# Specifying the initial nonequilibrium distribution $f(\textbf{r},\textbf{v},t)$ in Boltzmann equation?

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?

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.