The pseudo-Maxwell molecules

The pseudo-Maxwell molecules model is defined through its Boltzmann-Enskog equation, and not in terms of a true particle dynamics. This equation reads:

$$\begin{multline} \left( \frac{\partial}{\partial t}+ v_i \frac{\partial}{\partia... ...)-P(\mathbf{r},\mathbf{v},t)P(\mathbf{r},\mathbf{v}_2,t) \right ] \end{multline}$$

The prefactor $S(\mathbf{r},t)$ accounts for various terms, e.g. it can be taken proportional to $\Xi(\sigma,n(\mathbf{r},t))\sqrt{T(\mathbf{r},t)}$ if the derivation of Bobylev et al. is followed (see paragraph 5.1.5). In this case the Enskog correction $\Xi(\sigma,n)$ is the pair correlation function at distance $\sigma$ and density $n$, which averagely takes into account correlations between the positions (scattering angle) of colliding particles.

The assumption of homogeneity changes the above equation in the following way:

$$\frac{\partial P(\mathbf{v},t)}{\partial t} =S(t)\int d\mathbf{v}... ...^2}P(\mathbf{v}',t)P(\mathbf{v}_2',t)-P(\mathbf{v},t)P(\mathbf{v}_2,t) \right ]$$ (5.42)

When the velocities are scalar and the $S(t)$ dependence (e.g. the temperature) is absorbed by a time reparametrization $t \to \tau$, the equation simplifies:

$$\frac{\partial P(v,\tau)}{\partial\tau} +P(v,\tau)= \frac{1}{1-\gamma^*}\int duP(u,\tau)P\left(\frac{v-\gamma^* u}{1-\gamma^*},\tau\right)$$ (5.43)

with $\gamma^*=(1-r)/2$.

This equation is the master equation for the scalar Ulam model discussed at the end of the paragraph 5.1.5: at each step a pair of velocities $v_i$ and $v_j$ from a set of $N$ velocities is chosen and transformed according to the usual rule of one-dimensional inelastic collisions. It should be noted that the time $\tau$ is proportional to the number of collisions per particles.