Here is the expanded map of statistical mechanics with subconcepts added for each part: Statistical Mechanics I. Fundamentals A. Probability theory 1. Random variables a. Discrete random variables b. Continuous random variables c. Joint probability distributions d. Conditional probability 2. Probability distributions a. Binomial distribution b. Poisson distribution c. Gaussian distribution d. Exponential distribution 3. Expectation values a. Mean b. Variance c. Covariance d. Correlation functions 4. Moments a. Raw moments b. Central moments c. Moment-generating functions d. Cumulants 5. Central limit theorem a. Convergence of sum of random variables b. Gaussian approximation c. Applications in statistical physics B. Thermodynamics 1. Laws of thermodynamics a. Zeroth law i. Thermal equilibrium ii. Temperature b. First law i. Internal energy ii. Heat and work iii. Conservation of energy c. Second law i. Entropy ii. Irreversibility iii. Clausius inequality d. Third law i. Absolute zero temperature ii. Residual entropy 2. Thermodynamic potentials a. Internal energy i. Definition and properties ii. Differential form b. Enthalpy i. Definition and properties ii. Legendre transform of internal energy c. Helmholtz free energy i. Definition and properties ii. Legendre transform of internal energy d. Gibbs free energy i. Definition and properties ii. Legendre transform of enthalpy 3. Equations of state a. Ideal gas equation b. Van der Waals equation c. Virial equation d. Redlich-Kwong equation 4. Phase transitions a. First-order phase transitions i. Latent heat ii. Discontinuities in thermodynamic quantities b. Second-order phase transitions i. Continuous transitions ii. Critical exponents c. Ehrenfest classification d. Landau theory C. Classical mechanics 1. Hamiltonian mechanics a. Generalized coordinates and momenta b. Hamilton's equations of motion c. Poisson brackets d. Canonical transformations 2. Lagrangian mechanics a. Generalized coordinates and velocities b. Lagrangian function c. Euler-Lagrange equations d. Noether's theorem 3. Poisson brackets a. Definition and properties b. Canonical commutation relations c. Jacobi identity d. Symplectic geometry 4. Liouville's theorem a. Phase space volume preservation b. Incompressible flow in phase space c. Relation to ergodicity d. Implications for statistical mechanics D. Quantum mechanics 1. Postulates of quantum mechanics a. State vectors and Hilbert space b. Observables and Hermitian operators c. Measurement and Born's rule d. Time evolution and Schrödinger equation 2. Schrödinger equation a. Time-dependent Schrödinger equation b. Time-independent Schrödinger equation c. Stationary states and energy eigenvalues d. Boundary conditions and normalization 3. Heisenberg uncertainty principle a. Position-momentum uncertainty b. Energy-time uncertainty c. Generalized uncertainty relations d. Implications for measurements 4. Operators and observables a. Position and momentum operators b. Angular momentum operators c. Spin operators d. Commutation relations 5. Density matrix formalism a. Pure and mixed states b. Density operator c. Von Neumann equation d. Reduced density matrices and partial trace II. Ensembles A. Microcanonical ensemble 1. Definition and properties a. Isolated systems b. Fixed energy, volume, and particle number c. Equal a priori probability d. Ergodicity and thermalization 2. Entropy and temperature a. Boltzmann entropy b. Microcanonical temperature c. Relation to thermodynamic entropy d. Negative temperature and inverted populations 3. Applications a. Ideal gas in microcanonical ensemble b. Spin systems c. Gravitational systems d. Quantum isolated systems B. Canonical ensemble 1. Definition and properties a. Systems in contact with a heat bath b. Fixed temperature, volume, and particle number c. Boltzmann distribution d. Detailed balance 2. Partition function a. Definition and properties b. Relation to thermodynamic quantities c. Factorization for non-interacting systems d. Classical and quantum partition functions 3. Thermodynamic quantities a. Internal energy b. Entropy c. Specific heat d. Fluctuations and response functions 4. Applications a. Ideal gas in canonical ensemble b. Harmonic oscillator c. Magnetic systems d. Polymers C. Grand canonical ensemble 1. Definition and properties a. Systems in contact with a heat bath and particle reservoir b. Fixed temperature, volume, and chemical potential c. Grand canonical probability distribution d. Fluctuations in particle number 2. Grand partition function a. Definition and properties b. Relation to thermodynamic quantities c. Factorization for non-interacting systems d. Fugacity and activity 3. Chemical potential a. Definition and physical interpretation b. Relation to Gibbs free energy c. Equilibrium condition for particle exchange d. Electrochemical potential 4. Applications a. Ideal gas in grand canonical ensemble b. Adsorption and desorption c. Semiconductor doping d. Quantum gases and Bose-Einstein condensation D. Other ensembles 1. Isothermal-isobaric ensemble a. Fixed temperature, pressure, and particle number b. Gibbs free energy and enthalpy c. Volume fluctuations d. Applications in chemistry and biology 2. Isobaric-isoenthalpic ensemble a. Fixed pressure, enthalpy, and particle number b. Legendre transform of Gibbs free energy c. Temperature fluctuations d. Applications in atmospheric physics and geophysics 3. Generalized ensembles a. Tsallis statistics and non-extensive entropy b. Rényi statistics and generalized entropies c. Superstatistics and fluctuating intensive parameters d. Applications in complex systems and anomalous diffusion III. Statistical Mechanics of Interacting Systems A. Ideal gas 1. Classical ideal gas a. Maxwell-Boltzmann distribution b. Equation of state c. Specific heat and entropy d. Equipartition theorem 2. Quantum ideal gas a. Fermi-Dirac statistics i. Fermi-Dirac distribution ii. Fermi energy and chemical potential iii. Sommerfeld expansion iv. Applications in metals and semiconductors b. Bose-Einstein statistics i. Bose-Einstein distribution ii. Bose-Einstein condensation iii. Critical temperature and condensate fraction iv. Applications in superfluidity and lasers B. Real gases 1. Van der Waals equation a. Hard-core repulsion and attractive interactions b. Critical point and phase diagram c. Law of corresponding states d. Limitations and improvements 2. Virial expansion a. Pressure as a power series in density b. Virial coefficients and cluster integrals c. Relation to intermolecular potentials d. Convergence and radius of convergence C. Liquids 1. Radial distribution function a. Definition and physical interpretation b. Relation to structure factor c. Kirkwood-Buff theory d. Experimental measurements and simulations 2. Integral equations a. Ornstein-Zernike equation i. Direct and indirect correlations ii. Closure relations iii. Hypernetted chain approximation iv. Percus-Yevick approximation b. Percus-Yevick equation i. Hard-sphere fluid ii. Analytical solution for hard spheres iii. Extension to other potentials iv. Thermodynamic consistency 3. Perturbation theory a. Reference system and perturbation potential b. Zwanzig's free energy perturbation theory c. Barker-Henderson perturbation theory d. Weeks-Chandler-Andersen theory D. Solids 1. Lattice dynamics a. Harmonic approximation b. Dispersion relations and phonon modes c. Brillouin zones and reciprocal lattice d. Anharmonic effects and thermal expansion 2. Phonons a. Quantization of lattice vibrations b. Phonon dispersion and density of states c. Phonon heat capacity and Debye model d. Phonon-phonon interactions and thermal conductivity 3. Specific heat a. Einstein model b. Debye model c. Electronic specific heat and Fermi gas d. Magnetic specific heat and spin waves 4. Thermal expansion a. Grüneisen parameter b. Anharmonic potential and asymmetry c. Thermal stress and thermoelasticity d. Negative thermal expansion materials E. Magnetic systems 1. Ising model a. Spin-1/2 lattice and interaction Hamiltonian b. Exact solution in one dimension c. Mean-field approximation and Curie-Weiss law d. Critical behavior and universality 2. Heisenberg model a. Quantum spin operators and exchange interaction b. Ferromagnetic and antiferromagnetic ordering c. Spin waves and magnon excitations d. Quantum phase transitions and spin liquids 3. Mean-field theory a. Weiss molecular field approximation b. Landau theory of phase transitions c. Ginzburg-Landau theory and order parameter d. Limitations and beyond mean-field approaches 4. Spin waves a. Holstein-Primakoff transformation b. Dispersion relation and gap c. Magnon-magnon interactions d. Experimental detection and spin wave spectroscopy F. Quantum fluids 1. Superfluidity a. Landau criterion and critical velocity b. Vortices and quantized circulation c. Two-fluid model and second sound d. Josephson effect and phase coherence 2. Bose-Einstein condensation a. Macroscopic occupation of ground state b. Off-diagonal long-range order and coherence c. Gross-Pitaevskii equation and nonlinear dynamics d. Experimental realization in ultracold atoms 3. Fermi liquids a. Landau's Fermi liquid theory b. Quasiparticles and effective mass c. Landau parameters and stability conditions d. Fermi liquid instabilities and phase transitions G. Polymers and soft matter 1. Polymer chain models a. Freely jointed chain b. Worm-like chain c. Gaussian chain d. Excluded volume interactions 2. Flory-Huggins theory a. Lattice model and mixing entropy b. Interaction parameter and phase separation c. Upper and lower critical solution temperatures d. Applications in polymer blends and block copolymers 3. Reptation theory a. Tube model and topological constraints b. Primitive path and entanglements c. Diffusion and viscoelasticity d. Nonlinear rheology and shear thinning 4. Colloidal suspensions a. Brownian motion and diffusion b. Electrostatic interactions and DLVO theory c. Depletion forces and phase behavior d. Gelation and glass transition IV. Non-equilibrium Statistical Mechanics A. Linear response theory 1. Fluctuation-dissipation theorem a. Connection between fluctuations and response b. Kubo formula and susceptibility c. Einstein relation and mobility d. Generalized fluctuation-dissipation relations 2. Green-Kubo relations a. Transport coefficients and time correlation functions b. Shear viscosity and stress autocorrelation c. Thermal conductivity and heat current autocorrelation d. Electrical conductivity and current autocorrelation B. Transport phenomena 1. Diffusion a. Fick's laws and diffusion equation b. Einstein relation and Brownian motion c. Stokes-Einstein relation and hydrodynamic radius d. Anomalous diffusion and fractional dynamics 2. Viscosity a. Newton's law of viscosity and shear stress b. Navier-Stokes equations and hydrodynamics c. Shear thinning and shear thickening d. Viscoelastic fluids and complex rheology 3. Thermal conductivity a. Fourier's law and heat equation b. Phonon and electron contributions c. Wiedemann-Franz law and Lorenz number d. Thermal boundary resistance and Kapitza effect 4. Electrical conductivity a. Ohm's law and conductivity tensor b. Drude model and relaxation time approximation c. Boltzmann transport equation and scattering mechanisms d. Quantum corrections and weak localization C. Stochastic processes 1. Markov processes a. Memoryless property and transition probabilities b. Chapman-Kolmogorov equation c. Continuous-time Markov chains d. Hidden Markov models and inference 2. Master equation a. Probability evolution and gain-loss terms b. Detailed balance and equilibrium solution c. Birth-death processes and population dynamics d. Chemical reaction networks and enzyme kinetics 3. Fokker-Planck equation a. Drift and diffusion terms b. Kramers-Moyal expansion and truncation c. Stationary solution and potential landscape d. First-passage time problems and escape rates 4. Langevin equation a. Stochastic differential equation and noise terms b. Overdamped and underdamped limits c. Fluctuation-dissipation relation and temperature d. Numerical integration and stochastic algorithms D. Irreversible thermodynamics 1. Onsager reciprocal relations a. Linear response and phenomenological coefficients b. Symmetry of cross-coefficients c. Microscopic reversibility and detailed balance d. Thermoelectric and thermomagnetic effects 2. Prigogine's minimum entropy production principle a. Entropy production rate and dissipation b. Stationary states and variational principle c. Linear and nonlinear regimes d. Applications in chemical reactions and pattern formation E. Glassy systems and spin glasses 1. Structural glasses a. Supercooled liquids and glass transition b. Kauzmann paradox and entropy crisis c. Fragility and Angell plot d. Dynamical heterogeneity and cooperative rearrangements 2. Spin glasses a. Frustrated interactions and random couplings b. Edwards-Anderson model and order parameter c. Replica