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At high energy with $x=-t/s\in (0,1)$ fixed, reflection and Stirling formulas for the gamma function give the displayed exponential envelope, away from real-axis resonance poles or after suitable analytic continuation. It is softer than the power-law fixed-angle behavior of typical point-particle interactions and exposes the finite string length. This limit differs from the fixed-$t$ Regge limit and does not by itself prove finiteness of every loop amplitude.

A positive-energy standard-representation Dirac spinor has upper component $\xi$ and lower component $(\mathbf{\sigma }\cdot \mathbf{p})\xi /(E+m)$. To be an eigenstate of $\mathrm{diag}({\sigma }_x,{\sigma }_x)/2$, both components must have the same ${\sigma }_x$ eigenvalue. With $\xi ={\chi }_{\pm }$, the commutator obstruction $2i(p_y{\sigma }_z-p_z{\sigma }_y){\chi }_{\pm }$ has squared norm $4(p_y^2+p_z^2)$. It vanishes precisely for momentum parallel to x, including zero. Exact transverse-spin eigenstates therefore fail at nonzero transverse momentum; the obstruction is not a universal failure at every relativistic energy.

In a slowly deepening mixed layer driven by a rotating boundary, a phenomenological balance between energy input, turbulent loss and acceleration of entrained fluid can keep total kinetic energy approximately fixed. For a tank of radius $R$ this gives $u^{2}h=A{\Omega }^2{R}^3$, so velocity decreases as $h^{-1/2}$. The closure does not make the speed constant and is an additional physical assumption rather than a direct consequence of mass conservation.

For a weakly depleted homogeneous Bose gas with repulsive contact interaction, set $g=uN/V$ and ${ϵ}_k={\hslash }^2{k}^2/(2m)$. A zero-mode coherent state times a two-mode squeezed vacuum for each opposite-momentum pair has mean particle number $N=N_0+{\sum }_{k\ne 0}{\sinh }^2{\theta }_k$. Substitution of $N_0=N-\sum {\sinh }^2{\theta }_k$ into the leading condensate interaction energy, followed by a consistent quadratic truncation, gives the displayed variational energy. Omitting this substitution loses the subtraction embodied in the number-conserving Bogoliubov quadratic Hamiltonian.

For each mode set $A_k={ϵ}_k+g$ and $E_k=\sqrt{A_k^2-g^2}$. If $A_k>|g|$, choose $\tanh (2{\theta }_k^{\ast })=g/A_k$. The mode contribution equals $\frac{1}{2}{E}_k\cosh [2({\theta }_k-{\theta }_k^{\ast })]-A_k/2$, proving a unique minimum $(E_k-A_k)/2$. At fixed ultraviolet cutoff this lowers the energy relative to the uncorrelated condensate. The approximation requires small quantum depletion of a Bose condensate; an unregulated three-dimensional ground-state energy also requires renormalization of a three-dimensional dilute-gas contact coupling.

All $N$ bosons occupy the same normalized one-particle superposition with $|C{|}^2+|S{|}^2=1$. Its occupation distribution in the original two modes is binomial, not two independent Poisson distributions. In particular, $⟨N_1⟩=N|S{|}^2$ and $⟨N_0{N}_1⟩=N(N-1)|C{|}^2|S{|}^2$. These factorial moments give finite-particle corrections to interaction energies. The state is also the two-mode realization of a spin coherent state with spin $N/2$.

The fixed-target production threshold concerns the incident accelerator beam, whereas antiproton mass from momentum and velocity concerns an outgoing secondary beam. To see the distinction, consider a free proton hitting a stationary free proton, producingThe final electric charge is $+2e$ and final baryon number is $3-1=2$, as initially. Approximate all four final rest masses by the same proton rest mass $m$. With metric signature $(+---)$, define the squared centre-of-mass energy by $W^2=c^2(P_1+P_2{)}^2$. The incident proton has total energy $E=m{c}^2+K$, and the target four-momentum is $(mc,0)$. Four-momentum conservation and the mass shell relation giveAt threshold the four final particles have zero relative momentum in their centre-of-momentum frame, so $W_{\mathrm{thr}}=4m{c}^2$. HenceOnly $2m{c}^2$ of new rest energy is created. The larger incident kinetic energy is required because the final system must also carry the initial laboratory momentum. The incident total energy at threshold is $7m{c}^2$; neither this nor $K_{\mathrm{thr}}$ is the selected secondary momentum.

The 1955 authors explicitly qualified the stationary-nucleon threshold: target nucleon motion inside an atomic nucleus can lower the required incident energy.^[ref]

The kinematic reason is visible for two on-shell incoming nucleons:A target momentum directed against the beam increases $W^2$. Applying this illustration to a bound nucleon also requires binding energy and residual-nucleus recoil; the free stationary-target $6m{c}^2$ result is therefore not a universal sharp threshold for a copper target. A low observed secondary momentum is entirely compatible with a much larger accelerator kinetic energy.

For a quasistatic change of capacitance at fixed terminal electric potential difference, the capacitor energy is $U=C{V}^2/2$ and plate electric charge is $Q=CV$. Electrical work supplied to the electric field by the battery is $VdQ$, so mechanical work done on the capacitor is $d{W}_{\mathrm{mech}}=dU-VdQ=-V^{2}dC/2$. Increasing the separation of a parallel-plate capacitor makes $dC<0$: mechanical work is positive even though electrostatic energy decreases, because electrical energy returns to the source. Conversely, when $dC>0$, the battery supplies $V^{2}dC$: half increases capacitor energy and half is available as mechanical work done by the electric field. Differentiating capacitor energy alone at fixed terminal electric potential difference omits this exchange with the battery.

Work in natural units with $c=1$, neglect the electron mass, and put the initial proton at rest with mass $M$. Let $k$ and $k^{\prime }$ be the initial and final electron four-momenta, $q=k-k^{\prime }$, and $Q^2=-q^2>0$. Define the laboratory energy transfer ${\nu }_{\mathrm{lab}}=E-E^{\prime }=P\cdot q/M$. This ${\nu }_{\mathrm{lab}}$ has units of energy; the invariant $P\cdot q$ has units of energy squared. The Bjorken scaling variable and the final hadron invariant mass obeyThese follow by squaring $k-k^{\prime }$ and $P+q$, using $k^2=k^{\prime 2}=0$ and $P=(M,0)$. Solving the Bjorken x definition gives the outgoing energy and then the scattering angle:Alternatively, substituting $Q^2=4E{E}^{\prime }{\sin }^2(\theta /2)$ before solving gives $E^{\prime }=E/[1+2E{\sin }^2(\theta /2)/(Mx)]$. A chosen $x,E,Q^2$ must satisfy $E^{\prime }>0$ and $0\le {\sin }^2(\theta /2)\le 1$. In addition, $W$ must exceed the lightest allowed inelastic hadron threshold; positive $W^2$ alone is insufficient.

For an original numerical example, take $M=0.938\mathrm{GeV}$, $x=0.3$, $E=15\mathrm{GeV}$ and $Q^2=3{\mathrm{GeV}}^2$. Then ${\nu }_{\mathrm{lab}}\simeq 5.330\mathrm{GeV}$, $E^{\prime }\simeq 9.670\mathrm{GeV}$, $\theta \simeq 8.25^{\circ }$, and $W\simeq 2.807\mathrm{GeV}$. Varying $2\le Q^2/{\mathrm{GeV}}^2\le 5$ and $10\le E/\mathrm{GeV}\le 20$ at this same Bjorken x keeps every setting physical and gives $W^2>5.54{\mathrm{GeV}}^2$, above the proton-pion threshold and the stricter chosen continuum selection $W>2\mathrm{GeV}$. These are calculated settings, not historical measurements.

A flippase catalyzes directional movement of selected lipids between membrane leaflets. Such activity can establish or maintain lipid asymmetry, subject to specificity and energy coupling.

Let $E={\hslash }^2{k}^2/(2m)>0$ for the Kronig-Penney model, and propagate the column $y=(\psi ,{\psi }^{\prime }{)}^T$ by one lattice cell. Free propagation through distance $a$ and passage across one delta potential respectively give the propagation and jump matricesFor a cell beginning just after a barrier, $M=DP$ takes it to just after the next barrier. Both matrices have determinant one, so their product has reciprocal eigenvalues. Multiplication gives the half-trace, or Floquet discriminant,A Bloch state has Floquet matrix eigenvalues $e^{\pm iqa}$, hence $\cos (qa)=\Delta (E)$. Real crystal wavenumber $q$ therefore requires $|\Delta (E)|\le 1$. The values $\Delta =\pm 1$ give periodic or antiperiodic band-edge conditions. When $|\Delta |>1$, the reciprocal eigenvalues have unequal magnitudes and one cell solution grows while the other decays, producing a band gap in the infinite periodic system. The displayed positive-energy formula continues to negative energy by replacing trigonometric functions with their hyperbolic counterparts; the limit at $k=0$ uses $\sin (ka)/k\to a$.

For a beam at a specified surface, fluence is the time-integrated incident light intensity: $F=\int I(t)dt$, with units of energy per area. It records how much energy arrived during an exposure rather than the instantaneous rate of delivery.

For a constant rectangular pulse, $F=I_0\tau$. Two pulses can therefore have equal fluence and different peak intensities if their durations differ. This matters in nonlinear optics, where the response can depend strongly on instantaneous intensity, and in comparing exposure protocols. Photon fluence instead counts incident photons per area; converting it to energy fluence requires the photon energies.

A fluid dynamicist studies how fluids move and how their motion exchanges mass, momentum and energy. The work can combine mathematical models, controlled measurements and computation. Choosing what to resolve, what to approximate and which observations test the result is as central as solving an equation. A fluid dynamicist may work on applications ranging from aerodynamic drag force to transport in a blood.

For an original example, consider steady water flow through a duct with negligible density changes. Mass conservation gives $\rho A_1{U}_1=\rho A_2{U}_2$, with $U$ the cross-sectionally averaged speed.^[ref] If a circular duct doubles its radius while maintaining the same volume flow rate, its area becomes four times larger and its mean speed falls to one quarter. A researcher can compare that prediction with measured flow and velocity. Predicting the required pressure difference is another question, requiring information about losses and the velocity field.

In turbulence research, direct numerical simulation and large eddy simulation resolve different ranges of motion. A useful validation compares the same observable and averaging convention rather than simply comparing two pictures of complicated flow. Parviz Moin's institutional research profile describes this combination of computational methods with questions about turbulent-flow physics.^[ref]

Fluorescence is photon emission following photon absorption by matter: an excited state releases some of its energy as light. A fluorophore can therefore mark where something is, while its fluorescence emission spectrum and fluorescence lifetime report different aspects of its environment. Fluorescein is one example; FRET exploits a competing route for excitation energy rather than requiring a second emitted photon.

The usual molecular sequence is excitation, relaxation, then emission, as in the Jablonski diagram. The emitted photon commonly has lower energy and longer wavelength, giving a Stokes shift; the exceptions are discussed below.^[ref]

For one directed spin-resolved channel, the density of longitudinal states per unit length and wave number is $1/(2\pi )$. Since $v={\hslash }^{-1}dE/dk$, velocity times density per energy is $1/h$. A reservoir occupation imbalance therefore contributes particle flux $\int \delta f(E)dE/h$; in zero-temperature linear response it is $\delta \mu /h$. Multiplying by charge and transmission gives the Landauer current per mode.

Flux pinning is the trapping of Abrikosov vortices by spatial variations in a superconductor, such as defects or regions where superconductivity is already weakened. A vortex can lower its energy by placing its suppressed-order core in such a region, producing a barrier to displacement.

This matters for transport because an electric current exerts a force on vortices. Moving vortices produce an electric field and hence dissipative electrical response; pinning resists that motion and supports useful superconducting critical current density. Pinning also explains magnetic hysteresis and trapped magnetic flux. Thermal activation can still permit slow motion through flux creep, while stronger drive can produce vortex depinning. Thus pinning improves practical current carrying capacity without making every finite-temperature relaxation rate exactly zero.

The self-consistent one-electron operator obtained by varying a Slater determinant. It contains a local hartree potential and a nonlocal exchange action. Its occupied eigenfunctions define a stationary determinant; its eigenvalues are Lagrange multipliers and are not by themselves the total many-electron energy.

The focusing power wave equation is $u_{tt}-\Delta u=u|u{|}^{p-1}$. Its conserved energy has the negative potential term $-|u{|}^{p+1}/(p+1)$ and is not positive definite. Localized ordinary differential equation blowup for a wave equation produces smooth compactly supported data with finite maximal lifespan for suitable focusing powers.

The number of target chemical atoms per incident area along the beam. For a uniform foil at normal incidence, number density $n$ and thickness $t$ give $n_s=nt$, in ${\mathrm{m}}^{-2}$. At oblique incidence the traversed path must replace $t$. The single-encounter count model requires the chance of two significant deflections to be negligible and energy loss through the foil to be small. A small probability in one detector patch alone does not establish either condition.

For a normalized probability density $P$ and a dimensionless potential $V$, define the Fokker--Planck free-energy functionalAssume $Z=\int e^{-V}dx$ is finite and positive, and write $\pi =e^{-V}/Z$. Then $\mathcal{F}[P]=H(P\mid \pi )-\log Z$. Nonnegativity of relative entropy proves that the unique normalized minimizer is $\pi$, up to equality almost everywhere. The integrability assumption matters: a constant potential on the whole real line has no normalized density proportional to $e^{-V}$.

For a sufficiently regular solution of ${\partial }_{t}P=\nabla \cdot (PD\nabla (V+\log P))$ with symmetric positive-definite $D$ and no-flux boundaries, integration by parts givesThe free energy is thus a Lyapunov functional for the diffusion. Decrease alone is not a quantitative convergence rate; an inequality relating the energy gap to its dissipation, such as a logarithmic Sobolev inequality in the Gaussian case, supplies additional control.

Food is material consumed to supply energy, nutrients or other biologically usable components. Its physical and chemical properties affect digestion and absorption. The nutritional contribution depends on what an organism can process and use, rather than only the total amount swallowed.