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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:

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

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


next up previous contents
Next: The ``Baldassarri solution'' in Up: The exact distribution of Previous: The exact distribution of   Contents
Andrea Puglisi 2001-11-14