\documentstyle[aps,epsf,multicol,graphicx]{revtex}
\begin{document}
\title{Impurity in a Maxwellian Unforced Granular Fluid}
\author{E.~Ben-Naim$^1$ and P.~L.~Krapivsky$^2$}
\address{$^1$Theoretical Division and Center for Nonlinear Studies,
Los Alamos National Laboratory, Los Alamos, NM 87545, USA}
\address{$^2$Center for Polymer Studies and Department of Physics,
Boston University, Boston, MA 02215, USA}
\maketitle
\begin{abstract}
We investigate velocity statistics of an impurity immersed in a
uniform granular fluid. We consider the cooling phase, and obtain
scaling solutions of the inelastic Maxwell model analytically.
First, we analyze identical fluid-fluid and fluid-impurity collision
rates. We show that light impurities have similar velocity
statistics as the fluid background, although their temperature is
generally different. Asymptotically, the temperature ratio increases
with the impurity mass, and it diverges at some critical mass.
Impurities heavier than this critical mass essentially scatter of a
static fluid background. We then analyze an improved inelastic
Maxwell model with collision rates that are proportional to the
{\it average} fluid-fluid and fluid-impurity relative velocities.
Here, the temperature ratio remains finite, and the system is always
in the light impurity phase. Nevertheless, ratios of sufficiently
high order moments $\langle v^n_{\rm impurity}\rangle/\langle
v^n_{\rm fluid}\rangle$ may diverge, a consequence of the
multiscaling asymptotic behavior.
\\ {\bf PACS.} 05.20.Dd, 02.50.-r, 47.70.Nd, 45.70.Mg
\end{abstract}
\begin{multicols}{2}
\section{Introduction}
Granular media are typically polydisperse. For example, sand and
grains have a broad range of particle sizes and shapes. Such granular
mixtures exhibit size segregation, a ubiquitous collective phenomena
that underlies diverse processes including for example production and
transport of powders in industry, sand dunes propagation and volcanic
flows in geophysics \cite{jcw,csc}. The ``Brazil Nut'' problem where
an impurity is immersed in a uniform granular media is an extreme
realization of a granular mixture as it corresponds to the vanishing
volume fraction limit of a binary mixture. While this problem has been
extensively studied, dynamics of such impurities are not fully
understood \cite{rsps,al,jmp,drc,kjn,dj,hql,th}.
Understanding the velocity statistics of granular mixtures is a
necessary step in describing multiphase granular flows. Experimental
and theoretical studies show that in general, different components of
a granular mixture are characterized by different typical speeds,
i.e., granular temperatures are usually distinct \cite{wp}. However,
analytical treatment of this case is difficult given the coupling
between the different components of the mixture. An additional
complication arises from the large number of parameters, including
several restitution coefficients, governing the dynamics.
In contrast, the impurity problem is more amenable to analytical
treatment. First, the impurity is directly enslaved to the fluid and
second, there are fewer parameters \cite{sd}. In this paper, we study
the impurity problem in the framework of the Maxwell
model\cite{max,true} that assumes that the collision rate is uniform,
i.e., independent of the relative velocity of the colliding
particles. This simplifies the Boltzmann collision operator and thence
this model is widely used in kinetic theory\cite{true,e,bob}. The
Maxwell model is analytically tractable even when the collisions are
inelastic as shown in a number of recent studies of uniform and
polydisperse granular gases \cite{bk,bcc,bc,bmp,kb,eb,mp,santos}.
We obtain analytic results for the velocity distributions valid for
arbitrary spatial dimension and collision parameters. We consider two
versions of the Maxwell model. In the first, termed the Inelastic
Maxwell Model (IMM), the collision rates are completely independent of
the relative velocities of the colliding particles. In the second,
termed the Improved Inelastic Maxwell Model (IIMM), the collision
rates are proportional to the {\it average} relative velocity of the
colliding particles.
In the IMM, there are two phases separated by a critical impurity
mass $m_*(r_p,r_q)$, determined by the restitution coefficients $r_p$ and
$r_q$ characterizing fluid-fluid and fluid-impurity particle collisions,
respectively. When the impurity mass $m$ is smaller than the critical mass,
$m2p(1-p)$,
the impurity temperature is proportional to the fluid temperature
asymptotically, ${\Theta(t)\over T(t)}\to c$ as $t\to \infty$. In the
complementary region $2p(1-p)>1-q^2$ an extreme violation of equipartition
occurs, as the ratio of the fluid temperature to the impurity temperature
vanishes. The impurity is very energetic compared with the fluid and it
practically sees a static fluid. {}From Eq.~(\ref{c}) we find that at the
transition point $q=\sqrt{1-2p(1-p)}$. Employing relations
(\ref{rp})--(\ref{rq}) between the restitution coefficients and the collision
parameters we obtain the critical mass
\begin{equation}
\label{mstar}
m_*={r_q+\sqrt{(1+r_p^2)/2}\over 1-\sqrt{(1+r_p^2)/2}}.
\end{equation}
The heavy impurity phase arises when the impurity is a bit heavier than the
fluid particles: Even when the fluid-fluid collisions are completely
inelastic ($r_p=0$), the critical mass satisfies $m_*>1+\sqrt{2}$. For
weakly dissipative fluids ($r \to 1$), the critical mass diverges,
$m_*\propto (1-r_p)^{-1}$ (see Fig.~1).
Note now a few features of the light impurity phase. First, Eq.~(\ref{c})
generalizes the elastic fluid ($p=0$) result $c=(1-q)/(1+q)$\cite{p,mp1}.
That result was actually established for a hard sphere fluid, so at least
asymptotically both the IMM and the improved model predict the same
impurity temperature when the fluid is elastic. Further, the initial
impurity temperature becomes irrelevant and the impurity is governed by the
fluid background. The average energies of the impurity and fluid particles
are asymptotically equal when $m\Theta/T\to 1$, or when $mc=1$. Thus, energy
equipartition occurs on a particular surface in the three dimensional space
$(m,r_p,r_q)$ where $m(1-q)^2=1-q^2-2p(1-p)$. Using relations
(\ref{rp})--(\ref{rq}) between the collision parameters and the restitution
coefficients, energy equipartition occurs when the impurity mass $m_{\rm eq}$ is
given by
\begin{equation}
\label{m1}
m_{\rm eq}={1+r_p^2-2r_q^2\over 1-r_q^2}.
\end{equation}
As expected, this mass equals unity when $r_p=r_q$. Curiously, $m_{\rm eq}$ vanishes
when $r_q^2=(1+r_p^2)/2$, indicating that for $r_q>\sqrt{(1+r_p^2)/2}$,
energy equipartition does not occur, as shown in Fig.~2. However, in a
generic point in the parameter space $(m,r_p,r_q)$ equipartition does break
down \cite{wp,mp}. This is a signature of the dissipative and nonequilibrium
nature of the system.
\begin{figure}
%\centerline{\epsfxsize=8cm \epsfbox{fig1.eps}}
\centerline{\includegraphics[width=8cm]{fig1}}
\caption{The critical mass $m_*$ versus the restitution coefficients
$r_p$ and $r_q$.}
\end{figure}
In summary, when the impurity mass is smaller than the critical mass
$mk$. For given fluid collision
parameter $p$, the above relation has the solution $q_k$ only for
sufficiently small $k$. Thus, there are a few special values of the
impurity collision parameter $q_k$ for which the scaled impurity
velocity distribution is a linear combination of simple rational
functions $\left(1+w^2\right)^{-n}$ with $n=2,\ldots,k+1$.
While the scaling functions underlying the impurity and the fluid are
similar, more subtle features may differ. In particular, the full time
dependent behavior, as characterized by the moments of the impurity
distribution, exhibits rich behavior. Let $L_n(t)=\int dv\, v^n\,P(v,t)$ be
the moments of the fluid velocity distribution. Multiplying the equations
(\ref{bep})--(\ref{beq}) by $v^n$ and integrating, the moments obey the
recursive equations
\begin{eqnarray}
\label{l2n}
{d\over dt}L_n+a_nL_n
&=&\sum_{j=2}^{n-2} {n\choose j} p^{j}(1-p)^{n-j}L_{j}L_{n-j},\\
\label{m2n}
{d\over dt}M_n+b_nM_n
&=&\sum_{j=0}^{n-2} {n\choose j} q^{j}(1-q)^{n-j}M_{j}L_{n-j},
\end{eqnarray}
with
\begin{equation}
\label{abn}
a_n(p)=1-p^n-(1-p)^n, \qquad
b_n(q)=1-q^n.
\end{equation}
Asymptotically, the fluid moments decay exponentially according to\cite{bk}
\begin{equation}
\label{Ln}
L_{n}(t)\propto e^{-a_n(p)\,t}.
\end{equation}
Using this asymptotics we analyze the behavior of the impurity moments. The
second moment, i.e. the impurity temperature, was already shown to behave
similar to the fluid temperature when $a_2(p)b_{n}(q)$ for
sufficiently large $n$. Such moments are no longer governed by the fluid and
\begin{equation}
\label{mn-heavy}
M_n(t)\propto e^{-(1-q^n)t}.
\end{equation}
The corresponding moment ratio diverges asymptotically: $M_n/L_n\to\infty$ as
$t\to\infty$. Interestingly, the same behavior (\ref{mn-heavy}) is found in
the heavy impurity phase, as will be shown below. Therefore, the two phases
are not entirely distinct. A series of transitions affecting moments of
decreasing order occurring at increasing masses,
\begin{equation}
\label{masses}
m_1>m_2>\cdots>m_{\infty},
\end{equation}
signals the transition to the heavy impurity phase. When $m\geq m_n$, the
ratio $M_{2k}/L_{2k}$ diverges asymptotically for all $k\ge n$. This
generalizes the second moment transition occurring at $m_1\equiv m_*$. The
transition masses
\begin{equation}
m_n={r_q+{1\over 2}\left[(1-r_p)^{2n}+(1+r_p)^{2n}\right]^{1\over 2n}\over
1-{1\over 2}\left[(1-r_p)^{2n}+(1+r_p)^{2n}\right]^{1\over 2n}}
\end{equation}
are found from $q^{2n}=p^{2n}+(1-p)^{2n}$ and Eq.~(\ref{rq}). In
particular, $m_\infty=\lim_{n\to\infty}m_n=(1+r_p+2r_q)/(1-r_p)$, and
sufficiently light impurities ($mm_\infty\geq 1$, the impurity
must be heavier than the fluid for any transition to occur.
The above analysis of the light impurity phase suggests that in higher
dimensions, the impurity velocity distribution might be similar to
that of the fluid. In higher dimensions we again assume that the
Lorentz-Boltzmann equation admits a scaling solution. The
corresponding equation in Fourier space (\ref{gkt-rate}) then
considerably simplifies. Following the earlier treatments of the
fluid case\cite{kb,eb} we extract the high-energy tail from the
small-$k$ behavior of the Fourier transform of the scaled velocity
distribution. The outcome is that both velocity distributions have the
same algebraic high-energy tail
\begin{equation}
\label{taild}
{\cal Q}(w)\sim{\cal P}(w)\sim w^{-\sigma},
\end{equation}
with the exponent $\sigma$ calculated in \cite{kb,eb}. Such analysis
also yields the ratio of the prefactors governing this algebraic decay.
\subsection{The heavy impurity phase}
When the mass of the impurity is equal to or larger than the critical
mass, $m\geq m_*$, the velocities of the fluid particles are
asymptotically negligible compared with the velocity of the
impurity. Hence, in the $t\to\infty$ limit, fluid particles become
stationary as viewed by the impurity. Therefore, one can set ${\bf
u}_2\equiv 0$ in the collision rule (\ref{ruleq}):
\begin{equation}
\label{lorcol}
{\bf v}={\bf u}-(1-q)\,({\bf u}\cdot{\bf n})\, {\bf n}.
\end{equation}
This process is somewhat analogous to a Lorentz gas\cite{Lor}.
However, in the granular impurity system, a heavy particle scatters of
a static background of lighter particles, while in the Lorentz gas the
scatterers are infinitely massive. Despite this difference, the
mathematical descriptions of the two problems are
similar. Specifically, the collision rule for the inelastic Lorentz
gas \cite{p,mp1} is obtained from (\ref{lorcol}) by a mere replacement
of the factor $(1-q)$ with $(1+r_q)$.
Let us first consider the one-dimensional case where an explicit solution of
the velocity distribution is possible. Setting $u_2\equiv 0$ in the delta
function in the Lorentz-Boltzmann equation (\ref{beq}), integration over the
fluid velocity $u_2$ is trivial, $\int d u_2\,P(u_2,t)=1$, and integration
over the impurity velocity $u_1$ gives
\begin{equation}
\label{lorentz}
{\partial\over \partial t}Q(v,t)+Q(v,t)={1\over q}
Q\left({v\over q}\,,\,t\right).
\end{equation}
This equation can be solved directly by considering the stochastic
process the impurity particle experiences. In a sequence of
collisions, the impurity velocity changes according to $v_0\to q
v_0\to q^2 v_0\to \cdots$ with $v_0$ the initial velocity. After $n$
collisions the impurity velocity decreases exponentially, $v_n=q^n
v_0$. The collision rate is unity, and hence, the average number of
collisions experienced till time $t$ equals $t$. Furthermore, the
collision process is random, and therefore, the probability that the
impurity undergoes exactly $n$ collisions up to time $t$ is Poissonian
$t^ne^{-t}/n!$. Thus, the velocity distribution function reads
\begin{equation}
\label{pivt}
Q(v,t)=e^{-t}\sum_{n=0}^{\infty}{t^n\over n!}\,
{1\over q^n}\,Q_0\left({v\over q^n}\right),
\end{equation}
where $Q_0(v)$ is the initial velocity distribution of the impurity.
Indeed, one can check that this properly normalized solution satisfies
Eq.~(\ref{lorentz}).
Interestingly, the impurity velocity distribution function is a
time-dependent combination of ``replicas'' of the initial velocity
distribution. Since the corresponding argument is stretched, compact velocity
distributions display an infinite set of singularities, a generic feature of
the Maxwell model\cite{e,bk}.
The impurity velocity distribution exhibits interesting asymptotic behaviors.
Consider for simplicity the uniform initial velocity distribution: $Q_0(v)=1$
for $|v|<1/2$ and $Q_0(v)=0$ otherwise. The solution (\ref{pivt}) reduces to
a finite sum, with $n\leq N={\ln(2v)\over \ln q}$. In the physically
interesting limits $t\to\infty$ and $v\to 0$, the sum on the right-hand side
of Eq.~(\ref{pivt}) simplifies to respectively $e^{t/q}$ or $(t/q)^N/N!$ when
the number of terms is above or below the threshold value $N=t/q$. The
magnitude of $Q(v,t)$ above the threshold greatly exceeds the magnitude below
the threshold, so $Q(v,t)$ appears to approach a step function. A refined
analysis shows that the width of the front widens diffusively so the front
remains smooth although its relative width vanishes. Specifically, one finds
the following traveling-wave like scaling solution
\begin{equation}
\label{pivt1}
Q(v,t)\to e^{-t+t/q}\,\Psi(\eta),
\end{equation}
with the following wave form and coordinate
\begin{equation}
\label{pivt2}
\Psi(\eta)={1\over \sqrt{\pi}}\int_{-\infty}^\eta dx\,e^{-x^2}, \quad
\eta={{q\over \ln q}\,\ln(2v)-t\over
\sqrt{2qt}}.
\end{equation}
Note, however, that the large velocity tail ($\eta\to\-\infty$),
ignored in Eq.~(\ref{pivt2}), provides actually the dominant
contribution to the moments. This is an unusual traveling wave
form in the sense that the argument is the logarithm of the
velocity rather then the velocity itself, a reflection of the
exponentially decaying velocity.
In contrast to the velocity distribution, the moments $M_n(t)=\int dv\, v^n
Q(v,t)$ exhibit a much simpler behavior. Indeed, from Eq.~(\ref{lorentz})
one finds that every moment is coupled only to itself, ${d\over dt}
M_n=-(1-q^n)M_n$. Solving this equation we recover Eq.~(\ref{mn-heavy}); in
the heavy impurity phase, however, it holds for all $n$. Therefore the
moments exhibit multiscaling asymptotic behavior. The decay coefficients,
characterizing the $n$-th moment, depend on $n$ in a nonlinear fashion. This
multiscaling behavior excludes scaling solutions with sharp tails (stretched
exponentials decays and faster). While collisions with the fluid are
sub-dominant, they still lead to corrections to the leading asymptotic
behavior.
We turn now to arbitrary spatial dimensions $d$. The impurity velocity
changes according to Eq.~(\ref{lorcol}) with the impact direction ${\bf n}$
chosen randomly. The corresponding Lorentz-Boltzmann equation reads
\begin{eqnarray}
\label{dlorentz}
{\partial\over \partial t}Q({\bf v},t)+Q({\bf v},t)&=&\int d{\bf n}\int
d{\bf u}\,Q({\bf u},t)\nonumber\\
&& \times\delta\Big[{\bf v}-{\bf u}+(1-q)({\bf u}\cdot{\bf
n})\, {\bf n}\Big].
\end{eqnarray}
Moments of the velocity distribution
\begin{equation}
\label{mnt-def}
M_n(t)=\int d{\bf v}\, v^n\, Q({\bf v},t),
\end{equation}
can be obtained directly. We focus on the even moments of the
distribution. Indeed, they satisfy the following evolution equation
\begin{equation}
\label{mnt-eq}
{d\over dt}M_{2n}=-\big(1-\langle \xi^n\rangle\big)M_{2n}
\end{equation}
where $\mu=\cos^2\theta=(\hat {\bf u}\cdot {\bf n})^2$, $\xi\equiv
\xi(q,\mu)=1-(1-q^2)\mu$, and $\langle \cdot \rangle$ is the shorthand
notation for the angular integration: $\langle \,f
\,\rangle\equiv\int_0^1 {\cal D}\mu\, f(\mu)$. Since $d{\bf n}\propto
\sin^{n-2}\theta\,d\theta$, the (normalized) integration measure
${\cal D}\mu$ is
\begin{equation}
\label{dmu}
B\left({1\over 2},{d-1\over 2}\right){\cal D}\mu
=\mu^{-{1\over 2}}(1-\mu)^{d-3\over 2} d\mu
\end{equation}
where $B(a,b)$ is the beta function. For example, $\langle 1\rangle =1$, and
$\langle \mu\rangle=1/d$.
{}From the evolution equations (\ref{mnt-eq}), the moments are found
to decay exponentially with time
\begin{equation}
\label{mnt-sol}
M_{2n}(t)=M_{2n}(0)\,e^{-\left(1-\langle \xi^n\rangle\right)t}.
\end{equation}
In particular, $1-\langle \xi\rangle=(1-q^2)/d$, and thus, the temperature
decay $\Theta(t)=\Theta(0)\,e^{-(1-q^2)t/d}$ is recovered. There is a little
discrepancy with the exact temperature of Eq.~(\ref{theta-sol}) which depends
on the initial fluid temperature, $T_0$. This is a remnant of a transient
regime where the two velocity scales are comparable. Generally, the time
dependence is correct. Note also that in one dimension $\xi=q^2$ and hence
Eq.~(\ref{mnt-sol}) reduces to our earlier result. In the infinite dimension
limit, $\mu\to 0$, and all moments decay according to $e^{-t}$. However, in
general, the moments exhibit multiscaling asymptotic behavior, and knowledge
of the typical velocity is insufficient to fully characterize the entire
velocity distribution. Indeed, writing $M_{2n}\sim M_2^{\alpha_n}$, the
exponents \hbox{$\alpha_n=(1-\langle\xi^n\rangle)/(1-\langle\xi\rangle)$}
have a nontrivial spectrum.
The moments directly give a formal exact solution of the Fourier
transform of the impurity velocity distribution (\ref{gkt-def}). We
consider isotropic situations where $G\left({\bf k},t\right)\equiv
G(k^2,t)$ with $k\equiv |{\bf k}|$. Expanding the transform in powers
of $k^2$ and substituting the moment result (\ref{mnt-sol}) yields
\begin{equation}
\label{gkt-sol}
G({\bf k},t)=e^{-t}\sum_{n=0}^\infty
{\left(-k^2\right)^n\langle\mu^n\rangle\over (2n)!}\,M_{2n}(0)\,
e^{\langle \xi^n\rangle t}\,.
\end{equation}
While this is an explicit solution, it is not too illuminating. First,
it is in Fourier space, and second, it involves the complicated
angular averages $\langle\xi^n\rangle$.
Nevertheless, it can be shown that the solution remains a time
dependent combination of properly modified replicas of the initial
distribution. Indeed, either from Eq.~(\ref{gkt-sol}) or directly
from the Fourier transform equation \hbox{${\partial\over\partial
t}G({\bf k},t)+G({\bf k},t)= \langle G({\bf k}\xi,t)\rangle$}, the
solution can be rewritten in the form
\begin{equation}
\label{fkt-alt}
G(k^2,t)=e^{-t}\sum_{n=0}^\infty {t^n\over n!}\,G_n(k^2).
\end{equation}
Here, $G_0(k^2)$ is the initial Fourier transform, and the ``building
blocks'' $G_n$ are obtained from a recursive procedure of angular
integration $G_{n+1}(k^2)=\langle G_n(k^2\xi)\rangle$.
\section{The Improved Inelastic Maxwell Model}
In the IMM, the rates for fluid-fluid and fluid-impurity collisions
were identical (and therefore, set to unity for convenience). For
granular fluids, however, the collision rate is proportional to the
relative velocity \cite{cb,my,neg,dufty}. Therefore, one can improve
the Maxwell model by replacing the actual collision rate with an {\em
average} collision rate proportional to the average relative velocity.
The simplest choice of the average relative velocity is $\sqrt{\langle
({\bf v}_1-{\bf v}_2)^2\rangle }\propto \sqrt{(T_1+T_2)/2}$. For
fluid-fluid and impurity-fluid collisions we thus obtain $\sqrt{T}$
and $\sqrt{(T+\Theta)/2}$, respectively. Thus, different collision
rates multiply the collision integrals in the Boltzmann and
Lorentz-Boltzmann equations (\ref{bep})--(\ref{beq}). In the fluid
case, this overall prefactor merely affects the time dependence of the
temperature. As will be shown below, in the impurity case, the above
phase transition is suppressed in the IIMM, although secondary
transitions corresponding to higher order moments remain.
Let us again start with the behavior of the temperature.
The fluid temperature satisfies
\begin{equation}
\label{t-eq-mf}
{d \over dt}T=-\sqrt{T}\left[{2p(1-p)\over d}\,T\right].
\end{equation}
Solving this equation, we recover Haff's cooling law
$T(t)=T_0[1+t/t_0]^{-2}$, with $T_0$ the initial temperature and
$t_0=d/[p(1-p)T_0^{1/2}]$ \cite{pkh}.
The corresponding rate equation for the impurity temperature $\Theta$
is
\begin{equation}
\label{theta-eq-mf}
{d \over dt}\,\Theta=\sqrt{{T+\Theta\over 2}}
\left[-{1-q^2\over d}\,\Theta +{(1-q)^2\over d}\, T\right].
\end{equation}
Since we are primarily interested in the temperature ratio, $S=\Theta/T$, we
study this quantity directly. It evolves according to
\begin{eqnarray}
\label{ratio-mf}
{1\over \sqrt{T}}\,{d \over dt}\,S &=&\sqrt{{1+S\over2}}
\left[-{1-q^2\over d}\,S+{(1-q)^2\over d}\right]\nonumber\\
&+&{2p(1-p)\over d}\,S.
\end{eqnarray}
In the inelastic Maxwell model, the gain and the loss terms were
comparable, both increasing linearly with $S$. Here, in contrast, the
loss term, which grows as $S^{3/2}$, eventually overtakes the gain
term that grows only linearly with $S$. Therefore, $S\to c$ where $c$
is the root of the cubic equation
\begin{equation}
\label{root-mf} \sqrt{{1+c\over 2}}\left(c-{1-q\over 1+q}\right)
={2p(1-p)\over 1-q^2}\,c.
\end{equation}
Consequently, there is only one phase, the light impurity phase. We
note that the ratio $c$ is independent of the spatial dimension
$d$. Intuitively, since the impurity collision rate relatively
increases with the impurity temperature, the impurity energy
dissipation rate increases, thereby limiting the (relative) growth of
the impurity temperature.
\begin{figure}
%\centerline{\epsfxsize=8cm \epsfbox{fig3.eps}}
\centerline{\includegraphics[width=8cm]{fig3}}
\caption{The equipartition mass $m_{\rm eq}$, given by
Eq.~(\ref{m1-mf}), versus the restitution coefficients $r_p$ and
$r_q$.}
\end{figure}
Generally, there is no equipartition of energy except for a particular
surface in the space $(m,r_p,r_q)$. Energy equipartition occurs when
$mc=1$. Using the relations (\ref{rp})--(\ref{rq}),
we find the equipartition
mass
\begin{equation}
\label{m1-mf}
m_{\rm eq}=-{1\over 2}+{1\over 2}
\sqrt{1+8\left({1-r_q^2\over 1-r_p^2}\right)^2}.
\end{equation}
Figure 3 plots $m_{\rm eq}=m_{\rm eq}(r_p,r_q)$.
Qualitatively, our findings in the light impurity phase of the IMM extend to
the IIMM. For example, both velocity distributions follow scaling forms and
the large-velocity tails of both distributions are the same. In the one
dimensional case, explicit expressions for the impurity scaling function are
possible, and as the treatment follows closely that outlined in the light
impurity phase, we briefly outline the results. In 1D, the impurity velocity
distribution approaches a scaling solution $Q(v,t)\to T^{-1/2}{\cal Q}\left(v
T^{1/2}\right)$. The corresponding Fourier transform reads
$G(k,\tau)=g\left(|k|\,T^{1/2}\right)$. The only difference with the above
inelastic Maxwell model is that the collision terms are proportional to
$\beta^{-1}=\sqrt{(1+c)/2}$. Consequently, Eq.~(\ref{gz}) generalizes as
follows
\begin{equation}
\label{gz-mf}
-\beta p(1-p)z g'(z)+g(z)=(1+az)\,e^{-az}\,g(q z).
\end{equation}
Seeking a series solution of the form (\ref{gsol}), leads to the following
recursion relations for the coefficients
\begin{equation}
\label{An-mf}
A_n={(1-q)q^{n-1}-\beta p(1-p)\over 1-q^n-n\beta p(1-p)}\,A_{n-1},
\end{equation}
with $A_0=A_1=1$. Again, the velocity distribution is a combination
of powers of Lorentzians as in Eq.~(\ref{qw}):
\begin{equation}
\label{qw-mf}
{\cal Q}(w)={2\over \pi}\sum_{n=2}^{\infty}B_n
\left(1+w^2\right)^{-n},
\end{equation}
where $B_n$ are linear combinations of the coefficients $A_n$'s given
by the same expressions as in Sec.~II. In particular, the
large-velocity tail is generic ${\cal Q}(w)\sim w^{-4}$.
Given the algebraic form of the velocity distributions, we examine the
asymptotic behavior of moments of the velocity distribution. Moments
of the fluid and the impurity, $L_n$ and $M_n$, respectively, evolve
according to a straightforward generalization of
Eqs.(\ref{l2n})--(\ref{m2n}),
\begin{eqnarray}
\label{ln-mf}
{d\over d\tau_p}L_n+a_nL_n
&=&\sum_{j=2}^{n-2} {n\choose j} p^{j}(1-p)^{n-j}L_{j}L_{n-j},\\
\label{mn-mf}
{d\over d\tau_q}M_n+b_nM_n
&=&\sum_{j=0}^{n-2} {n\choose j} q^{j}(1-q)^{n-j}M_{j}L_{n-j}.
\end{eqnarray}
Here $a_n(p)$ and $b_n(q)$ are given by Eq.~(\ref{abn}) and
the collision counters,
\begin{eqnarray*}
\tau_p=\int_0^t dt'\,\sqrt{T}, \qquad
\tau_q=\int_0^t dt'\,\sqrt{(T+\Theta)/2},
\end{eqnarray*}
play the role of time in Eq.~(\ref{ln-mf}) and (\ref{mn-mf}),
respectively. Since the two temperatures are asymptotically
proportional to each other, $\tau_q\to\tau_p/\beta$. The fluid
moments decay according to $L_n\propto e^{-a_n(p)\tau_p}\propto
t^{-2a_n/a_2}$ \cite{bk}. Inserting the asymptotics $L_n\propto
e^{-a_n(p)\beta\tau_q}$ into Eq.~(\ref{mn-mf}) and performing the same
analysis as in the IMM we find that when $\beta a_n(p)