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For an incompressible Newtonian fluid of constant mass density $\rho$ and dynamic viscosity $\mu >0$, let $\mathbf{u}$ be velocity, $\mathbf{f}$ body-force density and $\mathbf{\sigma }=-pI+2\mu e$ the Cauchy stress tensor. Here $e=(\nabla \mathbf{u}+(\nabla \mathbf{u}{)}^T)/2$ is the symmetric rate-of-strain tensor. On a fixed domain $\mathcal{D}$,The nonnegative viscous dissipation rate is ${\mathcal{D}}_{\mu }=2\mu \int e:edV$; it enters the kinetic energy balance with a minus sign because it converts mechanical energy into heat.

To derive the identity, take the inner product of the Navier-Stokes equations with $\mathbf{u}$ and use incompressibility. The stress term becomes $\mathbf{u}\cdot (\nabla \cdot \mathbf{\sigma })=\nabla \cdot (\mathbf{\sigma }\mathbf{u})-\mathbf{\sigma }:\nabla \mathbf{u}$. Symmetry of $e$ gives $\mathbf{\sigma }:\nabla \mathbf{u}=2\mu e:e$, and the divergence theorem supplies the surface power.

For a fixed closed vessel with stationary no-slip walls and no body-force work, both surface integrals vanish. Its kinetic energy can only decrease through viscous dissipation. For an open domain, advected kinetic energy and stress work must also be retained.

For an isolated decay of a parent of rest mass $M$ into products of rest masses $m_i$, the difference between total final and initial kinetic energies is $(M-{\sum }_i{m}_i)c^2$ in every inertial frame. Conservation of total energy and inclusion of all products are essential.

In a dilute gas whose mean free path $\lambda$ is short compared with macroscopic gradient lengths, particles carry properties from their previous collisions over distances of order $\lambda$. A first-gradient isotropic estimate gives a transported flux $-n\overline{v}\lambda \nabla a/3$ for a property $a$ carried per particle. Applying this reasoning to species abundance, directed momentum $m{u}_x$ and thermal energy with per-particle heat capacity $c_v$ gives diffusivity $D\sim \overline{v}\lambda /3$, dynamic viscosity $\eta \sim nm\overline{v}\lambda /3$ and thermal conductivity $\kappa \sim n{c}_v\overline{v}\lambda /3$. These numerical prefactors are estimates, not exact collision-integral results.

For a constant hard-sphere diameter $d$, $\lambda \propto 1/(n{d}^2)$ and $\overline{v}\propto \sqrt{T/m}$. Consequently $\eta \propto \sqrt{mT}/d^2$, $\kappa \propto c_v\sqrt{T/m}/d^2$ and $D\propto \sqrt{T/m}/(n{d}^2)$. At fixed pressure the last scales as $T^{3/2}/p$. The near pressure-independence of viscosity and thermal conductivity applies only while gas-gas collisions, rather than apparatus walls, determine the mean free path. Temperature-dependent cross-sections and internal molecular excitations change the simple power laws.

Inductive energy associated with the inertia of current-carrying particles. In a superconducting ring it supplies the current term in the quantized fluxoid, so geometric flux need not itself be an exact integer flux quantum when this term is appreciable.

For an isotropic nonrelativistic Maxwellian particle population, kinetic temperature describes random translational motion, with mean kinetic energy $3k_B{T}_k/2$ per particle after subtracting bulk motion. Different species can have different kinetic temperatures outside thermal equilibrium. A rotational excitation temperature equals the kinetic temperature only when the excitation is sufficiently thermalized; a radiation brightness temperature is another distinct quantity.

A microscopic account of gas properties using the motion and collisions of its constituents. For a dilute monatomic ideal gas, translational kinetic energy gives $p=Nm⟨v^2⟩/(3V)$ and $U=3N{k}_{\mathrm{B}}T/2$. Interactions and additional molecular modes require extensions.

In an isotropic near-equilibrium quasiparticle transport picture, an angular average of squared velocity components supplies the factor1/3. Mode-resolved heat capacities times $v_g^2\tau$ then sum to thermal conductivity. Replacing them by $C⟨v⟩\ell /3$ is a characteristic-scale approximation, not an exact product of separately chosen averages. A solid can use this kinetic structure because mobile phonon and electron excitations transport energy even when the constituent atoms remain localized.

In a sufficiently strong electrostatic step, the positive-energy incident channel overlaps a negative-kinetic-energy channel on the far side. For a charged scalar field the outgoing antiparticle branch has negative charge current, allowing reflected particle current larger than the incident current while signed charge is conserved. The external source supplies energy for particle-antiparticle creation. This requires a multiparticle interpretation; negative transmitted charge current is not a negative probability.

Displacement of an atom when a collision transfers enough momentum to exceed its displacement threshold. Its energy dependence differs from radiolysis.

The Kronig-Penney model is an exactly solvable quantum mechanics model showing how a periodic potential produces energy bands and band gaps. R. de L. Kronig and W. G. Penney began with rectangular potential barriers and then took a narrow-barrier limit. This makes the relation between repeated quantum scattering and electronic band structure explicit.^[ref]

In the delta potential version, a particle of mass $m$ moves inThe lattice spacing $a$ has units of length; the strength $V_0$ has units of energy times length. The wavefunction is continuous, but integrating the Schrödinger equation across a barrier givesBetween barriers the wavefunction is a combination of plane waves. Bloch's theorem requires translation by one cell to multiply it by a phase $e^{iqa}$, where $q$ is the crystal wavenumber. Matching across one cell yields the Floquet discriminant of the delta-comb Kronig-Penney model, which decides whether a real $q$ is possible. With $V_0=0$ the band gaps close and the free particle dispersion is recovered; finite barriers open forbidden intervals through coherent repeated scattering.

A Kurie plot removes known electron phase-space and Coulomb factors from a beta spectrum and takes a square root. For negligible neutrino mass/recoil and constant nuclear shape factor, its ordinate is proportional to $Q-T$, giving a straight line whose kinetic-energy intercept is the beta endpoint. The precise factors depend on whether the measured spectrum is per momentum or per energy. Forbidden transitions generally need an additional shape correction.

In 1953, David Noel Ferguson Dunbar and collaborators used accelerated deuterons on nitrogen-14, populating carbon-12 through ${}^{14}\mathrm{N}(d,\alpha ){}^{12}\mathrm{C}$. Magnetic analysis resolved outgoing alpha particle groups. At $620\mathrm{keV}$ incident energy and laboratory angle $90$ degrees, they reported excitation $7.68\pm 0.03\mathrm{MeV}$, with yield $6\%$ of the $4.43\mathrm{MeV}$ group's yield. The stronger group supplied an internal comparison. The paper's opening explicitly relates the search to Fred Hoyle's abundance argument. This is evidence for a nuclear level from a different entrance reaction than stellar fusion.^[ref]

The inference uses energy and momentum conservation: at fixed incident energy and observation angle, more energy retained as carbon-12 excitation leaves less outgoing kinetic energy. A resolved particle group therefore identifies a nuclear level. This laboratory nuclear line supports an input required by a stellar synthesis model. It does not reproduce core helium burning, measure a complete stellar carbon yield, or by itself determine the stabilizing decay probability.

A lagging power factor describes an absorbing sinusoidal electrical load whose electric current lags its terminal electric potential difference. With the passive sign convention, chooseThe current maximum occurs later by $\phi /\omega$. The RMS phasors give complex electrical powerIn this equation the supply phasor is the real number $V$, while $I^{\ast }$ denotes the conjugate of the current phasor. Positive $Q$ is inductive reactive power under this convention. The primary MIT notes derive the conjugation and the peak-versus-RMS factor.^[ref]

For an original $230\mathrm{V}$, $10\mathrm{A}$ example with $\cos \phi =0.8$, active power is $1840\mathrm{W}$ and reactive power is $1380\mathrm{var}$. At $50\mathrm{Hz}$, $\phi \approx 0.6435$ radians corresponds to a current delay of about $2.05\mathrm{ms}$. The factor $0.8$ gives the magnitude of the phase relationship, while the word lagging supplies the direction lost by the cosine.

An ideal inductor is the limiting case $\phi =\pi /2$: it alternately stores and returns magnetic energy, with zero mean active power. An absorbing resistive-inductive load usually has both positive active power and positive reactive power. An ideal parallel capacitor can reduce the source's quadrature electric current; sufficient extra capacitance eventually crosses into leading power factor rather than improving the factor indefinitely.

For distorted waveforms, specify whether lagging refers to the fundamental displacement power factor; one phase angle does not characterize arbitrary harmonic components. A signed meter display is also not a universal lead/lag label. The manufacturer's sign-convention explanation distinguishes phase direction from the convention used for displayed factor signs.^[ref]

The Lamb-Mössbauer factor $f$ is the fraction in the coherent elastic line. In the harmonic-lattice approximation it is a Debye–Waller factor. Write the displacement of the emitting or absorbing atomic nucleus relative to its equilibrium site as $\mathbf{u}$, put $u_{\parallel }=\hat{\mathbf{k}}\cdot \mathbf{u}$, and define the angular wavenumber $k=E_{\gamma }/(\hslash c)$. Choose the equilibrium site so that $⟨u_{\parallel }⟩=0$. Its directional displacement variance is ${\sigma }_{\parallel }^2=⟨u_{\parallel }^2⟩$. A harmonic solid in thermal equilibrium has a Gaussian distribution for this displacement, whose characteristic function gives the elastic amplitudeTaking the squared magnitude gives the intensity fractionThe factor $1/2$ belongs to the amplitude exponent; it disappears when the amplitude is squared. For an isotropic solid, $⟨u_{\parallel }^2⟩=⟨|\mathbf{u}{|}^2⟩/3$, so an equivalent expression is $f=\exp [-k^2⟨|\mathbf{u}{|}^2⟩/3]$. Directional and three-dimensional mean squared displacements must not be substituted for one another without this factor.

An Einstein solid approximation replaces the vibrational spectrum by one positive angular frequency $\omega$. For the directional quantum harmonic oscillator, with the same effective mass $M$ as in $E_R$,Here $k_B$ is the Boltzmann constant and $T$ is temperature. This follows by taking the expected value of $u_{\parallel }^2$ in each oscillator energy eigenstate, $\hslash (n+1/2)/(M\omega )$, and averaging with the Boltzmann distribution. Substitution givesThe zero-temperature limit retains zero-point energy and a nonzero displacement variance. Cooling therefore increases the recoil-free fraction towards a limit below one for finite $E_R/(\hslash \omega )>0$. At high temperature, the exponent tends to $-2E_R{k}_{B}T/(\hslash \omega {)}^2$. This is a single-frequency approximation to a harmonic lattice's coherent elastic weight, not a complete description of all zero-net-energy processes of an isolated oscillator. Real vibrational spectra, directional binding and anharmonic motion can change its predictions.

Extend the stipulated teaching system by choosing $\hslash \omega =0.020\mathrm{eV}$, so $\eta =E_R/(\hslash \omega )=0.1$. Then $f(0)=0.90484$, while $f=0.80542$ at $k_{B}T=\hslash \omega$. If source and absorber are independent, the fraction of events eligible for recoil-free emission followed by recoil-free absorption contains the product $f_s{f}_a$; this gives $0.81873$ for two zero-temperature copies. It is not the total probability of absorption, which also depends on the number of resonant atomic nuclei, geometry and spectral line overlap.

A quantum electrodynamics correction separates hydrogen atom levels such as $2s_{1/2}$ and $2p_{1/2}$ that coincide in the simplest model.

The Lamb shift measures an energy level displacement beyond a specified Dirac equation reference. In the point-proton, infinite-nuclear-mass Coulomb potential model, $2S_{1/2}$ and $2P_{1/2}$ share principal quantum number $n=2$ and total electronic angular momentum $j=1/2$, hence energy, despite differing orbital angular momentum values.^[ref]

The Lamb-Retherford experiment converts a small energy difference into a large detector change: prepare a long-lived quantum state, transfer population into a short-lived one, and read the surviving beam.

For a simple original detector model, let ${\Phi }_S$ be the incident rate of surviving $2S$ atoms, $e$ the positive elementary charge, $\eta$ the mean number of collected electrons per incident $2S$ atom, and $I_{\mathrm{bg}}$ a stable background current. Take current magnitudes as positive. If microwaves remove a fraction $p$ before the target,The equality needs the same beam flux, background and collection efficiency in both measurements. It is an ideal interpretation of a normalized current loss, not a claim that an actual galvanometer directly counts individual transitions. Varying microwave amplitude tests whether the drive distorts the resonance; a field calibration is separately needed to turn a current-versus-field minimum into an energy difference.

The metastable atomic level matters because an electric-dipole selection rule suppresses the direct $2S\to 1S$ electric-dipole interaction, whereas the opposite-parity $2S\leftrightarrow 2P$ transition can be driven. A long lifetime can store the prepared population until interrogation. After transfer to $2P$, a short lifetime supplies the loss mechanism and a broad natural linewidth. A forbidden strong decay does not mean absolute stability: weaker decay channels and unwanted electric fields still matter.

At a stress-free natural reference state, an objective isotropic hyperelastic material with energy $W$ per unit mass has ${\rho }_0[W(I+H)-W(I)]=\lambda (\mathrm{tr}e{)}^2/2+\mu e:e+O(|H{|}^3)$, where $e=\mathrm{sym}H$. Twice differentiating gives the displayed Lamé parameters and the usual elastic stiffness tensor. A prestressed reference requires initial-stress corrections.

The Landau criterion asks whether creating an excitation can lower a moving superfluid's energy. For a weakly coupled, heavy obstacle moving through a uniform liquid initially at zero temperature, let $ϵ(p)>0$ be the excitation energy in the fluid rest frame, with momentum magnitude $p$. Transferring momentum $\mathbf{p}$ from an obstacle moving at velocity $\mathbf{v}$ supplies, to leading order in its negligible recoil, energy $\mathbf{v}\cdot \mathbf{p}$. Emission requiresFor an isotropic dispersion relation, the easiest direction is parallel to $\mathbf{v}$, giving the necessary kinematic conditionAn infimum includes cases in which the limiting slope occurs only as $p\to 0$. A finite-mass obstacle of mass $M$ instead loses $\mathbf{v}\cdot \mathbf{p}-p^2/(2M)$, so the corresponding emission inequality includes the extra positive recoil term $p^2/(2M)$ on the left. This is an energy and momentum balance, not a calculation of an emission rate.

The Landau criterion excludes spontaneous emission into the specified spectrum below its threshold. Above it, kinematic permission alone does not guarantee strong coupling or rapid decay. It is not a universal observed flow threshold: walls, channel geometry and quantum vortex nucleation can enable other decay routes, and finite temperature permits scattering from existing excitations. Using only bulk phonons and rotons leaves those processes outside the model.

Landau's 1941 energy and momentum argument used a roton approximation different from the finite-momentum minimum illustrated below.^[ref]

Landau received his Nobel Prize in Physics in 1962 while hospitalized in Moscow after a car accident.^[ref]

For a deliberately simple Bose-Einstein condensation comparison, stipulate an ideal free-particle excitation law $ϵ=p^2/(2m)$. Then $ϵ/p=p/(2m)$ has infimum zero. Macroscopic one-particle occupation therefore supplies no positive emission threshold by this argument alone. By contrast a linear phonon law $ϵ=cp$ yields $v_L=c$. The dispersion relation and the collective response contain information that the occupation criterion alone does not supply; Bose-Einstein condensation and thermodynamic superfluidity must be tested separately.

To check the picture algebraically, on the stipulated roton branch,Thus tangency occurs at $q_L=\sqrt{4.3}$, slightly above the roton energy minimum. At an energy minimum $e^{\prime }=0$, but a tangent through the origin needs $e^{\prime }=e/q>0$. Minimizing energy and minimizing the energy-to-momentum ratio are different operations. This geometric construction makes the criterion useful even when the full dispersion relation is supplied numerically rather than by a simple formula.

A uniform electric field perpendicular to a uniform magnetic field shifts each cyclotron oscillator's guiding centre and makes its energy depend linearly on the conserved momentum. The formerly degenerate Landau levels therefore become tilted bands.

A Landau quasiparticle is a fermionic excitation connected continuously to a Fermi gas particle when interactions preserve the phase. Near an isotropic Fermi surface, its energy relative to the chemical potential is ${\xi }_p={ϵ}_p-\mu \simeq v_F(p-p_F)$, with Fermi momentum $p_F$ and Fermi velocity $v_F$.^[ref]

The Pauli exclusion principle restricts decay. For a three-dimensional normal Fermi liquid with regular short-range interactions, a representative low-energy lifetime model isHere $E_{\ast }$ is an energy scale and $A$ is dimensionless; sharpness requires $\hslash /{\tau }_p\ll |{\xi }_p|$ at zero temperature.^[ref]

For an original example, stipulate $A=1$ and $T=0$. At $|{\xi }_p|/E_{\ast }=0.02$, the decay energy is $0.0004E_{\ast }$, just $2\%$ of the excitation energy; halving $|{\xi }_p|$ reduces that ratio to $1\%$. Approaching the Fermi surface improves the particle description even though the microscopic interaction has not weakened. This quadratic illustration is three-dimensional; low-dimensional corrections require separate analysis.