%------------------------macros------------------ \def\jpa #1 #2 #3 {{\sl J. Phys.\ A} {\bf #1}, #2 (#3)} \def\jcp #1 #2 #3 {{\sl J.\ Chem.\ Phys.} {\bf #1}, #2 (#3)} \def\jdep #1 #2 #3 {{\sl J.\ de Physique I} {\bf #1}, #2 (#3)} \def\pra #1 #2 #3 {{\sl Phys.\ Rev.\ A} {\bf #1}, #2 (#3)} \def\prb #1 #2 #3 {{\sl Phys.\ Rev.\ B} {\bf #1}, #2 (#3)} \def\pre #1 #2 #3 {{\sl Phys.\ Rev.\ E} {\bf #1}, #2 (#3)} \def\prl #1 #2 #3 {{\sl Phys.\ Rev.\ Lett.} {\bf #1}, #2 (#3)} \def\gtwid{\mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}}} \def\ltwid{\mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}}} \def\eg{{\it e.\ g.}}\def\ie{{\it i.\ e.}}\def\etc{{\it etc.}} \def\vs{{\it vs.}}\def\vv{{\it vice versa}}\def\ea{{\it et al.}} \def\apr{{\it a priori}}\def\apo{{\it a posteriori}} \def\pd#1#2{{\partial #1\over\partial #2}} %partial derivative \def\p2d#1#2{{\partial^2 #1\over\partial #2^2}} %second partial derivative \def\ith{{$i^{\rm th}$}} \def\nth{{$n^{\rm th}$}} \newcount\refnum\refnum=0 %automatic, non-labelled references \def\refi{\smallskip\global\advance\refnum by 1\item{\the\refnum.}} \newcount\rfignum\rfignum=0 %automatic, non-labelled figure labels in captions \def\rfigi{\medskip\global\advance\rfignum by 1\item{Figure \the\rfignum.}} \def\pvt{{P(v,t)}} \def\pinf{{P_{\infty}(v)}} \def\pxvt{{P(x,v,t)}} \def\pxvi{P_0(x,v)} \def\pnvt{{P_m(v,t)}} \def\pnt{{P_m(t)}} \def\qnvt{{Q_m(v,t)}} \def\pvi{{P_0(v)}} \def\pv{{\psi(V)}} \def\pn{{\phi(M)}} \def\pnv{{\Phi(M,V)}} \def\pvim{{P_0(v)}\sim v^{\mu}} \def\svt{S(v,t)} \def\rvt{R(v,t)} \def\stvt{\tilde S(v,t)} \def\pg{P_+} \def\ps{P_-} \def\pvp{P_0(v')} \def\fz{f(z)} \def\ct{c(t)} \def\cxt{c(X,t)} \def\cinf{c_{\infty}} \def\a{{\alpha}} \def\b{{\beta}} \def\m{\mu} \def\cta{{c\sim t^{-\a}}} \def\nta{{m\sim t^{\a}}} \def\vtb{{v\sim t^{-\b}}} \def\dx{\Delta x} \def\dv{\Delta v} \def\mo{(\m+1)} \def\mt{(\m+2)} \def\va{\mo/\mt} \def\vb{{1/\mt}} \def\apb{{\a+\b=1}} \def\inv{{\int dv}} \def\fint{{\int\limits_{-\infty}^\infty}} \def\nav{\langle m(v)\rangle} \def\lr#1#2#3{\left#1 #2 \right#3} \def\l{\lambda} \def\e{\epsilon} %----------------------------------------------------- \documentstyle[aps,multicol,epsf]{revtex} \begin{document} \title{Kinetics of Clustering in Traffic Flows} \author{E.~Ben-Naim, P.~L.~Krapivsky and S.~Redner} \address{Center for Polymer Studies and Department of Physics, Boston University, Boston, MA 02215} \maketitle \begin{abstract} We study a simple aggregation model that mimics the clustering of traffic on a one-lane roadway. In this model, each ``car'' moves ballistically at its initial velocity until it overtakes the preceding car or cluster. After this encounter, the incident car assumes the velocity of the cluster which it has just joined. The properties of the initial distribution of velocities in the small velocity limit control the long-time properties of the aggregation process. For an initial velocity distribution with a power-law tail at small velocities, $\pvim$ as $v \to 0$, a simple scaling argument shows that the average cluster size grows as $n \sim t^{\va}$ and that the average velocity decays as $v \sim t^{-\vb}$ as $t\to \infty$. We derive an analytical solution for the survival probability of a single car and an asymptotically exact expression for the joint mass-velocity distribution function. We also consider the properties of spatially heterogeneous traffic and the kinetics of traffic clustering in the presence of an input of cars. \end{abstract} \section{Introduction} A variety of approaches have been applied to describe the collective properties of traffic flows [1]. For example, to mimic congested traffic flow in two dimensions, cellular automation models have been proposed [2,3]. Asymmetric hopping processes have also been applied to model traffic flow on a one-dimensional road [4-6]. When the number of cars is large, traffic flows can be modelled phenomenologically in terms of a one-dimensional compressible gas [7-9]. Such an approach predicts the appearance of shock waves, where hydrodynamic quantities, such as the average density and velocity, become discontinuous. However, the hydrodynamic approach does not naturally describe the behavior of traffic flows in the low-density limit where there are large heterogeneities in traffic density. For this situation, a microscopic model may provide a more appropriate description. In this article, we introduce a ballistic aggregation process to model the kinetics of clustering in one-dimensional traffic flows. Our approach is inspired, in part, by the recent interesting results that have been obtained for a variety of reaction processes which involve ballistic particles including: ballistic agglomeration, $A_i+A_j \to A_{i+j}$, with momentum conserving collisions [10,11]; ballistic annihilation, $A+A \to 0$, [12,13]; and several nucleation and ballistic growth processes [14-16]. In our model, cars move ballistically in a one direction, say to the right, according to an initial velocity distribution. Clusters form whenever a faster car overtakes a slower car or cluster. The overtaking car then assumes the velocity of the lead car in the cluster. This model is an idealized description for one-lane traffic flow. While there are obvious shortcomings in our model, it is exactly soluble and permits a thorough understanding of the kinetics of the aggregation process. This paper is organized as follows. In section II, we present the model and postulate the scaling behavior for the velocity and the concentration of the clusters. This approach makes use of the statistical properties of the minimal random variable within a large sample. In section III, we investigate the distribution of cluster velocities. For this distribution, the cluster size is irrelevant and this feature allows us to consider a simpler ``coalescence only'' model. For this reduced problem, the velocity distribution is obtained exactly in terms of the initial distribution of car velocities and then evaluated for general continuous distributions. Building on these results, the general clustering process is solved in section IV and an asymptotically exact expression for the joint cluster mass-velocity distribution is obtained. In section V, we present a formal solution for the velocity distribution function for an inhomogeneous initial distribution of particles. We examine the temporal behavior that arises for a simple step function initial spatial distribution. In section VI, we investigate another generalization of the model to the situation with a spatially and temporally homogeneous input of cars. Depending on the functional form of the input velocity distribution in the low-velocity limit, the input can give rise to a steady state or to a system which continues to evolve indefinitely. We give our conclusions in section VII. The details of specific calculations are given in the Appendices. \section{Scaling Analysis} We consider an idealized one-dimensional traffic flow in which the size of each car is zero. This is appropriate for describing clustered traffic in the low-density limit, a situation which is often encountered on rural secondary roads. In the following, we will refer to such sizeless cars as particles. We consider the initial condition when there are only isolated particles (``monomers'') in the system with a random spatial distribution of density $c_0$. The initial velocity distribution, $P(v,t=0)$, can generally be written in the scaling form \begin{equation} P(v,t=0)={c_0\over v_0}P_0\lr ( {v\over v_0} ) \qquad v>0, \end{equation} with $\int_0^{\infty}P_0(z)dz=1$. Here we have tacitly subtracted the assumed finite value of the velocity of the slowest car from all velocities. In what follows, it is often convenient to introduce the dimensionless density $c/c_0\to c$, velocity $v/v_0\to v$ and time $c_0v_0t\to t$. This yields a rescaled initial concentration which is equal to unity. In our model, particles move at their initial velocities and whenever a particle overtakes a cluster aggregation occurs. The aggregation rule is simply that two colliding clusters form a new cluster with a velocity equal to the smaller of the two incident cluster velocities and with a mass equal to the sum of the two cluster masses (Figure 1). If we denote a cluster of mass $m$ and velocity $v$ by $A_{m,v}$, the process is described by the reaction scheme \begin{equation} A_{m_1,v_1} + A_{m_2,v_2}\to A_{m_1+m_2,{\rm min}\{v_1,v_2\}}. \end{equation} We present now a simple argument, based on the statistics of extremes, to predict the asymptotic time dependence of the typical cluster mass $m$ and typical cluster velocity $v$ at time $t$. Since the typical distance, $l$, between clusters grows with time as $l\sim vt$, the typical number of particles in a cluster is proportional to this distance, yielding $m \sim l \sim vt$. To find the typical velocity, one has to relate the mass of a cluster to its velocity. Such a relation may be found exactly for an auxiliary ``one-sided'' problem in which particles are placed with a fixed density $c_0$ to the left of a given ``target'' particle which moves at velocity $v$, and no particles are placed to the right. Eventually, this target particle will form a cluster that includes all consecutive particles to its left whose initial velocities are larger than $v$. The probability that there are exactly $k$ such particles is equal to $\ps\pg^k$, with $\pg(v)$ $\bigl(\ps(v)\bigr)$ defined as the probability that a particle has a velocity larger (smaller) than $v$, \ie, \hbox{$\pg(v)=\int_v^\infty P_0(v')dv'$}. Therefore, the average number of particles in the cluster that ultimately forms is given by \begin{equation} \nav=\sum_1^{\infty} k\ps\pg^k=\pg/\ps. \end{equation} Let us now assume a power-law behavior of the initial velocity distribution for small velocities, \begin{equation} \pvi\simeq a v^{\m}, \qquad v\ll1 \end{equation} with $\m>-1$ for normalizability. Imposing this power-law form in Eq.~(3) yields \begin{equation} \nav={\pg(v)\over\ps(v)}\propto {1\over v^{\mu+1}}, \end{equation} for sufficiently low velocities. For a particle which moves with the typical velocity, it is reasonable to expect that this result for the ``one-sided'' problem gives the correct behavior for the full ``two-sided'' problem. If we combine Eq.~(5) with our previous estimate $m\sim vt$, we find the following asymptotic relations, \begin{eqnarray} m&\sim t^{\a} \qquad{\rm with}\qquad \a&={\mu+1\over\mu+2}\nonumber\\ v&\sim t^{-\beta}\qquad{\rm with}\qquad \b&={1\over\mu+2} \end{eqnarray} Since the mass is conserved in the aggregation process, the typical cluster mass and the concentration of clusters $c$ are related by $c\sim 1/m\sim t^{-\a}$. Notice that in the limit $\mu\to \infty$, the mass grows linearly with time. In contrast, when $\mu\to -1$, the mass is roughly constant, since the velocity distribution becomes effectively unimodal and collisions are exceedingly rare. This qualitative dependence on the form of the initial velocity distribution is reminiscent of the ballistic annihilation process [13], where ballistically moving particles annihilate upon collision. In both processes, one finds that the fundamental exponents are related by $\apb$ as a consequence of the relation $c\sim 1/vt$. Moreover, for the two processes the decay exponents have similar functional dependences on the form of the initial velocity distribution. However, the values of the decay exponents are different: for example, for a uniform distribution (corresponding to $\m=0$) one finds $\a=1/2$ for the traffic model, while $\a\cong0.76$ is obtained in simulations of the annihilation process. Additionally, despite the qualitative similarity between these two models for continuous velocity distributions, different behaviors occur when the velocities are discrete. For such discrete distributions, the concentration typically decays algebraically in time for ballistic annihilation [13], while the concentration decays exponentially in time for the traffic model. Since both the typical mass and the velocity scale as power laws in time, the probability of finding a cluster of mass $m$ and velocity $v$, $\pnvt$, is expected to evolve toward a scaling distribution. Taking into account mass conservation, $\int_0^{\infty} dv\sum_m m \pnvt={\rm const.}$, we postulate the scaling form \begin{equation} \pnvt\simeq t^{\b-2\a}\pnv, \end{equation} for the joint distribution, where the scaled mass, $M$, and scaled velocity, $V$, are defined by \begin{equation} M=m/t^{\a}\qquad{\rm and}\qquad V=vt^{\b}. \end{equation} Note that while the mass $m$ is a discrete variable, the rescaled mass $M$ is continuous. Once the joint mass-velocity distribution is found, the single-variable mass and velocity distributions can be found by suitable integrations over the subsidiary variable. Thus the velocity distribution, $\pvt=\sum_m\pnvt$, should have the scaling form $$ \pvt\simeq t^{\b-\a}\pv,\eqno(9{\rm a}) $$ with $\pv=\int_0^{\infty}dM\pnv$, while the cluster-mass distribution, $\pnt=\int_0^{\infty} \pnvt$, should have the scaling form $$ \pnt\simeq t^{-2\a}\pn,\eqno(9{\rm b}) $$ with $\pn=\int_0^{\infty}dV\pnv$. \setcounter{equation}{9} \section{The Car Survival Probability} As a preliminary step in obtaining a full solution for traffic clustering, consider the velocity distribution $\pvt$. For this quantity, we can ignore the masses of each cluster and focus only on the survival probability of a given car. Thus the evolution of the velocity distribution is governed by the ``derived'' coalescence process \begin{equation} A_{v_1}+A_{v_2}\to A_{{\rm min}\{v_1,v_2\}}. \end{equation} In the coalescence process, the density of particles with velocity $v$ is identical to $\pvt$, the velocity distribution of clusters in the full traffic aggregation model defined by Eq.~(2). Let $\svt$ be the survival probability of particles of velocity $v$ at time $t$. Here ``survival'' means a car does not overtake any traffic, but an overtaken car is defined to have survived. Then the velocity distribution function is given by \begin{equation} \pvt=\pvi \svt. \end{equation} The survival probability $\svt$ can be found by considering the possible collisions of a particle with initial position and velocity $(x,v)$ with slower particles whose initial positions are to the right of $x$. A collision between the initial particle with co-ordinates $(x,v)$ and a slower $v'$-particle does not occur up to time $t$ if the interval $[x,x+(v-v')t]$ does not include the slower particle. For an initial velocity distribution, $\pvi$, and a Poissonian initial spatial distribution, the probability that there is no particle with velocity between $v'$ and $v'+dv'$ in the interval $[x,x+(v-v')t]$ is \begin{equation} \exp\lr [ {-dv'\pvp (v-v')t} ]. \end{equation} For a particle to survive to time $t$, this exclusion probability should be taken into account for every $v't'$. The probability for this composite event is simply the product of the probabilities of each event, \begin{equation} S_I(v,t,t')=\exp\Biggl[-\int_0^v dv'(v-v') \left\{P(v',t')(t-t')+\int_{t'}^t dt'' R(v',t'')(t-t'')\right\}\Biggr] . \end{equation} Here the first factor, obtained from Eq.~(13) by replacing $P_0(v')$ with $P(v',t')$, yields the probability of avoiding collisions with particles injected prior to time $t'$. The second factor represents the product of exclusions of the type off Eq.~(12) for times larger than $t'$ and velocities smaller than $v'$. This factor accounts for the probability that there are no collisions with particles injected at times $t'', t''>t'$. Note that the kernel of the second factor involves the input rate $R$ at time $t''$, which plays the role of the initial velocity distribution at this instant of time. Substituting Eq.~(34) into Eq.~(35) gives a nonlinear integral equation that describes the kinetics of traffic clustering in the presence of homogeneous particle input. In the following, we will assume that the flux is constant {\it both} in time and space, $\rvt=\pvi$ with $\int_0^{\infty} dv\pvi=1$, so that the governing integral equation is \begin{equation} \pvt=\pvi\int_0^t dt' \exp\lr [ {- {1 \over 2}t'^2\int_0^v dv'(v-v')\pvi -t'\int_0^v dv'(v-v')P(v',t-t')} ] . \end{equation} In parallel with the case of no input, consider again an initial velocity distribution with a power-law small-velocity tail, $\pvim$. Furthermore, let us assume that the concentration and velocity continue to vary as $\cta$ and $\vtb$, respectively, and that the velocity distribution continues to have the scaling form of Eq.~(9), $\pvt \sim t^{\b-\a}\pv$. Substituting these into Eq.~(36), one can extract consistency conditions for the exponents $\a$ and $\b$. For example, the powers of time on both sides of Eq.~(36) should be equal -- hence $\b-\a=-\b\m+1$. Furthermore, both terms in the exponential in the right-hand side of Eq.~(36) cannot depend on time explicitly -- hence $\b(\m+2)=2$ and $\a+\b=1$, respectively. These conditions yield $\a=\m/\mt$ and $\b=2/\mt$. Notice, however, that when $\m$ is positive the exponent $\a$ is also positive and therefore the concentration, $\cta$, decays to zero. This is in obvious contradiction with the nature of the problem: with a constant flux the concentration may grow indefinitely or a steady state concentration may be reached. Thus, one can expect that the above description holds only for $\m<0$. For positive $\m$, we anticipate that the system reaches a steady state with a constant concentration and a typical cluster mass $n\sim t$. At the transition $\m=0$, a logarithmic temporal dependence is anticipated to occur. While we cannot confirm the above picture rigorously, we can provide heuristic justification. First {\it assume} that the velocity distribution evolves towards a steady state $\pinf$. Then as $t \to \infty$ Eq.~(36) becomes \begin{equation} \pinf=\pvi\int_0^{\infty} dt' \exp\lr [ {-{1 \over 2}t'^2\int_0^v dv'(v-v')\pvi -t'\int_0^v dv'(v-v')P_{\infty}(v')} ] . \end{equation} If $P_0(v)\sim v^\m$ as $v\to 0$, Eq.~(37) suggests a similar behavior for the steady state velocity distribution function, \begin{equation} \pinf \simeq Av^{\nu} \quad v \ll 1. \end{equation} Since the concentration of clusters tends to the steady state limit $\cinf=\int_0^{\infty}\pinf\,dv$, the exponent $\nu$ must satisfy the inequality $\nu>-1$. If one substitutes the assumed power law behaviors for $\pvi$ and $\pinf$ into Eq.~(37), three possibility arise in the limit $v \to 0$ which depend on the sign of $\nu-{\m \over 2}+1$. In the case where $\nu>{\m \over 2}-1$, the first exponential factor in Eq.~(37) provides the dominant contribution. However, a simple calculation of the integral shows that $\nu={\m \over 2}-1$. Similarly, for $\nu<{\m\over 2}-1$, one again finds $\nu={\m \over 2}-1$. Only the last possibility, $\nu={\m \over 2}-1$, appears to be self-consistent. Since $\nu>-1$, we obtain $\m>0$. Therefore starting from the assumption that the system reaches the steady state we have obtained that the exponent $\m$ should be positive. This provides evidence for our conclusion that $\m=0$ demarcates the scaling and steady state behaviors. Notice also that for $0<\m \ll 1$, an asymptotic analysis of Eq.~(37) gives the numerical prefactor in Eq.~(38), $A \simeq \sqrt{a\m/2}$. This yields the estimate for the steady state concentration of clusters, \begin{equation} \cinf \simeq \sqrt{2a \over \m} \quad 0<\m \ll 1, \end{equation} Since $\cinf$ diverges as $\m \to 0$ this indicates that at the critical value $\m, \m=0$, the system is still evolving. Assuming the scaling form, obtained by the power counting analysis of Eq.~(36), let us examine the asymptotic behavior of the velocity distribution and the typical concentration. If we substitute the scaling assumptions \begin{equation} \pvt \simeq t^{\b-\a}\pv \quad{\rm with}\quad \a={\m \over \m+2}, \quad \b={2 \over \m+2}, \end{equation} into Eq.~(36), we arrive at the following equation for the scaling function $\pv$, \begin{equation} \pv=aV^\m \int_0^1d\tau \exp\lr[ {-{1 \over 2}b\tau^2 V^{\m+2} - \tau(1-\tau)^{\b-\a}\int_0^V\psi[V'(1-\tau)^\b](V-V')dV'} ], \end{equation} where $\tau=t'/t$ and $b=a/(\m+1)(\m+2)$. Although we are unable to solve this nonlinear integral equation in general, we can obtain interesting information regarding the interesting borderline case of $\mu=0$. This case corresponds to $\a=0$, suggesting that the concentration grows slower than algebraically in time. However, the concentration of clusters is given by \begin{equation} \ct \simeq t^{-\a}\int\pv dV, \end{equation} and for $\mu=0$ the integral diverges at the upper limit. To obtain the asymptotic behavior we consider Eq.~(41) for the case $\mu=0$, \begin{equation} \pv=a\int_0^1d\tau \exp\lr[ {-{1 \over 2}b\tau^2 V^2 - \tau(1-\tau)\int_0^V\psi[V'(1-\tau)](V-V')dV'} ]. \end{equation} If we temporarily ignore the second term in the exponent, we find $\pv \sim V^{-1}$ for $V \to {\infty}$. If we then include the second term in the exponent and apply the previous asymptotic behavior of $\pv \sim V^{-1}$, we find $\pv \sim (V\log{V})^{-1}$. These estimates suggest the ansatz \begin{equation} \pv \simeq C V^{-1} (\log V)^{-\l}, \end{equation} for $V \gg 1$. Upon substituting Eq.~(44) into Eq.~(43) one obtains the constants $\l=1/2$ and $C=1/\sqrt{2}$. With these values, the integral in the right-hand side of Eq.~(42) diverges as $\sqrt{2\log V}$, where $V$ now denotes the maximum value of the scaled velocity. Since $\beta=1$, this maximal velocity is proportional to $t$, which therefore suggests the logarithmic time dependence \begin{equation} \ct \simeq \sqrt{2\log t}. \end{equation} In the complementary case of $\m<0$, we have confirmed that the naive scaling ansatz is consistent with Eq.~(42). \section{Conclusion} We have introduced a simple ballistic aggregation process that mimics the kinetics of clustering in a single lane of traffic. Through direct probabilistic approaches, the analytical forms of the cluster velocity distribution and the joint mass-velocity distribution have been derived. For an initial velocity distribution of the form $P_0(v)\sim v^\mu$ as $v\to 0$, both the average velocity and the average cluster mass have power law time dependences with exponents that are rational functions of $\mu$. This qualitative behavior is similar to that observed in the closely related ballistic annihilation process. We are also able to determine the asymptotic form of the joint mass-velocity distribution. Our model can also be analyzed in the cases of a spatially heterogeneous particle distribution and continuous input of particles. For the simple case of an initial one-sided spatial distribution, the system evolves towards a scaling distribution both in velocity and spatial variables. We have thus found that the total number of clusters in the initially empty half-line grows logarithmically with time for all initial velocity distributions. When there is a steady input of particle in an initially empty system, we have found that there is a transition between steady state behavior for $\m>0$ and transient behavior for $\m<0$ which is similar to that found when there is no in put. For the borderline case of the uniform distribution, $\m=0$, we have found that the total concentration of clusters grows as $\sqrt{\log t}$. The irreversible traffic model introduced in this paper leads to ever-growing clusters. To describe traffic flows more realistically, several mechanisms to induce a steady state can be envisioned. For example, the input model can be generalized to incorporate a flux out of the system. Another realistic direction is to allow a faster car to pass a slower car at a rate which is some increasing function of the velocity difference of the two cars. This would allow a fast car to traverse a cluster car-by-car and ultimately regain its intrinsic velocity once the cluster is completely passed. It may prove interesting to examine the steady-state properties for this class of models. \section*{Acknowledgements} One of us (SR) thanks Chris Myers for useful discussions. We also gratefully acknowledge ARO grant \#DAAH04-93-G-0021, NSF grant \#DMR-9219845, and the Donors of The Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research. \appendix \section{Derivation of Eq.~(24)} In this appendix, we derive the asymptotic form of the cumulative velocity-mass distribution which is valid for an arbitrary initial velocity distribution with a power-law small velocity tail. The starting point for calculation of $\qnvt$ is the formal expression of Eq.~(23). By interchanging the order of the velocity and spatial integrations, the expression can be rewritten as \begin{equation} \qnvt=\pvt\prod_{i=1}^{m-1} \int_0^\infty dx_i \exp(-x_i) \int_{v+(x_1+\cdots+x_i)/t}^{\infty}dv_i\, P_0(v_i). \end{equation} Since the details of the initial distribution for large velocities do not change the form of the scaling solution, we may treat the more general power-law case by choosing a specific initial distribution of velocities whose form is convenient for performing the integration over the velocity in the right-hand side of Eq.~(A1). Since the velocity distribution near $v=0$ has a power-law tail, the most convenient initial distribution is \begin{equation} P_0(v)=a v^\m \exp \lr ( {-a v^{\m+1}/\mo} ) . \end{equation} For this initial distribution, the integration over the velocity variables is immediate and one finds, \begin{equation} \qnvt=\pvt\prod_{i=1}^{m-1} \int_0^\infty dx_i \exp(-x_i) \exp\lr [ {-a\Bigl( v+(x_1+\cdots+x_i)/t \Bigr)^{\m+1}/\mo } ] . \end{equation} To evaluate the multiple spatial integral, we define $f(z)\equiv\exp\left(-a z^{\m+1}/\mo\right)$ and expand $f(z)$ to first order about the point $z=v$ and then exploit a number of simplifications associated with performing the integrals over the factors $e^{-x_i}$. This gives \begin{eqnarray} Q_m(v,t)&=&\prod_{i=1}^{m-1} \int_0^\infty dx_i\, e^{-x_i} f\bigl(v+\e(x_1+\cdots+x_i)\bigr)\nonumber\\ &&\approx\prod_{i=1}^{m-1} \int_0^\infty dx_i\, e^{-x_i} f(v)+\bigl(\e(x_1+\cdots+x_i)\bigr)f'(v),\nonumber\\ &&\approx\prod_{i=1}^{m-1} f(v+\e i) +O(\e^2), \end{eqnarray} where $\e=1/t$. By substituting the explicit functional form, $f(z)=\exp\bigl(-az^{\m+1}/\mo\bigr)$, we have \begin{equation} \qnvt=\pvt\prod_{i=1}^{m-1} \exp\lr( {-a(v+i/t)^{\m+1}/\mo} ). \end{equation} Finally, the product of the exponential factors is written as an exponent of a sum. In the limit of large $m$, this sum is equivalent to the integral $a\int_0^{m-1} dy (v+y/t)^{\m+1}/\mo$. Evaluating this integral, the asymptotic form of the cumulant mass-velocity density is obtained as \begin{equation} \qnvt \simeq a v^\m\exp\lr [ {-b(vt^\b+mt^{-\a})^{\m+2}} ] . \end{equation} In evaluating this asymptotic expression, the exponential factor of $\pvt$ cancels the factor that emerges from lower limit of the integration over $y$. In Eq.~(24), the above expression is written as a scaling function. \section{\bf Analytical Solution for the Exponential Initial Velocity Distribution} We outline here the explicit analytical solution for $Q_m(v,t)$ for the exponential initial velocity distribution, $\pvi=e^{-v}$, which corresponds to the special case $\m=0$ and $a=1$. For this exponential distribution, the integration of $P_0(v_i)$ over the velocity variables $v_i$, according to Eq.~(A1), equals $\exp[-\bigl(v+(x_1+\cdots+x_i)/t\bigr)]$. Hence, we obtain \begin{equation} \qnvt=\pvt e^{-(m-1)v}\prod_{i=1}^{m-1} \int_0^\infty dx_i \exp\biggl[-x_i\biggl(1+{m-i\over t}\biggr)\biggr]. \end{equation} Upon integration over the space variables, the following exact expression is found for $\qnvt$, \begin{equation} \qnvt=\pvt e^{-(m-1)v} t^{m-1} {\Gamma(t+1)\over \Gamma(t+m)}, \end{equation} where $\Gamma(z)=\int_0^{\infty} x^{z-1}\exp(-x)$ is the Euler gamma function and the velocity distribution obtained from Eq.~(14) is $\pvt=\exp\bigl[-v-t\bigl(e^{-v}-1+v\bigr)\bigr]$. The exact forms for the joint mass-velocity distribution and for the mass distribution can be evaluated from Eq.~(B2) by taking the appropriate limits $m \to \infty$ and $v \to 0$. The resulting asymptotic expressions are identical with the expressions of Eqs.~(25) and (26) respectively. \begin{thebibliography}{99} \bibitem{} See, \eg, {\sl Transportation and Traffic Theory}, edited by N.~H.~Gartner and N.~H.~M.~ Wilson (Elsevier, New York, 1987); W.~Leutzbach, {\sl Introduction to the Theory of Traffic Flow} (Springer-Verlag, Berlin, 1988). \bibitem{} O.~Biham, A.~Middleton, and D.~Levine, {\sl Phys. Rev. A} {\bf 46}, 6124 (1992). \bibitem{} T.~Nagatani, {\sl Phys. Rev. E} {\bf 48}, 3290 (1993). \bibitem{} K.~Nagel and M.~Schreckenberg, {\sl J. de Physique I} {\bf 2}, 2221 (1992). \bibitem{} A.~Schadschneider and M.~Schreckenberg, {\sl J. Phys. A} {\bf 26}, L679 (1993). \bibitem{} T.~Nagatani, {\sl J. Phys. 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A} {\bf 45}, 2263 (1992); P.~L.~Krapivsky, {\sl J. Chem. Phys.} {\bf 97}, 8817 (1992). \bibitem{} J.~L.~Spouge, {\sl Phys. Rev. Lett.} {\bf 60}, 871 (1988). \end{thebibliography} \bigskip \centerline{\bf FIGURE CAPTIONS} \bigskip Fig. 1 Schematic illustration of the irreversible traffic model. A faster car overtakes a slower car and after the encounter, the faster car assumes the velocity of the slower car. Fig. 2 Illustration of the initial configuration of a possible three car cluster. \end{document}