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A superconducting persistent current is an electric current circulating in a closed path with superconductivity after the external driving electromotive force has ceased. Its associated magnetic field makes the circulation observable without inserting a measuring instrument into the path. This article concerns macroscopic circulation in superconductivity. A finite experiment establishes persistence over its observation window with a stated measurement error, rather than directly observing an infinite lifetime.

The ideal circuit connection is flux conservation in a superconducting loop. In a fixed loop with self-inductance $L$, a single circulating electric current $I$, and externally imposed linked magnetic flux ${\Phi }_{\mathrm{ext}}$, the lumped circuit approximation givesWhen the electrical resistance $R$ vanishes, integrating gives $LI+{\Phi }_{\mathrm{ext}}=\text{constant}$. Removing the external linked magnetic flux therefore changes the circulating electric current; holding it fixed afterwards preserves that electric current. This is a circuit approximation with a specified self-inductance, not a derivation of the microscopic state of superconductivity. For thin paths where the carrier contribution matters, fluxoid and flux quantization in a superconductor supplies the more careful description.

For constant $L>0$, constant effective electrical resistance $R\ge 0$, and no changing external linked magnetic flux, the RL circuit instead predictsThis exponential decay follows by separating $dI/I=-(R/L)dt$. A long relaxation time tests extremely small electrical resistance without trying to resolve an equally tiny steady electric potential difference. The stored magnetic energy is $U=L{I}^2/2$, so the same model gives $dU/dt=-R{I}^2$, exactly the Joule heating loss. The finite energy stored in the loop supplies the magnetic readout; the circulating electric current is not a continuously powered source of useful work.

A coarse-grained description in which quadratic inflaton oscillations have zero mean pressure and lose energy at constant effective decay rate $\Gamma$ to a relativistic component. Adding the equations preserves total energy accounting with the appropriate expansion work. Efficient transfer occurs around $H\sim \Gamma$, but producing particles and achieving thermal equilibrium are different steps. The potential alone does not specify the decay rate or thermalization interactions.

For a cubic graph with $E$ external legs, $3V=2I+E$ and $L=I-V+1$. Its superficial degree of divergence is therefore $6L-2I=6-2E$. Divergences in vacuum, one-, two-, and three-point functions can be absorbed into vacuum energy, a linear term, mass, wave-function renormalization, and cubic-coupling counterterms. Higher-point proper diagrams are superficially convergent after subtraction of divergent subgraphs. This establishes perturbative renormalizability, without asserting a nonperturbative positive measure for the unstable real cubic potential.

Peter Debye explained the heat capacity of a solid by counting a spectrum of quantized elastic vibrations. In the Debye model, a cutoff ensures the correct number of modes, while the low-frequency acoustic spectrum gives heat capacity proportional to $T^3$ at sufficiently low temperature. At high temperature, the same construction approaches the classical Dulong-Petit law. Connecting elastic vibrations with quantum occupation explains why a solid stores much less thermal energy at low temperature than a classical mode count predicts.^[ref]

A physicist.^[ref]

Peter Hänggi coauthored a formulation of fluctuating work in a driven quantum system. Initial and final energy measurements define a distribution of energy differences, rather than one fixed work operator applicable to the whole process. The resulting characteristic function connects quantum dynamics with the Jarzynski equality. This makes the measurement protocol part of a precise statement about quantum work fluctuations.^[ref]

For the physical free energythe Boltzmann factor is $e^{-F_{\theta }/(k_{B}T)}$. At $T>0$, the Gaussian thermal phase fluctuations therefore obeyThe temperature factor is necessary when ${\rho }_s$ is the energy stiffness defined by the parent. Defining a dimensionless stiffness $K={\rho }_s/(k_{B}T)$ instead replaces $k_{B}T/{\rho }_s$ by $1/K$.

The difference cancels the spatially uniform phase mode. At any fixed separation, $1-\cos (\mathbf{k}\cdot \mathbf{x})$ removes the small-$k$ singularity. What grows in low dimensions is the variance as the separation tends to infinity.

In one dimension, the integral equals $|x|/2$, so the variance is $(k_{B}T/{\rho }_s)|x|$. In two dimensions, with microscopic cutoff $a$ and $r\gg a$, it grows asFor the Gaussian field, the order parameter correlation isIt decays exponentially in one dimension and algebraically in two. Thus nonzero-temperature Gaussian fluctuations do not leave conventional long-range order in these dimensions. This is the long-wavelength mechanism related to the Mermin-Wagner theorem; vortices and interactions require the fuller theory of the Berezinskii–Kosterlitz–Thouless transition in the two-dimensional case.

Take a positive comparison elastic stiffness tensor $C^0$, excess energies $F_r(e)=W_r(e)-e:C^{0}e/2$, and a phasewise constant elastic polarization. The lower conjugate inequality majorizes the local energy by a quadratic-plus-linear function. Minimizing that function over compatible strains gives the displayed bound, where $A_{rs}=\int \int ({\chi }_r-c_r)\Gamma ({\chi }_s-c_s)$. Stationarity in ${\tau }^r$ gives $c_r{e}^r+{\sum }_s{A}_{rs}{\tau }^s=c_{r}E$, with $e^r=D{F}_{r\ast }({\tau }^r)$. Without this stationarity, dual strains must not be identified with the comparison solution's phase means.

Two equally modulated charge sheets in identical full-space electrolyte have charging energy per area ${\alpha }^2[1+e^{-pd}\cos \theta ]/(4\epsilon p)$, where $p=\sqrt{k^2+{\kappa }_D^2}$. Its phase-dependent interaction favors a half-wavelength offset, with opposite charge patches facing each other. For an electrolyte gap with negligible exterior displacement response, the energy is instead ${\alpha }^2[\coth (pd)+\cos \theta /\sinh (pd)]/(2\epsilon p)$, with the same minimizing phase. Absolute energies require specified exterior electrostatic interface boundary conditions.

The dimensionless occupancy scale $n{\lambda }_T^3$, with number density $n$ and thermal de Broglie wavelength ${\lambda }_T=h/\sqrt{2\pi m{k}_{B}T}$. Increasing it approaches the regime where quantum statistics matters. This use of degeneracy must not be confused with degeneracy of an individual energy eigenspace.

For the static energythe kink joining the vacua $-c$ and $c$ isIt saturates the Bogomolny bound $E\ge 4c^3/3$ by satisfying the first-order equation ${\varphi }^{\prime }=c^2-{\varphi }^2$.

One linear Bogoliubov branch $E_k=\hslash c_{s}k$ in three dimensions gives thermal energy ${\pi }^{2}V(k_{B}T{)}^4/(30{\hslash }^3{c}_s^3)$ and the displayed cubic heat capacity. There is one branch, rather than the three polarizations in a simple crystal's Debye model. The regime requires $\hslash c_s/L\ll k_{B}T\ll \mu$ for a box: the lower inequality permits a continuum integral, and the upper one selects phonons. Fixed finite boxes ultimately have discrete-level behaviour.

For electron energy transfer $\Delta$ and phonon energy ${\Omega }_q=\hslash {\omega }_q$, the symmetrized pair amplitude is $V_{\mathrm{eff}}=2|M_q{|}^2{\Omega }_q/({\Delta }^2-{\Omega }_q^2)$. It is attractive for $|\Delta |<{\Omega }_q$. This interaction is retarded, and screened Coulomb repulsion must also be considered.

The allowed phonon energies, including their wavevector, branch and phonon polarization labels when resolved. A phonon dispersion relation records the branch-resolved spectrum, whereas the phonon density of states sums over those labels.

To isolate conversion losses, suppose a converter receives 100 blue photons at ${\lambda }_b=450\mathrm{nm}$, absorbs a fraction $a=0.60$, and has fluorescence quantum yield $\Phi =0.90$: 0.90 converted photons per absorbed blue photon. Stipulate that every converted photon has ${\lambda }_c=560\mathrm{nm}$, that unabsorbed blue is transmitted, and that both outputs escape without further losses. The output is then 40 blue and 54 converted photons. The total photon yield is $0.94$, while the retained optical energy fraction isThe lower energy of each converted photon matters even if $\Phi =1$. Combining this stipulated converter with the preceding chosen light-emitting diode gives $P_{\mathrm{after}}=16.5430\mathrm{mW}$ and an overall optical/electrical power ratio $0.258485$. These chosen narrow lines and photon counts illustrate conversion accounting; they do not establish a white appearance, color quality or luminous efficacy. Those require the full spectrum and a model of human visual sensitivity.

Escape following chemical reactions that produce atoms or molecules with energies above the local gravitational escape threshold. The production altitude, collision environment and reaction energy distribution matter.

Photocurrent is the electric current attributable to illumination. Here $I_{\mathrm{ph}}\ge 0$ denotes its magnitude after background subtraction. For a photodiode without internal multiplication, let $P$ be incident radiant flux concentrated at wavelength $\lambda$, and let $\eta$ be external quantum efficiency, including losses before collection. An incident photon has energy $E_{\gamma }=hc/\lambda$, with Planck constant $h$ and speed of light $c$. Dividing energy per second by energy per photon gives the incident photon rate ${\Phi }_{\gamma }$. Multiplying by $\eta$ gives the mean collected pair rate ${\dot{N}}_{\mathrm{pair}}$; multiplying by elementary charge $q$ givesThe factor is $q$: each collected pair replenishes an electron at one contact and removes an electron at the other. A cross-section of the external wire therefore counts one transferred electron per completed pair, consistent with charge conservation. The electron and electron hole are two internal charge carriers, not two independent transfers through that wire.

Choose an illustrative photodiode model with $\lambda =650\mathrm{nm}$, $\eta =0.75$ and $P=1.00\mu \mathrm{W}$. Using the exact SI values $h=6.62607015\times 10^{-34}\mathrm{J}\mathrm{s}$, $c=299792458\mathrm{m}{\mathrm{s}}^{-1}$ and $q=1.602176634\times 10^{-19}\mathrm{C}$ givesA mean fraction $0.75$ does not require each fourth photon to be lost in a repeating pattern. It specifies a mean conversion rate. Nor does $0.75$ mean $0.75\mathrm{A}{\mathrm{W}}^{-1}$: quantum efficiency is dimensionless, whereas spectral responsivity compares electric current with power.

At fixed radiant flux, longer-wavelength photons have less energy, so more arrive per second. This explains the rising straight line when quantum efficiency is held constant. A real photodiode's quantum efficiency varies with wavelength; the plotted assumption cannot be extended past its band gap absorption cutoff. At fixed incident photon rate instead, $I_{\mathrm{ph}}=\eta q{\Phi }_{\gamma }$ has no additional factor of $\lambda$.

Photodetachment removes an electron from an anion using a photon. For chlorine, the reaction is ${\mathrm{Cl}}^{-}(g)+\gamma ⟶\mathrm{Cl}(g)+e^{-}$. Measuring the minimum photon energy for the specified final quantum state gives a photodetachment threshold and hence an electron affinity.

Photodissociation separates a molecule into fragments after photon absorption. For an isolated parent with a single absorbed photon, conservation of energy givesInternal excitation energies here are relative to the parent and product ground states used to define $D_0$; $E_{\mathrm{recoil}}$ is the total product translational kinetic energy in the parent's centre of mass frame. Measuring one product's kinetic energy is insufficient by itself. For two fragments of masses $m_A,m_B$, conservation of momentum gives $E_{\mathrm{recoil}}=E_A(m_A+m_B)/m_B$ when $A$ is detected. Unknown product excitation can conceal part of the available energy.

A photon transfers its energy to a bound electron, ejecting it if energetically allowed. Subsequent atomic relaxation can emit characteristic radiation or additional electrons. Its material and energy dependence differs from Compton scattering, which matters when converting computed tomography attenuation to the higher photon energy of positron emission tomography.

The photoelectric effect is the release of electrons from matter following photon absorption. Here the focus is emission from a metal surface into vacuum: a photon supplies energy, the work function pays the escape cost, and the remainder can become the emitted electron's kinetic energy. A photocurrent counts collected electric charge; a stopping potential probes the upper kinetic energy limit. These are different measurements.

In 1905, Albert Einstein proposed that an individual light quantum could transfer its whole energy to one electron. His §8 allowed losses before escape, distinguished the fastest emitted electrons from the rest, and predicted a straight stopping potential versus frequency relation with a material-independent slope. He separately predicted that changing light intensity changes the number emitted without changing their velocity distribution under otherwise identical conditions. He also discussed incomplete transfer and limits to the account.^[ref]

The useful modern starting model is a clean, homogeneous metal with independent single-photon events, negligible initial thermal broadening, negligible lowering of the surface barrier by an applied electric field, and negligible electric fields from the emitted electrons themselves. Specify the surface and its work function; it is not enough to name the bulk material. The Einstein photoelectric equation describes the upper edge of the emission energy distribution. It neither predicts that every absorbed photon liberates an electron nor supplies the whole photocurrent curve. Its appeal is that a complicated surface process still exposes the universal Planck constant through a simple slope.