\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 $c<c_0$, which implies
the typical cluster size $\langle m\rangle=c_0/c$. If large clusters
form, $\langle m\rangle\gg 1$, then the overall escape rate can be
estimated by $\langle m\rangle t_0^{-1}$. Assuming that most collisions
involve fast cars and slow clusters, the typical collision rate is
$cv_0$. In the steady state, the number of cars joining and leaving
clusters should balance and thus, $c_0/(ct_0)=v_0 c$ or
$c=(c_0/v_0t_0)^{1/2}$.  This heuristic argument gives the leading
behavior of the average cluster size

\begin{equation}
\label{m}
\langle m\rangle\sim R^{1/2}\qquad {\rm when}\quad R\gg 1, 
\end{equation}
where $R$ is the ratio of the two elementary time scales, the escape
time $t_{\rm esc}=t_0$ and the collision time $t_{\rm col}=(c_0v_0)^{-1}$:

\begin{equation}
\label{R}
R={t_{\rm esc}\over t_{\rm col}}=c_0v_0t_0. 
\end{equation}
We term this dimensionless quantity the ``collision number''.  For
large collision numbers, large clusters occur according to
Eq.~(\ref{m}), while for small collision numbers the effect of
collisions is small $\langle m\rangle\cong 1+{\rm const.}\times R$.
Analysis of the master equations detailed below confirms this
heuristic picture under quite general conditions.

The rest of this paper is organized as follows. In Sec.~II, the master
equations are used to derive analytical expressions for various
velocity distributions in the steady state. The leading behavior in
the limiting cases of light and heavy traffic are highlighted in
Sec.~III. Explicit expressions are written for the special cases of
uniform initial and final velocity distributions as well as discrete
distributions in Sec.~IV. The theoretical predictions compare well with
actual traffic data presented in Sec.~V.  We close with some open
problems, a discussion, and possible applications.


\section{Theory}

In the following, it is convenient to introduce dimensionless velocity
$v/v_0\to v$, space $xc_0\to x$, and time $c_0v_0t \to t$ variables.
This rescales the escape rate $t_0^{-1}$ to the inverse collision
number $R^{-1}$. Let $P(v,t)$ be the density of clusters moving with
velocity $v$ at time $t$.  Initially, isolated single cars drive with
their intrinsic velocities drawn from the distribution $P_0(v)\equiv
P(v,t=0)$. This intrinsic velocity distribution is normalized to
unity, $\int dv P_0(v)=1$. The flow is invariant under a velocity
translation, and the minimal velocity is set to zero.

Initially, the velocities and the positions of the particles are
uncorrelated. Escape effectively mixes the positions and the
velocities.  Assuming that no spatial correlations develop, a closed
master equation for the velocity distribution of clusters $P(v,t)$ can
be written
\begin{eqnarray}
\label{pvt}
{\partial P(v,t)\over \partial t}=&&R^{-1}\left[P_0(v)-P(v,t)\right]\\
-&&P(v,t)\int_0^v dv' (v-v')P(v',t).\nonumber
\end{eqnarray}
The density of slowed down cars with intrinsic velocity $v$ is
$P_0(v)-P(v,t)$. Such cars escape their clusters with rate $R^{-1}$, and
thus the escape term.  Collisions occur with rate proportional to the
velocity difference as well as the product of the velocity
distributions.  The integration limits ensure that only collisions with
slower cars are taken into account.  

Steady state is obtained by taking the long time limit $t\to\infty$ or
$\partial/\partial t=0$.  Since we are primarily interested in the
steady state, we omit the time variable $P(v)\equiv
P(v,t=\infty)$.  Equating the right-hand side of the master equation to
zero, a relation between the intrinsic car distribution and steady state
cluster distribution emerges

\begin{equation}
\label{pv}
P(v)\left[1+R\int_0^v dv'(v-v')P(v')\right]=P_0(v).
\end{equation}
Given the intrinsic velocity distribution this relation gives the
final cluster velocity distribution only implicitly. In contrast, the
inverse problem is simpler as knowledge of the final distribution, the
observed quantity in real traffic flows, gives explicitly the
intrinsic distribution.  We confirm that in the limit $R\to\infty$,
all clusters move with the minimal velocity $P(v)\to \delta(v)$, while
in the limit $R\to 0$, all cars move with their intrinsic velocity
$P(v)\to P_0(v)$. 

It is convenient to transform the integral equation (\ref{pv}) into a 
differential one.  Consider the  auxiliary function
\begin{equation}
\label{Q}
Q(v)=R^{-1}+\int_0^v dv'(v-v')P(v'),
\end{equation}
which gives the cluster distribution by second differentiation 
\begin{equation}
\label{PQ}
P(v)=Q''(v). 
\end{equation}
Thence, the steady state condition (\ref{pv}) reduces to the second
order nonlinear differential equation 
\begin{equation}
\label{Qrho}
Q(v)Q''(v)=R^{-1}P_0(v). 
\end{equation}
The boundary conditions are $Q(0)=R^{-1}$ and $Q'(0)=0$. 
The cluster concentration is found from the cluster velocity
distribution using 
\begin{equation}
\label{cdef}
c=\int_0^{\infty}  dv\, P(v),
\end{equation}
and the average cluster mass is simply $\langle m\rangle=c^{-1}$.
Furthermore, the average cluster velocity is obtained from 
\begin{equation}
\label{vavdef}
\langle v\rangle=c^{-1}\int_0^{\infty} dv \, v P(v).
\end{equation}

Cars may drive with a velocity smaller than their intrinsic one, and
it is natural to consider the joint velocity distribution $P(v,v')$,
the density of cars of intrinsic velocity $v$ driving with velocity
$v'$.  The master equation for the joint distribution reads

\begin{eqnarray}
\label{pvvt}
{\partial P(v,v')\over \partial t}
=&-&R^{-1}P(v,v')+(v-v')P(v)P(v')\nonumber\\
&-&P(v,v')\int_0^{v'} dv'' (v'-v'')P(v'')\\
&+&P(v')\int_{v'}^v dv''(v''-v')P(v,v'').\nonumber
\end{eqnarray}
The first term accounts for loss due to escape, while the rest of the
terms represent changes due to collisions. For instance, the last term
describes events where a $v$-car driving with velocity $v''$ is
further slowed down after a collision with a $v'$-cluster.  One can
verify that the total number of $v$-cars, 
\begin{equation}
\label{norm}
P_0(v)=P(v)+\int_0^v dv' P(v,v'),
\end{equation}
is conserved by the evolution Eqs.~(\ref{pvt}) and (\ref{pvvt}).

At the steady state, the joint distribution satisfies 
\begin{equation}
\label{QP}
P(v,v')Q(v')=(v-v')P(v)P(v')+Q(v,v')P(v'), 
\end{equation}
obtained using the definition of $Q(v)$ and the joint auxiliary function 
\begin{equation}
\label{Qvv}
Q(v,v')=\int_{v'}^v dw(w-v')P(v,w).
\end{equation}
Although the collision number $R$ does not appear in Eq.~(\ref{QP})
explicitly, it enters through $Q(v)$ and $P(v)$.  

Combining (\ref{QP}) with Eqs.~(\ref{Qvv}), (\ref{PQ}), and using the
relationship $P(v,v')=\partial^2 Q(v,v')/\partial {v'}^2$ yields 
\begin{equation}
\label{QP1}
{\partial \over \partial v'}
\left[Q^2(v')\,{\partial \over \partial v'}\,
{Q(v,v')\over Q(v')}\right]=(v-v')P(v)P(v').
\end{equation}
Integrating twice over $v'$ gives the joint auxiliary function in
terms of the single variable functions 
\begin{equation}
\label{QP3'}
Q(v,v')=P(v)Q(v')\int_{v'}^v\!\!{du\over
  Q^2(u)}\int_u^v\!\!dw(v-w)P(w). 
\end{equation}
The boundary conditions $Q(v,v)={\partial \over \partial
  v'}Q(v,v')\big|_{v'=v} =0$ were used to obtain this expression. 
Furthermore, integration by parts of $\int_u^v dw (v-w)P(w)=\int_u^v
dw (v-w)Q''(w)$ gives

\begin{equation}
\label{QP3}
Q(v,v')=P(v)\left[Q(v)Q(v')\int_{v'}^v {du\over Q^2(u)}-(v-v')\right].
\end{equation}
Substituting Eq.~(\ref{QP3}) into (\ref{QP}) and then replacing $PQ$
with $R^{-1}P_0$ we find a relatively simple expression for the joint
velocity distribution
\begin{equation}
\label{pvv}
P(v,v')={P_0(v)P_0(v')\over Q(v')}\int_{v'}^v {du\over [RQ(u)]^2}. 
\end{equation}

Another interesting quantity is the flux or the average velocity given
by \hbox{$J=\int dv \left[vP(v)+\int_0^v dw\, w P(v,w)\right]$}. From
the definition of the joint auxiliary function, the second integral is
identified with $Q(v,0)$, implying 
\begin{equation}
\label{jdef}
J=\int_0^{\infty} dv\left[vP(v)+Q(v,0)\right].  
\end{equation}
The integrand can be considerably simplified using Eq.~(\ref{QP3}),
$Q(0)=R^{-1}$, and Eq.~(\ref{Qrho}). The term $vP(v)$ cancels and we
find a useful expression for the flux 

\begin{equation}
\label{j}
J=\int_0^\infty dv\,P_0(v) \int_0^v {du\over [RQ(u)]^2}.
\end{equation}

One can also ask for the actual velocity distribution of cars defined via
\begin{equation}
\label{gdef}
G(v)=P(v)+\int_v^{\infty}\!\! dw\,P(w,v). 
\end{equation}
Substituting the
joint velocity distribution allows us to express the car velocity
distribution via single variable distributions 

\begin{equation}
\label{g}
G(v)=P(v)\left[1+R\int_v^\infty dw\,P_0(w)\int_v^w 
{du\over [RQ(u)]^2}\right].
\end{equation}
The car velocity distribution satisfies the normalization conditions
\hbox{$1=\int\!dv\, G(v)$} and \hbox{$J=\int\! dv \, v G(v)$}.


In summary, for arbitrary intrinsic velocity distributions, the entire
steady state problem is reduced to the nonlinear second order
differential equation (\ref{Qrho}). Given $Q(v)$, steady state
characteristics such as $P(v)$, $P(v,v')$, $J$, and $G(v)$ can be
calculated using the {\rm explicit} formulae (\ref{PQ}), (\ref{pvv}),
(\ref{j}), and (\ref{g}), respectively.



\section{Limiting cases}

Although one cannot solve Eq.~(\ref{Qrho}) analytically in general, it
is still possible to obtain the leading behavior in the limits of 
$R\to 0$ and $R\to \infty$.

\subsection{Low Collision Numbers}

To analyze the flow characteristics in the collision-controlled
regime, $R\ll 1$, we use Eq.~(\ref{pv}) to write $P(v)$ as a perturbation
expansion in $R$:
\begin{equation}
\label{pv1}
P(v)\cong P_0(v)\left[1-R\int_0^v\!\!\!dv'(v-v')P_0(v')\right].
\end{equation}
In this limit, the auxiliary function is roughly constant $RQ(v)\cong 
1$,  and Eq.~(\ref{pvv}) gives the joint distribution to first
order in $R$ 

\begin{equation}
\label{pvv1}
P(v,v') \cong R(v-v')P_0(v)P_0(v').
\end{equation}
The final density and flux are

\begin{equation}
\label{cj1}
c\cong 1-c_1 R, \quad
J\cong J_0-J_1 R,
\end{equation}
with $c_1=\int dv P_0(v)\int_0^v dv'(v-v')P_0(v')$, $J_0=M_1$,
$J_1=M_2-M_1^2$ ($M_n$ are the moments of the intrinsic velocity
distribution $M_n=\int dv\,v^n P_0(v)$).  The coefficient $J_1\geq 0$
equals the width of the initial velocity distribution. This gives a
simple intuitive picture: the larger the initial velocity
fluctuations, the smaller the flux.  By either substituting the joint
velocity distribution into the definition of $G(v)$, or from
Eq.~(\ref{g}), the car velocity distribution is


\begin{equation}
\label{g1}
G(v)\cong P_0(v)\left[1+R\int_0^\infty dv' (v'-v)P_0(v')\right]. 
\end{equation}
As the integral is over the entire velocity range, the order $R$
correction is positive for small $v$ and negative for large $v$. In
other words $G(v)>P_0(v)$ when $v<v_c$. The crossover velocity equals
the average intrinsic velocity $v_c=J_0=M_1$, as seen from
Eq.~(\ref{g1}). 

We conclude that the collision-controlled limit is weakly interacting,
explicit expressions for the leading corrections of the steady state
properties are possible.

\subsection{Large Collision Numbers}

The analysis in the complementary escape-controlled regime, $R\gg 1$,
is more subtle since the condition $R\int_0^v dv'
(v-v')P_0(v')\ll 1$ is satisfied only for small
velocities.  No matter how large $R$ is, sufficiently slow cars
are not affected by collisions, and $P(v)$ is given by
Eq.~(\ref{pv1}) when $v\ll v^*$. The threshold velocity $v^*\equiv
v^*(R)$ is estimated  from \hbox{$R\int_0^{v^*} dv(v^*-v)P_0(v)\sim 1$}. 

It is useful to consider algebraic intrinsic distributions

\begin{equation}
\label{mu}
P_0(v)=(\mu+1)v^{\mu} \quad \mu>-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, $v<v^*$, the cluster velocity distribution is only slightly
affected by collisions, while in the outer region $v>v^*$, 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 $0<v<1$. This case appears to be the most relevant to
real traffic flows since the intrinsic velocity distribution should
be regular near  the minimal velocity.  Integrating $QQ''=R^{-1}$
subject to the boundary conditions $Q(0)=R^{-1}$ and $Q'(0)=0$ gives
$Q'=\sqrt{2R^{-1}\ln (RQ)}$. Second integration gives
\begin{equation}
\label{q}
\int_1^{RQ}{dq\over \sqrt{2\ln q}}=v\sqrt{R},
\end{equation}
and thus implicitly determines $Q(v)$.  Evaluating the leading
behavior when $R\gg 1$, we find 
\begin{equation}
\label{asympt}
\langle m\rangle \simeq \sqrt{R\over \ln R}, \quad
\langle v\rangle \simeq {1\over \ln R}, \quad
J\simeq \sqrt{\pi\over 2R}.
\end{equation}
Fig.~2 shows the velocity distribution obtained numerically using
Eqs.~(\ref{q}) and (\ref{g}) for $R=10$.  For $v\ll v^*$, $G(v)\gg
P_0(v)$, and for $v\gg v^*$, \hbox{$G(v)\cong P(v)\ll P_0(v)$}. The
calculated distributions are consistent with the predictions, $G(0)\sim
R^{1/2}$ and $v^*\sim R^{-1/2}$. The car velocity distribution is 
linear near the origin in agreement with Eq.~(\ref{g2}).



\subsection{Uniform Cluster Distribution}

Consider the uniform final cluster distribution $P(v)=c$.  This
inverse problem is simple as all quantities can be obtained
explicitly.  From Eq.~(\ref{Q}), the auxiliary function is
$Q(v)=R^{-1}+{1\over 2}cv^2$ and from Eq.~(\ref{Qrho}) the initial
distribution reads

\begin{equation}
P_0(v)=c\left[1+{1\over 2} Rcv^2\right].
\end{equation}
The overall initial concentration is unity, thereby relating 
$R$ and $c$ via $1=c+{1\over 6}Rc^2$. The flux is calculated from 
Eq.~(\ref{j}), 
\begin{equation}
\label{Jlambda}
J={(3+\lambda)\sqrt{\lambda}\tan^{-1}\sqrt{\lambda}+
\lambda-\ln(1+\lambda)\over 3R},
\end{equation}
with $\lambda={1\over 2}Rc=(3/2)\left[\sqrt{1+2R/3}-1\right]$.  These
explicit solutions agree with our low and high $R$ predictions.  For
instance, when $R\gg 1$ we find $\langle m\rangle \sim R^{1/2}$ and
$J\sim R^{-1/4}$.  If we look at the initial distribution,
$P_0(v)\cong (6/R)^{1/2}+3v^2$, then the constant part is negligible
and the distribution corresponds to the $\mu=2$ case of the power-law
distribution (\ref{mu}).  For this case the size exponent is
$\alpha=1/2$ and the flux exponent is $\gamma=1/4$, see (\ref{mav})
and (\ref{j2}), in agreement with our findings.



Substituting $P_0(v)$ and $Q(v)$ in Eq.~(\ref{pvv}) and performing the
integration gives the joint distribution

\begin{eqnarray}
\label{Plambda}
P(v,v')&=&
2R^{-1}\lambda^2{(v-v')(1-\lambda v v')\over 1+\lambda v'^2}\\
+2R^{-1}\lambda^{3/2}(1&+&\lambda v^2)
\left[\tan^{-1}(\lambda^{1/2}v)-\tan^{-1}(\lambda^{1/2}v')\right].\nonumber
\end{eqnarray}
A direct integration of the joint distribution confirms the
conservation law (\ref{norm}), thus providing a useful check of
self-consistency.  The joint velocity distribution is linear in the
velocity difference for small $v$ and $v'$. This is reminiscent of the
small collision number behavior of Eq.~(\ref{pvv1}).  As the
velocity difference increases, significant curvature develops (see
Fig.~3). 

\begin{figure}
\vspace{-.2in}
%\centerline{\epsfxsize=8.5cm \epsfbox{fig3.eps}}
\centerline{\includegraphics[width=8.5cm]{fig3}}
\noindent{\small {\bf Fig.3 }
The joint velocity distribution for the uniform final distribution 
case with $R=10$.}
\end{figure}


\subsection{Discrete Velocity Distribution}

The results formulated for continuous distributions can be used to study
the special case of discrete velocity distribution as well.  Here we
quote the results in terms of the original (non-dimensionless)
quantities. Consider the intrinsic velocity distribution

\begin{equation}
\label{rho-dis}
P_0(v)=\sum_{i=1}^n c_i\delta(v-v_i),
\end{equation}
with $v_1<v_2<\cdots<v_n$.  We denote by $p_i$ the discrete
counterpart of the cluster velocity distribution, e.g., $P(v)=\sum_{i=1}^n
p_i\delta(v-v_i)$.  The steady state condition of Eq.~(\ref{pv}) reads
\begin{equation}
\label{pi}
p_i\left[1+t_0\sum_{j=1}^{i-1}(v_i-v_j)p_j\right]=c_i.
\end{equation}
Substituting the intrinsic velocity distribution and solving
iteratively, we get 
\begin{eqnarray}
\label{p123}
p_1&=&c_1\nonumber\\
p_2&=&{c_2\over 1+c_1(v_2-v_1)t_0}\\
p_3&=&{c_3\over 1+c_1(v_3-v_1)t_0+{c_2(v_3-v_2)t_0\over
    1+c_1(v_2-v_1)t_0}}\nonumber
\end{eqnarray}
etc. Rather than a solution to a differential equation, the steady state
solution is in the form of an explicit continued fraction. This
expression involves the initial distribution and the velocity
differences, and can be useful to analyze data in a histogram form. In a
similar way, explicit expressions can be obtained for the rest of the
steady state properties.

\section{Rural Traffic Observations}

To compare the theoretical predictions with actual traffic flows, we
collected data in a rural one lane road where passing is allowed.  We
chose a road near Los Alamos that was as uniform as possible: over a
long stretch it did not contain junctions, stop signs or stop lights.
The number of cars and the number of clusters passing a given point in
each direction in a fixed time interval was recorded, thereby measuring
the flux and the average cluster size, respectively.  The data was then
histogramed, and the cluster size $\langle m\rangle$ was plotted as a
function of the flux $I$ (the observed flux $I$ should be distinguished
from the flux $J$ which is in a reference frame moving with the slowest
car). We verified that the behavior was independent of the traffic
direction as well as the time of day. The former test confirms that the
road is indeed uniform.

\begin{figure}
%\centerline{\epsfxsize=8.5cm \epsfbox{fig4.eps}}
\centerline{\includegraphics[width=8.5cm]{fig4}}
\noindent{\small {\bf Fig.4 }
  The average cluster size $\langle m\rangle$ as a function of the
  flux $I$. The data was obtained from 20 hours of 
  observations of over 5,000 cars.  Each data point represents an
  average over roughly 500 cars, and the error bars account for the
  standard deviation between different measurements.}
\end{figure}

We were able to collect data primarily in the dilute limit.  According
to Eq.~(\ref{cj1}), $\langle m\rangle\cong 1+{\rm const.}\times R$
when $R\ll 1$. The velocity range $v_0$ was much smaller than the
minimal velocity $v_{\rm min}$, and consequently $I=c_0(v_{\rm
  min}+v_0J)\propto c_0v_{\rm min}$. Thus, the main quantity which
varies with the flux is the concentration, and the collision number,
$R=c_0v_0t_0\propto c_0 \propto I$ is proportional to the flux.  In
other words, the theory predicts that in the low flux (or collision
number) limit, the average mass grows linearly with the flux, $\langle
m\rangle\cong 1+{\rm const.}\times I$.  The observations agree with
this prediction as for sufficiently small fluxes, $I<4$ cars/minute,
there is a linear dependence between the average cluster size and the
flux (see Fig.~4).  We conclude that at least for low collision
numbers, the theoretical predictions concerning the cluster size agree
with actual traffic data.



\section{Discussion}

An important property, the cluster size distribution is absent from
our treatment so far \cite{Gavrilov}. Naturally, the size and the
velocity of a cluster are strongly correlated and one must consider
$P_m(v)$, the distribution of clusters of size $m$ and velocity $v$.
The joint cluster size-velocity distribution obeys the master equation

\begin{eqnarray}
\label{pmvt}
{\partial P_m(v)\over\partial t}
&=&R^{-1}[mP_{m+1}(v)-(m-1)P_m(v)]\nonumber\\
&+&R^{-1}\delta_{m,1}[P_0(v)-P(v)]-F(v)P_m(v)\\
&+&\int_v^{\infty} dv' (v'-v)
\sum_{j=1}^{m} P_j(v')P_{m-j}(v)\nonumber
\end{eqnarray}
which applies for all $m\geq 1$.  Terms proportional to $R^{-1}$ account
for escape, while the rest represent collisions.  The factor
$F(v)=\int_0^{\infty} dv'|v-v'| P(v')$ measures the overall collision
rate experienced by a $v$-cluster, and is reminiscent of kinetic theory.
Summing Eqs.~(\ref{pmvt}), one recovers the rate equation (\ref{pvt})
for $P(v)=\sum_m P_m(v)$.  On the other hand, integration over the
entire velocity range does not reduce Eqs.~(\ref{pmvt}) to a closed
system of rate equations for the cluster size distribution $P_m=\int dv
P_m(v)$. Therefore, the entire joint distribution is needed to determine
$P_m$.  Additionally, we note that in Eqs.~(\ref{pvt}) and (\ref{pvvt}),
the integration limits include only slower velocities, a feature that
considerably simplifies the analysis.  This property is lost for
Eqs.~(\ref{pmvt}), therby making analytical treatment harder.

Nevertheless, a leading order analysis is still possible for low collision
numbers. For example, the density of single cars is given by the
expansion 
\begin{equation}
P_1(v)=P_0(v)-RP_0(v)\int_0^{\infty}dv'\,|v-v'|P_0(v')+\ldots 
\end{equation}
In general, one can see that 
\begin{equation}
P_m(v)=R^{m-1}\tilde P_m(v)+{\cal O}(R^m).
\end{equation}
Heuristically, clusters with $m$ cars are created by $m-1$ collisions
and a factor $R$ is generated in each collision. The perturbation
expansion functions $\tilde P_m(v)$ can be obtained recursively using 
\begin{equation} 
\label{recursive}
(m-1)\tilde P_m(v)=\int_v^\infty dv'(v'-v) \sum_{j=1}^{m-1}\tilde
P_j(v')\tilde P_{m-j}(v). 
\end{equation}
For example, the first term reads 
\begin{equation} 
\tilde P_2(v)=P_0(v)\int_v^{\infty} dv' (v-v')P_0(v'). 
\end{equation}
We also note that the infinite set of recursive equations
(\ref{recursive}) can be transformed into a closed equation 
\begin{equation} 
{\partial^2\over \partial v^2}{\partial\over \partial z}\,
\ln \tilde P(z,v)=\tilde P(z,v),
\end{equation}
for the generating functions $\tilde P(z,v)=\sum z^{m-1} \tilde
P_m(v)$. Although the functions $\tilde{P}_m(v)$ become quite
complicated, the overall prefactor $R^{m-1}$ suggests an exponential
cluster size distribution in the dilute limit.

In the special case of a bimodal velocity distribution, a solution is
possible. The structure of clusters here is simple: A cluster of size
$m$ consists of a leading slow car and $m-1$ fast cars behind it.  The
rate equation (\ref{pmvt}) simplifies considerably, and a Poisson size
distribution is found $P_m\propto e^{-f}f^{m-1}/(m-1)!$.  The
collision rate $f$ is equal to the product of the escape time, the
velocity difference, and the fast car concentration. This steady state
distribution satisfies a detailed balance condition as the escape rate
and the collision rate are equal microscopically, $(m-1)P_m=fP_{m-1}$.
Thus, an equilibrium steady state is reached.  However, in general, a
nonequilibrium steady state is approached with the collision rate and
the escape rate balancing only macroscopically. This is seen by noting
that the cluster size may increase by an arbitrary number due to
collisions, but can decrease only by one due to escape.
  
Further investigation of the collision term in the rate equation
will be useful as well. In the no escape case $R^{-1}=0$, the exact
Boltzmann equation
\begin{equation}
{\partial P(v,t)\over\partial t}=-P(v,t)\int_0^v dv' (v-v')P_0(v') 
\end{equation}
is different from our master equation as $P_0(v')$ replaces $P(v',t)$
in the integrand\cite{eps}. This seemingly small difference is
 important as it shows that the system remembers the initial
state. We argue that escape, no matter how small, induces mixing and
acts to erase this memory, and therefore Eq.~(\ref{pvt}).  It still
remain, however, to establish quantitatively how appropriate is this
mean field assumption. 

The model and the results presented above can be generalized to study
other traffic situations.  First, a multilane flow can be treated as a
system of coupled one lane flows. Escape naturally couples neighboring
lanes.  Second, a natural generalization is to heterogeneous situations
where passing is allowed only in a fraction $r$ of the road.  We expect
that for regular distribution of these passing segments the problem
should reduce to the homogeneous case with a renormalized collision
number $R/r$. The most challenging question appears to be the role
played by the escape mechanism.  We considered the case where all cars
are equally likely to escape.  This assumption simplified the master
equation considerably as the escape term is linear in $P(v)$, and thus,
is exact. The complementary case where only the first car in the cluster
can escape is interesting as well.  For low collision numbers, large
clusters are unlikely, and the behavior is independent of the escape
mechanism. However, for high collision numbers the escape mechanism
becomes weaker and larger clusters should form.  Indeed, a scaling
argument along the lines of Eq.~(\ref{m}) gives $\langle m \rangle \sim
R$ in the car-cluster dominated regime.

In conclusion, despite the simplifying assumptions made, the suggested
model results in realistic behavior.  The overall picture is both
familiar and intuitive: due to the presence of slower cars, clusters
form and the overall flux is reduced.  Our theory is in qualitative
agreement with rural traffic observations in the dilute case.  For heavy
traffic, the characteristics of the flow are solely determined by the
distribution of slow cars. A single dimensionless parameter, the
collision number $R$, ultimately determines the nature of the steady
state. The stationary distributions obtained analytically provide a
simple practical recipe for calculating the flow properties for
arbitrary intrinsic distributions. It will be interesting to analyze
velocity distributions from actual traffic data using these theoretical
tools.

\vspace{.1in}

We thank I.~Daruka and S.~Redner for useful discussions. PLK thanks
the CNLS for its hospitality and the ARO for financial support.

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\end{multicols}

\end{document}

