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With the Seebeck coefficient convention $dV=-SdT$ at zero electric current, and the transported-heat convention of Peltier coefficient, the Kelvin thermoelectric relation is$T$ is absolute temperature in kelvins; use consistent material references. This scalar linear response relation assumes local thermal equilibrium and reciprocal transport without time-reversal-breaking fields or order. Onsager reciprocal relations supply the symmetry; conservation of energy alone is insufficient. A magnetic field requires a reversed-field reciprocity comparison.^[ref]

In particular, the A-minus-B thermocouple coefficient in Seebeck effect is $S_A-S_B$, whereas absorbed junction heat for conventional current A to B uses $S_B-S_A$. There is no contradiction: terminal voltage order and crossing direction are separate choices. Multiplying a measured voltage by an absolute temperature without specifying these choices does not determine junction heating.

A short reciprocity calculation makes the factor $T$ visible. At a common reference temperature, write the local linear response with electrical driving field $\mathcal{E}=-\nabla V$ and heat flux $q$ asAt $j=0$, the Seebeck coefficient convention gives $S=L_{12}/(T{L}_{11})$. At $\nabla T=0$, the Peltier coefficient is $q/j=L_{21}/L_{11}$. Thus $L_{21}=L_{12}$ gives $\Pi =TS$. The chosen forces are conjugate to the fluxes in the local entropy production, $j\mathcal{E}/T+q(-\nabla T)/T^2$; arbitrary rescaling of just one force would spoil this argument.

The relation is local. Varying $S(T)$ permits distributed Thomson effect heating, with $\tau =TdS/dT$; the constant-coefficient model below omits it. Endpoint substitutions into a constant-property formula can therefore break its energy accounting.

Lars Onsager formulated reciprocal relations connecting cross-coupled transport coefficients near thermodynamic equilibrium. The Onsager reciprocal relations link microscopic reversibility with symmetry properties of macroscopic linear response, subject to the relevant time-reversal and field conventions.^[ref]

An Onsager reciprocal relation connects cross-couplings between near-equilibrium transport processes. Choose thermodynamic forces $X_j$ and fluxes $J_i$ so that their products give the entropy production, and write $J_i={\sum }_j{L}_{ij}{X}_j$. With microscopic reversibility, no imposed time-reversal-breaking field and variables with the appropriate matching time-reversal parity, the coefficients obey $L_{ij}=L_{ji}$. The choice of conjugate force and flux matters: arbitrary rescaling of just one coefficient need not preserve its numerical symmetry.

For coupled thermal and electrical transport, a thermal driving force can induce charge flow while an electrical driving force induces heat flow. Reciprocity relates those cross effects, reducing the number of independent transport measurements. Lars Onsager's 1931 paper develops this reasoning from microscopic reversibility and distinguishes it from thermodynamic positivity alone.^[ref] External magnetic fields and odd time-reversal variables require a generalized parity-sensitive relation rather than an unconditional symmetric matrix.