\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)