The Boltzmann hierarchy and the Boltzmann equation

In a rarefied gas $N$ is a very large number and $\sigma$ is very small; let us say, to fix ideas, that we have a box whose volume is $1 \quad cm^3$ at room temperature and atmospheric pressure. Then $N \simeq 10^{20}$ and $\sigma \simeq 10^{-8}cm$ and (from Eq. (2.70)) for small $s$ we have $(N-s)\sigma^2 \simeq N\sigma^2 \simeq 1m^2$; at the same time the difference between $\mathbf{r}_i$ and $\mathbf{r}_i+\sigma \hat{\mathbf{n}}$ can be neglected and the volume occupied by the particles ( $N\sigma^3 \simeq 10^{-4}cm^3$) is very small so that the collision between two selected particles is a rather rare event. In this spirit, the Boltzmann-Grad limit has been suggested as a procedure to obtain a closure for Eq. (2.70): $N \to \infty$ and $\sigma \to 0$ in such a way that $N \sigma^2$ remains finite. We stress the fact that (as seen in section 2.1.2) the total number of collisions in the unit of time is given by the total scattering cross section multiplied by $N$, which for a system of hard spheres gives $N \pi \sigma^2$. The Boltzmann-Grad limit, therefore, states that the single particle collision probability must vanish, but the total number of collisions remains of order $1$.

Within this limit, the BBGKY hierarchy reads:

$$\frac{\partial P_s}{\partial t} + \sum_{i=1}^s \mathbf{v}_i \cdot... ... \vert\mathbf{V}_i \cdot \hat{\mathbf{n}}\vert d\hat{\mathbf{n}} d \mathbf{v}_*$$ (2.69)

where the arguments of $P_{s+1}'$ and of $P_{s+1}$ are the same as above, except that the position of the $(s+1)-th$ particle ( $\mathbf{r}_*'$ and $\mathbf{r}_*$) is equal to $\mathbf{r}_i$ (as $\sigma \to 0$). Eq. (2.72) gives a complete description of the time evolution of a Boltzmann gas (i.e. the ideal gas obtained in the Boltzmann-Grad limit), usually called the Boltzmann hierarchy.

Finally the Boltzmann equation is obtained if the molecular chaos assumption is taken into account:

$$P_2(\mathbf{r}_1,\mathbf{v}_1,\mathbf{r}_2,\mathbf{v}_2,t)=P_1(\mathbf{r}_1,\mathbf{v}_1)P_1(\mathbf{r}_2,\mathbf{v}_2)$$ (2.70)

for particles that are about to collide (that is when $\mathbf{r}_2=\mathbf{r}_1-\sigma\hat{\mathbf{n}}$ and $\mathbf{V}_{12} \cdot \hat{\mathbf{n}}<0$). This assumption naturally stems from the Boltzmann-Grad limit, as it is reasonable that, in the limit of vanishing single-particle collision rate, two colliding particles are uncorrelated. The lack of correlation of colliding particles is the essence of the molecular chaos assumption. We underline that nothing is said about correlation of particles that have just collided.

With the assumption (2.73) we can rewrite the first equation of the hierarchy (2.72), omitting the $_1$ subscript for simplicity:

$$\frac{\partial P(\mathbf{r},\mathbf{v})}{\partial t}+\mathbf{v} \... ...}_*))\vert\mathbf{V} \cdot \hat{\mathbf{n}}\vert d\mathbf{v}_*d\hat{\mathbf{n}}$$ (2.71)

with $\mathbf{v}'=\mathbf{v}-\hat{\mathbf{n}}(\mathbf{V} \cdot \hat{\mathbf{n}})$, $\mathbf{v}_*'=\mathbf{v}_*+\hat{\mathbf{n}}(\mathbf{V} \cdot \hat{\mathbf{n}})$, $\mathbf{V}=\mathbf{v}-\mathbf{v}_*$. This represents the Boltzmann equation for hard spheres. We also observe that the integral in Eq. (2.74) is extended to the hemisphere $S_+$ but could be equivalently extended to the entire sphere $S^2$ provided a factor $1/2$ is inserted in front of the integral itself, as changing $\hat{\mathbf{n}} \to -\hat{\mathbf{n}}$ does not change the integrand.

From a rigorous point of view [58], the molecular chaos has to be assumed and cannot be proved. However it has been demonstrated that if the Boltzmann hierarchy has a unique solution for data that satisfy for $t=0$ a generalized form of chaos assumption:

$$P_s(\mathbf{r}_1,\mathbf{v}_1,...,\mathbf{r}_s,\mathbf{v}_s,t)=\prod_{j=1}^sP_1(\mathbf{r}_j,\mathbf{v}_j,t)$$ (2.72)

than Eq. (2.75) holds at any time and therefore the Boltzmann equation is fully justified. Otherwise it has also been proved that if Eq. (2.75) is satisfied at $t=0$ and the Boltzmann equation (2.74) admits a solution for the given initial data, then the Boltzmann hierarchy (2.72) has at least a solution which satisfy (2.75) at any time $t$.