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The action $(2\pi {)}^{-1}\oint pdq$ is the area enclosed by a periodic phase-plane orbit divided by $2\pi$.

For a finite duct segment of phase length $\lambda$ joined to a semi-infinite medium, continuity of pressure and velocity gives the outgoing potential amplitudeAt half-integer $\lambda /\pi$, a large impedance mismatch can strongly enhance transmitted flux for a prescribed piston displacement.

A Beltrami flow has velocity parallel to vorticity.

A spurion is a nondynamical background field assigned transformation properties so that couplings or symmetry-breaking parameters can be treated as if they arose from symmetry-covariant fields.

Supergravity is a theory with local supersymmetry and necessarily contains gravity.

Soft supersymmetry breaking adds operators of positive mass dimension that break supersymmetry without reintroducing new ultraviolet quadratic divergences.

Four-dimensional $\mathcal{N}=1$ supersymmetry has one Weyl-spinor supercharge and four real supercharges.

Gauge/gravity duality relates a quantum field theory without dynamical gravity to a gravitational or string theory in one additional emergent dimension.

Bosonic string theory quantizes a relativistic string using only bosonic worldsheet fields. Lorentz-invariant quantization requires 26-dimensional target spacetime and contains a tachyonic ground state.

A superfluid exhibits superfluidity, with phase stiffness and a phase-gradient current. In an appropriate ordered or quasi-ordered regime it supports low-speed flow without viscous dissipation, and its long-wavelength Goldstone boson has a sound-like dispersion. True thermodynamic phase selection and Bose-Einstein condensation need separate infrared conditions.

Quantum magnetism studies collective phases and excitations of interacting quantum spins.

A phase-space distribution function gives the occupation density of particles as a function of position and momentum. For a spatially homogeneous gas it depends only on momentum.

A critical phenomenon is scale-invariant behavior near a continuous phase transition, described by universal critical exponents.

Up to volume-independent terms, $F=-N{k}_{B}T\log V$, so $-\partial F/\partial V=N{k}_{B}T/V$.

An isothermal noninteracting gas in potential $mgz$ has an exponential vertical density and scale height $k_{B}T/(mg)$.

The canonical ensemble is the probability distribution for a system in thermal equilibrium with a heat bath at fixed temperature $T$, with fixed particle number and external parameters. It allows energy exchange, so individual energy measurements fluctuate. With Boltzmann constant $k_B$ and inverse temperature $\beta =1/(k_{B}T)$,The heat bath is idealized as large enough that exchanges barely change its temperature; coupling must permit equilibration while contributing negligible interaction energy. A finite heat bath approximates this description only when its temperature change is negligible over the relevant exchanges. Strong coupling or a small heat bath can require a different description.

For one two-level system at positive temperature, the Boltzmann weights are $1,e^{-\beta \Delta }$, henceAt $\beta \Delta =\ln 3$, $Z_1=4/3$ and $p=1/4$. This is a probabilistic thermal mixed state. For quantum levels, its density matrix is $\rho =(1-p)|0⟩⟨0|+p|1⟩⟨1|$. The pure state $\sqrt{1-p}|0⟩+\sqrt{p}|1⟩$ has the same energy measurement probabilities, but also has quantum coherence between the energy eigenstates. Matching those probabilities does not make the states equivalent.

For an additive system with short-range interactions and a regular extensive entropy, microcanonical, canonical, and grand canonical predictions agree for bulk observables in the thermodynamic limit because relative fluctuations vanish. Equivalence can fail for finite systems, long-range interactions, or at singular phase-coexistence points.

At fixed temperature, volume, and chemical potential, the grand canonical ensemble permits both energy and particle number to fluctuate and weights a state by $e^{-\beta (E-\mu N)}$.

The microcanonical ensemble assigns equal probability to accessible states in a narrow energy shell at fixed energy, volume, and particle number.

For a discrete probability distribution over microstates, the Gibbs entropy is