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An electrochemical capacitor storing charge through high-area electric double layers, pseudocapacitance, or a combination. Its device energy depends on usable voltage, inactive mass and rate capability as well as electrode capacitance.

In this fully gapped isotropic BCS quasiparticle spectrum, $\Delta$ is the minimum single-quasiparticle excitation energy. The interval between $-\Delta$ and $+\Delta$ has width $2\Delta$. A process that creates two quasiparticles from the BCS ground state has an energetic threshold $2\Delta$, subject also to its transition-selection rules.^[ref]

For example, two independently created excitations with ${\xi }_1={\xi }_2=0$ cost $E_1+E_2=2\Delta$. With the chosen cutoff example this is $1.46574\mathrm{meV}$, while adding one excitation costs at least $0.732871\mathrm{meV}$. This factor of two matters when comparing photon absorption with the Giaever superconducting tunnelling experiment. Quantum tunnelling from a normal metal can add one excitation to a sample exhibiting superconductivity; with two gapped electrodes, their excitation costs add. A minimum excitation cost is an energetic condition; the measured onset can additionally depend on temperature and on the available transitions.

A transition of part of a current-carrying superconducting system into a resistive state. Joule heating can spread the transition, and the stored magnetic energy must be handled safely by the system's protection design. Zero steady winding resistance does not remove the finite magnetic energy or the refrigeration requirement.

Superfluidity is a quantum-fluid response with phase stiffness and flow without viscous dissipation in an appropriate low-speed regime. A superfluid supports a phase-gradient current; defects and excitations can limit that flow. Bose-Einstein condensation concerns macroscopic one-particle occupation and is a distinct criterion: the Gaussian phase stiffness theory can have divergent absolute-phase fluctuations while retaining a cost for gradients.

Why does cooling a liquid make it flow more freely through a tiny gap? Helium-4 supplies an experimental entrance to this question: the helium slit-flow experiment compares the motion of liquid levels on opposite sides of the transition into superfluidity. A flow measurement probes resistance to motion; a measurement of Bose-Einstein condensation probes occupation of a quantum state. The two-fluid model of superfluidity and the Landau criterion connect these questions to entropy transport and to the energy required to create an excitation.

Supergravity is a theory in which supersymmetry is a local symmetry, with transformation parameters depending on spacetime position. Making supersymmetry local introduces a spin-$3/2$ gauge field, the gravitino, alongside the graviton describing gravity.

The reason gravity appears is structural: two supersymmetry transformations include a translation, and local translations require a dynamical spacetime geometry. The number of supercharges and the spacetime dimension determine the allowed multiplets and interactions. Eleven-dimensional supergravity and type IIA supergravity are examples with different field content and dimensional settings.

Supergravity often serves as a low-energy effective field theory of a superstring or M-theory description. That role does not make a generic finite-derivative supergravity action a complete quantum theory at arbitrarily high energy. Its validity depends on the scales at which neglected higher-derivative and additional-state effects become important.

Super-Kamiokande used $535$ days ($33.0$ kiloton-years) and $22.5$ fiducial kilotons of water. Inner photomultiplier times and charges reconstructed interactions; the outer anticoincidence detector vetoed entering particles and marked exits. Contained single-ring events used likelihood classification: electron particle showers blur rings; muons give sharper edges. The $1.33\mathrm{GeV}$ visible-energy sample division is not a universal detection threshold. Simulation estimated classification efficiency above about $98\%$, distinct from total detection efficiency; stopping muons and decay electrons checked classification.^[ref]

Downward neutrinos travel about $15\mathrm{km}$; upward ones about $13000\mathrm{km}$. The multi-GeV muon-like up/down asymmetry was $-0.296\pm 0.048\pm 0.010$; electron-like events lacked a comparable effect. Calibration constrained directional bias. This was disappearance evidence; tau neutrino appearance was untagged and sterile conversion remained possible.^[ref]

Writing $U$ and $D$ for the selected upward and downward populations, the asymmetry is $A=(U-D)/(U+D)$: a negative value means fewer upward events. An absolute source-normalization error alone rescales both populations and leaves this ratio unchanged. A direction-dependent response error need not cancel.

The Super Proton-Antiproton Synchrotron used the converted SPS ring for counterrotating proton and antiproton beams. In the discovery running, two $270\mathrm{GeV}$ beams provided $\sqrt{s}=540\mathrm{GeV}$, unlike a stationary-target collision at the same incident energy. Antiproton production, accumulation and stochastic cooling supplied the collisions used in the W and Z discovery experiments.^[ref]

Why can opposite electric charges share the bending magnets? Reversing both $q$ and velocity $\mathbf{v}$ leaves the Lorentz force $q\mathbf{v}\times \mathbf{B}$ unchanged at a given point. Both circulating species can therefore bend toward the same ring center in the same magnetic field, with equal magnetic rigidity magnitudes. This argument explains the bending geometry; actual injection, focusing, timed acceleration and collisions still require a complete particle accelerator system.

In a canonical Wess–Zumino model, complete the static transverse energy into $|{\varphi }^{\prime }-e^{i\alpha }\overline{W^{\prime }}{|}^2+2\mathrm{Re}(e^{-i\alpha }dW/dz)$. Choosing the phase of $\Delta W$ gives the tension bound. A solution of the displayed first-order equation attains it and preserves two of four real supercharges. For $W={\mu }^2\Phi /g-g{\Phi }^3/3$, $\varphi (z)=(\mu /g)\tanh (\mu z)$ interpolates between the two vacua and has tension $8{\mu }^3/(3g^2)$.

The candidate zero modes are $e^{-{\int }^{x}w}$ in the minus sector and $e^{+{\int }^{x}w}$ in the plus sector; this follows from the positive squared-norm expressions for their energy. Their membership in the Hilbert space determines whether supersymmetry is unbroken. For $w=x$ only the first is normalizable. For $w=x^2+a$, $a>0$, neither is normalizable, while both partner potentials grow quartically and have discrete spectra. Thus their strictly positive lowest energy is paired and supersymmetry is broken, with zero Witten index.

A vacuum of a globally supersymmetric gauge theory preserves supersymmetry exactly when every auxiliary F-field and D-field can vanish simultaneously. Its vacuum energy is then zero because the scalar potential is a sum of nonnegative squares.

For a finite physical supermultiplet at fixed positive energy, fermion parity anticommutes with each odd supercharge. Trace cyclicity makes $\mathrm{Tr}[(-1{)}^F\{Q,Q^{†}\}]=0$. Summing the Super-Poincaré algebra diagonal entries replaces the anticommutator by $4E$, giving equal bosonic and fermionic state counts. A zero-energy vacuum is an exception to the division by $E$, so it need not have a partner. The argument requires an invariant fixed-momentum representation, which is the step lost under explicit supersymmetry breaking.

A supply rate specifies the instantaneous quantity whose time integral is compared with a change of storage function in dissipativity. With control input $u$, plant output $y$ and state $x$, a chosen function $w(u,y)$ enters the balanceThe supply rate can be negative: it then represents net withdrawal in the chosen accounting. Its physical interpretation depends on the variables and sign convention, rather than on the word supply alone.^[ref]

For a mechanical terminal with applied force $F$ and inward-convention velocity $v$, choose $w=Fv$, the entering power. For an electrical terminal with electric current entering its positive-potential terminal, the corresponding choice is $w=Vi$, with $V$ an electric potential difference. Both have units of watts when $S$ is energy. Reversing one terminal sign without changing the other reverses the apparent power flow.

An original capacitor example illustrates the sign. Let $C=2\mathrm{F}$, terminal electric potential difference $V=3\mathrm{V}$, and charging electric current $i=1\mathrm{A}$. Its stored electrostatic energy is $S=C{V}^2/2=9\mathrm{J}$, and at that instant $\dot{S}=Vi=3\mathrm{W}$. With the same potential and $i=-1\mathrm{A}$, the supply rate is $-3\mathrm{W}$ and storage decreases. This is consistent with nonnegative stored energy; negative supply does not mean negative storage.

A different supply rate can express a signal-performance requirement:Nonnegative final storage and zero initial storage then bound accumulated output-square by ${\gamma }^2$ times accumulated input-square. These squares need not be physical power. Scaling or changing the units of the signals requires corresponding changes in the coefficients. A general real quadratic choice can be written as $w=y^{T}Qy+2y^{T}Lu+u^{T}Ru$, with symmetric $Q,R$; the selected quadratic form states which input–output relation is being tested.

For radiatively efficient release of Newtonian surface binding energy, $L\sim GM\dot{M}/R_{\ast }$ and $L_E=4\pi GMc/\kappa$ imply the displayed rate. Its efficiency is $\eta \sim GM/(R_{\ast }{c}^2)$, so at fixed opacity the maximum rate scales with stellar radius. This differs from defining an Eddington supply rate with an arbitrarily fixed reference efficiency. Geometry, advection and mass loss can modify the simple limit.

The 1935 primary paper The Spin-Isomerization of Hydrogen at Paramagnetic and Diamagnetic Surfaces, pp. 1251–1252, describes evacuated catalyst samples cooled before admitting molecular hydrogen. Gas samples were analyzed in a glass cell in liquid air: a heated platinum wire and Wheatstone bridge converted changes in thermal conductivity into electrical resistance readings, calibrated against normal hydrogen.^[ref]

Its comparison of oxide surfaces supplied a control: specimens exhibiting strong paramagnetism converted the gas much faster than the comparison exhibiting diamagnetism despite all adsorbing molecular hydrogen. Contact time, pressure, and temperature were recorded. This supports a role for surface magnetic properties beyond adsorption alone; it does not identify every microscopic surface site or directly measure nuclear-spin orientations.^[ref]

The inference chain is preparation, contact with a specified surface, then a calibrated bulk transport readout of composition. It is distinct from a direct calorimetry measurement of heat capacity: transport and heat capacity can both depend on the spin populations, but they are different measured quantities. Changing the catalyst's own surface structure, introducing magnetic impurities, or changing the detector's gas pressure can confound an uncalibrated comparison. The free-rotor partition functions predict equilibrium populations and stored energy, not the conversion rate on a particular surface.

An rf field restricted to a metal's skin layer can transfer energy coherently to electrons whose orbits return to that layer. With magnetic field parallel to the surface, resonance can occur at $\omega =n{\omega }_c$, producing harmonics at $B_n=m_c\omega /(ne)$. Surface loss or cavity impedance replaces simple bulk transmission as the measured signal.

For a liquid interface with constant surface tension $\gamma$, the mechanical surface energy is the energy associated with its surface area $A$:The additive constant $C$ does not affect the work needed to change surface area. For several interfaces with the same surface tension, $A$ means their total surface area; the two faces of a thin soap film must both be counted. ^[ref]

With assumed $\gamma =0.072\mathrm{N}{\mathrm{m}}^{-1}$, creating $1\mathrm{c}{\mathrm{m}}^2=10^{-4}{\mathrm{m}}^2$ of one interface requires $7.2\mu \mathrm{J}$ of reversible work. In the sliding-rim example illustrated under surface tension, moving a $20\mathrm{mm}$ rim outwards by $1\mathrm{mm}$ creates $4.0\times 10^{-5}{\mathrm{m}}^2$ across the two faces and requires $2.88\mu \mathrm{J}$. Independently, multiplying its $2.88\mathrm{mN}$ holding force by the displacement gives the same work.

This makes the preference for a round isolated droplet precise. At fixed volume $V$, the three-dimensional isoperimetric inequality giveswith equality for a sphere. Thus a sphere minimizes surface energy among sufficiently regular closed shapes enclosing the same volume, when surface tension is uniform and other energy contributions are negligible. This is an application of the isoperimetric inequality, rather than a proof of it. Walls, wetting, and gravity can change the shape by adding constraints or other energy contributions. ^[ref]

The differential rule $d{E}_{\mathrm{s}}=\gamma dA$ integrates directly to $E_{\mathrm{s}}=\gamma A+C$ because surface tension is constant. At fixed liquid volume, decreasing surface area can drive viscous fluid flow; the released surface energy can supply viscous dissipation. This mechanical bookkeeping is useful for predicting which deformations can release energy. If surface tension depends on temperature or composition, distinguish this mechanical area-work relation from the complete interfacial thermodynamic free energy and internal energy.

A linear deep-water gravity wave of surface-displacement amplitude $a$ has equal mean kinetic energy and potential energy, and total mean energy per horizontal area $\mathcal{E}={\rho }_{w}g{a}^2/2$. Its energy transport speed is the group velocity. This amplitude is half the crest-to-trough height.

Surface tension $\gamma$ is the tensile force per unit length across a line drawn in a liquid interface. For a clean interface with isotropy, it has the same value in every tangential direction: each side pulls the other along the surface, perpendicular to the line. A straight segment of length $\ell$ therefore transmits a force of magnitude $\gamma \ell$ from one interface. Here we assume constant $\gamma >0$, with fixed temperature and composition. The same coefficient measures the reversible work required to create surface area:The equality of units follows from $1\mathrm{J}=1\mathrm{N}\mathrm{m}$; the equality of the mechanical coefficients follows from the moving-rim argument below. ^{[ref] [ref]}

For a concrete example, take an ideal flat soap film spanning a rectangular wire frame with a straight sliding rim of length $\ell$. Pull the rim outwards in the plane of the film, perpendicular to the rim. Each of the film's two liquid–gas interfaces pulls the rim back towards the film with force $\gamma \ell$, so the outward holding force is $F=2\gamma \ell$. The two faces are two interfaces, each with its own surface tension. With the assumed toy value $\gamma =0.072\mathrm{N}{\mathrm{m}}^{-1}$ and $\ell =20\mathrm{mm}=0.020\mathrm{m}$, the required force is $2.88\mathrm{mN}$; one interface would contribute $1.44\mathrm{mN}$. This value of $\gamma$ is a stipulated model parameter, rather than a measured value for a particular soap solution.

Why does an interface resist being enlarged? Its molecules have a different environment from those in the bulk liquid; changing that environment has an energy cost. In the constant-surface tension model, reducing surface area lowers surface energy. This connects an easily measured rim force to the shape of a droplet, the Young–Laplace equation for its pressure, and the restoring mechanism of capillary waves. It is useful because one coefficient links forces, energy and geometry. ^[ref]

The moving-rim argument also explains the factor of two without memorizing a special formula. An outward displacement $dx$ creates surface area $\ell dx$ on each face, hence $d{A}_{\mathrm{total}}=2\ell dx$. Slow pulling without viscous dissipation requires work $Fdx=\gamma d{A}_{\mathrm{total}}$, giving $F=2\gamma \ell$. For one interface, the same argument gives $F=\gamma \ell$. A surface tension gradient instead produces Marangoni stress; the constant-$\gamma$ assumption excludes that additional effect.

A surface gravity wave breaks when its smooth free-surface motion becomes unstable and overturns or forms a strongly dissipative front. In the nondispersive shallow-water approximation, intersecting compressive characteristic curves signal failure of a smooth simple wave; a hydraulic bore replaces it at the integral-model level. Dispersion can instead produce an oscillatory bore, while physical breaking involves turbulence and energy loss beyond the inviscid equations.

If a nonconserved surfactant polarization has local energy $\nu |\mathbf{p}{|}^2/2+\lambda \mathbf{p}\cdot \nabla \varphi$, eliminating $\mathbf{p}$ changes the composition-gradient coefficient from $\kappa$ to $\tilde{\kappa }=\kappa -{\lambda }^2/\nu$. This expresses the surfactant's reduction of interfacial energy.