The quantum Boltzmann equation, also known as the Uehling–Uhlenbeck equation, is the quantum mechanical modification of the Boltzmann equation, which gives the nonequilibrium time evolution of a gas of quantum-mechanically interacting particles. Typically, the quantum Boltzmann equation is given as only the “collision term” of the full Boltzmann equation, giving the change of the momentum distribution of a locally homogeneous gas, but not the drift and diffusion in space. It was originally formulated by L.W. Nordheim (1928), and by and E. A. Uehling and George Uhlenbeck (1933).
Application to semiconductor physics
A typical model of a semiconductor may be built on the assumptions that: • The electron distribution is spatially homogeneous to a reasonable approximation (so all x-dependence may be suppressed) • The external potential is a function only of position and isotropic in p-space, and so \mathbf{F} may be set to zero without losing any further generality • The gas is sufficiently dilute that three-body interactions between electrons may be ignored. Considering the exchange of momentum \mathbf{q} between electrons with initial momenta \mathbf{k} and \mathbf{k_1}, it is possible to derive the expression \begin{aligned} \mathcal{Q}[f](\mathbf{k}) &= \frac{-2}{\hbar (2\pi)^5}\int d\mathbf{q} \int d\mathbf{k_1} &\quad\times \delta\!\left(\frac{\hbar^2}{2m}(|\mathbf{k-q}|^2+|\mathbf{k_1+q}|^2-\mathbf{k}_1^2-\mathbf{k}^2)\right) \\ &\quad\times \left[f_{\mathbf{k}} f_{\mathbf{k_1}} (1-f_{\mathbf{k-q}})(1-f_{\mathbf{k_1+q}}) - f_{\mathbf{k-q}} f_{\mathbf{k_1+q}} (1-f_{\mathbf{k}})(1-f_{\mathbf{k_1}})\right] \end{aligned} == References ==