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According to Ashcrift & Mermin (chapter 13):

If the electrons about [itex]r[/itex] have equilibrium distribution appropriate to local temperature [itex]T(r)[/itex],

[tex]g_n (r,k,t)=g_n^o (r,k)=\frac {1}{ exp^{(\epsilon_n (k) -\mu (r))/kT} +1} (formula 13.2)[/tex] then collisions will not alter the form of distribution function. We know in the time interval dt a fraction [itex]\frac{dt}{\tau_n(r,k)}[/itex] 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:

[tex]g_n (r,k,t)=\frac{dt}{\tau_n(r,k)} g_n^o (r,k) (formula 13.3)[/tex]

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 [itex]g_n (r,k,t)=g_n^o (r,k)[/itex] which is 13.2. on the other hand using this interpretation, near position r at time interval dt due to collisions[itex] \frac{dt}{\tau_n(r,k)} [/itex] electrons will leave the point (r,n,k) to another point say [itex](r,n^\prime,k^\prime)[/itex] so that [tex]dg_n (r,k,t)=g^0_n{\prime} (r,k^\prime,t^\prime)- g_n^0 (r,k,t)[/tex] where [itex]t^\prime=t+dt[/itex].

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

Could anyone please help me?

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