 3. Quantum information theory 4. Quantum error correction and fault-tolerant computation B. Econophysics and social systems 1. Financial markets 2. Wealth distribution 3. Opinion dynamics 4. Complex networks C. Neuroscience and brain dynamics 1. Neural networks 2. Attractor neural networks 3. Criticality in brain dynamics 4. Information processing in the brain D. Ecology and evolutionary dynamics 1. Population dynamics 2. Evolutionary game theory 3. Speciation and extinction 4. Ecological networks E. Quantum gravity and holography 1. AdS/CFT correspondence 2. Black hole thermodynamics 3. Entanglement entropy 4. Emergent spacetime