A synthesis of the cooling granular gas problem

In this section we want to offer a quick reference table on the existing results on the kinetics of cooling granular gas in $d=1$ and $d=2$. We review the results on energy decay, velocity distributions and spatial correlations, indicating if they have been obtained analytically or numerically and specifying the models used. In particular we have focused our attention on the usual Inelastic Hard Object Model in $d=1$ (rods) and $d=2$ (disks), the pseudo-Maxwell (Ulam) model and finally the Inelastic Lattice Gas models that we have introduced. We have also considered the diffusion model (Equation (5.66)) and the sticky gas model, i.e. the Burgers equation in the inviscid limit (Equation (5.32)).

The energy decay for $d=1$

	Homogeneous	Non-homog.
1d model	regime	regime
Hard Rods	$t^{-2} \equiv e^{-\tau}$	$t^{-2/3}$
Lattice Gas	$e^{-\tau}$	$\tau ^{-1/2}$
Diffusion equation	$/$	$t^{-1/2}$
Sticky gas (Burgers)	$/$	$t^{-2/3}$

The Haff regime has been predicted analytically [103] and verified in all Hard Objects simulations. The energy decay for the correlated regime has been verified numerically by Ben-Naim et al. [22] and is consistent with the hypothesis of a sticky gas universality class. The fact that the inviscid Burgers equation and the sticky gas model are equivalent has been demonstrated [57]. The diffusion equation and the Burgers equation can of course have transient from the initial state to their correlated asymptotic regime.

The energy decay for $d=2$

	Homogeneous	Non-homog.
2d model	regime	regime
Hard Disks	$t^{-2} \equiv e^{-\tau}$	$\tau^{-1}$
Lattice Gas	$e^{-\tau}$	$\tau^{-1}$
Diffusion equation	/	$t^{-1}$
Sticky gas (Burgers)	/	$t^{-1}$

The $\tau^{-1}$ law has been predicted analytically [45], simulations of the Hard Disks model in the correlated regime are quite rare and have not verified this prediction in a satisfactory manner [45,175].

The velocity distributions for $d=1$

	Homogeneous	Non-homog.
1d model	regime	regime
Hard Rods	2 peaks	Gaussian
Lattice Gas	2 peaks	Gaussian
Diffusion equation	/	Gaussian
Sticky gas (Burgers)	/	$\exp(-(v/v_0(t))^3)$
Boltzmann	2 peaks	/
Ulam	$\pi/(1+(v/v_0(t))^2)^2$	/

The velocity distribution for the Hard Rods system has been obtained numerically (in the homogeneous regime [158,197], in the correlated regime [14,12]). The velocity distribution for the Burgers equation has been predicted analytically [89]. The two peaks in the solution of the 1d Boltzmann equation have been predicted analytically by Caglioti et al [25], in the framework of an asymptotic solution for the quasi-elastic limit of this equation. An analytic asymptotic solution of the homogeneous pseudo-Maxwell (Ulam) scalar model has been obtained by us (``Baldassarri solution'' [14,12,13]), and very well verified by numerical solutions with Gaussian or uniform initial conditions.

The velocity distributions for $d=2$

	Homogeneous	Non-homog.
2d model	regime	regime
Hard Disks	$\simeq \exp(-v/v_0(t))$	Gaussian ???
Lattice Gas	$\simeq \exp(-v/v_0(t))$ ?	Gaussian
Diffusion equation	/	Gaussian
Sticky gas (Burgers)	/	$\exp(-(v/v_0(t))^3)$
Boltzmann	$\simeq \exp(-v/v_0(t))$	/
Ulam	$\simeq (v/v_0(t))^{-\beta(r)}$	/

Numerical simulations of the Hard Disks model have not verified in a satisfactory manner the exponential tails of the velocity distributions (an interesting 3D experimental measurement of the exponential tails has been obtained in [144]). These have been predicted solving the homogeneous Boltzmann equation [211]. The Gaussian asymptotic distribution is a pure conjecture, based on some evidence (e.g. Figure 9 of the article by Huthmann, Orza and Brito [114]). The asymptotic solution of the (vectorial) Ulam model has not been obtained analytically (calculation can be found in previous chapter, as well as in Bobylev et al. [33]).

The time behavior of the correlation length for $d=1$ and $d=2$, in the correlated stage

	1d	2d
Hard Objects	$\sim t^{-2/3}$	$\sim \tau^{-1}$ ?
Lattice Gas	$\sim \tau^{-1/2}$	$\sim \tau^{-1}$
Diffusion equation	$t^{-1/2}$	$t^{-1}$
Sticky gas (Burgers)	???	???

The growth of the correlation length for the Hard Rods has been obtained numerically [12,14], while in 2d has been predicted analytically [214], with no reliable numerical checks.