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

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.