1. Sep 8, 2014

### hokhani

I have seriously stocked in the subject below.

According to Ashcrift & Mermin (chapter 13):

If the electrons about $r$ have equilibrium distribution appropriate to local temperature $T(r)$,
$$g_n (r,k,t)=g_n^o (r,k)=\frac {1}{ exp^{(\epsilon_n (k) -\mu (r))/kT} +1} (formula 13.2)$$ then collisions will not alter the form of distribution function. We know in the time interval dt a fraction $\frac{dt}{\tau_n(r,k)}$ of electrons in band n with wave vector k near position r will suffer a collision that does alter their band index and/or wave vector. If the above form of distribution function is nevertheless to be unaltered, then the distribution of those electrons that emerge from collisions into band n with wave vector k during the same interval must precisely compensate for this loss. Thus:
$$g_n (r,k,t)=\frac{dt}{\tau_n(r,k)} g_n^o (r,k) (formula 13.3)$$

I don’t know how is 13.3 obtained.

In fact I don’t know what the text means by form of distribution function in the expression “collisions will not alter the form of distribution function”? On one hand if the electrons near r which left the point (n,k) due to collision are precisely compensated then at a specific (r,k) the distribution function doesn’t have to change and the expression means that $g_n (r,k,t)=g_n^o (r,k)$ which is 13.2. on the other hand using this interpretation, near position r at time interval dt due to collisions$\frac{dt}{\tau_n(r,k)}$ electrons will leave the point (r,n,k) to another point say $(r,n^\prime,k^\prime)$ so that $$dg_n (r,k,t)=g^0_n{\prime} (r,k^\prime,t^\prime)- g_n^0 (r,k,t)$$ where $t^\prime=t+dt$.

However if we were to accept this, how can we consider the $dg_n$ as the number of electrons which have left $(r,n,k)$ toward $(r,n\prime,k\prime)$ due to collision at time interval dt namely$\frac{dt}{\tau_n(r,k)} g_n^o (r,k)$? If so, we necessarily must have just prior to collision $g^0_n(r,k,t)= g^0_n\prime(r,k\prime,t)$ so that this amount of electrons that enter there, perform a change in distribution function as much as$\frac{dt}{\tau_n(r,k)} g_n^o (r,k)$!