The Enskog correction

The Boltzmann-Grad limit (see paragraph 2.2.3) restricts the validity of the Boltzmann equation to rarefied gases. This conditions is necessary to consider valid the Molecular Chaos which states the independence of colliding particles. In principle, in fact, two colliding particles can be correlated due to an intersection of their collisional histories: one simple possibility is that they may have collided some time before or, alternatively, they may have collided with particles that have collided before. Moreover, the spatial extension of particles (i.e. the fact that they are not really pointlike) restricts the possibilities of motion and as a consequence the degree of independence (this is the so called excluded volume effect). All these kinds of correlations become relevant when the gas is not in the situation considered by the Boltzmann-Grad limit, that is when the gas is not rarefied but (either moderately or highly) dense.

The first approach to the problem of not rarefied gases was introduced by Enskog [62]: he did not consider the effects of velocity correlations due to common collisional histories, but simply added to the Boltzmann equation an heuristic correction to take into account short range correlations on positions only. In general the two-body probability distribution function can be written in terms of the one-body functions:

$$P_2(\mathbf{r}_1,\mathbf{v}_1,\mathbf{r}_2,\mathbf{v}_2,t)=g_2(\m... ...r}_2, \mathbf{v}_2)P_1(\mathbf{r}_1,\mathbf{v}_1)P_1(\mathbf{r}_2,\mathbf{v}_2)$$ (2.88)

where $g_2$ is the pair correlation function. The Molecular Chaos assumption states that $g_2(\mathbf{r}_1,\mathbf{r}_1+\sigma \hat{\mathbf{n}},\mathbf{v}_1,\mathbf{v}_2) \equiv 1$.

In the Enskog theory the Molecular Chaos assumption is modified in the following way:

$$P_2(\mathbf{r}_1,\mathbf{v}_1,\mathbf{r}_1+\sigma \hat{\mathbf{n}... ...\mathbf{r}_1,\mathbf{v}_1)P_1(\mathbf{r}_1+\sigma\hat{\mathbf{n}},\mathbf{v}_2)$$ (2.89)

i.e. $g_2 \equiv \Xi(\sigma,n)$ for particles entering or coming out from a collision, and the existence of a well defined coarse-grained density $n(\mathbf{r}_1)$ is assumed. The term $\Xi(\sigma,n)$ becomes a multiplicative constant in front of the collisional integral $Q(P,P)$, giving place to the so-called Boltzmann-Enskog equation. Of course, in a general non-homogeneous situation, the density is a spatially and temporally non-uniform quantity which can be described by a macroscopic field: one may assume (as it is in kinetic theory) that this field changes slowly in space-time, so that the Boltzmann-Enskog equation can be locally solved with constant $n$ as it was a Boltzmann equation with an effective total scattering cross section $\Xi(\sigma,n)N\sigma^2$.

For elastic hard disks or hard spheres, spatial correlations are described by the formulas of Carnahan and Starling [56]:

where $\phi$ is the solid fraction ( $\phi=n\pi\sigma^2/4$ in $d=2$, $\phi=n\pi \sigma^3/6$ in $d=3$). This formula is expected to work well with solid fractions below $\phi_c$, where a phase transition takes place [3], with $\phi_c=0.675$ in $d=2$.

The Enskog correction produces, for example, important corrections to the transport coefficients and to the pressure term in the hydrodynamic description (see paragraph 2.3.9).

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