\documentstyle[multicol,epsf,aps]{revtex}
%\documentstyle[epsf]{ioplppt}
\begin{document}
\title{Steady State Properties of 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, USA}
\address{$\ddag$Center for Polymer Studies and Department of Physics,
Boston University, Boston, MA 02215, USA}
\maketitle
\begin{abstract} 
  
  The role of the passing mechanism in traffic flows is examined.
  Specifically, we consider passing rates that are proportional to the
  difference between the velocities of the passing car and the passed
  car.  From a Boltzmann equation approach, steady state properties of
  the flow such as the flux, the average cluster size, and the
  velocity distributions are found analytically.  We show that a
  single dimensionless parameter determines the nature of the flow and
  helps distinguish between dilute and dense flows. For dilute flows,
  perturbation expressions are obtained, while for dense flows, a
  boundary layer analysis is carried out. In the latter case, extremal
  properties of the initial velocity distribution underly the leading
  scaling asymptotic behavior.  For dense flows, the stationary
  velocity distribution exhibits a rich ``triple deck'' boundary layer
  structure.  Furthermore, in this regime fluctuations in the flux may
  become extremely large.

\noindent{PACS numbers:  02.50-r, 05.40.+j,  89.40+k, 05.20.Dd}

\end{abstract}

\begin{multicols}{2} 
\section{Introduction}

Traffic flows display a variety of cooperative behaviors similar to
nonequilibrium driven systems such as gas and granular flows 
\cite{Gartner,hydr1,hydr2,Prigogine}. Typically, only  a few major 
characteristics of the interparticle interaction are responsible for
such collective phenomena. Therefore, it is important to use models
that are as simple as possible. Such a strategy has proven useful in
studying traffic flows, where the ultimate goal is understanding
complex phenomena such as slowing down, traffic jams, synchronized
flows, and phase transitions \cite{Gartner,kr}.

Theoretical approaches to modeling traffic flows are quite diverse and
include fluid mechanics  \cite{hydr1,hydr2,kerner,hel}, cellular
automata  \cite{Nagel,Biham,Nagatani,Nagel2,Brankov,np,sas,Ktitarev,Evansca},
particle hopping  \cite{Der,Benjamini,krug,Evans}, kinetic
theory  \cite{Prigogine,hel,Pav}, and ballistic
motion  \cite{bal1,bal2,eps,ep,jap,Nagatani1}.  In the hydrodynamic
description, space and time are both continuous variables, while in
cellular automata they are both discrete. Moreover, the former is a
macroscopic approach, while the latter is microscopic.  Kinetic theory
and especially ballistic motion models can help bridge this gap as
they are formulated on a microscopic level, but lead naturally to a
macroscopic theory.  In our previous studies, a Boltzmann equation was
derived for traffic flows in no-passing zones of one lane roadways 
 \cite{eps}, and then generalized to passing zones as well  \cite{ep}.  A
transition from a low-density ``laminar'' flow to a high-density
``congested'' flow was generally found.  This transition as well as
other statistical properties are well described by a single
dimensionless parameter, $R$, termed the collision number.

In our former study \cite{ep}, a constant passing rate was assumed.
However, a passing rate that increases with velocity difference is more
realistic as faster drivers tend to pass more often than slower drivers.
Therefore, we study the complementary case where the passing rate is
linear in the velocity difference.  Our goal is to examine the role
played by the passing mechanism. While the governing equations become
more complicated, a formal analytical solution is still possible.
Although certain quantitative features such as the scaling exponents
change, the qualitative picture remains the same.  Interestingly, this
accelerated passing mechanism leads to a triple deck (a boundary layer
accompanied by an additional inner layer) structure of the car velocity
distribution.

\section{Theory}

In the ballistic motion model, fluctuations in the velocity of a single
isolated car are neglected. We thus consider a one-dimensional traffic
flow where size-less cars (``particles'') move with a constant velocity.
Initially, cars are randomly distributed in space and they move with
their intrinsic velocities.  The presence of slower cars forces some
cars to drive behind a slower car and therefore leads to the formation
of clusters.  When a cluster overtakes a slower cluster, a larger
cluster forms and its velocity is the smaller of the two velocities.
Meanwhile, all cars in a given cluster may escape their respective
clusters and resume driving with their intrinsic velocity.  We consider
the case where the escape rate is proportional to $\Delta v$, the
difference between the velocity of the passing car and lead (slowest)
car in the cluster, i.e., $\ell^{-1}\Delta v$, where $\ell$ has
dimensions of length.

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.  Consequently, the
escape rate $\ell^{-1}\Delta v$ becomes $R^{-1}\Delta v$.  The
dimensionless number $R=t_{\rm esc}/t_{\rm col}$ is simply equal to the
ratio of the two elementary time scales, the escape time scale $t_{\rm
  esc}=\ell/v_0$ and the collision time scale $t_{\rm
  col}=(c_0v_0)^{-1}$ \cite{ep}.  We term $R=c_0 \ell$ the ``collision
number''.

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 $P_0(v)\equiv P(v,t=0)$.  The
dimensionless intrinsic velocity distribution is thus normalized to
unity, $\int dv P_0(v)=1$.  The flow is invariant under a finite
velocity translation, and the minimal velocity is set to zero.

Neglecting correlations between the velocities and the positions of
particles, a mean-field equation for the cluster velocity distribution
$P(v)$ can be written
\begin{eqnarray}
\label{pvt}
{\partial P(v)\over \partial t}&=&R^{-1}\int_0^v dv' (v-v')P(v,v')\\
&-&P(v)\int_0^v dv' (v-v')P(v'),\nonumber
\end{eqnarray}
where the time variable $t$ has been suppressed.  This master equation
involves $P(v,v')$, the density of cars of intrinsic velocity $v$
driving with actual velocity $v'<v$.  Such slowed down cars escape their
clusters with rate proportional to the velocity difference,
$R^{-1}(v-v')$, 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.

The master equation is complemented by a second rate equation
corresponding to the joint velocity distribution $P(v,v')$,
\begin{eqnarray}
\label{pvvt}
{\partial P(v,v')\over \partial t}=
&-&R^{-1}(v-v')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}
In this equation as well, the escape term is proportional to the velocity
difference. A useful check of self-consistency is that the total density of
$v$-cars, $P_0(v)=P(v)+\int_0^v dv' P(v,v')$ is conserved by these master
equations.

We restrict attention to the steady state properties, which can be obtained
by taking the limit $t\to\infty$ or $\partial/\partial t\equiv 0$.  We will
express the joint distribution via the single cluster densities, and then
insert it into (\ref{pvt}) to get a {\em closed} equation for the cluster
velocity distribution.  Once the cluster distribution is found, average
quantities such as the average cluster concentration, $c$, and the average
cluster velocity, $\langle v\rangle$, will easily follow
\begin{equation}
\label{cvdef}
c=\int_0^{\infty}  dv\, P(v),\qquad 
\langle v\rangle=c^{-1}\int_0^{\infty} dv \, v P(v). 
\end{equation}
Furthermore, knowledge of the joint velocity distribution, $P(v,v')$, will
enable computation of $J$, the average flux
\begin{equation}
\label{jdef}
J=\int_0^{\infty} dv \left[vP(v)+\int_0^v dw\, w P(v,w)\right],  
\end{equation}
and $G(v)$, the actual car velocity distribution 
\begin{equation}
\label{gdef}
G(v)=P(v)+\int_v^{\infty}\!\! dw\,P(w,v). 
\end{equation}
This latter quantity satisfies the normalization conditions 
\hbox{$1=\int\!dv\, G(v)$} and \hbox{$J=\int\! dv \, v G(v)$}.

We turn now to solving the steady state master equations.  It proves
useful to define two auxiliary functions, $Q(v,v')$ and $T(v,v')$:
\begin{eqnarray}
\label{aux}
Q(v,v')&=&R^{-1}(v-v')+\int_0^{v'} dv'' (v'-v'')P(v''),\nonumber\\
T(v,v')&=&\int_{v'}^v dv''(v''-v')P(v,v'').
\end{eqnarray} 
The densities $P(v')$ and $P(v,v')$ can be obtained from these
auxiliary functions by differentiation
\begin{equation}
\label{dQv} 
P(v')={\partial^2 Q(v,v')\over \partial {v'}^2},\qquad
P(v,v')={\partial^2 T(v,v')\over \partial {v'}^2}.
\end{equation}
To obtain $P(v,v')$ let us first re-write (\ref{pvvt}) as
\begin{equation}
\label{QP}
P(v,v')Q(v,v')=(v-v')P(v)P(v')+T(v,v')P(v'). 
\end{equation}
Using the definition of the auxiliary functions (\ref{aux}), this
equation is rewritten as 
\begin{equation}
\label{QP1}
{\partial \over \partial v'}
\left[Q^2(v,v')\,{\partial \over \partial v'}\,
{T(v,v')\over Q(v,v')}\right]=(v-v')P(v)P(v').
\end{equation}
Integrating twice over $v'$ gives the joint auxiliary function $T(v,v')$
in terms of $Q(v,v')$ and
the single variable functions:
\begin{equation}
\label{QP2}
{T(v,v')\over P(v)}=Q(v,v')\int_{v'}^v\!\!{du\over
  Q^2(v,u)}\int_u^v\!\!dw(v-w)P(w).
\end{equation}
The boundary conditions $T(v,v)={\partial \over \partial v'}
T(v,v')\big|_{v'=v}=0$ were used to obtain (\ref{QP2}).  Replacing
$P(w)$ with ${\partial^2 Q(v,w)\over \partial w^2}$ and integrating by
parts gives
\begin{equation}
\label{QP3}
{T(v,v')\over P(v)}=Q(v)Q(v,v')\int_{v'}^v{du\over Q^2(v,u)}-(v-v'). 
\end{equation} 
Here, $Q(v)=Q(v,v)=\int_0^{v} dv' (v-v')P(v')$.  Substituting Eq.~(\ref{QP3})
into (\ref{QP}), we find a relatively simple expression for the joint
velocity distribution
\begin{equation}
\label{pvv}
P(v,v')=P(v)P(v')Q(v)\int_{v'}^v {du\over Q^2(v,u)}.
\end{equation}
Furthermore, combining this joint velocity distribution with the
normalization condition $P_0(v)=P(v)+\int_0^v dv' P(v,v')$, we arrive
at
\begin{equation}
\label{final}
P_0(v)=P(v)\left[ 1+Q(v)\int_0^v\!\! dv'P(v')\int_{v'}^v \!\!{du\over
    Q^2(v,u)}\right].  
\end{equation}
Additional simplification can be achieved by using the relationship
$P(v')={\partial^2 Q(v,v')\over \partial {v'}^2}$ and performing the
integration by parts.  This yields
\begin{equation}
\label{integ}
P_0(v)= Q(v) Q''(v)\left[{R\over v}+{1\over R}
\int_0^v {du\over Q^2(v,u)}\right]. 
\end{equation}
Since $Q(v,u)=R^{-1}(v-u)+Q(u)$, the function appearing in the integrand
contains only $Q(u)$; therefore, the integro-differential equation
(\ref{integ}) is a closed equation for $Q(v)$.  Given the auxiliary
function $Q(v)$, the cluster velocity distribution is calculated from
\begin{equation}
\label{pv}
P(v)=Q''(v).  
\end{equation}
Eq.~(\ref{gdef}) then gives the actual car velocity distribution
\begin{equation}
\label{gv}
G(v)=P(v)\left[1+\int_v^\infty\!\!\! dw P(w)Q(w)\int_v^w \!\!\!
{du\over Q^2(w,u)}\right].
\end{equation}
We now compute the flux, or the average car velocity given by
Eq.~(\ref{jdef}).  From the definition of the joint auxiliary function,
the second integral in (\ref{jdef}) equals $T(v,0)$, thereby implying
\hbox{$J=\int_0^{\infty} dv\left[vP(v)+T(v,0)\right]$}.  The integrand
can be further simplified by using Eq.~(\ref{QP3}).  The term $vP(v)$
cancels and we find a useful expression for the flux
\begin{equation}
\label{j}
J=\int_0^{\infty} dv\,P(v)Q(v){v\over R}\int_0^v {du\over Q^2(v,u)}. 
\end{equation}

We conclude that for arbitrary intrinsic velocity distributions, $P_0(v)$,
the entire steady state problem is reduced to the closed integro-differential
equation (\ref{integ}) for the auxiliary function $Q(v)$. Once this function
is known, steady state characteristics such as $P(v)$, $P(v,v')$, $G(v)$, and
$J$ are given by Eqs.~(\ref{pv}), (\ref{pvv}), (\ref{gv}), and (\ref{j}),
respectively.  Despite the complicated nature of the equations, a leading
order analysis is still generally possible, as detailed below.


\section{Leading Behavior}

In the physically relevant limits of small and large collision numbers, a
leading order analysis is possible. When $R\ll 1$, a perturbation expansion
in the small parameter $R$ shows that the flow is almost uninterrupted. In
the complementary case, $R\gg 1$, a boundary layer analysis is needed for the
velocity distributions.  The leading scaling behavior shows that a transition
from a ``laminar'' low-density flow into a ``congested'' high-density flow
occurs.


\subsection{Laminar Flows}

In the collision-controlled regime, the velocity distributions can be
obtained perturbatively. Let us write $P(v)=\sum_{n\geq 0} R^n
P^{(n)}(v)$ and $P(v,v')=\sum_{n\geq 0} R^n P^{(n)}(v,v')$. The zeroth order
terms are trivial $P^{(0)}(v)=P_0(v)$ and $P^{(0)}(v,v')=0$.  The first
order correction for the joint velocity distribution can be computed by
inserting the zeroth order approximation in the right-hand side of
Eq.~(\ref{pvv}).  This gives
\begin{displaymath}
P^{(1)}(v,v')=\cases{RP_0(v)P_0(v')&$v-v'\gg R$;\cr
RP_0(v)P_0(v'){(v-v')\over v-v'+RQ_0(v)}&$v-v'\ll R$;\cr}
\end{displaymath}
where the notation $Q_0(v)=\int_0^v dv' (v-v')P_0(v')$ was used.  One
can check that $P^{(1)}(v,v')$ satisfies the master Eq.~(\ref{pvvt}) to
first order in $R$. This joint velocity distribution together with
Eq.~(\ref{gdef}) determines the cluster and car velocity distributions
to first order
\begin{eqnarray}
\label{pv1}
P(v)&\cong& P_0(v)\left[1-R\int_0^v dv' P_0(v')\right],\\
G(v)&\cong& P_0(v)\left[1-R\int_0^v dv' P_0(v')+R\int_v^{\infty}
  dv'P_0(v')\right].
\nonumber
\end{eqnarray}
Furthermore, the cluster density and the flux can be evaluated to
first order in $R$ by integrating Eq.~(\ref{pv1}) 

\begin{equation}
\label{cj1}
c\cong 1-{R\over 2}, \qquad
J\cong J^{(0)}-J^{(1)} R,
\end{equation}
with $J^{(0)}=\int dv\,vP_0(v)$ and $J^{(1)}=\int dv\,P_0(v)Q_0(v)$.
The order $R$ corrections are obtained by writing the integrand using
derivatives of $Q_0(v)$ and integrating by parts. For example, $c=\int\,
dv\, P(v)\cong 1-R\int\, dv\, Q_0''(v)Q_0'(v)$, and similarly for the
flux.  Interestingly, the first order correction to the density is
universal in that it does not depend on the details of the intrinsic
velocity distribution.  We conclude that as the collision-controlled
limit is weakly interacting, explicit expressions for the leading
behavior are possible for the steady state properties.

\subsection{Congested Flows}

When $R\gg 1$, slow and fast cars exhibit very different behaviors.
Small enough velocities are not affected by the presence of faster cars,
and the perturbative expression (\ref{pv1}) holds for $P(v)$, i.e.,
$P(v)\cong P_0(v)$ for $v\ll v^*$. At the threshold velocity $v^*$, the
correction to the initial velocity distribution in Eq.~(\ref{pv1})
becomes of order unity, $1\sim R\int_0^{v^*} dv' P_0(v')$.

We consider (without loss of generality) algebraic intrinsic velocity
distributions 
\begin{equation}
\label{mu}
P_0(v)=(\mu+1)v^{\mu} \quad \mu>-1,
\end{equation}
in the velocity range [0:1]. Thus, the threshold velocity decreases
with growing $R$ according to 
\begin{equation}
\label{vstar}
v^* \sim R^{-1/(\mu+1)}.  
\end{equation}
We expect that the velocity distributions are strongly suppressed for
velocities much larger than this threshold velocity.  We further assume
an algebraic behavior, $P(v)\sim R^{\delta}v^{\sigma}$. As
$P(v)=Q''(v)$, the following auxiliary function $Q(v)\sim
R^{\delta}v^{\sigma+2}$ is implied.  Substituting this expression into
Eq.~(\ref{integ}) and noting that the term $R/v$ dominates over the
integral in the square brackets, we find $\delta=-1/2$ and
$\sigma=(\mu-1)/2$.  The behavior of the cluster velocity distribution
is therefore
\begin{equation}
\label{pvlead}
P(v)\sim\cases{
v^{\mu}&$v\ll v^*$;\cr
R^{-1/2}v^{(\mu-1)/2}&$v\gg v^*$.}
\end{equation}
Of course, the small and large velocity components of the cluster
velocity distribution match at the threshold velocity, $P(v^*)\sim
P_0(v^*)$.  Notice that behavior of the cluster velocity distribution is
different than the one found in the case of constant escape
rates \cite{ep} where $\sigma=\mu-1$ when $\mu<0$ and $\sigma=\mu/2-1$
when $\mu>0$. We conclude that in the case of linear escape rate the
scaling exponents change, and the behavior becomes generic in $\mu$.

Using the leading behavior of $P(v)$, we determine the average cluster
concentration, $c=\int dv P(v)\sim R^{-1/2}$, and the average number
of cars per cluster, $\langle m\rangle=c^{-1}\sim R^{1/2}$. This
behavior is in accordance with the following heuristic argument.  Let
the initial car concentration be $c_0$, the final cluster
concentration be $c\ll c_0$, and the typical velocity be $v_0$.  Thus,
the typical cluster size is $\langle m\rangle=c_0/c$. When large
clusters form, $\langle m\rangle\gg 1$, and the overall escape rate
can be estimated by $\langle m\rangle\ell^{-1} v_0 $. On the other
hand, the typical collision rate is $cv_0$.  In the steady state the
number of cars entering and leaving clusters must balance, and
therefore the overall collision and escape rates are equal, $(c_0/c)
\times\ell^{-1} v_0=v_0 c$.  Therefore, $c\sim (c_0/\ell)^{1/2}$ and
$\langle m\rangle=c_0/c \sim\sqrt{c_0\ell}\sim R^{1/2}$ is recovered.

In the case of velocity independent escape rates, similar behavior
occurred only when the intrinsic velocity distribution did not have a
strong fast car component, namely when $\mu>0$. In the present case,
the flow is dominated by large and slow clusters. We conclude that as
fast cars are more likely to pass when the escape rate is linear, the
cluster velocities are only slightly reduced due to collisions. This
is consistent with the fact that the average cluster velocity $\langle
v\rangle$, defined in Eq.~(\ref{cvdef}), remains of order unity.

To study the actual velocity distribution $G(v)$, the integral
$I=\int_v^w du [R^{-1}(w-u)+Q(u)]^{-2}$ should be evaluated. We note
that at $v=v^{**}\sim R^{-1/(\mu+3)}$, the two terms in the integrand
become comparable. The leading order behavior is thus $I\sim w^{-2}R^2
v^{**}$ for $v\ll v^{**}$, and $I\sim Rv^{-(\mu+2)}$ for $v\gg v^{**}$.
Combining this leading behavior with $P(v)$ given by Eq.~(\ref{pvlead})
and substituting into Eq.~(\ref{gv}) one finds $G(v)\sim
R^{(\mu+2)/(\mu+3)}P(v)$ for $v\ll v^{**}$, and $G(v)\sim
v^{-\mu-2}P(v)$ for $v\gg v^{**}$.  Using Eq.~(\ref{pvlead}), we arrive
at the leading order behavior of the actual car velocity distribution
\begin{equation}
\label{gvlead}
G(v)\sim\cases{R^{-1/2}v^{-(\mu+5)/2}            &$v^{**}\ll v$;\cr 
               R^{(\mu+1)/(2\mu+6)}v^{(\mu-1)/2} &$v^{*}\ll v\ll v^{**}$;\cr
               R^{(\mu+2)/(\mu+3)} v^{\mu}       &$v\ll v^*$.}
\end{equation}
It is simple to verify that these expressions match at the two threshold
velocities, $v^*$ and $v^{**}$.  In the case of constant escape rates,
only a single threshold velocity was found.  Here, in contrast, an
interesting triple deck structure with two marginal velocities emerges.
Besides the threshold at $v^*\sim R^{-1/(\mu+1)}$ we observe an
additional threshold located at $v^{**}\sim R^{-1/(\mu+3)}$.  The latter
threshold velocity is proportional to the flux, obtained from $J=\int\,
dv\, v G(v)$,
\begin{equation}
\label{j2}
J\sim v^{**} \sim R^{-1/(\mu+3)}.
\end{equation}
Unlike the scaling behavior of the average cluster mass, the flux scaling law
depends on $\mu$.  No flux reduction occurs in the limit where the intrinsic
velocity distribution is dominated by fast cars ($\mu\to\infty$), while
maximal flux reduction, $J\sim R^{-1/2}$, occurs in the limit when the
distribution is dominated by slow cars ($\mu\to -1$).

It is instructive to study velocity fluctuations by evaluating the moments of
the actual velocity distribution defined via $G_n=\int dv\,v^n G(v)$.  These
moments are calculated from Eq.~(\ref{gvlead}) to give
\begin{equation}
\label{gn}
G_n\sim\cases{R^{-1/2}       &$n>(\mu+3)/2$;\cr 
              R^{-n/(\mu+3)} &$n<(\mu+3)/2$.}
\end{equation}
Interestingly, fluctuations in the flux become very large ($G_2\gg
G_1^2$) when $-1<\mu<1$. This suggests that slowing down due to
collisions dominates when the intrinsic velocity distribution has a
strong small velocity component. This results in a broadening of the
actual velocity distribution.


One may wonder whether this behavior is restricted to purely algebraic
distributions. Since both of the threshold velocities approach zero as
the collision number diverges, for sufficiently large $R$, a vanishingly
small component of the velocity distribution is sampled.  Thus, if the
limit $\mu=\lim_{v\to 0} v{\partial\over\partial v} \ln P_0(v)$ exists,
then $\mu$ determines the behavior when $R\to\infty$ as detailed above.
We conclude that our previous analysis applies to a broad range of
intrinsic distributions, not only purely algebraic ones.


We also see that when $R\to\infty$ the solution to Eq.~(\ref{integ})
exhibits a two layer structure, a result reminiscent of classical
boundary layer theory\cite{nayfeh}.  Inside the boundary layer, $v\ll
v^*$, the cluster velocity distribution is only slightly affected by
collisions, while in the outer region, $v\gg v^*$, the cluster velocity
distribution is much smaller than the intrinsic velocity distribution.
The threshold velocity $v^*$ is determined by the condition that below
$v^*$ the intrinsic velocity distribution remains unaffected, $P(v)\cong
P_0(v)$.  The actual velocity distribution, however, exhibits a richer
three layer structure. The first layer (referred to as the lower deck
in fluid mechanics) is followed by the middle deck
$v^*\ll v\ll v^{**}$ and then the upper deck completes the structure. 
Such triple deck structures were originally found within the framework 
of classical fluid dynamics and applied mathematics \cite{nayfeh}.

\section{Conclusions}

In summary, we have studied traffic flows with a passing mechanism
which favors faster cars over slower cars.  Despite the complicated
structure of the master Boltzmann equation, a reduction to an
integro-differential equation and a complete formal solution is
possible. The overall behavior agrees qualitatively with the constant
escape rate case.  For example, a single dimensionless parameter, the
collision number $R$, ultimately determines the nature of the flow in
the steady state. In the ``laminar'' flow regime, $R\ll 1$,
corrections due to collisions are of order $R$. Here, large clusters
are rare, and the exact form of the passing mechanism plays a
secondary role.  In agreement with rural traffic observations, the
average cluster mass grows linearly with flux in this dilute limit. In
the ``congested'' regime, i.e., when $R\gg 1$, we found a boundary
layer structure for the cluster velocity distribution.  The flux is
reduced by an algebraic function of $R$.

However, quantitative differences do arise including changes in the flux
reduction exponent, and the behavior of the velocity distribution in the
limit of large velocities. Furthermore, the actual velocity distribution
exhibits a triple deck structure with two threshold velocities in
contrast with the constant escape rate.  However, as both of the
threshold velocities vanish when $R\to\infty$, extreme statistics of the
velocity distribution still dominate the flow in this congested phase.
This latter result, valid for a broad class of intrinsic velocity
distributions, is reminiscent of intermittency in turbulent flows.
Additionally, we showed under what conditions do fluctuations in flux
can become significant.

As one would intuitively expect, a linear escape mechanism more
effectively suppresses clustering because extremely fast cars will spend
less time in clusters. Indeed, we found that the average cluster
velocity remains of order unity which implies that situations where the
flow consists solely of slow clusters are avoided.

It is remarkable that the underlying master equations are still
solvable even under more complicated escape rules.  Unlike in
Boltzmann equations arising in kinetic theory, the integration limits
are restricted thereby enabling a reduction to an ordinary
differential equation and a formal solution.  We expect that further
analytical progress may be possible. For example, it will be useful to
apply the above formalism to other realistic situations such as
multi-lane traffic.


\vspace{.1in}
\noindent
We thank S.~Redner for useful discussions.  The work of PLK has been
supported by NSF (grant DMR-9632059) and ARO (grant DAAH04-96-1-0114).


\vspace{.1in}

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     M.~J.~Lighthill and G.~B.~Whitham, {\sl Proc. R. Soc. London
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\end{thebibliography}
\end{multicols}
\end{document}