Burgers hypothesis, TDGL hypothesis and a note on pseudo-Maxwell molecules models

The Burgers hypothesis

In a recent paper, Ben-Naim et al. [22] have studied the cooling granular gas in $d=1$, discovering that, after the homogeneous (Haff) phase, it asymptotically becomes independent of the value of inelasticity $\epsilon$ and maps onto the sticky gas model (that in $d=1$ is equivalent to the cooling granular gas with $r=0$, i.e. $\epsilon=\gamma_0/2=1$). For such a sticky gas the temperature decays as $t^{-2/3}$. Moreover, this gas shows instability in the density and velocity profiles: density clusters and velocity shocks form. The cooling gas in $d=1$ become asymptotically equivalent to a sticky gas [57], for any value of $\epsilon$, because of the formation of clusters: when the cluster size (number of particles belonging to it) has reached the critical value $N_c(\epsilon) \sim \epsilon^{-1}$, a particle colliding with the cluster is absorbed, i.e. never emerges from it. The $t^{-2/3}$ regime lasts until all the particles have fallen in one big cluster and the kinetic energy becomes constant. The authors have verified their idea with MD simulations, using a trick to avoid inelastic collapse: every time two particles collide with an absolute relative velocity lesser than $\delta$, the collision is assumed to be elastic. They have verified that the results of MD simulations are independent of the value of $\delta$. As the sticky gas is described by the inviscid (the kinematic viscosity $\nu=\eta/mn \to 0$) limit of the Burgers equation [199]:

$$\frac{\partial u}{\partial t}=-u \frac{\partial u}{\partial x}+\nu \frac{\partial ^2 u}{\partial x^2}$$ (5.29)

the velocity profile can be predicted to be discontinuous, with shocks (i.e. asymmetric discontinuities shown in Fig. fig_shocks) in correspondence with the clusters of the gas. The relation to the Burgers equation is useful also to establish an estimate of the tails of the asymptotic velocity distribution $P(v,t) \sim t^{1/3}\Phi(vt^{1/3})$ (independent of $r$), which reads $\Phi(z) \sim \exp(-\vert z\vert^3)$. The MD simulations have revealed very slight deviations from the Gaussian, and the authors have imputed the discrepancy from the expected behavior to the smallness of the constant in front of $\vert z\vert^3$. Finally, Ben-Naim et al. have conjectured that the Burgers equation describes the flow velocity $\mathbf{u}(\mathbf{r},t)$ of a cooling granular gas for arbitrary values of the inelasticity $\epsilon$ in generic $d$ dimensions, i.e.:

$$\frac{\partial u_i}{\partial t}=-u_j \frac{\partial u_i}{\partial r_j}+\nu \frac{\partial^2 u_j}{\partial r_j^2}$$ (5.30)

where $\nu$ is the kinematic viscosity and the solutions are looked in the rotation-free class $\boldsymbol{\nabla} \times \mathbf{u}=0$. This conjecture implies that the asymptotic behavior of the cooling inelastic gas is independent of $\epsilon$ (as it always falls in the universality class of the sticky gas) and that the upper critical dimension for the disappearance of the inelastic collapse is $d_c=4$. The simplest objection to this conjecture comes from the consideration that in $d=1$ the $r=0$ case is equivalent to the sticky gas, while in $d>1$ the $r=0$ is not equivalent, as normal component of the relative velocity are in principle different from zero after a collision.

The TDGL hypothesis

In another interesting paper, Wakou et al. [219] have demonstrated that the evolution of the flow field of a granular fluid, neglecting the convective term $\mathbf{u}\cdot \boldsymbol{\nabla}\mathbf{u}$, can be cast in the form of a Time Dependent Ginzburg-Landau equation for a non-conserved order parameter

$$\frac{\partial \mathsf{S}}{\partial \tau} =(\gamma_0+D \nabla^2)\mathsf{S}-\frac{2}{d}D\overline{\vert S\vert^2}S=-V \frac{\delta\mathcal{H}[\mathsf{S}]}{\delta \mathsf{S}^\dagger}$$ (5.31)

$$\mathcal{H}[\mathsf{S}] =-\frac{1}{2}\gamma_0\overline{\vert\mathsf{S}\vert^2}+\frac{1}{2... ...mathsf{S}\vert^2}+\frac{1}{2d}D\left(\overline{\vert\mathsf{S}\vert^2}\right)^2$$ (5.32)

with $D=\nu/\omega$ independent of time, and the tensorial order parameter $\mathsf{S}=\boldsymbol{\nabla}\tilde{\mathbf{u}}$ (e.g. tr$S=\nabla \cdot \tilde{\mathbf{u}}=0$) $\tilde{\mathbf{u}}=\mathbf{u}/v_0(t)$ the rescaled velocity field.

It is useful to report Equation (5.33) in Fourier space:

$$\frac{\partial \mathsf{S}_\mathbf{k}}{\partial \tau} = \left\{\gamma_0-Dk^2-\frac{2D}{dV^2}\sum_{\mathbf{q}}\vert\math... ...}=-V^2\frac{\delta\mathcal{H}[\mathsf{S}]}{\delta\mathsf{S}^\dagger_\mathbf{k}}$$ (5.33)

$$\mathcal{H}[\mathsf{S}] =\frac{1}{2V^2}\sum_\mathbf{k}(-\gamma_0+Dk^2)\vert S_\mathbf{k}\... ...2d}\left(\frac{1}{V^2}\sum_\mathbf{k}\vert\mathsf{S}_\mathbf{k}\vert^2\right)^2$$ (5.34)

The typical form of the Ginzburg-Landau equation of motion for a non-conserved order parameter can be recognized: $\partial_t \phi=\nabla^2 \phi-V'(\phi)$ with $V(\phi)=A(1-B\phi^2)^2$. The Landau free energy functional $\mathcal{H}$ has a quartic term $\mathsf{S}^4$ and a quadratic term $\mathsf{S}^2$. It differs from the standard Landau free energy in that the quartic term contains summations over two totally independent wave numbers.

The energy functional has a continuous set of degenerate minima, having the shape of a Mexican hat. However the above considerations have been made neglecting the convective term of the hydrodynamics equations. When this term is added, only subsets of admissible solutions are selected out of this infinite set of minima. In two dimensions only two distinct minima survive: therefore the $d=2$ cooling granular gas, during the formation of instabilities, greatly resembles the spinodal decomposition for a non-conserved order parameter (Model A universality class, in the Hohenberg-Halperin classification [110]).

The pseudo-Maxwell molecules model

Another approach to the study of cooling granular gas, as we have already discussed, is the study of the kinetic equation, assuming that some form of (Boltzmann or Boltzmann-Enskog) Molecular Chaos hypothesis holds. Some authors (Ben-Naim et al [24], Bobylev et al. [33]) have conjectured the possibility of get an insight on the kinetics of granular gas using a modification of the Boltzmann equation, i.e. using the Maxwell molecules model with inelasticity, usually referred to as the pseudo-Maxwell model. The connection between Maxwell model and granular kinetics has been conjectured by Bobylev et al. on the basis of heuristic considerations. We quote a passage from their article appeared on Journal of Statistical Physics [33]: the Equation (2.98) ``is often considered as the basic kinetic equation for granular media. On the other hand, this equation is a fairly rough mathematical model of a real physical process. Even without mentioning rotational effects and deviations from spherical shape of particles, one can realize that the formula'' (2.98) ``is very rough outside of the Boltzmann-Grad limit. This formula certainly does not account for non-Markovian effects of repeating collisions. Moreover, the model of inelastic scattering based on empirical restitution coefficient is also a very rough approximation. Thus we have the equation'' (2.98) ``which is (a) a very rough approximation from physical point of view and (b) quite complicated equation from mathematical point of view. These obvious considerations lead to the idea of using a simplified version of Eq. `` (2.98). It must be noted that Bobylev et al. refer to the Enskog-Boltzmann equation, i.e. to Eq. (2.98) with the Enskog corrections to the Molecular Chaos hypothesis (and with exact dependence on positions).

The simplification suggested is the following:

$$\vert v-v'\vert \simeq S \sqrt{T(\mathbf{r},t)}$$ (5.35)

to be put in the Boltzmann (or Enskog-Boltzmann) equation ($S$ is an appropriate constant). The assumption (5.37) leads to a dramatic simplification of the collisional integral $Q(P,P)$, which (in the absence of Enskog corrections) becomes:

$$\int d\mathbf{v}_2 \int_{\mathbf{V}_{12} \cdot \hat{\mathbf{n}} >... ...{v}_2',t)-P(\mathbf{r}_1,\mathbf{v}_1,t)P(\mathbf{r}_1,\mathbf{v}_2,t) \right ]$$ (5.36)

In the Boltzmann equation it appears multiplied by a factor $\sigma^2 S \sqrt{T(\mathbf{r},t)}$. As discussed in section 2.2, this is analogous to the collisional integral of the Maxwell molecules model, as there is no dependence on the relative velocity in the scattering cross section. This model of kinetic equation is called pseudo-Maxwell model, as it concerns inelastic hard-core collisions.

Bobylev and co-workers have studied this model obtaining some theorems of existence and uniqueness of self-similar homogeneous asymptotic solutions and interesting relations among coefficients of Taylor expansion of the characteristic function for the asymptotic solution. They have also discussed the hydrodynamics (Euler) equations and their linear stability, finding again long wave-length instabilities. We discuss some of their result in paragraph 5.3.

It is also noticeable that the pseudo-Maxwell model with the (ideal) assumption of homogeneous density and perfectly uncorrelated colliding particles is analogous to another model, introduced some years ago by S. Ulam [208], in order to clarify the emergence of Maxwellian statistics in perfectly elastic gases. The model consists of a set of $N$ ``energies'' with the following discrete evolution: at every time step a pair of energies is chosen at random and their kinetic energy is randomly re-distributed with the constraint that the sum is conserved. The asymptotic distribution of energies, independently of the starting distribution, is Maxwellian. The homogeneous pseudo-Maxwell model can be considered the inelastic counterpart of this Ulam model.

Ben-Naim and Krapivsky [24] have studied the same model considering scalar velocities (as in $d=1$) in an infinitely connected topology (every particle can collide with every other particle, as in Ulam model), discovering that the asymptotic solution exhibits multiscaling, i.e. the moments of the solving distribution increase in time with different exponents, coming to the wrong conclusion that there is not a scaling solution $P(v,t)=\frac{1}{v_0(t)}P(\frac{v}{v_0(t)})$. We have found a correct scaling asymptotic solution for this model in $d=1$ and verified, with numerical simulations, that it is reached by a very general class of initial distribution. This is discussed in the third section of the chapter.