# Non-equilibrium electronic distribution in the time-relaxation approximation - Which is the boundary condition?

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In Chapter 13 of Ashcroft-Mermin - "Solid State Physics", the following non equilibrium electronic phase-space distribution for the semiclassical electrons in a periodic crystal is derived: $$g(\mathbf r , \mathbf k , t)= g_0(\mathbf r ,\mathbf k )-\intop _{-\infty} ^{t} \text {d} t' \dfrac{\text d g_0 (\mathbf r(t'),\mathbf k (t'))}{\text d t'}\exp [-\intop _{t'} ^t \frac{\text {d} s}{ \tau (\mathbf r (s) , \mathbf k(s))} ],$$ where $g_0$ is Fermi-Dirac's distribution with a local $T(\mathbf r)$ and $\mu (\mathbf r)$, and $\mathbf r(t'),\mathbf k (t')$ is the semiclassical phase space trajectory which passes through $\mathbf r, \mathbf k$ at time $t$.

I understand that this is a solution of Boltzmann's transport equation for the semiclassical motion, in the time relaxation approximation i.e.:$$\dfrac{\partial g}{\partial t}_\text{coll.}=-\dfrac{g-g_0}{\tau}.$$

But which is the appropriate boundary condition to reproduce this solution? A possibility is: $$g(\mathbf r , \mathbf k ,-\infty) = g_0 (\mathbf r , \mathbf k),$$ where the stationary $g_0(\mathbf r ,\mathbf k)$ is itself a solution if the temperature gradient and the electromagnetic field are zero at $t=-\infty$.

However, Mermin's derivation does not mention this (nor any) initial condition, so I suppose that the solution is somewhat more general. Indeed, I'm wondering if for any initial condition the solution must tend asimptotically to this one.

Any help is appreciated, thank you.

This post imported from StackExchange Physics at 2016-02-07 12:59 (UTC), posted by SE-user pppqqq

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