The Boltzmann equation without collision operator $\Omega$ is as follows: $$\dfrac{\partial f}{\partial t} + \mathbf v \cdot \nabla f = 0 \quad\quad\quad\quad\quad(1)$$ Where $\mathbf v$ is the velocity, and $f$ is the distribution function which is a function of position, velocity, and time.

The Maxwell-Boltzmann distribution function is: $$f(\mathbf v) = 4\pi v^2\left( \dfrac{m}{2k\pi T}\right)^{3/2}e^{-\dfrac{mv^2}{2kT}}\quad\quad\quad\quad\quad(2)$$ My question is: what is the relation between the equation $(1)$ and the function $(2)$? is this latter function a solution to the equation $(1)$? if it is, I do not see explicitly the time and the position in the function $f$

Thank you

The Boltzmann equation without collision operator $\Omega$ is as follows: $$\dfrac{\partial f}{\partial t} + \mathbf v \cdot \nabla f = 0 \tag{1}$$ Where $\mathbf v$ is the velocity, and $f$ is the distribution function which is a function of position, velocity, and time.

The Maxwell-Boltzmann distribution function is: $$f(\mathbf v) = 4\pi v^2\left( \dfrac{m}{2k\pi T}\right)^{3/2}e^{-\dfrac{mv^2}{2kT}}\tag{2}$$ My question is: what is the relation between the equation $(1)$ and the function $(2)$? is this latter function a solution to the equation $(1)$? if it is, I do not see explicitly the time and the position in the function $f$