diff --git a/lattice-boltzmann.com/content/1-dimensionless-lbm.md b/lattice-boltzmann.com/content/1-dimensionless-lbm.md
index 1aab1ae6cb08bfdf62fa6fe424b9927c06c78580..50df1a6c4e503ce4a636f316ac8b3a01947da777 100644
--- a/lattice-boltzmann.com/content/1-dimensionless-lbm.md
+++ b/lattice-boltzmann.com/content/1-dimensionless-lbm.md
@@ -97,6 +97,11 @@ $$
 \left[M \cdot L^{-2D}\cdot T^D\right]
 $$
 By defining the characteristic velocity of our particles by $\xi_0$ (due to thermal agitation)
+$$
+\begin{equation}
+\xi_0^2=\frac{k_BT}{m},
+\end{equation}
+$$
 the non-dimensional quantities of interest become
 $$
 \begin{equation}
@@ -120,11 +125,24 @@ $$
 \mathrm{Kn}=\frac{\tau\xi_0}{L},
 \end{equation}
 $$
-is the Knudsen number.
-
-As for the Navier-Stokes equations we see that the non-dimensional BGK equation is parametrized
-with an unique transport coefficient which is the Knudsen number. In the next episodes of this series
+is the Knudsen number. As for the Navier-Stokes equations we see that the non-dimensional BGK equation is parametrized
+with an unique transport coefficient which is the Knudsen number in this case. In the next episodes of this series
 we will make the link between the BGK and the Navier--Stokes equations and show how we discretize
 the BGK equation in order to simulate weakly compressible fluid flows.
 
+Finally we want to express the non-dimensional Maxwell-Boltzmann distribution as a function of only non-dimensional quantities and it reads
+$$
+\begin{equation}
+{f^{eq}}^\ast(\bm{x}^\ast, \bm{\xi}^\ast, t^\ast) = \frac{\rho^\ast(\bm{x}^\ast, \bm{\xi}^\ast, t^\ast)}{(2\pi\theta^\ast(\bm{x}^\ast, \bm{\xi}^\ast, t^\ast))^{D/2}}\exp\left(-\frac{\left(\bm{\xi}^\ast-\bm{u}^\ast(\bm{x}^\ast, \bm{\xi}^\ast, t^\ast)\right)^2}{2\theta^\ast(\bm{x}^\ast, \bm{\xi}^\ast, t^\ast)}\right),
+\end{equation}
+$$
+where $\theta^\ast$ is given by
+$$
+\begin{equation}
+\theta^\ast = \frac{k_B T}{m\cdot \xi_0^2}
+\end{equation}
+$$
+In the next episode we will discuss in more details, the link between the description of a fluid in the Boltzmann framework
+and in the Navier--Stokes framework.
+