Microstates and Phase Space
Ensembles
- Boltzmann entropy in the microcanonical ensemble
- Canonical ensemble
- Canonical ensemble from the microcanonical ensemble
- Generalized Gibbs ensemble (GGE)
- Grand canonical ensemble
- Isothermal–isobaric (NPT) ensemble
- Isothermal–isobaric partition function
- Microcanonical measure
- Microcanonical shell
- Negative heat capacity in the microcanonical ensemble
Partition Functions
Entropy and Free Energy
Averages and Fluctuations
- Ensemble Covariance of Two Observables
- Ensemble Variance of an Observable
- Ensemble average
- Fluctuation of an Observable
Correlation Functions
- Connected Correlation Function
- Connected correlations as cumulants
- Connected correlations as derivatives of expectations
- Correlation length
- Quantum correlation function
- Specific heat from energy fluctuations
- Susceptibility
- Two-Point Correlation Function
Constructions
- Bogoliubov variational bound (Peierls–Bogoliubov inequality)
- Chemical potential from entropy
- Cluster expansion (construction)
- Constructing free energies from partition functions
- Constructing observables from log partition functions
- Constructing the canonical partition function
- Constructing the grand canonical partition function
- Cumulant generating function
- DLR specification (Gibbsian conditional probabilities)
- Fluctuation formulas from log Z
- Infinite-volume Gibbs measure as a weak limit
- Legendre transform from Helmholtz free energy to grand potential
- Legendre transform from entropy to Helmholtz free energy
- Legendre transform from internal energy to Gibbs free energy
- Legendre transform from internal energy to enthalpy
- Mayer expansion (construction)
- Mean-field variational construction
- Pressure from entropy
- Quantum thermal correlation function (construction)
- Reduced density matrix (construction)
- Temperature from entropy
- Thermal state from entropy maximization
- Transfer-matrix construction in 1D
Theorems and Results
- Boltzmann H-theorem
- Breakdown of ensemble equivalence
- Canonical energy fluctuation identity
- Canonical energy identity
- Carnot theorem
- Chernoff bounding lemma
- Cluster expansion theorem (analyticity from convergent polymer expansions)
- Concavity of the von Neumann entropy
- Crooks fluctuation theorem
- Crooks fluctuation theorem
- Dobrushin uniqueness theorem
- Energy fluctuations and heat capacity in the canonical ensemble
- Ensemble inequivalence for long-range (nonadditive) systems
- Entropy maximization implies equality of chemical potential
- Entropy maximization implies equality of pressure
- Entropy maximization implies equality of temperature
- Equipartition theorem (classical canonical ensemble)
- Equivalence of microcanonical and canonical ensembles (thermodynamic limit)
- Euler relation for extensive thermodynamic potentials
- Fekete’s lemma
- Fluctuation–dissipation theorem (FDT)
- Fluctuation–response equivalence (canonical covariance identities)
- Gibbs
- Gibbs variational principle (canonical ensemble)
- Gibbs–KMS theorem (finite quantum systems)
- Golden–Thompson lemma
- Grand-canonical number fluctuation identity
- Griffiths monotonicity lemma
- High-temperature exponential decay of correlations
- Hydrodynamic Limit Theorem
- Integrating factor lemma
- Isentropic processes are reversible adiabatic processes
- Jarzynski equality (theorem)
- Jensen
- KMS imaginary-time periodicity
- Kosterlitz–Thouless theorem (2D XY transition via vortex unbinding)
- Laplace Principle
- Lee–Yang circle theorem
- Legendre transform swaps conjugate variables
- Liouville theorem (invariance of phase-space volume)
- Maxwell relations from equality of mixed partial derivatives
- Mermin–Wagner theorem (no continuous symmetry breaking in d ≤ 2)
- Mixed energy–number fluctuation identity (grand canonical)
- Monotonicity of quantum relative entropy
- Multiple Gibbs states and spontaneous symmetry breaking
- Phase coexistence implies nondifferentiability of the pressure
- Positivity of $C_V$ from thermodynamic stability
- Positivity of isothermal compressibility from stability
- Pressure identity in the canonical ensemble
- Saddle-Point (Laplace) Method in One Dimension
- Stability implies concavity/convexity of thermodynamic potentials
- Standard derivative identities from Maxwell relations
- Stirling
- Susceptibility equals variance of magnetization
- Uniqueness implies analyticity (no phase transition in the uniqueness region)
- Vanishing of relative fluctuations in the thermodynamic limit
Other
- Boltzmann H-functional
- Boltzmann equation (kinetic theory)
- Bose–Einstein condensation (ideal Bose gas)
- Carnot efficiency formula
- Central Limit Theorem in the High-Temperature Gibbs Regime
- Classical harmonic oscillator in the canonical ensemble
- Continuous symmetries in spin systems
- Convergence of the Virial Expansion
- Convergence of the cluster expansion
- Criterion for an exact differential
- Critical exponents
- Critical points in phase diagrams
- Curie–Weiss model
- Debye model
- Degenerate Fermi gas (ideal Fermi gas at low temperature)
- Detailed balance
- Donsker–Varadhan Large Deviation Principle
- Einstein solid
- Equilibrium as a large-deviation minimizer
- Example: classical monatomic ideal gas
- Example: two-level paramagnet (noninteracting spins)
- Example: van der Waals gas
- Exponential decay of correlations in the uniqueness regime
- FKG inequality
- Free-energy difference from nonequilibrium work
- GKS inequalities (Griffiths–Kelly–Sherman)
- Golden–Thompson inequality in statistical mechanics
- Green–Kubo relations
- Griffiths inequalities
- Jarzynski equality
- KMS implies Gibbs in finite quantum systems
- Kosterlitz–Thouless (BKT) transition
- Kubo formula (linear response)
- Landau free-energy functional
- Lee–Yang Theorem and Extensions
- Legendre duality between free energy and entropy
- Markov chain (discrete time)
- Markov semigroup (continuous time)
- Master equation
- Maximum entropy with constraints (Gibbs/exponential family)
- Mayer cluster integrals
- Mean-field approximation
- Metastable state
- Nonequilibrium steady state
- Nonequilibrium work distribution
- Ornstein–Zernike form
- Peierls argument
- Peierls–Bogoliubov inequality
- Pirogov–Sinai theory (low-temperature phase diagrams with multiple ground states)
- Pressure / log-partition density
- Quantum harmonic oscillator in thermal equilibrium
- RG fixed point
- Rate function for magnetization
- Renormalization-group (RG) transformations
- Sackur–Tetrode entropy (monatomic ideal gas)
- Scaling relations among critical exponents
- Structure factor
- Surface tension and interface free energy
- TFAE: Characterizations of Gibbs Measures
- TFAE: Formulations of the Second Law
- TFAE: Legendre Duality Between Entropy and Free Energy
- TFAE: quantum equilibrium at inverse temperature β
- Topological defect: vortex
- Unattainability formulation of the third law
- Universality class
- Virial coefficients