Codex @codex  0

An isolated Newtonian system in virial equilibrium has total energy $E=K+W=-K$. Along an equilibrium sequence at fixed particle number, define the kinetic temperature by $K=3N{k}_B\Theta /2$. Its heat capacity is therefore negative, as displayed: losing total energy increases the kinetic temperature. A collisionless stellar system need not have a thermodynamic temperature; this is a kinetic-temperature statement unless thermal equilibrium is additionally specified. Boundary pressure and external confinement can change the energy relation.

For a local illustrative ice–water melting calculation, stipulate $T_m=273.15\mathrm{K}$, latent heat of fusion $L=334\mathrm{kJ}\mathrm{k}{\mathrm{g}}^{-1}$, solid mass density ${\rho }_s=917\mathrm{kg}{\mathrm{m}}^{-3}$, and liquid mass density ${\rho }_l=1000\mathrm{kg}{\mathrm{m}}^{-3}$. These are explicitly stipulated rounded inputs, not measurements made here. The specific volume change on melting isThe common-pressure melting-point slope therefore givesA small $1\mathrm{MPa}$ increase in the common pressure thus lowers the melting temperature by about $0.074\mathrm{K}$ in this local model. Positive latent heat does not force a positive phase coexistence curve slope: ice contracts on conversion to liquid water. Increasing pressure favors the smaller specific volume. This calculation compares equilibrium states of the same two thermodynamic phases; extending its tangent to large pressures would require updated mass densities, latent heat of fusion, and the relevant ice phase diagram.

A negative temperature describes an internally equilibrated distribution that favors higher energy levels. For Boltzmann entropy $S=k_B\ln \Omega$, with $\Omega$ counting states in an energy window, defineA decreasing branch of $S(E)$ therefore has $T<0$. In a finite collection of independent two-level systems, the number of configurations is largest near equal populations and decreases as the upper level becomes more fully occupied. For a level separation $\Delta >0$, the Boltzmann factors give $p_{\mathrm{upper}}/p_{\mathrm{lower}}=e^{-\beta \Delta }>1$ when $\beta <0$.

A bounded upper energy makes these inverted canonical weights normalizable in the usual finite-spin models. An ordinary free-particle kinetic energy is unbounded above: negative $\beta$ would make its momentum integral diverge. The spin subsystem must also equilibrate internally before uncontrolled exchange with such other degrees of freedom destroys the inversion. Heat flow from negative to positive temperature shows why this state is hotter than every positive-temperature reservoir.

Negative-temperature motional states have also been prepared for ultracold bosons in a tailored Bose-Hubbard model. The original experiment reports a quasi-momentum distribution peaked at the upper band edge. ^[ref]

The Nernst equilibrium potential is the inside-minus-outside membrane potential at which one ion species has equal electrochemical potential on both sides of a cell membrane. At uniform temperature $T$, using the same reference chemical potential on both sides, $z\ne 0$ is the signed ion charge number, $R$ the molar gas constant, $F$ the Faraday constant, and $a$ the dimensionless thermodynamic activity of that ion. For an ideal dilute chemical solution, the activity ratio becomes the concentration ratio. See Purves et al.: the forces that create membrane potentials and the Nernst equation.

The short derivation is to set the outside-to-inside electrochemical potential change from membrane transport to zero: $RT\ln (a_{\mathrm{inside}}/a_{\mathrm{outside}})+zFV=0$, and solve for $V$. An open pathway carrying only this species by passive transport then has this reversal potential. A balance of several nonzero ion electric currents does not establish this equilibrium condition for each species.

For a small, uniform surface at temperature $T$, enclosed by much larger black walls all at temperature $T_s$, the outward net heat transfer by thermal radiation isAssume an opaque grey body that is a diffuse thermal emitter, with the same emissivity $ϵ$ and radiometric absorptance at the surface's current temperature. Its emitting area $A$ sees only the enclosure: the view factor to the walls is one, with no self-viewing surface, opening or intervening obstruction. The intervening medium must neither absorb nor emit appreciable thermal radiation; a vacuum supplies a useful idealization. The enclosure is maintained at uniform $T_s$. The small-body assumption makes that reservoir description physically reasonable; genuinely black isothermal walls supply the incoming irradiance even without an infinite area ratio. Positive ${\dot{Q}}_{\mathrm{rad}}$ means net loss by the surface; negative means net gain. This expression counts thermal radiation alone, not all heat transfer.^[ref]

The reasoning is a two-way energy balance. Black walls supply irradiance $E=\sigma T_s^4$, so the surface absorbs $P_{\mathrm{abs}}=ϵAE$ and emits $P_{\mathrm{emit}}=ϵA\sigma T^4$. Subtract these to obtain ${\dot{Q}}_{\mathrm{rad}}$. For an opaque surface the remaining fraction $1-ϵ$ of incoming radiant flux is reflected; counting emitted plus reflected outgoing radiant flux minus total incoming radiant flux gives the same result. Kirchhoff's law of thermal radiation justifies equal spectral absorption and emission factors; the grey body assumption lets one factor be taken outside both spectral integrals.

Choose $ϵ=0.8$, $A=0.01{\mathrm{m}}^2$, $T=500\mathrm{K}$ and $T_s=300\mathrm{K}$. These are stipulated model inputs, not measured material properties. Using $\sigma \approx 5.670374419\times 10^{-8}\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-4}$ givesThe incoming irradiance is $459.300\mathrm{W}{\mathrm{m}}^{-2}$, corresponding to $4.5930\mathrm{W}$ incident on the surface; only $80\%$ is absorbed. At $T=T_s=300\mathrm{K}$, emitted and absorbed powers are both $3.6744\mathrm{W}$, so radiative equilibrium has zero net exchange despite continuing emission. Changing only emissivity to $0.2$ reduces both one-way powers and the net loss by a factor of four, giving $6.1694\mathrm{W}$ net. A low-emissivity coating therefore suppresses both radiative cooling to colder walls and radiative heating from hotter walls.

Near equal temperatures, write $T=T_s+\Delta T$. Taylor's theorem givesFor constant emissivity, the radiative heat transfer coefficient $h_{\mathrm{rad}}$ is approximately $4.8992\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$ at $T_s=300\mathrm{K}$ and $ϵ=0.8$. This local linear approximation is useful for small temperature differences; the $200\mathrm{K}$ difference in the worked example needs the full fourth-power expression.

A cooling experiment requires a wider first law of thermodynamics balance. In a uniform-temperature model without a heater,where $C(T)$ is the sample's heat capacity and the last two terms are signed outward convection and thermal conduction losses. A vacuum suppresses convection but does not eliminate thermal conduction through supports or sensor leads. A cooling slope therefore cannot establish emissivity unless these paths and heat capacity are independently constrained. Multiple wall temperatures, incomplete view factors, wavelength-selective materials or appreciable intervening absorption require a more complete radiative transfer calculation.

Neuronal paired-pulse recovery probes the effect of an earlier action potential with a later test stimulus. First establish a conditioning action potential; then vary the delay $\Delta$ to a test pulse of specified amplitude, duration and delivery site. Record whether a second action potential occurs, its timing and, where relevant, its waveform. Repeating the test across amplitudes distinguishes recovery of responsiveness to one pulse from recovery of the minimum effective pulse. Direct current clamp stimulation can probe the receiving neuron's excitability; stimulation through a synapse additionally allows synaptic plasticity to alter the input.

A documented whole-nerve teaching experiment keeps a bundle of axons in physiological solution, with its proximal end drawn into a stimulating tube and its distal end into a recording tube. The suction electrodes use short plastic tubes and $1\mathrm{ml}$ syringe-plunger terminals, held by O-rings. Longitudinal electric currents create a voltage drop in the thin solution layer between bundle and tube, read by a cathode-ray oscilloscope. Electrode fit affects amplitude. An immediate stimulation artifact remains even without the nerve, whereas the neural signal arrives after a delay.^[ref][ref]

The paired-stimulus exercise sets $0.1\mathrm{ms}$ pulses to obtain a maximal undistorted A wave of the compound action potential, then increases the spacing. The described first response remains constant while the second grows with spacing. Different axons recover at different times; the summed response therefore grows as more contribute. A smaller second compound action potential does not mean every individual action potential has shrunk. The source contrasts this with a single axon's response or failure. Its response-amplitude recovery criterion is not a universal single-cell threshold criterion.^[ref]

Useful additional controls would include single-pulse baselines interleaved with paired trials, a verified conditioning action potential, enough rest between trials, and fixed temperature, electrode geometry, pulse width and initial membrane potential. Confirm stimulus delivery and amplifier recovery, separate the stimulus artifact from a second neural event, and check whether waveforms overlap before measuring amplitudes. These are proposed checks, not additional results claimed for that teaching experiment. Even a clean compound action potential does not uniquely identify sodium availability: recruitment, propagation and synchronization of different axons influence the summed signal. Nor does failure at a distant recording site alone distinguish failed initiation from failed propagation.

A neuronal refractory period is the change in excitability following an action potential: recent firing changes whether, and how readily, a neuron can fire again. The absolute neuronal refractory period prevents a second regenerative action potential; during the relative neuronal refractory period, some stimuli that work after full recovery still fail. These are useful distinctions, but their durations are not universal constants. The preparation, temperature, starting membrane potential, stimulation protocol and criterion for detecting an action potential must accompany a measured interval.

The mechanism continues the opposing sodium and potassium electric currents described under action potential and the Hodgkin-Huxley model. After a spike, ion-channel inactivation leaves fewer sodium-selective voltage-gated ion channels available for regenerative depolarization, while persisting potassium electrical conductance opposes that depolarization. In the Hodgkin-Huxley model, recovery of $h$ restores sodium availability and relaxation of $n$ reduces potassium electrical conductance. Closed but available channels differ from inactivated channels: opening and recovery from ion-channel inactivation are different processes.^[ref]

This short memory matters for spike trains: a stimulus-only Poisson process can predict very short interspike intervals that a refractory neuron cannot produce. Neuronal paired-pulse recovery makes that memory visible. The phenomenological paired-pulse recovery example below separates a chosen recovery rule, pulse strength and event readout; refractory limits on neuronal firing then shows why an absolute bound is not a prediction of sustained firing rate.

The surface flux of a uniformly heated cylinder is $Ha/2$. Equating it to the Newton-cooling flux $\alpha [T(a)-T_{\mathrm{air}}]$ gives the displayed surface temperature. The full temperature profile adds the internal parabolic excess to this surface value. Electrical power per length equals the surface heat loss per length in steady state.

Newton's law of cooling models an object's temperature $y$ using a positive constant rate $k$ and ambient temperature $T_a$. The displayed ordinary differential equation assumes one temperature represents the object, the environment remains at $T_a$, and the proportional rate stays fixed over the modeled range. It also describes warming when $y<T_a$. These are modeling assumptions: ^[ref] explain the proportional law through a tangent line approximation and caution about large temperature differences.

Dimensional analysis requires $k$ to have units of inverse time. The product $k(t-t_0)$ in the exponential function solution is therefore dimensionless. The equilibrium point $y=T_a$ attracts both warmer and colder starting states: the excess temperature undergoes exponential decay. See initial value problem for computed cooling times, direction field for the family of trajectories, and separable differential equation for a derivation.

Nickel-56 and iron-54 have the same neutron number and differ by two protons. Dividing their nuclear statistical equilibrium abundance formulas cancels the neutron density and leaves the displayed ratio, with $\Delta B=B_{56}-B_{54}$ and $K$ containing temperature-independent statistical and mass factors in the simplified model. For material with equal total proton and neutron numbers and only these nuclei plus free protons, charge balance gives $n_p=2n_{54}$. Thus $n_{56}/n_{54}^3\propto T^{-3}{e}^{\Delta B/(kT)}$. This has a finite temperature maximum only if the signed energy difference is negative; conventional positive total nuclear binding energies for these two actual isotopes instead give a positive difference.

Sadi Carnot investigated what limits the work obtainable by transferring heat between hot and cold bodies. His reversible-engine argument compares machines without tying the ideal performance to a particular working substance. A reversible reference cycle can run backward, so a supposedly better engine could be combined with it to produce an impossible net result. This reasoning underlies Carnot's theorem and the modern bound $\eta \le 1-T_c/T_h$ for reservoirs at absolute temperatures $T_h>T_c$. The temperature notation is modern, rather than Carnot's original formulation.^[ref]

A molecular nitrogen experiment used a $0.5\mathrm{mm}$ pulsed nozzle and two skimmers. A $5\mathrm{cm}$ tube raised the rotational temperature. Opposed vacuum-ultraviolet laser pulses excited selected molecular levels and, about $10\mathrm{ns}$ later, selectively ionized excited nitrogen fragments. A microchannel plate detector, phosphor screen and camera recorded radii.^[ref]

The measured squared radii were fitted against molecular excitation energies, referenced to the parent ground state:The intercept at zero radius gave the excited-product threshold; the fitted slope absorbed the detector scale. A wavemeter monitored the excitation source; scanning the probe covered the recoil Doppler shift. Subtracting the atomic excitation energies converted the intercept to ground-product $D_0$.^[ref]

An apertured grounded plate and detector timing rejected faster competing channels. Selecting fragment electronic states distinguished nearby thresholds. Agreement of $D_e$ across isotopes after zero-point energy correction tested consistency. The method still requires correctly assigned levels, adequate near-threshold signal and stable imaging; a signal drop alone can reflect another channel or a barrier.^[ref]

This measurement determines a state-resolved ground-state dissociation energy. Obtaining a finite-temperature bond-dissociation enthalpy requires the thermal increments above; obtaining a bond potential-well depth requires the zero-point energy correction. The three quantities describe different starting conditions for the same specified separation.

Wilhelm Wien received the award for discoveries concerning the laws of thermal radiation.^[ref] Wien's displacement law connects a blackbody spectrum's peak wavelength to its temperature.

Awarded to Heike Kamerlingh Onnes for investigations of matter at low temperatures, including helium liquefaction. The award's scope includes the experimental means of reaching those conditions.^[ref]

A noise temperature expresses received or internally added noise power on an equivalent temperature scale. For one matched receiving channel in the classical thermal limit, an ideal rectangular bandwidth $\Delta \nu$ gives $P=k_{\mathrm{B}}{T}_N\Delta \nu$, with Boltzmann constant $k_{\mathrm{B}}$. An equivalent noise temperature need not equal a component's physical temperature.

A non-Fermi liquid is an interacting fermionic state whose low-energy behavior is not described by the long-lived quasiparticles of a Fermi liquid. In the conventional Fermi liquid picture, excitations close enough to the Fermi surface become sharp compared with their excitation energy. If that organizing principle fails, extending the same quasiparticle approximation toward zero energy can give the wrong response.

The term describes a failure of one framework, rather than one unique replacement theory. A one-dimensional Luttinger liquid provides a controlled example: its long-wavelength excitations are collective density modes, and its correlation functions need not have the free-fermion form. An anomalous finite-temperature transport curve alone does not establish a particular zero-temperature non-Fermi liquid. Spectral and thermodynamic measurements must be interpreted alongside the relevant dimensionality, interactions and accessible energy range.

A single optical shift represents pressure only to the extent that the calibration's mechanical and thermal conditions apply. Under hydrostatic stress, all directions have the same compressive normal stress. Under deviatoric stress, differently oriented ruby grains can respond differently; spatially varying pressure can also superpose shifted lines. These mechanisms can produce different optical signals at equal mean pressure.^[ref]

The 2001 account describes scanning local line positions and measuring line width over an extended region. The former resolves spatial differences; the latter supplies a rapid warning.^[ref]

Neither a broadened line nor a narrow one uniquely determines the stress tensor. Instrument resolution, temperature variation and unresolved overlapping signals can also affect width. A small observed region can have a narrow line while differing from another region. Compare locations, reference traces and instrument settings before treating width as evidence for a particular mechanical explanation. These are proposed diagnostic checks, not additional historical controls attributed to the 1975 experiment.

Temperature is another independent input. A useful local expansion is $\Delta \lambda \simeq s_{p}p+s_T(T-T_0)$: one measured shift cannot determine both unknowns. Measure or control temperature, or supply a second independently calibrated observable. The coefficients need not remain constant across a large range. Even when a pressure sensitivity transfers over some temperatures, the reference line, line width and practical readout quality can change.

The historical high-temperature account reports degrading pressure measurement quality above about $300{}^{\circ }\mathrm{C}$.^[ref]

At zero temperature, a rigid spin-degenerate transmission spectrum with symmetric source/drain bias has ${\mu }_{S,D}=E_F\pm eV/2$ and the displayed differential conductance. Half-integer plateaus in units $2e^2/h$ occur when the two bias-window edges sample different channel counts. The bias convention and electrostatic response matter.

When radiative rates compete with or dominate collisions, level populations need statistical equilibrium coupled to radiative transfer, rather than thermal population laws at the local temperature. Radiation from other depths can make the local radiation field differ from the Planck function. This changes ionization and spectral-line strengths, so a geometrically thin atmosphere may still require non-LTE abundance modelling.

Normal hydrogen denotes the $3:1$ orthohydrogen-to-parahydrogen mixture, the high-temperature equilibrium limit. Near room temperature the equilibrium ratio is already close to this limit. Cooling an isolated sample need not change its composition on the measurement timescale: ordinary collisions can redistribute rotational populations within each spin species much faster than they interconvert the spin species. A kinetically frozen composition can therefore coexist with rotational thermal equilibrium separately within each species.

The distinction was historically decisive. The institutional account Die Entdeckung des para-Wasserstoffs, in its discussion of the rotational heat capacity, explains that assuming an equilibrating composition missed the observed behavior, whereas freezing the high-temperature ratio reproduced it. It also records Karl Friedrich Bonhoeffer and Paul Harteck's 1929 preparation using a catalyst and thermal conductivity detection.^[ref]