\documentstyle[graphicx,multicol,epsf,aps]{revtex}
\begin{document}
\title{Shock-Like Dynamics of Inelastic Gases}
\author{E.~Ben-Naim$\dag$, S.~Y.~Chen$\dag$, G.~D.~Doolen$\dag$,
and S.~Redner$\ddag$}
\address{$\dag$Theoretical Division and Center for Nonlinear Studies,
Los Alamos National Laboratory, Los Alamos, NM 87545}
\address{$\ddag$Center for Polymer Studies and Department of Physics,
Boston University, Boston, MA 02215}
\maketitle
\begin{abstract}
We provide a simple physical picture which suggests that the
asymptotic dynamics of inelastic gases in one dimension is
independent of the degree of inelasticity. Statistical
characteristics, including velocity fluctuations and the velocity
distribution are identical to those of a perfectly inelastic sticky
gas, which in turn is described by the inviscid Burgers equation.
Asymptotic predictions of this continuum theory, including the $t^{-2/3}$
temperature decay and the development of discontinuities in the
velocity profile, are verified numerically for inelastic gases.
\end{abstract}
{PACS:} 47.70.Nd, 45.70.Mg, 05.40.-a, 81.05.Rm
\begin{multicols}{2}
Gases of inelastically colliding particles model the dynamics of
granular materials \cite{pkh,lpk}, geophysical flows \cite{csc}, and
large-scale structure of matter in the universe \cite{sz}.
Typically, a fraction of the kinetic energy is dissipated in each
collision, leading to interparticle velocity
correlations, a clustering instability \cite{gz,dlk,gzb,kwg,sg}, and in
the absence of external energy input, an inelastic collapse
\cite{bm,my,db,ernst,vn,brey}. The last feature presents an obstacle
to long-time simulations, as an infinite number of collisions occur
in a finite time.
In this Letter, we propose that a freely evolving inelastic gas is
asymptotically in the universality class of a completely inelastic,
sticky gas. Specifically, the temperature decreases in time as $t^{-2}$
over an intermediate range, but asymptotically decays as $t^{-2/3}$.
To test this hypothesis, we employ a simulation in which collisions
between particles with sufficiently small relative velocities are
perfectly elastic. This method allows us to bypass the inelastic
collapse and probe the asymptotic regime.
We consider $N$ identical point particles undergoing inelastic
collisions in a one-dimensional periodic system of length $L$. The
particles have typical interparticle spacing $x_0=L/N$ and their
typical velocity is $v_0$. We employ dimensionless space and time
variables, $x\to x/x_0$, and $t\to t v_0/x_0$, thereby rescaling the
ring length to $N$. Inelastic and momentum conserving collisions
are implemented by changing the sign of the relative velocity and
reducing its magnitude by a factor $r=1-2\epsilon$, with $0\leq
r\leq 1$, after each collision. It is convenient to view the
particle identities as ``exchanged'' upon collision, so that in a
perfectly elastic collision the particles merely pass through each
other, while for a small inelasticity each particle suffers a small
deflection. The outcome of a collision between a particle with
velocity $v$ and another particle with velocity $u$ is therefore
\begin{equation}
\label{rule}
v\to v-\epsilon(v-u).
\end{equation}
The granular temperature, or velocity fluctuation,
$T(t)=\langle v^2(t)\rangle-\langle v(t)\rangle^2$, can be estimated
in the intermediate time regime by considering the outcome of a single
collision under the assumption that the system remains homogeneous.
In each such collision, the energy lost is $\Delta
T\propto-\epsilon(\Delta v)^2$, with $\Delta v$ the relative
velocity, while the time between collisions is $\ell/\Delta v$.
Assuming homogeneity, we neglect fluctuations in the mean-free path
$\ell\cong 1$ and posit a single velocity scale so that $v\sim \Delta
v\sim T^{1/2}$. The temperature therefore obeys the rate equation
$dT/dt\propto-\epsilon\, T^{3/2}$ giving
\begin{equation}
\label{t2}
T(t)\sim (1+A\epsilon t)^{-2},
\end{equation}
with $A$ a constant of order unity \cite{pkh}. For small times $t\ll
t_{\rm dissip}\sim \epsilon^{-1}$ dissipation is negligible and the
temperature does not evolve -- the gas is effectively
elastic. For larger times, the dissipation leads to a $\epsilon^{-2}
t^{-2}$ temperature decay.
However, this behavior {\em cannot\/} be valid asymptotically, as the
temperature must decrease monotonically with increasing dissipation.
Moreover, the temperature is bounded from below by that of the
perfectly inelastic gas with a vanishing restitution coefficient,
$r=0$. For such a sticky gas, the temperature decays as $t^{-2/3}$
and the typical cluster mass grows as $t^{2/3}$ \cite{cpy}. This
behavior is reminiscent of diffusion-controlled two species
annihilation, where a small reaction probability results in a
homogeneous intermediate time regime in which the density follows a
$t^{-1}$ mean-field decay, even for low spatial dimension $d$.
However, at long times single-species domains which are opaque to
opposite-species particles form and a slower $t^{-d/4}$ density decay
follows\cite{kr}.
For the inelastic gas, we argue that the role of the reaction
probability is played by $\epsilon$. For small $\epsilon$, a particle
can penetrate through a domain of $N0$ cases. As shown in Fig.~4, the normalized
velocity distribution
\begin{equation}
\label{pvscl}
P(v,t)\sim t^{1/3}\Phi(vt^{1/3}),
\end{equation}
is described by an identical scaling function $\Phi(z)$ for these
widely different values of $r$. This universality provides further
confirmation that the asymptotic behavior for any $r<1$ is governed by
the $r=0$ ``fixed point''.
Further insights about the behavior of the inelastic gas are provided
by the connection to the Burgers equation \cite{sz,jmb}. Since sticky
gases are described by the inviscid ($\nu\to 0$) limit of the Burgers
equation
\begin{equation}
\label{bur}
v_t+vv_x=\nu v_{xx},
\end{equation}
supplemented by the continuity equation \hbox{$\rho_t+(\rho v)_x=0$},
we conclude that this continuum theory also describes the asymptotics
of the inelastic gas in the thermodynamic limit. The Burgers equation
may be reduced to the diffusion equation by the Hopf-Cole
transformation $v=-2\nu(\ln u)_x$, and therefore is solvable. In our
case, the relevant initial condition is delta-correlated velocities
$\langle v_0(x)v_0(x')\rangle=\delta(x-x')$. The resulting velocity
profile is discontinuous, and the corresponding shocks can be
identified with clusters in the sticky gas. Indeed, both shock
coalescence processes and cluster-cluster collisions in the sticky gas
conserve mass and momentum.
The relation to the Burgers equation is useful in several ways.
First, statistical properties of the shock coalescence process have
been established analytically \cite{fm}. For example, the tail of the
particle velocity distribution (\ref{pvscl}) is suppressed according
to
\begin{equation}
\label{pvtail}
\Phi(z)\sim \exp(-{\rm const.}\times |z|^3\,),\qquad |z|\gg 1.
\end{equation}
This behavior can be understood by considering the density of the
fastest (order unit velocity) particles. For such a particle to
maintain its velocity to time $t$, it must avoid collisions. This
requires that an interval of length $\propto t$ ahead of the particle
must be initially empty \cite{bkr}. For an initially random spatial
distribution, the probability of finding such an interval decays
exponentially with length; thus $P(1,t)\sim \exp(-{\rm const.}\times
t)$. Using \hbox{$\Phi(z)\sim \exp(-{\rm const.}\times
|z|^{\gamma})$} and $z=vt^{1/3}$ then yields $\gamma=3$.
Interestingly, over most of the range of scaled velocities, the
numerically obtained velocity distribution deviates only slightly from
a Gaussian, reflecting the small constant in (\ref{pvtail}) \cite{fm}.
Another important prediction of Eq.~(\ref{bur}) is that the velocity is
linear in the Eulerian coordinate $x$ and the Lagrangian coordinate
$q(x,t)$
\begin{equation}
\label{vx}
v(x,t)={x-q(x,t)\over t}.
\end{equation}
This form also characterizes the asymptotic velocity profile of
inelastic gases. Fig.~5 shows such a sawtooth velocity profile from
an inelastic gas simulation. The slopes of the linear segments of the
profile are consistent with the $t^{-1}$ prediction of Eq.~(\ref{vx}).
The inelastic collapse is simply a finite time singularity characterized
by the development of a discontinuity in the velocity profile, {\it i.e.},
a shock.
\begin{figure}
\narrowtext
%\centerline{\epsfxsize=9cm \epsfbox{fig5.eps}}
\centerline{\includegraphics[width=9cm]{fig5}}
\caption{Shock profile of an inelastic gas.
Density and velocity are plotted at time $t=10^5$ in a system with
$N=2\times 10^4$ particles, $r=0.99$ and $\delta=10^{-4}$. A line with slope
$t^{-1}$ is plotted for reference. The number of shocks is consistent with
the expected number $Nt^{-2/3}\protect\cong 9$.}
\end{figure}
In higher dimensions as well, the temperature of an inelastic gas is a
monotonically increasing function of $r$ and hence, it is bounded from
below by the $r=0$ case. Therefore, we speculate that $r=0$ remains
the fixed point in higher dimensions. On the other hand, the Burgers
equation \hbox{${\bf v}_t+{\bf v}\cdot \nabla{\bf v}=\nu\nabla^2 {\bf
v}$} approximately describes the sticky gas in the limit $\nu\to
0$ \cite{sz}. The known $t^{-d/2}$ temperature decay of the Burgers
equation \cite{sz}, valid for $2\leq d\leq 4$ (with possible
logarithmic corrections at the crossover dimensions), then yields
\begin{equation}
T(t)\sim \cases{1&$t\ll \epsilon^{-1}$;\cr
\epsilon^{-2}t^{-2}&$\epsilon^{-1}\ll t\ll \epsilon^{-{4/(4-d)}}$;\cr
t^{-d/2}&$\epsilon^{-{4/(4-d)}}\ll t\ll N^{2/d}$;\cr
N^{-1}&$N^{2/d}\ll t$.}
\end{equation}
Interestingly, both the decay exponents \cite{be,cdnt}, the formation
of string-like clusters \cite{gz,my,lm}, and even the possibility of a
percolating network of clusters \cite{jt}, features that were found
primarily numerically, are all predicted by the Burgers equation.
Additionally, the critical cluster size increases with the dimension
according to $N_c(\epsilon)\sim \epsilon^{-2d/(4-d)}$, suggesting that
the inelastic collapse is avoided when $d>d_c=4$, and that the
homogeneous gas behavior $T\sim\epsilon^{-2}t^{-2}$ holds indefinitely
above this critical dimension.
In summary, our results suggest that the asymptotic behavior of a
one-dimensional inelastic gas with many particles is governed by the
$r=0$ sticky gas fixed point, and that the appropriate continuum
theory is the inviscid Burgers equation. This connection provides
several exact statistical properties of inelastic gases. Conversely,
inelastic gases may provide a useful tool to study shock dynamics.
The suggestive behavior of the inelastic gas in higher dimensions
deserves careful investigation.
This research is supported by the Department of Energy under contract
W-7405-ENG-36 (at LANL), and by the NSF under grant DMR9632059 (at
BU). We thank G.~P.~Berman and P.~L.~Krapivsky for useful discussions.
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\end{multicols}
\end{document}