Within the single relaxation time approximation, the collision term in the Boltzmann equation is approximated as $$\Big(\frac{\partial f}{\partial t}\Big)_{\rm coll}=-\frac{(f-f_{\rm eq})}{\tau}$$ where $f\equiv f(\textbf{r},\textbf{v},t)$ is the distribution out of equilibrium and $f_{\rm eq}=f_{\rm eq}(\textbf{v})$ is the Maxwell-Boltzmann distribution, for example.

To show that at $t\to \infty$ the $f$ relaxes to $f_{\rm eq}$, one needs to specify the nonequilibrium distribution $f(\textbf{r},\textbf{v},t)$ at some initial time $t=t_0$. How does one specify that?


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Note that this is really a non-linear equation, because $f_{eq}$ depends on the 0'th moment (particle number=chemical potential), 1st moment (mean velocity) and 2nd moment (mean energy=temperature) of $f$. As a result, establishing convergence is not entirely trivial. Also note that $f$ has non-trivial $x$ dependence, so you really expect convergence to local equilibrium, with $T(x,t)$, $\mu(x,t)$ and $\vec{u}(x,t)$ governed by solutions to the Navier-Stokes equation.

Regarding initial conditions we expect that any positive, reasonably smooth $f(x,0)$ provides an acceptable initial condition. Note that existence and uniqueness of the full Boltzmann equation (with a 2-body collision kernel) has been established (the corresponding problem for Navier-Stokes is a Millenium Prize Problem). There is a fair amount of work on the convergence of the latticized version of the Boltzmann equation with a BGK kernel (known as the lattice Boltzmann equation), see, for example, the book by Succi. A little bit of googling also provides references on existence and uniqueness for the continuum case, see here.


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