Thermoelectric effect

Definition

Let an isotropic conductor carry an electric current density $J$ and a heat flux $J_Q$ at absolute temperature $T(x)$. Let $E$ be the effective electric field, the electric field plus the gradient of the chemical potential of the carriers divided by their charge. In the linear regime the conductor obeys $$J = \sigma (E - S \nabla T), \qquad J_Q = \Pi J - \kappa \nabla T,$$ where $\sigma > 0$ is the electrical conductivity and $\kappa > 0$ the thermal conductivity at zero current. The thermoelectric effect is the coupling of $J$ and $J_Q$ through the coefficients $S$ and $\Pi$, in three forms.

1. Seebeck effect: at zero current, a temperature gradient produces the field $E = S \nabla T$; $S$ is the Seebeck coefficient.
2. Peltier effect: at uniform temperature, a current carries the heat flux $J_Q = \Pi J$; $\Pi$ is the Peltier coefficient.
3. Thomson effect: a steady current along a temperature gradient produces heat reversibly at the rate $-\mathcal{K}\, J \cdot \nabla T$ per unit volume. This adds to the Joule heating $\abs{J}^2/\sigma$, and $\mathcal{K}$ is the Thomson coefficient.