\documentstyle[graphicx,multicol,epsf,aps]{revtex} \begin{document} \title{Stationary Velocity Distributions in Traffic Flows} \author{E.~Ben-Naim$\dag$ and P.~L.~Krapivsky$\ddag$} \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 introduce a traffic flow model that incorporates clustering and passing. We obtain analytically the steady state characteristics of the flow from a Boltzmann-like equation. A single dimensionless parameter, $R=c_0v_0t_0$ with $c_0$ the concentration, $v_0$ the velocity range, and $t_0^{-1}$ the passing rate, determines the nature of the steady state. When $R\ll 1$, uninterrupted flow with single cars occurs. When $R\gg 1$, large clusters with average mass $\langle m\rangle\sim R^{\alpha}$ form, and the flux is $J\sim R^{-\gamma}$. The initial distribution of slow cars governs the statistics. When $P_0(v)\sim v^{\mu}$ as $v\to 0$, the scaling exponents are $\gamma=1/(\mu+2)$, $\alpha=1/2$ when $\mu>0$, and $\alpha=(\mu+1)/(\mu+2)$ when $\mu<0$. \noindent{PACS numbers: 02.50-r, 05.40.+j, 89.40+k, 05.20.Dd} \end{abstract} \begin{multicols}{2} \section{Introduction} Traffic flows are strongly interacting many-body systems. They also present a natural testbed for theories and techniques developed for physical systems such as kinetic theory and hydrodynamics. Traffic systems have been receiving much attention recently \cite{Gartner}, and a number of approaches were suggested including fluid mechanics \cite{hydr1,hydr2,Prigogine,kerner}, cellular automata \cite{Nagel,Biham,Nagatani,Schadschneider,Nagel1,Nagel2,Brankov,Ktitarev}, particle hopping \cite{Der,Benjamini,Evans,Popkov}, and ballistic motion \cite{eps,jap1,jap2,Helbing,Nagatani1}. The diversity of the approaches reflects the rich phenomenology which includes shock waves, clustering, and slowing down. Traffic networks can be viewed as low dimensional systems. For example, rural traffic is intrinsically one-dimensional and urban grid traffic is two-dimensional. This important simplifying feature makes analytical treatment possible. Ballistic models are harder to simulate than cellular automata and particle hopping models. However, they are quite realistic since time and space are treated as continuous variables. They can also prove useful for theoretical treatment. An exactly solvable clustering process shows that extremal properties of the velocity distribution determine the kinetic behavior \cite{eps}. However, it results in ever-growing and ever-slowing jams with a trivial steady state in a finite system. In this study, we investigate more realistic situations where fast cars can pass slow cars. This is motivated by and should be applicable to passing zones of one lane roadways as well as multilane highways. Our goal is to determine analytically statistical properties of the flow such as the flux, and characterize their dependence on the intrinsic velocity distribution. We start by formulating the model. Consider a one-dimensional traffic flow with sizeless cars (``particles'') moving with a constant velocity. We assume that cars have intrinsic velocities by which they would drive on an empty road. Initially, cars are randomly distributed in space and they drive with their intrinsic velocities. However, the presence of slower cars forces some cars to drive behind a slower car and therefore leads to the formation of clusters. Simple collision and escape mechanisms are implemented. When a cluster overtakes a slower cluster, a larger cluster forms. It moves with the smaller of the two velocities. Meanwhile, all cars in a given cluster may escape their respective cluster and resume driving with their intrinsic velocity (see Fig.~1). We assume a constant escape rate $t_0^{-1}$. The actual collision and escape times are proportional to the car size and thus set to zero (these time scales should become important in heavy traffic). \begin{figure} %\centerline{\epsfxsize=7cm \epsfbox{fig1.eps}} \centerline{\includegraphics[width=7cm]{fig1}} \noindent{\small {\bf Fig.1} Space time diagram of the traffic model. Formation of a cluster with two fast cars is shown to the left and formation of a one car cluster and its breakup due to escape is shown to the right.} \end{figure} A heuristic argument suggests that a single dimensionless parameter underlies the steady state. Consider a state where the car concentration is $c_0$, and the typical intrinsic velocity range is $v_0$. Let the steady state cluster density be $cP_0(v)$ when $v-1, \end{equation} in the velocity range [0:1] with the prefactor ensuring unit normalization. For such distributions, the threshold velocity decreases with growing $R$ according to $v^* \sim R^{-{1\over \mu+2}}$. For $v\gg v^*$, the integral in Eq.~(\ref{pv}) dominates over the constant factor and \hbox{$R P(v)\int_0^v dv' (v-v') P(v')\sim v^{\mu}$}. Anticipating an algebraic behavior for the cluster velocity distribution, $P(v)\sim R^{\sigma}v^{\delta}$ when $v\gg v^*$, gives different answers for positive and negative $\mu$. The leading behavior for $v\gg v^*$ can be summarized as follows \begin{equation} \label{pvlead} P(v)\sim\cases{(v^*)^\mu(v/v^*)^{\mu-1}&$\mu<0$;\cr (v/v^*)^{-1}[\ln(v/v^*)]^{-{1\over 2}}&$\mu=0$;\cr (v^*)^\mu (v/v^*)^{{\mu\over 2}-1}&$\mu>0$.} \end{equation} The small and large velocity components of $P(v)$ match at the threshold velocity, $P(v^*)\sim P_0(v^*)$. Careful analysis, detailed in the following section, is needed to get the logarithmic corrections in the borderline case $\mu=0$. Substituting the leading asymptotic behavior of Eq.~(\ref{pvlead}) into Eq.~(\ref{cdef}), the average cluster size is found \begin{equation} \label{mav} \langle m\rangle\sim\cases{R^{(\mu+1)/(\mu+2)} &$\mu<0$;\cr (R/\ln R)^{1/2} &$\mu=0$;\cr R^{1/2} &$\mu> 0$.} \end{equation} Similarly, the average cluster velocity defined in Eq.~(\ref{vavdef}) is evaluated \begin{equation} \label{vav} \langle v\rangle\sim\cases{R^{\mu/(\mu+2)} &$\mu<0$;\cr 1/\ln R &$\mu=0$;\cr {\rm const} &$\mu> 0$.} \end{equation} Two distinct regimes of behavior emerge. For $\mu>0$, car-cluster collisions dominate while for $\mu<0$ cluster-cluster collisions dominate. The scaling argument given in the introduction assumes the former picture, and thus it does not hold in general. {\it A posteriori}, one can extend the scaling argument to the $\mu<0$ regime. The argument becomes involved, and we do not present it here. Interestingly, in the cluster-cluster dominated regime, the scaling behavior for the average cluster size, $\langle m\rangle\sim R^{\alpha}$ with $\alpha=(\mu+1)/(\mu+2)$, is identical to the {\em kinetic} scaling, $\langle m\rangle\sim (c_0v_0t)^{\alpha}$ with the same $\alpha$, found in the no passing limit \cite{eps}. This suggests an analogy between the dimensionless collision number $R=c_0v_0t_0$ and the dimensionless time $c_0v_0t$. On the other hand, the steady state behavior is much richer as it is characterized by two regimes of behavior and different exponents. The flux can be evaluated in a similar fashion using Eq.~(\ref{j}), \begin{equation} \label{j2} J\sim v^* \sim R^{-{1\over \mu+2}}. \end{equation} Interestingly, the flux is proportional to the threshold velocity $v^*$. As a result, the flux exponent $\gamma=1/(\mu+2)$ is a regular function of $\mu$ unlike the cluster size exponent $\alpha$. Eq.~(\ref{j2}) is also consistent with identification of the crossover velocity $v_c$ with the marginal velocity $v^*$. No flux reduction occurs when the intrinsic distribution is dominated by fast cars, i.e., in the limit $\mu\to\infty$. In the other extreme, the maximal flux reduction $J\sim R^{-1}$ is realized when $\mu\to -1$. The car velocity distribution is strongly enhanced in the low velocity limit, as seen by evaluating Eq.~(\ref{g}) \begin{equation} \label{g2} G(v)\sim R^{\mu+1\over \mu+2}v^{\mu}(1-{\rm const.}\times v^{\mu+1}), \quad v\ll v^*. \end{equation} On the other hand, for $v\gg v^*$ we get \begin{equation} \label{g3} {G(v)\over P(v)}\sim \cases{ 1+R^{\mu/(\mu+2)}\,v^{-1}(1-v^{\mu+1}) &$\mu<0$;\cr 1+(v^{-1}-1)[\ln(v/v^*)]^{-1} &$\mu=0$;\cr v^{-\mu-1} &$\mu>0$.\cr} \end{equation} As a check of self-consistency, one can easily verify that \hbox{$\int dv\,G(v)\sim 1$}, and \hbox{$J=\int dv\,v\,G(v) \sim v^*$}. The car velocity distribution is useful for studying velocity fluctuations. More generally, one can consider the moments of the velocity distribution, $G_n=\int dv v^n G(v)$. Naively, one would expect $G_n\sim (G_1)^n$, which together with $G_1\equiv J$ and Eq.~(\ref{j2}) implies $G_n\sim R^{-n/(\mu+2)}$. Using Eq.~(\ref{g3}) one computes the moments to confirm this expectation for sufficiently small $n$, namely for $n<1-\mu$ if $\mu<0$, $n<1$ if $\mu=0$, and $n<1+\mu/2$ if $\mu>0$. When the index $n$ exceeds above thresholds, more interesting behavior is found: \begin{equation} \label{moments} G_n \sim \cases{R^{(\mu-1)/(\mu+2)} &$-1<\mu<0, n>1-\mu$;\cr R^{-1/2}(\ln R)^{-3/2} &$\mu=0, n>1$;\cr R^{-1/2} &$\mu>0, n>1+\mu/2$.\cr} \end{equation} For sufficiently small $\mu$, the fluctuations in flux are very large, $G_2\gg G_1^2$. Thus in the most interesting region $-1<\mu<2$ fluctuations dominate. In summary, as $R\to\infty$ the solution to the differential equation (\ref{Qrho}) exhibits a boundary layer structure. Inside the boundary layer, $vv^*$, the cluster velocity distribution is much smaller than the intrinsic velocity distribution. The threshold velocity $v^*$ is determined by the small velocity behavior of the intrinsic velocity distribution, and for the algebraic distributions (\ref{mu}) we have found $v^*\sim R^{-{1\over \mu+2}}\to 0$. The behavior detailed above in the escape controlled limit is not restricted to purely algebraic distributions but is quite general. We conclude that a single parameter \begin{equation} \label{mudef} \mu=\lim_{v\to 0} v{\partial\over\partial v } \ln P_0(v) \end{equation} determines the behavior as $R\to\infty$. In short, extreme statistics underly the escape-limited flow properties. Additionally, an interesting transition between a slow and a fast velocity dominated flow occurs at $\mu=0$. \section{Examples} Although the above analysis is quite general, it applies only to the limiting values of $R$. To examine intermediate behavior, it is also useful to obtain explicit solutions for some special cases. Below, we consider two relevant cases: uniform $P_0(v)$ and $P(v)$. We also obtain explicit expressions in the case of discrete velocity distributions. \subsection{Uniform Intrinsic Distribution} \begin{figure} %\centerline{\epsfxsize=9cm \epsfbox{fig2.eps}} \centerline{\includegraphics[width=9cm]{fig2}} \noindent{\small {\bf Fig.2 } Velocity distributions in the case of a uniform initial distribution $P_0(v)=1$, for $R=10$.} \end{figure} We now consider the case of a uniform intrinsic distribution, $P_0(v)=1$ for $0