heat generation through muscle activity associated with shivering.

heat production through mechanisms other than shivering. A mechanism found in living placentals need not have originated with early synapsid endothermy.

A thermoelectric generator converts part of a heat flow driven by a temperature difference into electrical output through the Seebeck effect. Thermoelectric legs connect hot and cold sides; their electrical arrangement combines the generated voltages while their thermal paths carry heat between the sides.

A load draws electric current, so delivered power depends on both open-circuit voltage and internal resistance. Shorting the output gives current but zero terminal voltage, while an open circuit gives voltage but no useful delivered power. Thermal contacts and heat leakage alter the actual leg temperatures, so a material's thermoelectric figure of merit alone does not specify the complete generator's efficiency.

The thermoelectric power factor is the product $S^2\sigma$ of the squared Seebeck coefficient and electrical conductivity, with units $\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-2}$. It combines a material's voltage response to a temperature gradient with its ability to carry electric current. For example, $S=200\mu \mathrm{V}{\mathrm{K}}^{-1}$ and $\sigma =10^5\mathrm{S}{\mathrm{m}}^{-1}$ give $S^2\sigma =0.004\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-2}$.

At fixed geometry and imposed temperature difference, a constant-property thermoelectric generator behaves as a voltage source with internal electrical resistance. Matching the external electrical resistance to it maximizes output power; the material dependence of that maximum is proportional to the thermoelectric power factor. Maximizing thermal efficiency also requires controlling thermal conductivity, which is why the thermoelectric figure of merit includes the thermal denominator.^[ref]

Band convergence brings distinct extrema of an electronic band structure close enough in energy that several valleys contribute to charge carrier transport at the operating temperature. Increasing the effective valley degeneracy can improve the balance between Seebeck coefficient and electrical conductivity, rather than merely increasing the number of carriers in one valley. Composition and semiconductor doping can tune this energy separation. A primary example is the lead telluride-selenide alloy study that combined measured transport with a multiband model.^[ref]

The thermoelectric power factor is not determined by valley degeneracy alone. Additional inter-valley scattering can reduce charge carrier mobility, so a larger number of populated valleys need not improve transport. The spacing of the valleys in both energy and wave vector, and the relevant electron-phonon interactions, matter.^[ref]

For a constant-property thermoelectric module consisting of two material legs, let ${\alpha }_{np}=S_p-S_n$ be their Seebeck coefficient difference, $R$ their total series electrical resistance, and $K$ their total parallel thermal conductance. The couple's thermoelectric figure of merit isIts units are ${\mathrm{K}}^{-1}$; multiply by an absolute temperature to form a dimensionless figure. This expression assumes that contact losses are absent or consistently included in $R$ and $K$.

For equal leg lengths $l$ and areas $A$, with equal electrical resistivity $\rho$ and thermal conductivity $\kappa$, one has $R=2\rho l/A$ and $K=2\kappa A/l$. Consequently $Z_{\mathrm{pair}}={\alpha }_{np}^2/(4\rho \kappa )$. The factor four follows from combining two series resistances and two parallel conductances; it disappears only if the symbols have already been defined as pair properties. With opposite equal Seebeck coefficients $S_p=-S_n=S$, the result reduces to $S^2/(\rho \kappa )$.

Natural variables are the independent variables appearing in the differential that directly identifies a potential's conjugate quantities. For $U(S,V,N)$, $dU=TdS-pdV+\mu dN$. Legendre transforms exchange a variable for its conjugate: $F=U-TS$ has natural variables $T,V,N$, and $G=F+pV$ has $T,p,N$. Equilibrium minimum principles require the corresponding environmental constraints and fixed conserved amounts.

The ideal Atkinson cycle considered here has reversible adiabatic compression, isochoric heat addition, a longer reversible adiabatic expansion and isobaric heat rejection. Its expansion ends at the initial pressure rather than at the initial volume. With constant ideal-gas heat capacities, $r=V_1/V_2$ and $r_p=p_3/p_2$, the temperatures are $T_2=T_1{r}^{\gamma -1}$, $T_3=r_p{T}_2$, and $T_{4^{\prime }}=T_1{r}_p^{1/\gamma }$. This ideal cycle is distinguished from the detailed valve timing and losses of a practical engine.

The composition coordinate at the critical endpoint where coexisting phases become identical. Its value depends on the system and control parameters.

For $p=n{k}_{B}T-b{n}^2/2+c{n}^3/6$ with $b,c>0$, the first two density derivatives vanish at $n_c=b/c$, $k_B{T}_c=b^2/(2c)$ and $p_c=b^3/(6c^2)$. The scaled equation is $p/p_c=x^3-3x^2+3tx$, with $x=n/n_c$ and $t=T/T_c$.

A thermal radiation shield interposes a surface to reduce radiative heat exchange between an apparatus and its surroundings. temperature, emissivity and view factors determine its effectiveness; conductive supports and gas transport remain separate heat paths. A thermal radiation shield is not automatically an ionizing-radiation shield.

For one quantum harmonic oscillator of frequency $\Omega$, $Z=e^{-x/2}/(1-e^{-x})$ with $x=\hslash \Omega /(k_{B}T)$. Differentiating $\ln Z$ gives $U=\hslash \Omega [1/2+1/(e^x-1)]$ and $C_V=k_B{x}^2{e}^x/(e^x-1{)}^2$. Zero-point energy leaves $U\to \hslash \Omega /2$ as $T\to 0$ but contributes no heat capacity. Excitation freeze-out makes $C_V$ exponentially small, while high temperature gives $U\sim k_{B}T$ and $C_V\to k_B$.

The tensor relating a small unconstrained thermal strain to a temperature change: ${\mathbf{\epsilon }}^{\mathrm{th}}=\mathbf{\alpha }\Delta T$. A crystal orientation rotates this tensor into laboratory coordinates.

The emission coefficient per volume, frequency and solid angle of a Maxwellian plasma has $j_{\nu }\propto n_e{\sum }_i{Z}_i^2{n}_i{T}^{-1/2}{e}^{-h\nu /(k_{B}T)}{\bar{g}}_{\mathrm{ff}}$. Define the angular convention before supplying a numerical coefficient. The high-frequency cutoff measures temperature, while the normalization measures a density-squared emission integral. The radio spectrum is almost flat when optically thin and rises as ${\nu }^2$ when optically thick at fixed projected area.

For a hydrogen-dominated fully ionized plasma, the line-of-sight emission measure is $\mathrm{EM}=\int n_e^{2}dl$. Its units distinguish it from a volume-integrated emission measure $\int n_e^{2}dV$. Thermal free-free optical depth is proportional to $\mathrm{EM}{T}^{-3/2}{\nu }^{-2}{\bar{g}}_{\mathrm{ff}}$ in the radio limit. Inferring density also requires depth, ionization and clumping assumptions.^[ref]

The fraction $f_V$ of a geometric volume occupied by a specified emitting gas phase is its volume filling factor. For approximately uniform density within that phase, a density-squared luminosity scales as $n_e^2{f}_{V}V$. Without knowledge of $f_V$, emission gives an rms-density combination rather than a unique mean gas density. This filling factor is distinct from the fraction of a telescope beam occupied on the sky.

For a uniform ionized hydrogen column with known temperature and fixed projected angular area, free-free brightness and optical depth depend on $\mathrm{EM}=n_e^{2}L$. The displayed rescaling leaves the whole ideal spectrum unchanged. density and path length cannot be separated without additional information. A known geometry/distance, or a distinct column diagnostic proportional to $n_{e}L$, can break the degeneracy; gas clumping factor and temperature structure complicate that inference.

In an ideal axisymmetric volume-conserving crystal grain model, directional growth is proportional to $1-3⟨{\cos }^2\theta ⟩$, where $\theta$ is the angle between that direction and the crystal c-axis. Random crystallographic texture has $⟨{\cos }^2\theta ⟩=1/3$.