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The membrane bending modulus sets the energy cost of curvature. A larger $k_c$ suppresses thermal undulations and resists wrapping around small objects.

This coefficient multiplies the squared mean-curvature term $\frac{\kappa }{2}\int (2H-C_0{)}^{2}dA$ in the Helfrich membrane energy. It has units of energy, unlike a beam's flexural rigidity. For small height modes of a nearly flat membrane of area $A$, with Fourier normalization $h_q=A^{-1}\int h(\mathbf{x})e^{-i\mathbf{q}\cdot \mathbf{x}}{d}^{2}x$, quadratic bending and tension energies give $⟨|h_q{|}^2⟩=k_{B}T/[A(\kappa q^4+\sigma q^2)]$ by the equipartition theorem. Thermal-flicker measurements can therefore separate short-wavelength bending from longer-wavelength tension effects.

A membrane-mediated interaction potential is the separation-dependent energy produced when deformation fields around two membrane-bound objects overlap. Its sign and range depend on membrane tension, bending stiffness, inclusion geometry, and which side of the membrane each object occupies.

Membrane transport is the passage of substances across a cell membrane. For the small solutes considered here, distinguish the available route from its driving force: a lipid bilayer, ion channel, or carrier protein determines how passage can occur, while differences in chemical potential and membrane potential determine which direction is energetically downhill. A large driving force cannot make an unavailable route conduct. See Alberts et al.: principles of membrane transport.

For an ideal dilute chemical solution at uniform temperature and pressure, with the same reference molar chemical potential on both sides, the molar electrochemical potential change for transferring a solute from compartment A to B isHere $c_{\mathrm{A}},c_{\mathrm{B}}$ are concentrations, $T$ is absolute temperature, $R$ is the molar gas constant, $z$ is signed ion charge number, $F$ is the Faraday constant, and $V_{\mathrm{B}}-V_{\mathrm{A}}$ is the electric potential difference. The natural logarithm contains a dimensionless ratio. The electrical term vanishes for an uncharged solute. More generally, use thermodynamic activities in place of the ideal concentration ratio. This combines the ideal-solution chemical potential difference with electrical energy per mole; see IUPAC Gold Book: electrochemical potential. For a biological cell, taking B inside and A outside makes the voltage term use the existing membrane potential convention.

A membrane transport protein enables selected substances to cross a cell membrane. A membrane channel forms a permeation pathway, whereas a carrier binds its cargo and changes conformation. Transport can follow an electrochemical gradient or be coupled to an energy source to move material against it. For example, GLUT4 supports facilitated glucose transport, whereas the sodium-potassium ATPase couples transport to ATP hydrolysis. Selectivity and transport mechanism are separate properties. ^[ref]

Light initiates methane chlorination by photodissociation of $\mathrm{C}{\mathrm{l}}_2$ through homolysis. The elementary bookkeeping isHere $h\nu$ denotes the photon energy. The first propagation step replaces a chlorine radical with a methyl radical; the second regenerates a chlorine radical. Adding the two propagation equations cancels both free radicals and gives $\mathrm{C}{\mathrm{H}}_4+\mathrm{C}{\mathrm{l}}_2⟶\mathrm{C}{\mathrm{H}}_3\mathrm{Cl}+\mathrm{HCl}$: chloromethane and hydrogen chloride are made while the chain carrier is regenerated.^[ref]

For example, recombination $2{\mathrm{C}{\mathrm{H}}_3}^{\bullet }⟶{\mathrm{C}}_2{\mathrm{H}}_6$ of methyl radicals makes ethane and a covalent bond from two unpaired electrons, removing two carriers. Recombination of two ${\mathrm{Cl}}^{\bullet }$ also terminates carriers; in the gas phase, stabilization of the product requires disposal of released energy, for example to a third collision partner. The figure omits that partner. Termination competes with propagation, and further chlorination can consume chloromethane, so this cycle alone does not predict a pure product or a measured reaction rate.^[ref]

M. Gustafsson coauthored research on electron affinity.^[ref]

For chlorine, a published photodetachment threshold is $A_{\mathrm{ea}}/(hc)=29138.59(22)\mathrm{c}{\mathrm{m}}^{-1}$, approximately $3.612725\mathrm{eV}$ per chemical atom. Here $h$ is the Planck constant, $c$ the speed of light, and the inverse-length unit expresses an energy through a spectroscopic wavenumber. The parentheses give the reported uncertainty in the last digits, not a range of possible sign conventions.

For constant-density cylindrical rotation $r\Omega (r){\mathbf{e}}_{\varphi }$ and a purely toroidal magnetic field, an axisymmetric radial displacement gives $b_{\varphi }=(B_{\varphi }/r-B_{\varphi }^{\prime }){\xi }_r$. Combining its curvature force with conservation of angular momentum gives the restoring coefficient $\mathcal{D}$ above. A local incompressible radial-vertical mode has ${\omega }^2=(k_z^2/k^2)\mathcal{D}$. Positive $\mathcal{D}$ gives stability in this ideal interchange setting; negative $\mathcal{D}$ permits growth. This is distinct from weak-field magnetorotational instability: a steeply increasing toroidal field can release current energy even without differential rotation.

Michael Dopita modeled the physical processes producing emission from ionized astronomical gas. With David C. Nicholls and Ralph S. Sutherland, he investigated why different spectral lines can give discrepant electron temperatures and chemical abundances in H II regions and planetary nebulae.

Their proposed explanation changed the assumed electron kinetic energy distribution from a Maxwell-Boltzmann distribution to a Kappa distribution, which has a more populated high-energy tail. Because different atomic transitions sample different electron energies, this change affects their inferred temperatures differently. The paper presented a testable interpretation of the diagnostic discrepancy.^[ref]

For an unpolarized parent charged lepton undergoing a three-body leptonic V-A interaction decay, neglecting daughter masses and radiative corrections, the normalized charged-daughter energy distribution in the parent rest frame is $f(x)=2x^2(3-2x)$, where $x=2E_{\ell }^{\ast }/m_{\mathrm{parent}}$ lies in $[0,1]$. Direct integration gives ${\int }_0^{1}f(x)dx=1$ and ${\int }_0^{1}xf(x)dx=7/10$. A specified parent spin orientation also produces angular terms, so this one-variable law alone does not describe polarized samples.

The microcanonical ensemble describes an isolated equilibrium system with fixed volume, particle number and energy within a narrow shell. It assigns equal probability to the accessible microstates inside that shell. For a discrete count $\Omega (E)$, the entropy is $S=k_B\log \Omega$; classical phase-space counting also requires a specified measure and shell width.

An idealized insulated rigid container motivates the constraints. Its particles exchange energy internally, but the whole container exchanges neither heat nor particles with its surroundings. A small subsystem can still have fluctuating energy because the rest acts as a reservoir. In the thermodynamic regime where that reservoir has a well-defined temperature, the subsystem approaches a canonical ensemble. Equal shell probabilities are an ensemble prescription; whether a particular dynamics explores the accessible shell on an experimental timescale is a separate equilibration question.

A microgrid is a bounded group of interconnected electrical loads and distributed energy resources that can be controlled as one entity. Its defining boundary and coordinated operation matter: a collection of nearby generators is not sufficient by itself. A microgrid may exchange power with an external electric power system, while islanded microgrid operation supplies its connected electrical loads without that external connection. The DOE overview identifies generation, storage, and the controller as interacting equipment, rather than treating storage capacity alone as a guarantee of service.^[ref]

For a worked operating point, suppose a microgrid must supply $100\mathrm{kW}$ of electrical load plus $5\mathrm{kW}$ of losses. Its solar supply delivers $40\mathrm{kW}$ and an electrical generator supplies $30\mathrm{kW}$. Storage must deliverIf the storage system can continuously deliver $40\mathrm{kW}$ and has $70\mathrm{kWh}$ of usable output energy, it can support this unchanged operating point for $70/35=2\mathrm{h}$. Here usable output energy already accounts for the chosen discharge convention; no further conversion loss is silently subtracted.

Now let a cloud reduce the solar supply to $10\mathrm{kW}$. The storage demand becomes $65\mathrm{kW}$, above its $40\mathrm{kW}$ rating even with a full store. The controller must obtain more generation or reduce electrical load. This power-rating limit versus energy autonomy in a microgrid explains why an energy budget and an instantaneous power constraint must both be checked. A test of islanded microgrid operation therefore records the external connection state, generation and storage power, and the response of the connected electrical loads, rather than merely checking that a battery is present.

A microstate is a complete microscopic specification within the chosen model. For two distinguishable, noninteracting two-level systems, label the units $A,B$ and their states $0,1$, with energies $0,\Delta$ and $\Delta >0$. The four microstates arewith total energies $0,\Delta ,\Delta ,2\Delta$. Distinguishability matters: $(0,1)$ and $(1,0)$ describe different assignments to the labeled units. Noninteraction matters too: their energies add without an extra coupling term.

For a classical particle model, a microstate can instead be a point in phase space, specifying every position and momentum. In the quantum canonical ensemble, the sum uses energy eigenstates; a general quantum state may also contain quantum coherence, which an energy label alone does not specify.

A single photon thermalized in a macroscopic body adds energy $hc/\lambda$. If its heat capacity is sufficiently large that the temperature change is negligible, the entropy increase is $hc/(\lambda T)$. The Boltzmann entropy then gives the displayed ratio of accessible microstates. The exponential can be enormous even though the deposited energy is macroscopically tiny.

A microwave is a short-wavelength radio wave, commonly on millimetre-to-centimetre scales. Like other electromagnetic waves, it carries energy through coupled electric fields and magnetic fields; its scale makes conducting waveguides and resonant cavities practical.

A source, conducting guide, movable probe and detector can map a standing wave produced by reflection. Neighboring amplitude maxima are separated by half the guide wavelength, which need not equal the free-space wavelength because guided propagation has its own dispersion relation. Microwaves also drive suitable molecular or spin transitions, linking cavity apparatus to spectroscopy.

Researcher in anaerobic cell biology. With Donald G. Lindmark, Müller characterized the hydrogenosome and its role in pyruvate metabolism, connecting an unusual organelle with energy production without oxygen.^[ref]

Robert Andrews Millikan's 1916 experiment used a magnetically turned wheel and rotary knife to expose sodium or lithium in vacuum. A surrounding copper-gauze/solid-copper collector caught electrons; an electrometer recorded accumulated electric charge over thirty seconds at each applied voltage. He located endpoints graphically from photocurrent curves. Mercury spectral lines, selected by a monochromator and filters, controlled frequency; rejecting shorter-wavelength contamination protected the endpoint. An oxidized collector suppressed reverse emission for selected lines. A copper test plate measured the contact potential difference by a displacement-null method; a certified voltmeter set the electrical scale. He compared stable-surface sweeps with rapid alternating-frequency measurements to control drift. The slope tested $h/e$; the threshold required the contact correction. His gauge did not measure mercury vapor. Confirmation of the equation coexisted with his continued skepticism about localized light quanta.^[ref]

For a modern repetition of the logic, first establish electrical zero with the illumination blocked, characterize leakage, and verify the voltmeter scale independently of the presumed Planck constant. Scan several voltages near each endpoint rather than calling one noisy zero a cutoff. Hold the illuminated area and surface state fixed when varying light intensity; vary frequency independently when testing the slope. Repeating a reference frequency between other readings reveals time-dependent offsets. Agreement of a slope with $h/e$ tests an energy relation; it does not alone identify every microscopic step of photon absorption and escape.

For positive masses, let $r$ be the actual minimum pairwise separation and use positive binding energy $U={\sum }_{j<k}G{m}_j{m}_k/r_{jk}$. If $m_{(1)},m_{(2)}$ are the smallest masses, then $G{m}_{(1)}{m}_{(2)}\le rU\le G{\sum }_{j<k}{m}_j{m}_k$. Thus $U^{-1}$ is comparable to $r$ with constants depending only on the masses.

For an isolated uniform sphere with fixed positive cosmological constant, energy conservation and the virial theorem give $2\eta x^3-(2+\eta )x+1=0$, $x=R_v/R_m$. The physical root starts at $x=1/2$ when $\eta =0$ and decreases for $0<\eta <1$. At the limiting value $\eta =1$, the cubic factors as $(x-1)(2x^2+2x-1)$; the collapsing branch approaches $(\sqrt{3}-1)/2$. Finite turnaround therefore gives a strict lower bound, whose infimum is approached rather than attained. Compared with zero vacuum energy at the same turnaround data, the negative vacuum potential and its virial contribution require stronger final gravitational binding.

A long-baseline neutrino oscillation experiment using a Fermilab beam, near and far detectors, and magnetized iron/scintillator sampling calorimeters. A charged current muon neutrino event gives a negative muon and a hadronic shower; their reconstructed energies estimate the incident neutrino energy.