3.5.5.3 Energy Balance Equation

For the weight $\ensuremath{X}=\ensuremath{\mathcal{E}}$ , the energy balance equation is obtained. Starting from (3.29) with a microscopic relaxation time approximation, one obtains

$$\ensuremath{\ensuremath{\partial_{t} \ensuremath{w}}} + \ensurema... ...\rangle}}{\ensuremath{{\tau}}} + \ensuremath{G_\nu^\ensuremath{\mathcal{E}}}\,.$$ (3.64)

While the Poisson bracket in the second term vanishes according to (A.8), the average in the third term can be identified as the particle current. $\ensuremath{G_\nu^\ensuremath{\mathcal{E}}}$ represents the net energy generation rate by recombination processes. Assuming Boltzmann statistics in the diffusion approximation, the averages on the right side can be expressed as $3/2 \ensuremath{\nu}k_\ensuremath{\mathrm{B}} \ensuremath{T_\ensuremath{\nu}}$ and $3/2 \ensuremath{\nu}k_\ensuremath{\mathrm{B}}\ensuremath{T_{\mathrm{L}}}$ , respectively. Thus, the final energy balance equation reads

$$\frac{3}{2} k_\ensuremath{\mathrm{B}}\ensuremath{\ensuremath{\par... ...L}}}}{\ensuremath{{\tau}}} - \ensuremath{G_\nu^\ensuremath{\mathcal{E}}} = 0\,.$$ (3.65)

Having a closer look at the final result, one can easily recognize the total energy conservation character of this equation. The change of total energy within an infinitesimal small volume is equal to the energy influx minus the energy exchanged with the lattice. In the homogeneous case without external driving forces, the last term ensures the relaxation of the carrier temperature $\ensuremath{T_\ensuremath{\nu}}$ to $\ensuremath{T_{\mathrm{L}}}$ .

M. Wagner: Simulation of Thermoelectric Devices