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Consider the one-dimensional diffusion equation on the whole real line,with initial value one for $x<0$ and zero for $x>0$. This is an idealized Heaviside step function profile, not a measured material or an apparatus specification. The heat kernel gives a convolution with the initial profile:The change of variables $s=(x-y)/(2\sqrt{Dt})$ turns the convolution into a Gaussian integral tail. The resulting similarity variable $\eta =x/(2\sqrt{Dt})$ shows that the transition width grows like $\sqrt{Dt}$: multiplying elapsed time by four doubles every fixed-profile distance from the original interface.

At every positive time, $u(0,t)=1/2$, while $u(x,t)\to 1$ as $x\to -\infty$ and $u(x,t)\to 0$ as $x\to +\infty$. For every fixed $x\ne 0$, the limit as $t\to 0^{+}$ returns the specified initial value. The value assigned at the single initial discontinuity does not change this convolution. The error function therefore explains smoothing without pretending that a discontinuous initial profile remains sharp.

Direct differentiation checks the partial differential equation. Writing $\eta =x/(2\sqrt{Dt})$ givesThe model assumes constant diffusion coefficient, no advection, and no finite-domain boundary effects. For an interpretation as heat conduction, the heat equation applies to a normalized temperature difference under its own material assumptions.

For conductivity $\lambda ={\lambda }_{0}T$, no cooling in the hot phase and a zero-temperature boundary at $m=0$, the mass-coordinate equation is linear diffusion. Its similarity profile solves $f^{\prime \prime }+\xi f^{\prime }/2=0$, $f(0)=0$, $f(\infty )=1$, giving the displayed error function. The thermal layer broadens as the square root of time. Conducted energy enters the cold radiating reservoir at rate per area $L={\lambda }_{0}A{T}_m(0)=A{T}_0\sqrt{{\lambda }_0}/\sqrt{\pi \tau }$, hence $L\propto t^{-1/2}$.

Leo Esaki's 1958 report describes alloy-fabricated germanium p-n junctions, with acceptor impurity concentration $1.6\times 10^{19}{\mathrm{cm}}^{-3}$ and donor impurity concentration approximately $10^{19}{\mathrm{cm}}^{-3}$. Its readout was the current-voltage characteristic: a forward peak occurred at $0.035\pm 0.005\mathrm{V}$. The report says specimens could be reproduced; above $0.3\mathrm{V}$ the forward branch approximately followed the ordinary Shockley diode equation.^[ref]

In his 1973 lecture, Leo Esaki recalls an earlier junction whose strong room-temperature reverse conduction preceded a negative-slope forward segment revealed by cooling. Its estimated zero-bias width was about $200$ ångströms ($20\mathrm{nm}$), supported by capacitance measurements. Increasing semiconductor doping further made the negative slope visible across the measured temperatures, including room temperature. Thus preparation and temperature variation were experimental controls, not merely changes to a theoretical diagram. The low onset voltage argued against avalanche production of electrons and electron holes; the energy band explanation connected the forward decline to loss of available final states.^[ref]

The historical readouts were junction electric current versus applied electric potential difference and a separate capacitance check; preparation and temperature were varied. The cited descriptions do not identify a meter model or give a complete wiring diagram.

For a modern repetition, measure junction voltage directly while stepping the supply and recording electric current, repeat both sweep directions, and compare temperatures and differently doped specimens. These are proposed controls, not a reconstruction of the historical apparatus. Lead electrical resistance, Joule heating, contact quality and voltage resolution can obscure the result. The tunnel-diode load-line stability calculation also predicts that an external resistor can make a negative-slope operating point inaccessible in a slow sweep. A jump in terminal readings then need not mean the underlying current-voltage characteristic is discontinuous.

A graduated tube for confining a gas, measuring its volume, and often examining changes after a chemical reaction. The reading requires temperature, pressure and vapor pressure corrections.

The transformation $L\to \alpha +\beta$ in which one liquid produces two solid phases at a binary eutectic composition and temperature under fixed pressure. The resulting two-phase constituent must be distinguished from either individual thermodynamic phase.

An eutectic system has a composition and temperature at which a liquid is in equilibrium with two distinct solid phases. Cooling liquid of that composition through the eutectic temperature produces both solid phases together.

The eutectic temperature is the temperature of simultaneous liquid–two-solid phase coexistence in an eutectic system. In directional freezing of a solution away from the eutectic composition, a mushy layer can end at this temperature, where its remaining liquid solidifies.

Evaporation transfers molecules from a liquid surface into the surrounding vapour. It can occur below the boiling temperature, and competes with condensation from the vapour. Its net rate depends on the surrounding vapor pressure and transport conditions. Removing latent heat through evaporation can cool the remaining liquid.

The ratio of an evaporation cutoff to thermal energy. Keeping the cutoff fixed while lowering temperature increases this ratio and reduces population in the escaping tail.

The equal magnitudes of the opposing partial electrode-reaction current densities at equilibrium. Their net current is zero. The area convention, activities and temperature must be specified; an exchange current density alone does not specify a nonlinear current-potential curve.

For one mole, a simple repulsive-volume model uses $p(V-b)=RT$ with $b>0$ and domain $V>b$. The Maxwell relation and first law of thermodynamics give $(\partial U/\partial V{)}_T=T(\partial p/\partial T{)}_V-p=0$, so $U=U(T)$. If its molar heat capacity at constant volume $C_V$ is temperature independent, it is constant, $U=C_{V}T+U_{\ast }$ and $C_p=C_V+R$. Reversible adiabatic processes obey $T(V-b{)}^{R/C_V}=\mathrm{constant}$ or $p(V-b{)}^{1+R/C_V}=\mathrm{constant}$. All positive-temperature isotherms are strictly decreasing and convex; this single-phase model contains no attractive liquid-gas coexistence mechanism. The parameter $b$ is an effective unavailable molar volume, not automatically the literal geometric volume of the molecules.

An exergonic reaction has negative molar reaction change in Gibbs free energy under the specified conditions. At fixed temperature and pressure, its forward direction is thermodynamically favourable; its reaction rate remains a separate question. See OpenStax Biology 2e, 6.2: exergonic reactions.

For a closed system with fixed composition in an environment at temperature $T_R>0$ and pressure $p_R$, define $A=U+p_{R}V-T_{R}S$, where $U$ is internal energy, $V$ is volume and $S$ is entropy. The exergy is this availability relative to the equilibrium reference state, $A-A_{\mathrm{eq}}$. When the combined system and environment are isolated except for the specified exchanges, $\Delta S_{\mathrm{tot}}=-\Delta A/T_R$, so the Second law of thermodynamics makes $A$ decrease toward its constrained minimum. Away from equilibrium, the environmental $T_R,p_R$ must not be replaced by the system's instantaneous values.

An exoplanet atmosphere controls the planet's spectrum, heat redistribution, chemistry, clouds, and atmospheric escape.

An exoplanet cloud is a suspended condensate aerosol that scatters or absorbs radiation. Its particle sizes, composition, vertical extent, and horizontal coverage shape transmission, reflection, and thermal-emission spectra.

For a closed system with one specified chemical reaction, the extent of reaction $\xi$ records progress from $\xi =0$ through $n_i=n_{i,0}+{\nu }_i\xi$. Here $n_i$ is the amount of substance of species $i$, and the signed stoichiometric vector has negative entries for reactants and positive entries for reaction products. For the water chemical equation in the order ${\mathrm{H}}_2,{\mathrm{O}}_2,{\mathrm{H}}_2\mathrm{O}$, it is $(-2,-1,2)$. The normalization of $\xi$ depends on the chosen coefficients. See IUPAC Gold Book, extent of reaction.

With particle numbers instead of molar amounts, a molecular extent ${\xi }_N$ obeys $d{N}_i={\nu }_{i}d{\xi }_N$, and the chemical potentials are measured per particle. At fixed temperature and volume, the Helmholtz free energy then satisfies $dF=({\sum }_i{\nu }_i{\mu }_i)d{\xi }_N$. An interior chemical equilibrium requires ${\sum }_i{\nu }_i{\mu }_i=0$; the molar convention gives the same relation with chemical potentials per mole.

The Faraday effect is magnetic field-induced circular birefringence: opposite circular polarizations acquire different wave phases, rotating transmitted linear polarization. For a transparent medium with isotropy before applying the magnetic field, in the linear magnetic-response regime at fixed vacuum wavelength ${\lambda }_0$ and temperature $T$,Here $V$ is the Verdet constant, $L$ is the material path length, and $B_{\parallel }$ is the applied magnetic flux density component along the ray. This $\mathbf{B}$ is the external bias, distinct from the oscillating magnetic field of the light itself. Michael Faraday discovered the effect in 1845.^[ref]

Fix the sign before using the formula. On the forward path choose a right-handed Cartesian coordinate system, propagation along $+z$, and measure $\theta$ from $+x$ toward $+y$ for the electric field's oscillation line. Positive $\theta$ is counterclockwise in the $x$-right, $y$-up end view seen from $z>0$, looking toward the source. Then $B_{\parallel }=B_z$ and positive $V$ means positive rotation for positive $B_z$. This convention differs from the clockwise-positive convention in optical rotation. A linear polarization line is unchanged by $180^{\circ }$; a measured orientation determines accumulated rotation only modulo $\pi$, unless continuity selects a branch.

For uniform $B_{\parallel }$, the formula reduces to $\theta =VBL$. Its appeal is that a weak change of refractive index becomes a visible change of orientation. Magnetic reversal measurement of Faraday rotation turns this into an inference of $V$ or of the magnetic field path integral. Faraday rotation on a round trip explains the directional behavior used by an optical isolator. In an astronomical cold plasma, the high-frequency limit instead leads to the rotation measure law $\theta =\mathrm{RM}{\lambda }_0^2$; that wavelength dependence is not a universal law for transparent solids.

A William & Mary teaching experiment places a $10\mathrm{cm}$ glass rod inside a solenoid. Align a polarized helium-neon laser through the rod's centre onto a photodiode after a rotatable polarimeter analyzer. First scan the analyzer from maximum to minimum transmission in $5^{\circ }$ steps and record detector voltage, establishing the angular response. A function generator and electronic amplifier then drive sinusoidal electric current through the solenoid; a multimeter measures the electric current and an oscilloscope reads the photodiode. Trigger the oscilloscope from the generator, remove the large constant detector component by AC coupling, and average repeated traces. Convert the small synchronized voltage modulation to rotation using the calibrated angular slope. The manual fits rotation against magnetic flux density, using its stated coil calibration and converting measured root mean square current to peak current. This is a specified laboratory procedure, rather than measured results reproduced here.^[ref]

To test the inference, add matched zero-electric current readings before and after each reversal, hold the analyzer fixed during paired readings, and compare equal positive and negative electric currents. A blocked beam tests electrical pickup in the detector channel. These are proposed controls. Joule heating scales as current squared, so reversing electric current at equal magnitude helps separate an odd rotation from even heating effects; it does not remove drift between readings. Record temperature and allow repeatable settling. Changes in transmission, mechanical motion, stressed-glass birefringence and detector pickup can imitate a signal. Synchronized averaging suppresses unsynchronized noise, but cannot reject an artifact synchronized to the drive.

Quantitative extraction of the Verdet constant also needs the integral of magnetic flux density through the rod: a finite solenoid can have a nonuniform axial magnetic field. Use its calibrated path integral, or justify a uniform-field approximation; do not silently multiply a central field by the whole rod length. A signed reversal or phase-sensitive readout tests the part odd in applied magnetic field. Its agreement with the simple rotation law supports the model under these controls, rather than proving that every change of transmitted light intensity is Faraday rotation.

Michael Faraday's 1845 experiment combined solid carbon dioxide, diethyl ether and reduced pressure. Removal of vapour cooled the bath further. He reported a low alcohol-thermometer indication near $-166{}^{\circ }\mathrm{F}$, while explicitly discussing lag and uncertainty in the graduation at low temperature.^[ref]