2.3.3 Principle of Detailed Balance

In equilibrium the distribution function is known and the left hand side of the Boltzmann equation is equal to zero^2.19:

$$(1-f_\mathrm{eq}(\vec{r},\vec{k},t))\int S(\vec{k}^{'},\vec{k},\vec{r},t)f_\mathrm{eq}(\vec{r},\vec{k}^{'},t)\,d\vec{k}^{'}-$$

$$-f_\mathrm{eq}(\vec{r},\vec{k},t)\int (1-f_\mathrm{eq}(\vec{r},\vec{k}^{'},t))S(\vec{k},\vec{k}^{'},\vec{r},t)\,d\vec{k}^{'}=0,$$ (2.60)

which is valid for any quasi-momenta $\hbar\vec{k}$. To satisfy this equation for all quasi-momenta $\hbar\vec{k}$ the following equality must be valid:

$$f_\mathrm{eq}(\vec{r},\vec{k}^{'},t)(1-f_\mathrm{eq}(\vec{r},\vec... ...{k},t)(1-f_\mathrm{eq}(\vec{r},\vec{k}^{'},t))S(\vec{k},\vec{k}^{'},\vec{r},t).$$ (2.61)

Using the explicit form of the Fermi-Dirac distribution function one obtains from (2.61):

$$S(\vec{k}^{'},\vec{k},\vec{r},t)\exp\bigg[\frac{\epsilon(\vec{k})... ...k},\vec{k}^{'},\vec{r},t)\exp\bigg[\frac{\epsilon(\vec{k}^{'})}{k_{B}T}\biggr].$$ (2.62)

Equation (2.62) is called the principle of detailed balance and relates the probabilities of forward and backward processes. For elastic processes, $\epsilon(\vec{k})=\epsilon(\vec{k}^{'})$, (2.62) gives:

$$S(\vec{k}^{'},\vec{k},\vec{r},t)=S(\vec{k},\vec{k}^{'},\vec{r},t),$$ (2.63)

that is for elastic processes the scattering probabilities of forward and backward processes are equal. S. Smirnov: