# Macroscopic limits of non-local kinetic descriptions of vehicular traffic

F. A. Chiarello  A. Tosin

###### Abstract

We study the derivation of macroscopic traffic models out of optimal speed and
follow-the-leader particle dynamics as hydrodynamic limits of non-local
Povzner-type kinetic equations. As a first step, we show that optimal speed
vehicle dynamics produce a first order macroscopic model with non-local flux.
Next, we show that non-local follow-the-leader vehicle dynamics have a
universal macroscopic counterpart in the second order Aw-Rascle-Zhang traffic
model, at least when the non-locality of the interactions is sufficiently
small. Finally, we show that the same qualitative result holds also for a
general class of follow-the-leader dynamics based on the headway of the
vehicles rather than on their speed. We also investigate the correspondence
between the solutions to particle models and their macroscopic limits by means
of numerical simulations.

Keywords: stochastic particle models, optimal speed, follow-the-leader, non-
local kinetic equations, hydrodynamic limits

Mathematics Subject Classification: 35Q20, 35Q70, 90B20

##  1 Introduction

One of the most celebrated macroscopic traffic models based on fluid dynamic
equations is the Lighthill-Whitham-Richards (LWR) model [32, 40], which
consists in a scalar equation expressing the conservation of the number of
cars on the road:

$\partial_{t}\rho+\partial_{x}(\rho\pazocal{V}(\rho))=0,$

$\rho=\rho(x,t)$ being the mean traffic density, i.e. the number of vehicles
per unit length of the road, in the point $x$ at time $t$ and $\pazocal{V}$
the mean traffic speed. Although widely used in traffic applications thanks to
its simplicity and to a well-established analytical theory, this model has
some limitations. We recall, in particular, that it allows for speed
discontinuities resulting in infinite accelerations. Great efforts have been
devoted to overcome these drawbacks, among which it is worth mentioning some
non-local versions of the LWR model proposed in very recent times, see e.g.,
[3, 8, 19, 22, 28]. Most of them share the idea to model the mean traffic
speed as a downstream convolution between the traffic density and a prescribed
decreasing kernel. The convolution introduces naturally a smooth non-locality
in the flux of vehicles, which produces Lipschitz-continuous speeds in $x$ and
$t$ ensuring bounded accelerations. Moreover, such a non-locality may describe
the anisotropic behaviour of drivers adapting the speed of their vehicles to
that of vehicles ahead, caring particularly of close ones.

Another option to cure the aforesaid limitations of the LWR model is to
consider second order macroscopic models, in which the vehicle density $\rho$
and mean speed $u=u(x,t)$ are regarded as two independent hydrodynamic
parameters. One of the most popular models in this class is the Aw-Rascle-
Zhang (ARZ) model [1, 45], which consists in a system of two scalar equations
expressing the conservation of the number of cars on the road and the balance
of linear momentum:

$\begin{cases}\partial_{t}\rho+\partial_{x}(\rho u)=0\\\ \partial_{t}u+(u-\rho
p^{\prime}(\rho))\partial_{x}u=0.\end{cases}$

Unlike other second order models, such as e.g., [34], which were affected by
physical inconsistencies due to a too strict link with the fluid dynamic
equations inspiring them, cf. [12], the ARZ model accounts correctly for the
anticipation ability of the drivers through a prescribed (pseudo-) pressure of
traffic $p=p(\rho)$. Notice that such an anticipation ability may be regarded
as a non-locality in vehicle interactions.

From this discussion, it is clear that non-local models are motivated by the
necessity to provide a description of car flow closer to the actual physics of
traffic. On the other hand, their construction relies largely on heuristic
ideas. As a matter of fact, the non-local traffic models mentioned above,
including the ARZ model, have been also obtained from microscopic descriptions
by means of many-particle limits, see e.g., [6, 13, 14, 21]. In these cases,
the adopted procedure consists in showing that the solutions of the
macroscopic models can be recovered as limits of the solutions of selected
microscopic models used as educated guesses for the discretisation of the
former.

The main contribution of this paper is instead to recover first and second
order traffic models as physical limits of fundamental non-local particle
dynamics. This way, we will establish structural links between microscopic and
macroscopic models genuinely grounded on first principles instead of simply
assessing the consistency of ad-hoc particle discretisations of macroscopic
models.

To this purpose, we will make use of concepts and tools from collisional
kinetic theory, which since almost twenty years has been systematically
rediscovered as a powerful and flexible mathematical approach to interacting
multi-agent systems [33] often very different from the gas molecules which
first inspired the work by Boltzmann. The pioneer of the application of
kinetic theory to the modelling of traffic flow was Prigogine [37, 38]. Years
later, other scholars started again to apply it to the mathematical
investigation of traffic flow phenomena [4, 23, 24, 25, 26, 29, 30]. Nowadays,
the literature includes several contributions touching also quite modern
applications, such as e.g., driver-assist and autonomous vehicles [10, 16, 42]
and the quantification of the uncertainty in traffic data [27, 44].

Combining classical methodologies of the kinetic theory with more modern ones
developed in the above-cited papers, we will formulate optimal speed and
Follow-the-Leader (FTL) non-local particle dynamics at the mesoscopic level,
thereby obtaining non-local Povzner-type collisional kinetic equations of
traffic. Passing then to the hydrodynamic limit, possibly under suitable
approximations of the non-locality of the interactions, we will recover
explicit closure relationships at the macroscopic scale, which will result in
self-consistent hydrodynamic models accounting for the non-locality in various
forms. In particular, we will show that optimal speed dynamics give rise to a
first order macroscopic model with a non-local flux slightly different from
the ones typically postulated in heuristic constructions; and that non-local
FTL dynamics have the ARZ model, or possible generalisations of it, as
“universal” macroscopic counterpart, at least when the non-locality of the
interactions is small enough.

In more detail, the paper is organised as follows. In Section 2, we show that
a class of first order non-local traffic models emerges as the hydrodynamic
limit of optimal speed particle dynamics. We also compare the obtained model
with other non-local first order models proposed in the literature. In Section
3, we show instead that the ARZ model arises as the hydrodynamic limit of FTL
particle dynamics for arbitrary non-local interaction kernels with
sufficiently small support. This result generalises the one obtained in [11],
which was based specifically on an Enskog-type kinetic description. In Section
4, we further generalise the result of Section 3 to a wider class of non-local
FTL particle dynamics characterised by quite arbitrary interaction functions
expressed in terms of the space headway between the vehicles instead of their
speed. Specifically, we show that an ARZ-like macroscopic model provides again
a “universal” macroscopic description in suitable parameter regimes including
the smallness of the support of the non-local interaction kernel. In Section
5, we extensively investigate and compare the solutions produced by the
particle models and by the hydrodynamic models by means of numerical
simulations. In Section 6, we finally outline some conclusions.

##  2 Optimal speed dynamics

We consider a sufficiently large ensemble of indistinguishable vehicles, each
of which is identified by the dimensionless position $X_{t}\in\mathbb{R}$ and
dimensionless speed $V_{t}\in[0,\,1]$ at time $t>0$. We assume the following
discrete-in-time dynamical model:

$X_{t+\Delta{t}}=X_{t}+V_{t}\Delta{t},\qquad
V_{t+\Delta{t}}=V_{t}+a\Theta\Bigl{(}\pazocal{V}(\rho(X^{\ast}_{t},t))-V_{t}\Bigr{)},$
(1)

where $\Delta{t}>0$ is a (small) time step and $a>0$ is a parameter. Moreover,
$\Theta\in\\{0,\,1\\}$ is a binary random variable describing whether during
the time step $\Delta{t}$ a randomly chosen vehicle with microscopic state
$(X_{t},V_{t})$ updates ($\Theta=1$) or not ($\Theta=0$) its speed by relaxing
it towards the optimal speed determined by a prescribed function
$\pazocal{V}$. The latter depends on the traffic density $\rho$ in a point
$X^{\ast}_{t}$ representing the position of another randomly picked vehicle
which the previous vehicle possibly interacts with. The interaction is
mediated by an interaction kernel $B=B(X^{\ast}_{t}-X_{t})$ fixing the rate at
which the two vehicles with relative position $X^{\ast}_{t}-X_{t}$ may
interact. We express this process by letting

$\Theta\sim\operatorname{Bernoulli}(B(X^{\ast}_{t}-X_{t})\Delta{t}),$ (2)

so that the probability for a speed update to happen is
$B(X^{\ast}_{t}-X_{t})\Delta{t}$. Notice that we need $0\leq
B(X^{\ast}_{t}-X_{t})\Delta{t}\leq 1$ for consistency and this requires some
assumptions.

######  Assumption 2.1 (Interaction kernel).

We assume that $B$ is non-negative and compactly supported in the interval
$[0,\,\eta]$ with $0<\eta<+\infty$. We also assume that $B$ is bounded.

######  Remark.

Assumption 2.1 implies in particular that the interaction kernel $B$ is
forward-looking. Indeed, $B(y)=0$ whenever $y<0$, thus
$B(X^{\ast}_{t}-X_{t})=0$ whenever $X^{\ast}_{t}<X_{t}$, i.e. if vehicle
$X^{\ast}_{t}$ is behind vehicle $X_{t}$. This is consistent with the idea
that interactions among vehicles are essentially anisotropic and mainly
addressed to vehicles in front. Assumption 2.1 mimics then the behaviour of
drivers who look ahead and adapt the speed of their vehicles to that of
vehicles in front of them.

Owing to Assumption 2.1, we may fix
$\Delta{t}\leq\frac{1}{\sup_{y\in[0,\,\eta]}B(y)}$ in order for the law (2) of
the random variable $\Theta$ to be well-defined. We point out that this is
actually not a limitation, as in a moment we will consider the continuous-time
limit $\Delta{t}\to 0^{+}$.

We also set some assumptions on the function $\pazocal{V}$:

######  Assumption 2.2 (Optimal speed).

We assume that $\pazocal{V}=\pazocal{V}(\rho)$ is bounded between $0$ and $1$
for all $\rho\geq 0$.

######  Remark.

Assumption 2.2 defines the minimal feature of $\pazocal{V}$ necessary for the
subsequent developments, specifically for the consistency of model (1), see
below. However, from the modelling point of view other characteristics may be
desirable, although in our case not strictly necessary for technical purposes.
Among them, we recall in particular the fact that $\pazocal{V}$ be a
decreasing function of $\rho$, so that the optimal speed diminishes as the
traffic gets more and more congested.

From Assumption 2.2 we obtain that $0\leq a\leq 1$ is a necessary and
sufficient condition for the physical consistency of the particle model (1).
By “physical consistency” we mean, in particular, that the post-interaction
speed belongs to the dimensionless interval $[0,\,1]$ for any pre-interaction
speed in the same interval. Writing
$V_{t+\Delta{t}}=(1-a\Theta)V_{t}+a\Theta\pazocal{V}(\rho(X^{\ast}_{t},t))$ we
recognise indeed that $V_{t+\Delta{t}}$ is a convex combination of
$V_{t},\,\pazocal{V}(\rho(X^{\ast}_{t},t))\in[0,\,1]$.

For completeness, we mention that the other vehicle participating in the
interaction is assumed to keep its speed unchanged, hence

$X^{\ast}_{t+\Delta{t}}=X^{\ast}_{t}+V^{\ast}_{t}\Delta{t},\qquad
V^{\ast}_{t+\Delta{t}}=V^{\ast}_{t}.$ (3)

###  2.1 Kinetic description

To address the aggregate trends emerging from the dynamics (1)-(3) we
reformulate the particle model along the lines of statistical mechanics and
kinetic theory.

Let $f=f(x,v,t):\mathbb{R}\times[0,\,1]\times(0,\,+\infty)\to\mathbb{R}_{+}$
be the kinetic distribution function of the microscopic state $(x,v)$ of the
vehicles at time $t$. Hence, $f(x,v,t)\,dx\,dv$ gives the probability that at
time $t>0$ a vehicle has a position comprised between $x$ and $x+dx$ and a
speed comprised between $v$ and $v+dv$. Averaging (1)-(3) and taking the
continuous-time limit $\Delta{t}\to 0^{+}$, by standard arguments (see e.g.,
[18, 33]) we formally obtain that $f$ satisfies the equation

$\partial_{t}f+v\partial_{x}f=Q(f,f),$ (4)

where

$Q(f,f)(x,v,t):=\frac{1}{2}\int_{\mathbb{R}}\int_{0}^{1}B(x_{\ast}-x)\left(\frac{1}{J}f(x,{}^{\prime}\\!v,t)f(x_{\ast},{}^{\prime}\\!v_{\ast},t)-f(x,v,t)f(x_{\ast},v_{\ast},t)\right)dv_{\ast}\,dx_{\ast}$
(5)

is the collisional operator111“Collisional” is a legacy from the jargon of
classical kinetic theory of gases. In this context, “collisions” have to be
meant in the abstract as “interactions”.. Here, ${}^{\prime}\\!v$,
${}^{\prime}\\!v_{\ast}$ denote the pre-interaction speeds generating the
post-interaction speeds $v$, $v_{\ast}$ when an interaction takes place
($\Theta=1$) in the dynamics (1)-(3). Specifically,

${}^{\prime}\\!v=v-\frac{a}{1-a}(\pazocal{V}(\rho(x_{\ast},t))-v),\qquad{}^{\prime}\\!v_{\ast}=v_{\ast},$

which is known as the inverse interaction; whereas $J=1-a$ is the modulus of
the Jacobian determinant of the direct interaction, namely the transformation

$v^{\prime}=v+a(\pazocal{V}(\rho(x_{\ast},t))-v),\qquad
v^{\prime}_{\ast}=v_{\ast}$ (6)

from pre- to post-interaction speeds222Note the change of notation with
respect to the inverse interaction, with $v$, $v_{\ast}$ denoting here the
pre-interaction speeds and $v^{\prime}$, $v^{\prime}_{\ast}$ the post
interaction speeds.. In order for the interaction not to be singular we
require $a<1$.

Due to the non-locality in space featured by $Q$, (4) is a Povzner-type
kinetic equation, cf. [17, 36], which formally reduces to a Boltzmann-type
equation when $B$ tends to the Dirac delta centred in $x$, cf. [31].

The traffic density $\rho$ appearing in (1) is expressed in terms of the
distribution function $f$ as its zeroth-order $v$-moment:

$\rho(x,t):=\int_{0}^{1}f(x,v,t)\,dv.$

###  2.2 Hydrodynamic limit

The statistical description provided by the kinetic equation (4) is the basis
to upscale the particle dynamics (1)-(3) to the macroscopic level. To this
purpose, we introduce a small scale parameter $0<\varepsilon\ll 1$, which in
this context plays the role of the Knudsen number of the classical kinetic
theory, and we perform the following hyperbolic scaling of time and space:

$t\to\frac{t}{\varepsilon},\qquad x\to\frac{x}{\varepsilon},$ (7)

whence $\partial_{t}\to\varepsilon\partial_{t}$ and
$\partial_{x}\to\varepsilon\partial_{x}$. Owing to this, (4) takes the form

$\partial_{t}f^{\varepsilon}+v\partial_{x}f^{\varepsilon}=\frac{1}{\varepsilon}Q(f^{\varepsilon},f^{\varepsilon}),$
(8)

$f^{\varepsilon}$ denoting now the kinetic distribution function parametrised
by $\varepsilon$.

Let $\varphi:[0,\,1]\to\mathbb{R}$ be an arbitrarily chosen observable
quantity (test function). From the expression (5) of $Q$ we compute

$\displaystyle\begin{aligned}
\int_{0}^{1}\varphi(v)&Q(f^{\varepsilon},f^{\varepsilon})(x,v,t)\,dv\\\
&=\frac{1}{2}\int_{\mathbb{R}}\int_{0}^{1}\int_{0}^{1}B(x_{\ast}-x)(\varphi(v^{\prime})-\varphi(v))f^{\varepsilon}(x,v,t)f^{\varepsilon}(x_{\ast},v_{\ast},t)\,dv\,dv_{\ast}\,dx_{\ast},\end{aligned}$
(9)

where $v^{\prime}$ is given by (6). Choosing $\varphi\equiv 1$ we obtain in
particular

$\int_{0}^{1}Q(f^{\varepsilon},f^{\varepsilon})(x,v,t)\,dv=0,\qquad\forall\,\varepsilon>0,$
(10)

therefore $\varphi\equiv 1$ is a collisional invariant. Conversely, for
$\varphi(v)=v$ we obtain

$\int_{0}^{1}vQ(f^{\varepsilon},f^{\varepsilon})(x,v,t)\,dv=\frac{a}{2}\rho^{\varepsilon}(x,t)\int_{\mathbb{R}}B(x_{\ast}-x)\rho^{\varepsilon}(x_{\ast},t)(\pazocal{V}(\rho^{\varepsilon}(x_{\ast},t))-u^{\varepsilon}(x,t))\,dx_{\ast},$
(11)

where $\rho^{\varepsilon}$ is the traffic density associated with the
distribution function $f^{\varepsilon}$ and

$u^{\varepsilon}(x,t):=\frac{1}{\rho^{\varepsilon}(x,t)}\int_{0}^{1}vf^{\varepsilon}(x,v,t)\,dv$

is its mean speed. Thus, $\varphi(v)=v$ is in general not a collisional
invariant.

Integrating (8) with respect to $v$ and taking (10) into account gives the
conservation law

${\color[rgb]{1,0,0}\partial_{t}}\int_{0}^{1}f^{\varepsilon}(x,v,t)\,dv+\partial_{x}\int_{0}^{1}vf^{\varepsilon}(x,v,t)\,dv=0.$
(12)

At the same time, passing to the hydrodynamic limit $\varepsilon\to 0^{+}$ in
(8) under the formal assumption that the left-hand side of the equation
remains bounded implies that the limit distribution $f^{0}$ solves

$Q(f^{0},f^{0})=0$ (13)

with, according to (11),

$u^{0}(x,t)=\frac{\displaystyle\int_{\mathbb{R}}B(x_{\ast}-x)\rho^{0}(x_{\ast},t)\pazocal{V}(\rho^{0}(x_{\ast},t))\,dx_{\ast}}{\displaystyle\int_{\mathbb{R}}B(x_{\ast}-x)\rho^{0}(x_{\ast},t)\,dx_{\ast}}.$

In conclusion, passing formally to the hydrodynamic limit in (12) we discover
that the traffic density $\rho^{0}$, which we may simply rename $\rho$ for
brevity, satisfies

$\partial_{t}\rho+\partial_{x}\left(\rho\frac{\tilde{B}\ast(\rho\pazocal{V}(\rho))}{\tilde{B}\ast\rho}\right)=0,$
(14)

where $\tilde{B}(y):=B(-y)$ and $\ast$ denotes convolution. This is a
conservation law with non-local flux providing the hydrodynamic counterpart of
the stochastic particle model (1)-(3).

######  Remark.

It is interesting to compare (14) with other non-local first order macroscopic
traffic models proposed in the literature (cf. [5] for a thorough overview).
They are all based on the continuity equation for the traffic density $\rho$
closed with different expressions of the flux. For instance, in [3, 8] the
following equation is proposed:

$\partial_{t}\rho+\partial_{x}\left(g(\rho)\pazocal{V}\left(\int_{x}^{x+\eta}B(x_{\ast}-x)\rho(x_{\ast},t)\,dx_{\ast}\right)\right)=0,$
(15)

where $g$ is some non-negative function while $\pazocal{V}$, $B$ are non-
decreasing functions and, in particular, $B$ is a kernel modelling the
behaviour of drivers adapting their speed to the density of vehicles in front
of them. Instead, in [19] the following variation of the previous model is
presented:

$\partial_{t}\rho+\partial_{x}\left(g(\rho)\int_{x}^{x+\eta}B(x_{\ast}-x)\pazocal{V}(\rho(x_{\ast},t))\,dx_{\ast}\right)=0,$
(16)

where now $\pazocal{V}$ is some assigned density-dependent speed. Ideally, we
may say that in these models the mean traffic speed is assumed to result from:
(i) the evaluation of the average traffic density ahead in the first case;
(ii) the evaluation of the average traffic speed ahead in the second case.

Although reasonable, these models are postulated heuristically. Our derivation
of (14) indicates instead that a non-local first order macroscopic model
consistent with simple, yet meaningful, microscopic first principles has a
mean speed in the form of a spatial average of the macroscopic flux
$\rho\pazocal{V}(\rho)$ normalised by the corresponding spatial average of the
macroscopic density $\rho$.

Finally, it is worth mentioning that in [41] a non-local first order
macroscopic traffic model is derived from a microscopic cellular automaton
implementing an asymmetric size exclusion process, which reproduces the front-
rear anisotropy of vehicle interactions, and a look-ahead strategy, which
accounts for non-local vehicle interactions. The resulting model reads:

$\partial_{t}\rho+\partial_{x}\left(\rho(1-\rho)\frac{1}{\tau}e^{-\int_{x}^{+\infty}B(x_{\ast}-x)\rho(x_{\ast},t)\,dx_{\ast}}\right)=0,$

where $\tau>0$ is a relaxation time. Here, the mean traffic speed is given by
the classical linear diagram $1-\rho$ modulated by a non-local exponential
term, which accounts for the density distribution ahead.

In Figure 1, we compare the traffic densities obtained numerically with our
model (14) (Model 1), model (15) (Model 2) and model (16) (Model 3). We
reproduce the same numerical experiment proposed in [19, Figure 6] with
constant interaction kernel and $\pazocal{V}(\rho)=1-\rho^{5}$. From these
numerical results we notice that our model produces less oscillations than the
other models in correspondence of the rightmost discontinuity of the initial
datum, namely the one which would evolve as a rarefaction wave in the local
LWR model. This is consistent with the fact that, as previously mentioned, our
model performs more averages of the hydrodynamic parameters compared to the
other models, which may give rise to less oscillating density profiles. On the
contrary, we observe that our model produces more oscillations than the other
models in correspondence of the leftmost discontinuity of the initial datum,
namely the one which would evolve as a shock wave in the local LWR model.

Figure 1: Numerical comparison of non-local first order macroscopic models
(14) (Model 1), (15) (Model 2), (16) (Model 3). Left: initial condition.
Right: density $\rho$ at time $t=0.05$ obtained with
$\pazocal{V}(\rho)=1-\rho^{5}$, $g(\rho)=\rho$, $B(y)=\frac{1}{\eta}$
(constant interaction kernel) with $\eta=0.1$ and spatial mesh size
$\Delta{x}=10^{-3}$. See [19, Figure 6]

##  3 Follow-the-Leader dynamics

In this section we consider instead Follow-the-Leader (FTL) interactions among
the vehicles, meaning that at time $t>0$ a vehicle in $X_{t}$ updates its
speed $V_{t}$ on the basis of the speed $V^{\ast}_{t}$ of another vehicle in
$X^{\ast}_{t}$. Unlike (1), now the speed $V^{\ast}_{t}$ plays a direct role
in determining the new speed $V_{t+\Delta{t}}$. Although the interacting
vehicle $(X^{\ast}_{t},V^{\ast}_{t})$ is chosen randomly in the traffic
stream, it is usually though of as a leading vehicle of vehicle
$(X_{t},V_{t})$.

The particle model (1) modifies as

$X_{t+\Delta{t}}=X_{t}+V_{t}\Delta{t},\qquad
V_{t+\Delta{t}}=V_{t}+\lambda\Theta(V^{\ast}_{t}-V_{t}),$ (17)

where $\lambda>0$ is a constant coefficient representing the sensitivity of
the drivers, cf. [20]. Again,

$\Theta\sim\operatorname{Bernoulli}(B(X^{\ast}_{t}-X_{t})\Delta{t})$

with the interaction kernel $B$ fulfilling Assumption 2.1. Rules (17) are
complemented also in this case with

$X^{\ast}_{t+\Delta{t}}=X^{\ast}_{t}+V^{\ast}_{t}\Delta{t},\qquad
V^{\ast}_{t+\Delta{t}}=V^{\ast}_{t},$ (18)

sticking to the idea that in an interaction the leading vehicle does not
change speed.

Condition $0\leq\lambda\leq 1$ is necessary and sufficient to guarantee the
physical consistency of (17), specifically the fact that
$V_{t+\Delta{t}}\in[0,\,1]$ for all $V_{t},\,V^{\ast}_{t}\in[0,\,1]$. Indeed,
$V_{t+\Delta{t}}$ is a convex combination of $V_{t}$, $V^{\ast}_{t}$ through
the coefficient $\lambda\Theta$.

###  3.1 Kinetic description

The kinetic description is provided by (4), (5), the inverse interaction being
now

${}^{\prime}\\!v=v-\frac{\lambda}{1-\lambda}(v_{\ast}-v),\qquad{}^{\prime}\\!v_{\ast}=v_{\ast}$

and the direct interaction

$v^{\prime}=v+\lambda(v_{\ast}-v),\qquad v^{\prime}_{\ast}=v_{\ast}$ (19)

with Jacobian determinant $J=1-\lambda$. In order for the interaction not to
be singular we require $\lambda<1$.

For purposes which will be clear in a moment, if the spatial non-locality of
the interactions is sufficiently small, namely if

$\eta=\left|\operatorname{supp}(B)\right|\ll 1,$ (20)

cf. Assumption 2.1, we may introduce an approximation of the collisional
operator $Q$, which will be useful in the sequel. Consider the first order
truncation of the Taylor expansion of $f$ about $x$:

$f(x_{\ast},v_{\ast},t)\approx
f(x,v_{\ast},t)+\partial_{x}f(x,v_{\ast},t)(x_{\ast}-x),$ (21)

which is justified by the fact that if $x_{\ast}-x\in\operatorname{supp}(B)$
then, owing to (20), $x,\,x_{\ast}$ are close to each other. Consequently,
from (5) we obtain:

$\displaystyle\begin{aligned}
Q(f,f)(x,v,t)&\approx\frac{\pazocal{B}_{0}}{2}\int_{0}^{1}\left(\frac{1}{J}f(x,{}^{\prime}\\!v,t)f(x,{}^{\prime}\\!v_{\ast},t)-f(x,v,t)f(x,v_{\ast},t)\right)dv_{\ast}\\\
&\phantom{=}+\frac{\pazocal{B}_{1}}{2}\int_{0}^{1}\left(\frac{1}{J}f(x,{}^{\prime}\\!v,t)\partial_{x}f(x,{}^{\prime}\\!v_{\ast},t)-f(x,v,t)\partial_{x}f(x,v_{\ast},t)\right)dv_{\ast}\\\
&=:\pazocal{B}_{0}Q_{\text{B}}(f,f)(x,v,t)+\pazocal{B}_{1}Q_{\text{B}}(f,\partial_{x}f)(x,v,t),\end{aligned}$
(22)

where

$\pazocal{B}_{0}:=\int_{\mathbb{R}}B(y)\,dy,\qquad\pazocal{B}_{1}:=\int_{\mathbb{R}}yB(y)\,dy.$
(23)

Notice that $Q_{\text{B}}(f,f)$ is a classical Boltzmann-type collisional
operator in which the distribution functions of the interacting vehicles are
computed in the same space position $x$. The local correction
$Q_{\text{B}}(f,\partial_{x}f)$ keeps instead track of the (small) non-
locality of the interactions. In weak form, for an arbitrary observable
quantity $\varphi=\varphi(v)$, we have (cf. (9)):

$\int_{0}^{1}\varphi(v)Q_{\text{B}}(f,g)(x,v,t)\,dv=\frac{1}{2}\int_{0}^{1}\int_{0}^{1}(\varphi(v^{\prime})-\varphi(v))f(x,v,t)g(x,v_{\ast},t)\,dv\,dv_{\ast}$
(24)

with $g=f$ and $g=\partial_{x}f$, respectively.

###  3.2 Hydrodynamic limit

By the hyperbolic scaling (7) of space and time we obtain the scaled kinetic
equation (8). Next, from (9) we may identify the collisional invariants, using
now the interaction rules (19). In particular, we get again that
$\varphi\equiv 1$ is a collisional invariant, while for $\varphi(v)=v$ we
obtain

$\int_{0}^{1}vQ(f^{\varepsilon},f^{\varepsilon})(x,v,t)\,dv=\frac{\lambda}{2}\rho^{\varepsilon}(x,t)\int_{\mathbb{R}}B(x_{\ast}-x)\rho^{\varepsilon}(x_{\ast},t)(u^{\varepsilon}(x_{\ast},t)-u^{\varepsilon}(x,t))\,dx_{\ast}.$

We notice that, even for $\varepsilon\to 0^{+}$ when (13) holds, from this
relationship it is not immediate to extract explicit analytical information on
the limit mean speed $u^{0}$. Thus it is difficult to proceed with the
classical hydrodynamic limit, which requires to identify precisely the
collisional invariants and the local equilibrium values of the non-conserved
quantities. To circumvent this difficulty, we assume (20) and look for the
best local hydrodynamic approximation of the non-local FTL particle dynamics.

####  3.2.1 Approximate hydrodynamics

Under assumption (20) we may rely on the approximation (22) of the collisional
operator $Q$. The hyperbolic scaling (7) causes the kinetic equation (4) to
take the form

$\partial_{t}f^{\varepsilon}+v\partial_{x}f^{\varepsilon}=\frac{\pazocal{B}_{0}}{\varepsilon}Q_{\text{B}}(f^{\varepsilon},f^{\varepsilon})+\pazocal{B}_{1}Q_{\text{B}}(f^{\varepsilon},\partial_{x}f^{\varepsilon}),$
(25)

considering that
$Q_{\text{B}}(f^{\varepsilon},\varepsilon\partial_{x}f^{\varepsilon})=\varepsilon
Q_{\text{B}}(f^{\varepsilon},\partial_{x}f^{\varepsilon})$ from the bi-
linearity of $Q_{\text{B}}$. Recalling (19) and (24), we easily see that

$\int_{0}^{1}Q_{\text{B}}(f^{\varepsilon},f^{\varepsilon})(x,v,t)\,dv=\int_{0}^{1}vQ_{\text{B}}(f^{\varepsilon},f^{\varepsilon})(x,v,t)\,dv=0,\qquad\forall\,\varepsilon>0,$

hence $\varphi\equiv 1$ and $\varphi(v)=v$ are collisional invariants.
Conversely,

$\int_{0}^{1}v^{2}Q_{\text{B}}(f^{\varepsilon},f^{\varepsilon})(x,v,t)\,dv=\lambda(1-\lambda){(\rho^{\varepsilon})}^{2}(x,t)\Bigl{(}{(u^{\varepsilon})}^{2}(x,t)-E^{\varepsilon}(x,t)\Bigr{)},$
(26)

where we have denoted by

$E^{\varepsilon}=E^{\varepsilon}(x,t):=\frac{1}{\rho^{\varepsilon}(x,t)}\int_{0}^{1}v^{2}f^{\varepsilon}(x,v,t)\,dv$

the energy of the distribution function $f^{\varepsilon}$. Parallelly, from
(24) we observe that

$\displaystyle\int_{0}^{1}Q_{\text{B}}(f^{\varepsilon},\partial_{x}f^{\varepsilon})(x,v,t)\,dv=0$
$\displaystyle\int_{0}^{1}vQ_{\text{B}}(f^{\varepsilon},\partial_{x}f^{\varepsilon})(x,v,t)\,dv=\frac{\lambda}{2}\left(\int_{0}^{1}f^{\varepsilon}(x,v,t)\,dv\cdot\partial_{x}\int_{0}^{1}v_{\ast}f^{\varepsilon}(x,v_{\ast},t)\,dv_{\ast}\right.$
$\displaystyle\phantom{\int_{0}^{1}vQ_{\text{B}}(f^{\varepsilon},\partial_{x}f^{\varepsilon})(x,v,t)\,dv=}\left.-\partial_{x}\int_{0}^{1}f^{\varepsilon}(x,v_{\ast},t)\,dv_{\ast}\cdot\int_{0}^{1}vf^{\varepsilon}(x,v,t)\,dv\right),\qquad\forall\,\varepsilon>0.$

Therefore, multiplying (25) by the two collisional invariants above and
integrating with respect to $v$ yields, for every $\varepsilon>0$, the
following system of balance laws:

(27)

Taking now to the limit $\varepsilon\to 0^{+}$ in (25) and assuming formally
that the left-hand side of the equation, as well as the second term at the
right-hand side, remain bounded we get that the limit distribution $f^{0}$
solves

$Q_{\text{B}}(f^{0},f^{0})=0$

with, owing to (26), $E^{0}(x,t)={(u^{0})}^{2}(x,t)$ at least for
$0<\lambda<1$ (notice that $\lambda<1$ by assumption while $\lambda>0$ is
necessary for meaningfulness of the model, as $\lambda=0$ would imply no
interactions among the vehicles). Consequently, passing formally to the limit
in (27) we obtain that the traffic density and mean speed $\rho^{0}$, $u^{0}$,
which we rename simply $\rho$, $u$ for convenience, satisfy

$\begin{cases}\partial_{t}\rho+\partial_{x}(\rho u)=0\\\ \partial_{t}(\rho
u)+\partial_{x}(\rho
u^{2})=\dfrac{\lambda\pazocal{B}_{1}}{2}\rho^{2}\partial_{x}u.\end{cases}$

Using the first equation, we rewrite the second equation as

$\partial_{t}u+\left(u-\rho\frac{\lambda\pazocal{B}_{1}}{2}\right)\partial_{x}u=0$

so that, on the whole, the hydrodynamic description of the particle model
(17)-(18) under the assumption (20) of small non-locality of the interactions
is provided by the system

$\begin{cases}\partial_{t}\rho+\partial_{x}(\rho u)=0\\\\[5.69054pt]
\partial_{t}u+\left(u-\rho\dfrac{\lambda\pazocal{B}_{1}}{2}\right)\partial_{x}u=0,\end{cases}$
(28)

i.e. the celebrated Aw-Rascle-Zhang (ARZ) traffic model [1, 45] with (pseudo-)
pressure $p=p(\rho)$ satisfying

$p^{\prime}(\rho)=\frac{\lambda\pazocal{B}_{1}}{2}.$

Hence, $p(\rho):=\frac{\lambda\pazocal{B}_{1}}{2}\rho$ up to an arbitrary
additive constant. It is immediate to check that (28) is a hyperbolic model
with eigenvalues $u$ and $u-\rho\frac{\lambda\pazocal{B}_{1}}{2}$. From
Assumption 2.1, in particular the fact that
$\operatorname{supp}(B)\subseteq\mathbb{R}_{+}$, and (23) we deduce
$\pazocal{B}_{1}>0$, hence no eigenvalue of (28) is greater than $u$.
Consequently, model (28) has the property that small disturbances in traffic
cannot propagate faster than the flow of vehicles, which reproduces correctly
the front-rear anisotropy of vehicle interactions. Fulfilling such a property
was the main motivation for the introduction of the ARZ model [1] as a cure
for some physical inconsistencies of previous second order macroscopic traffic
models [12].

Summarising, we have shown that:

> The hydrodynamic limit of _any_ non-local FTL particle model of the form
> (17)-(18) _with sufficiently small non-locality_ is the ARZ model (28) with
> a (pseudo-) pressure $p$ reminiscent of the non-local interaction kernel
> through the coefficient $\pazocal{B}_{1}$.

This result generalises the one obtained in [15], where the ARZ model is
recovered as the hydrodynamic limit of an Enskog-type kinetic description of
FTL vehicle dynamics. Notice that an Enskog-type description amounts to
(4)-(5) with the particular choice $B(y)=\delta(y-\eta)$, where $\delta$
denotes the Dirac delta distribution and $\eta>0$ is in this case the fixed
distance separating the interacting vehicles.

This result shows also that the ARZ model, originally conceived as a
macroscopic traffic model accounting for the anticipation ability of the
drivers in a physically consistent way, cf. [1, 12, 39], is in fact more in
general the prototype of any non-local FTL interaction model at least for a
sufficiently small non-locality.

##  4 Generalised Follow-the-Leader dynamics

In this section we consider the following generalisation of the non-local FTL
particle model (17)-(18):

$X_{t+\Delta{t}}=X_{t}+\pazocal{V}(S_{t})\Delta{t},\qquad
S_{t+\Delta{t}}=S_{t}+\Theta(\Psi_{\lambda}(S^{\ast}_{t})-\Psi_{\lambda}(S_{t})),$
(29)

and

$X^{\ast}_{t+\Delta{t}}=X^{\ast}_{t}+\pazocal{V}(S^{\ast}_{t})\Delta{t},\qquad
S^{\ast}_{t+\Delta{t}}=S^{\ast}_{t},$ (30)

where now $S_{t},\,S^{\ast}_{t}\geq 0$ are the space headways of the
interacting vehicles, i.e. the free space each of them has in front,
$\pazocal{V}:\mathbb{R}_{+}\to\ [0,\,1]$ is the speed of a vehicle given as a
function of the headway and $\Psi_{\lambda}:\mathbb{R}_{+}\to\mathbb{R}_{+}$
is an interaction function parametrised by the sensitivity $\lambda>0$ of the
drivers. Follow-the-Leader particle models that can be recast in the form (29)
have been considered e.g., in [10, 35, 43].

We set the following technical assumptions on $\Psi_{\lambda}$:

######  Assumption 4.1 (Interaction function).

We assume that $\Psi_{\lambda}$ is non-negative and such that
$\Psi_{\lambda}(s)\sim\lambda s$ when $\lambda\to 0^{+}$. In particular, we
assume that there exists $\psi_{\lambda}:\mathbb{R}_{+}\to\mathbb{R}_{+}$,
with $\psi_{\lambda}(s)\to 1$ for all $s\in\mathbb{R}_{+}$ when $\lambda\to
0^{+}$, such that

$\Psi_{\lambda}(s)=\lambda s\psi_{\lambda}(s).$

Moreover, we assume

$\sup_{\lambda>0}\sup_{s\in\mathbb{R}_{+}}\psi_{\lambda}(s)<+\infty.$

A reference family of functions $\Psi_{\lambda}$ complying with Assumption 4.1
is

$\Psi_{\lambda}(s)=\frac{\lambda s}{{(1+\lambda s^{\alpha})}^{\beta}}$

with $\alpha,\,\beta>0$. In the aforementioned papers [10, 35, 43] the case
$\alpha=\beta=1$ is considered with the interaction function written in the
equivalent form $\Psi_{\lambda}(s)=1-\frac{1}{1+\lambda s}$.

Unlike the cases discussed in the previous sections, now it is not possible to
identify a universal range of values of $\lambda$, valid for all possible
choices of $\Psi_{\lambda}$, which ensures the physical consistency of
interactions (29), namely the fact that $S_{t+\Delta{t}}\geq 0$ for all
$S_{t},\,S^{\ast}_{t}\geq 0$. Therefore it is necessary to neglect explicitly
possible unphysical interactions producing $S_{t+\Delta{t}}<0$. This may be
done by applying a cutoff to the interaction frequency parametrising the law
of $\Theta$:

$\Theta\sim\operatorname{Bernoulli}\left(\frac{1}{\lambda}\chi(S_{t}+\Psi_{\lambda}(S^{\ast}_{t})-\Psi_{\lambda}(S_{t})\geq
0)B(X^{\ast}_{t}-X_{t})\Delta{t}\right),$ (31)

where $\chi(\cdot)$ is the characteristic function of the event in
parenthesis. $S_{t}+\Psi_{\lambda}(S^{\ast}_{t})-\Psi_{\lambda}(S_{t})$ is the
value of the post-interaction headway if an interaction takes place. If the
pair $(S_{t},\,S^{\ast}_{t})$ produces an unphysical $S_{t+\Delta{t}}<0$ then
the corresponding interaction between the two vehicles is discarded, because
$\chi(S_{t}+\Psi_{\lambda}(S^{\ast}_{t})-\Psi_{\lambda}(S_{t})\geq 0)=0$
whence $\Theta\sim\operatorname{Bernoulli}(0)$. The further coefficient
$\frac{1}{\lambda}$ in (31) is used to normalise the interaction rate with
respect to the sensitivity of the drivers, in such a way that different models
corresponding to different $\Psi_{\lambda}$ are more comparable.

###  4.1 Kinetic description

The kinetic description of the particle model (29)-(31) is now provided by a
distribution function
$f=f(x,s,t):\mathbb{R}\times\mathbb{R}_{+}\times(0,\,+\infty)\to\mathbb{R}_{+}$
of the microscopic state $(x,s)$ of the vehicles. By the same guidelines
recalled in Section 2.1, we obtain that the kinetic equation satisfied by $f$
is of the form (4) with

$Q(f,f)(x,s,t)=\\\
\frac{1}{2\lambda}\int_{\mathbb{R}}\int_{\mathbb{R}_{+}}B(x_{\ast}-x)\left(\chi(s\geq
0)\frac{1}{J}f(x,{}^{\prime}\\!s,t)f(x_{\ast},{}^{\prime}\\!s_{\ast},t)-\chi(s^{\prime}\geq
0)f(x,s,t)f(x_{\ast},s_{\ast},t)\right)ds_{\ast}\,dx_{\ast},$ (32)

where:

  1. (i)

${}^{\prime}\\!s$, ${}^{\prime}\\!s_{\ast}$ are the pre-interaction headways
producing the post-interaction headways $s$, $s_{\ast}$ if an interaction
takes place ($\Theta=1$) in the dynamics (29)-(31). The inverse interaction
giving ${}^{\prime}\\!s$, ${}^{\prime}\\!s_{\ast}$ as functions of $s$,
$s_{\ast}$ is

${}^{\prime}\\!s-\Psi_{\lambda}({}^{\prime}\\!s)=s-\Psi_{\lambda}(s_{\ast}),\qquad{}^{\prime}\\!s_{\ast}=s_{\ast};$

more explicit expressions depend on the specific function $\Psi_{\lambda}$
used;

  2. (ii)

$J=\left|1-\Psi_{\lambda}^{\prime}({}^{\prime}\\!s)\right|$ is the modulus of
the Jacobian determinant of the direct interaction, i.e.

$s^{\prime}=s+\Psi_{\lambda}(s_{\ast})-\Psi_{\lambda}(s),\qquad
s_{\ast}^{\prime}=s_{\ast},$ (33)

which is also used in the term $\chi(s^{\prime}\geq 0)$;

  3. (iii)

the coefficient $\frac{1}{\lambda}$ in front of the expression of $Q$ comes
from normalisation factor of the interaction rate in (31).

The weak form of the collisional operator $Q$ reads

$\displaystyle\begin{aligned}
\int_{\mathbb{R}_{+}}\varphi(s)&Q(f,f)(x,s,t)\,ds\\\
&=\frac{1}{2\lambda}\int_{\mathbb{R}}\int_{\mathbb{R}_{+}}\int_{\mathbb{R}_{+}}B(x_{\ast}-x)\chi(s^{\prime}\geq
0)(\varphi(s^{\prime})-\varphi(s))f(x,s,t)f(x_{\ast},s_{\ast},t)\,ds\,ds_{\ast}\,dx_{\ast}\end{aligned}$
(34)

for every observable quantity $\varphi:\mathbb{R}_{+}\to\mathbb{R}$. This
expression is similar to (5) but now the full interaction kernel is
$B(x_{\ast}-x)\chi(s^{\prime}\geq 0)$. In particular, the cutoff
$\chi(s^{\prime}\geq 0)$ introduces a strong non-linearity in $Q$, which makes
the kinetic equation less amenable to analytical investigations. To circumvent
this difficulty of the theory, we consider the particle model (29)-(31) and
the corresponding kinetic description in the limit $\lambda\to 0^{+}$. The
motivation for taking $\lambda$ small is that (29) and Assumption 4.1 suggest
that the smaller $\lambda$ the easier to meet the requirement $s^{\prime}\geq
0$ because $s^{\prime}\approx s$. Consequently, we may expect the cutoff in
(34) to be less and less problematic. Since, as we mentioned before, a
universal maximum value of $\lambda$ cannot be fixed, the limit $\lambda\to
0^{+}$ is also representative of universal aggregate trends at least in the
regime of a small sensitivity of the drivers.

Since $\chi(s^{\prime}\geq 0)=1-\chi(s^{\prime}<0)$, we may rewrite (34) as

$\displaystyle\int_{\mathbb{R}_{+}}\varphi(s)$ $\displaystyle
Q(f,f)(x,s,t)\,ds$
$\displaystyle=\frac{1}{2\lambda}\int_{\mathbb{R}}\int_{\mathbb{R}_{+}}\int_{\mathbb{R}_{+}}B(x_{\ast}-x)(\varphi(s^{\prime})-\varphi(s))f(x,s,t)f(x_{\ast},s_{\ast},t)\,ds\,ds_{\ast}\,dx_{\ast}$
$\displaystyle\phantom{=}-\frac{1}{2\lambda}\int_{\mathbb{R}}\int_{\mathbb{R}_{+}}\int_{\mathbb{R}_{+}}B(x_{\ast}-x)\chi(s^{\prime}<0)(\varphi(s^{\prime})-\varphi(s))f(x,s,t)f(x_{\ast},s_{\ast},t)\,ds\,ds_{\ast}\,dx_{\ast}$
$\displaystyle=:\mathscr{Q}(f,f)[\varphi](x,t)+\mathscr{R}(f,f)[\varphi](x,t).$

Now observe from (33) that $s-\Psi_{\lambda}(s)\leq s^{\prime}$ and
furthermore, owing to Assumption 4.1, $s-\Psi_{\lambda}(s)=s-\lambda
s\psi_{\lambda}(s)\geq(1-\lambda C)s$, where $C>0$ is a constant such that
$\psi_{\lambda}(s)\leq C$ for all $\lambda,\,s\in\mathbb{R}_{+}$. Hence
$(1-\lambda C)s\leq s^{\prime}$, which implies
$\chi(s^{\prime}<0)\leq\chi((1-\lambda C)s<0)$. Consequently,

$\left|\mathscr{R}(f,f)[\varphi](x,t)\right|\leq\\\
\frac{1}{2\lambda}\int_{\mathbb{R}}\int_{\mathbb{R}_{+}}\int_{\mathbb{R}_{+}}B(x_{\ast}-x)\chi((1-\lambda
C)s<0)\left|\varphi(s^{\prime})-\varphi(s)\right|f(x,s,t)f(x_{\ast},s_{\ast},t)\,ds\,ds_{\ast}\,dx_{\ast}$

and since $\chi((1-\lambda C)s<0)=0$ for $\lambda\leq\frac{1}{C}$ we get

$\lim_{\lambda\to 0^{+}}\mathscr{R}(f,f)[\varphi](x,t)=0,$

whence

$\lim_{\lambda\to
0^{+}}\int_{\mathbb{R}_{+}}\varphi(s)Q(f,f)(x,s,t)\,ds=\lim_{\lambda\to
0^{+}}\mathscr{Q}(f,f)[\varphi](x,t).$

To compute the limit of $\mathscr{Q}(f,f)[\varphi]$ we assume that $\varphi$
is sufficiently smooth, say $\varphi\in C^{2}_{c}(\mathbb{R}_{+})$, and we
expand $\varphi(s^{\prime})-\varphi(s)$ in Taylor series up to the second
order with Lagrange remainder:

$\displaystyle\mathscr{Q}(f,f)$ $\displaystyle[\varphi](x,t)$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}}\int_{\mathbb{R}_{+}}\int_{\mathbb{R}_{+}}B(x_{\ast}-x)\varphi^{\prime}(s)\bigl{(}s_{\ast}\psi_{\lambda}(s_{\ast})-s\psi_{\lambda}(s)\bigr{)}f(x,s,t)f(x_{\ast},s_{\ast},t)\,ds\,ds_{\ast}\,dx_{\ast}$
$\displaystyle\phantom{=}+\frac{\lambda}{2}\int_{\mathbb{R}}\int_{\mathbb{R}_{+}}\int_{\mathbb{R}_{+}}B(x_{\ast}-x)\varphi^{\prime\prime}(\bar{s}){\bigl{(}s_{\ast}\psi_{\lambda}(s_{\ast})-s\psi_{\lambda}(s)\bigr{)}}^{2}f(x,s,t)f(x_{\ast},s_{\ast},t)\,ds\,ds_{\ast}\,dx_{\ast},$

where $\bar{s}:=\vartheta s+(1-\vartheta)s_{\ast}$ for some
$\vartheta\in[0,\,1]$. Since, owing to Assumption 4.1, $\psi_{\lambda}$ is
uniformly bounded with respect to $\lambda,\,s\in\mathbb{R}_{+}$, and, owing
to Assumption 2.1, $B$ is also bounded, we may pass to the limit under the
integral by dominated convergence up to assuming that $f$ has $x$-integrable
first and second $s$-moments, i.e.:

$\int_{\mathbb{R}}\int_{\mathbb{R}_{+}}s^{p}f(x,s,t)\,ds\,dx<+\infty\quad\text{for
}p=1,\,2.$

In such a case we obtain

$\displaystyle\lim_{\lambda\to 0^{+}}\mathscr{Q}(f,f)[\varphi](x,t)$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}}\int_{\mathbb{R}_{+}}\int_{\mathbb{R}_{+}}B(x_{\ast}-x)\varphi^{\prime}(s)(s_{\ast}-s)f(x,s,t)f(x_{\ast},s_{\ast},t)\,ds\,ds_{\ast}\,dx_{\ast}$
$\displaystyle=:\int_{\mathbb{R}_{+}}\varphi(s)\tilde{Q}(f,f)(x,s,t)\,ds,$

$\tilde{Q}(f,f)$ being the operator by which we may replace $Q(f,f)$ given by
(32) in the limit $\lambda\to 0^{+}$.

Under (20) and (21) we may further approximate

$\displaystyle\begin{aligned}
\int_{\mathbb{R}_{+}}\varphi(s)\tilde{Q}(f,f)(x,s,t)\,ds&\approx\frac{\pazocal{B}_{0}}{2}\int_{\mathbb{R}_{+}}\int_{\mathbb{R}_{+}}\varphi^{\prime}(s)(s_{\ast}-s)f(x,s,t)f(x,s_{\ast},t)\,ds\,ds_{\ast}\\\
&\phantom{\approx}+\frac{\pazocal{B}_{1}}{2}\int_{\mathbb{R}_{+}}\int_{\mathbb{R}_{+}}\varphi^{\prime}(s)(s_{\ast}-s)f(x,s,t)\partial_{x}f(x,s_{\ast},t)\,ds\,ds_{\ast}\\\
&=:\pazocal{B}_{0}\int_{\mathbb{R}_{+}}\varphi(s)\tilde{Q}_{\text{B}}(f,f)(x,s,t)\,ds+\pazocal{B}_{1}\int_{\mathbb{R}_{+}}\varphi(s)\tilde{Q}_{\text{B}}(f,\partial_{x}f)(x,s,t)\,ds,\end{aligned}$
(35)

where $\tilde{Q}_{\text{B}}(f,f)$ expresses local interactions while
$\tilde{Q}_{\text{B}}(f,\partial_{x}f)$ is a correction approximating the
small non-locality by means of a space derivative.

The kinetic equation (4) takes finally the form

$\partial_{t}f+\pazocal{V}(s)\partial_{x}f=\pazocal{B}_{0}\tilde{Q}_{\text{B}}(f,f)+\pazocal{B}_{1}\tilde{Q}_{\text{B}}(f,\partial_{x}f),$
(36)

considering that, according to (29), (30), the transport speed of the vehicles
is $\pazocal{V}=\pazocal{V}(s)$.

###  4.2 Hydrodynamic limit

The hyperbolic scaling (7) applied to (36) produces the equation

$\partial_{t}f^{\varepsilon}+\pazocal{V}(s)\partial_{x}f^{\varepsilon}=\frac{\pazocal{B}_{0}}{\varepsilon}\tilde{Q}_{\text{B}}(f^{\varepsilon},f^{\varepsilon})+\pazocal{B}_{1}\tilde{Q}_{\text{B}}(f^{\varepsilon},\partial_{x}f^{\varepsilon}),$
(37)

which in the hydrodynamic limit $\varepsilon\to 0^{+}$ can be tackled
similarly to Section 3.2.1. In particular, letting $\varphi(s)=1,\,s$ in (35)
we discover

$\int_{\mathbb{R}_{+}}\tilde{Q}_{\text{B}}(f^{\varepsilon},f^{\varepsilon})(x,s,t)\,ds=\int_{\mathbb{R}_{+}}s\tilde{Q}_{\text{B}}(f^{\varepsilon},f^{\varepsilon})(x,s,t)\,ds=0,\qquad\forall\,\varepsilon>0,$

whereas for $\varphi(s)=s^{2}$ we obtain

$\int_{\mathbb{R}_{+}}s^{2}\tilde{Q}_{\text{B}}(f^{\varepsilon},f^{\varepsilon})(x,s,t)\,ds={(\rho^{\varepsilon})}^{2}(x,t)\Bigl{(}{(h^{\varepsilon})}^{2}(x,t)-E^{\varepsilon}(x,t)\Bigr{)}$
(38)

with

$h^{\varepsilon}(x,t):=\frac{1}{\rho^{\varepsilon}(x,t)}\int_{\mathbb{R}_{+}}sf^{\varepsilon}(x,s,t)\,ds,\qquad
E^{\varepsilon}(x,t):=\frac{1}{\rho^{\varepsilon}(x,t)}\int_{\mathbb{R}_{+}}s^{2}f^{\varepsilon}(x,s,t)\,ds$

the mean headway and the energy of the distribution $f^{\varepsilon}$,
respectively.

Multiplying (37) by $\varphi(s)=1,\,s$ and integrating with respect to $s$
produces, for every $\varepsilon>0$, the following system of balance laws:

(39)

Taking instead the limit $\varepsilon\to 0^{+}$ in (37), while assuming
formally that the left-hand side and the second term at the right-hand side
remain bounded, yields that the limit distribution $f^{0}$ solves

$\tilde{Q}_{\text{B}}(f^{0},f^{0})=0$

with, owing to (38), $E^{0}(x,t)={(h^{0})}^{2}(x,t)$. This implies in
particular that $f^{0}$ has zero variance, hence that it is the monokinetic
distribution

$f^{0}(x,s,t)=\rho^{0}(x,t)\delta(s-h^{0}(x,t)).$

Consequently, passing formally to the limit in (39), we discover that the
traffic density and mean headway $\rho^{0}$, $h^{0}$, which we rename simply
as $\rho$, $h$, satisfy

$\begin{cases}\partial_{t}\rho+\partial_{x}(\rho\pazocal{V}(h))=0\\\
\partial_{t}(\rho h)+\partial_{x}(\rho
h\pazocal{V}(h))=\dfrac{\pazocal{B}_{1}}{2}\rho^{2}\partial_{x}h.\end{cases}$

Using the first equation, we can rewrite the second equation as

$\partial_{t}h+\left(\pazocal{V}(h)-\rho\dfrac{\pazocal{B}_{1}}{2}\right)\partial_{x}h=0,$

whereby the hydrodynamic description of the particle model (29)-(31) in the
limit $\lambda\to 0^{+}$ takes finally the form

$\begin{cases}\partial_{t}\rho+\partial_{x}(\rho\pazocal{V}(h))=0\\\\[5.69054pt]
\partial_{t}h+\left(\pazocal{V}(h)-\rho\dfrac{\pazocal{B}_{1}}{2}\right)\partial_{x}h=0.\end{cases}$
(40)

We notice that model (40) has evident formal analogies with the ARZ model
(28). In particular, it is easy to see that also this model is hyperbolic with
eigenvalues given by $\pazocal{V}(h)$ and
$\pazocal{V}(h)-\rho\frac{\pazocal{B}_{1}}{2}$. Since now $\pazocal{V}(h)$ is
the flow speed and, as already noticed, $\pazocal{B}_{1}>0$, we recover the
physically meaningful property that small disturbances in traffic do not
propagate faster than the flow of the vehicles.

In conclusion:

> In the regime of small driver sensitivity $\lambda$, the hydrodynamic limit
> of _any_ generalised non-local FTL particle model of the form (29)-(31) with
> sufficiently small non-locality is well approximated by the _ARZ-like model_
> (40) arising for $\lambda\to 0^{+}$.

##  5 Numerical tests

In this section, we compare the numerical results produced by the stochastic
particle models (1)-(3), (17)-(18) and (29)-(31) with those produced by the
corresponding hydrodynamic limits (14), (28) and (40) in some test cases. The
particle models are solved by means of spatially non-local Monte Carlo
algorithms, which will be detailed in the next subsections. Conversely, the
first order non-local macroscopic model is solved by means of an upwind-type
numerical scheme introduced in [19], see also [5, 7, 9]; while the second
order ARZ and ARZ-like models are solved by a splitting of the transport and
source contributions with a classical upwind scheme at every time step to
discretise the homogeneous system.

In all tests we consider as spatial domain the interval
$[-1,\,1]\subset\mathbb{R}$ with periodic boundary conditions and we prescribe
as initial conditions the following traffic density:

$\rho_{0}(x)=\begin{cases}0.8&\text{if }x<0\\\ 0.2&\text{if }x>0\end{cases}$

and, whenever needed, the following mean speed:

$u_{0}(x)=\begin{cases}0.5&\text{if }x<0\\\ 0.6&\text{if }x>0.\end{cases}$

Figure 2: Initial condition

To reproduce the initial traffic density at the particle level, if
$N\in\mathbb{N}$ is the total number of particles used in the Monte Carlo
algorithm we sample uniformly $0.8N$ particle positions in the interval
$[-1,\,0]$ and $0.2N$ particle positions in the interval $[0,\,1]$. Likewise,
to reproduce the initial mean speed, for $x<0$ we sample $0.8N$ speed values
uniformly distributed in the interval $[0,\,1]$, so that their (theoretical)
mean is $0.5$; while for $x>0$ we sample $0.2N$ speed values uniformly
distributed in the interval $[0.2,\,1]$, so that their (theoretical) mean is
$0.6$. We refer the reader to [11] for more details about the procedure to
link particles and hydrodynamic variables in the implementation of numerical
algorithms. Figure 2 depicts the initial condition just discussed.

Moreover, we fix the following interaction kernel:

$B(y)=\left(1-\frac{y}{\eta}\right)\chi(0\leq y\leq\eta),$

which complies with Assumption 2.1 and decreases linearly from $1$ to $0$ in
the interval $[0,\,\eta]$ (hence $\max_{y\in[0,\,\eta]}B(y)=1$). Such a kernel
models a stronger influence of closer vehicles and a weaker influence of
farther vehicles and is such that

$\pazocal{B}_{0}=\frac{\eta}{2},\qquad\pazocal{B}_{1}=\frac{\eta^{2}}{6}.$

###  5.1 Optimal speed dynamics

1 Fix $\Delta{t}=\frac{\varepsilon}{\max_{y\in[0,\,\eta]}B(y)}$;

2 Fix $\Delta{x}\leq\eta$ and consider a space mesh
$\\{E_{h}\\}_{h\in\mathbb{Z}}$ of pairwise disjoint cells with
$\left|E_{h}\right|=\Delta{x}$ and $\cup_{h\in\mathbb{Z}}E_{h}=\mathbb{R}$;

3 for  _$n=0,\,1,\,2,\,\dots$_ do

4 for  _$h\in\mathbb{Z}$_ do

5 Find the particles belonging to the cell $E_{h}$. Let
$\mathscr{E}^{n}_{h}:=\\{i\in\\{1,\dots,N\\}:x^{n}_{i}\in E_{h}\\}$;

6 for  _$i\in\mathscr{E}^{n}_{h}$_ do

7 Sample randomly $k\in\mathbb{Z}$ such that $h\leq k\leq
h+\lfloor{\frac{\eta}{\Delta{x}}}\rfloor$;

8 Sample randomly $j\in\mathscr{E}^{n}_{k}$;

9 Let $\rho^{n}_{k}$ be an approximation of the traffic density $\rho$ in
$E_{k}$ at time $t_{n}:=n\Delta{t}$;

10 Sample
$\Theta\sim\operatorname{Bernoulli}(B(x^{n}_{j}-x^{n}_{i})\frac{\Delta{t}}{\varepsilon})$,
$\Theta=\theta\in\\{0,\,1\\}$;

11 Update $v^{n}_{i}$ to $v^{n+1}_{i}$ according to (1):
$v^{n+1}_{i}=v^{n}_{i}+\theta a(\pazocal{V}(\rho^{n}_{k})-v^{n}_{i})$;

12

13

14 Update the particle positions according to (1):
$x^{n+1}_{i}=x^{n}_{i}+v^{n+1}_{i}\Delta{t}$, $i\in\\{1,\dots,N\\}$;

15

Algorithm 1 Non-local Monte Carlo algorithm for the particle model (1)-(3)

We begin by considering the stochastic particle model (1)-(3), to which there
corresponds the first order non-local macroscopic model (14). In particular,
we choose the following optimal speed function:

$\pazocal{V}(\rho)=\frac{1}{\tanh{(1)}}\tanh{\left(\frac{1}{1+\rho}\right)},$
(41)

as a prototype inspired by [2] which complies with Assumption 2.2. In
Algorithm 1 we detail the implementation of the Monte Carlo scheme that we use
to approach numerically the particle model.

Figure 3: Solution of the particle model (1)-(3) (markers) obtained with
$N=10^{6}$ particles and $\Delta{x}=10^{-2},$ and of the corresponding first
order non-local hydrodynamic model (14) (solid line) at the computational time
$t=1$ for fixed $\eta=10^{-2}$, $a=0.5,\,\Delta{x}=10^{-3}$ and scaling
parameter $\varepsilon$ decreasing from $10^{-3}$ to $5\cdot 10^{-5}$ (left to
right)

Figure 3 shows that the density profiles of the solutions produced by the
particle model and by the non-local hydrodynamic model tend to coincide
substantially as the scaling parameter $\varepsilon$ decreases from $10^{-3}$
to $5\cdot 10^{-5}$. This correctly reproduces the hydrodynamic limit
anticipated by the theory. We notice that for $\varepsilon=5\cdot 10^{-5}$
some oscillations still appear in the particle solution in correspondence of
the rear shock wave exhibited by the hydrodynamic solution. These
oscillations, which are the microscopic counterpart of the sharp localised
variation in the mass of vehicles, may be smeared out by further reducing the
scaling parameter $\varepsilon$, however at the price of an increased
computational cost of the Monte Carlo algorithm, where
$\Delta{t}=O(\varepsilon)$ (cf. Algorithm 1).

Figure 4: Solution of the particle model (1)-(3) (markers) obtained with
$N=10^{5}$ particles, $\Delta{x}=10^{-3}$ at left and $\Delta{x}=10^{-2}$ at
right, and of the corresponding first order non-local hydrodynamic model (14)
(solid line) at the computational time $t=1$ for fixed $\varepsilon=10^{-2}$,
$a=0.5,\,\Delta{x}=10^{-3}$ and size $\eta$ of the support of $B$ increasing
from $10^{-2}$ to $10^{-1}$ (left to right)

Figure 4 shows instead the effect of the size $\eta$ of the support of the
interaction kernel $B$ on the agreement between the particle and hydrodynamic
density profiles for fixed scaling parameter $\varepsilon$. We clearly notice
that the larger the support the more damped the oscillations of the particle
model about the average macroscopic solution even for a scaling parameter as
moderately small as $\varepsilon=10^{-2}$.

Figure 5: Solution of the particle model (1)-(3) (markers) with
$\Delta{x}=10^{-2}$ and of the first order non-local hydrodynamic model (14)
(solid line) at the computational time $t=1$ with $\Delta{x}=10^{-3}$ for the
following sets of parameters: (left) $a=\varepsilon=10^{-2}$, $\eta=10^{-1}$,
$N=10^{5}$; (right) $a=\varepsilon=10^{-3}$, $\eta=10^{-2}$, $N=10^{6}$

Now we investigate how the relaxation parameter $a$ affects the reliability of
the aggregate description provided by the non-local hydrodynamic model (14)
with respect to the original particle model (1)-(3). This is especially
important in practical situations where the scaling parameter $\varepsilon$
might not be taken arbitrarily small, such as e.g., in numerical simulations.
Although the limit model (14) is unaffected by the value of $a$, for finite
$\varepsilon>0$ one cannot rely on the exact limit condition (13) and has
instead to settle for an approximate condition of the form
$Q(f^{\varepsilon},f^{\varepsilon})\approx 0$, which implies (cf. (11))

$u^{\varepsilon}\approx\frac{\tilde{B}\ast(\rho^{\varepsilon}\pazocal{V}(\rho^{\varepsilon}))}{\tilde{B}\ast\rho^{\varepsilon}}.$

One may expect such an approximation to hold when $\varepsilon$ is
sufficiently small. In fact, the validity of the approximation does not depend
on $\varepsilon$ alone but on ratio $a/\varepsilon$. This becomes apparent
multiplying (8) by $v$ and integrating with respect to $v$, which, taking (11)
into account, yields

$\partial_{t}\int_{0}^{1}vf^{\varepsilon}\,dv+\partial_{x}\int_{0}^{1}v^{2}f^{\varepsilon}\,dv=\frac{a}{2\varepsilon}\rho^{\varepsilon}\left(\tilde{B}\ast(\rho^{\varepsilon}\pazocal{V}(\rho^{\varepsilon}))-(\tilde{B}\ast\rho^{\varepsilon})u^{\varepsilon}\right).$

If $a/\varepsilon$ is sufficiently large, i.e. $a\gg\varepsilon$, then the
left-hand side of this equation is formally negligible with respect to the
right-hand side, which makes the approximation above for $u^{\varepsilon}$
numerically acceptable. Conversely, if $a/\varepsilon$ is not large enough,
because either $a\sim\varepsilon$ or $a\ll\varepsilon$, then the left-hand
side of the previous equation is not actually negligible with respect to the
right-hand side. In this case, the approximation for $u^{\varepsilon}$ may not
be fully justified and the hydrodynamic description (14) may not provide a
reliable description of the particle dynamics (1)-(3) for the given value of
$\varepsilon$. Figure 5 illustrates two examples of disagreement between the
particle and the macroscopic solutions due to $a=\varepsilon$ for different
sets of the other parameters.

###  5.2 Follow-The-Leader dynamics

1 Fix $\Delta{t}=\frac{\varepsilon}{\max_{y\in[0,\,\eta]}B(y)}$;

2 Fix333In second order models, $\eta$ has to be small for the theory
developed in the previous sections to hold. For this reason, unlike Algorithm
1, in Algorithm 2 it is fair to take the mesh size $\Delta{x}$ directly equal
to $\eta$.$\Delta{x}=\eta$ and consider a space mesh
$\\{E_{h}\\}_{h\in\mathbb{Z}}$ of pairwise disjoint cells with
$\left|E_{h}\right|=\Delta{x}$ and $\cup_{h\in\mathbb{Z}}E_{h}=\mathbb{R}$;

3 for  _$n=0,\,1,\,2,\,\dots$_ do

4 for  _$h\in\mathbb{Z}$_ do

5 Find the particles belonging to the cell $E_{h}$. Let
$\mathscr{E}^{n}_{h}:=\\{i\in\\{1,\dots,N\\}:x^{n}_{i}\in E_{h}\\}$;

6 for  _$i\in\mathscr{E}^{n}_{h}$_ do

7 Sample randomly $j\in\mathscr{E}^{n}_{h}$;

8 Sample
$\Theta\sim\operatorname{Bernoulli}(B(x^{n}_{j}-x^{n}_{i})\frac{\Delta{t}}{\varepsilon})$,
$\Theta=\theta\in\\{0,\,1\\}$;

9 Update $v^{n}_{i}$ to $v^{n+1}_{i}$ according to (17):
$v^{n+1}_{i}=v^{n}_{i}+\theta\lambda(v^{n}_{j}-v^{n}_{i})$;

10

11

12 Update particle positions according to (17):
$x^{n+1}_{i}=x^{n}_{i}+v^{n+1}_{i}\Delta{t}$, $i\in\\{1,\dots,N\\}$;

13

Algorithm 2 Non-local Monte Carlo algorithm for the particle model (17)-(18)

We consider now the stochastic FTL particle model (17)-(18), to which there
corresponds the second order macroscopic ARZ model (28) at least in the regime
of small non-locality, cf. (20). Algorithm 2 reports the implementation of the
Monte Carlo scheme that we use in this case to solve the particle model
numerically.

Figure 6: Solution of the particle model (17)-(18) (markers) with $N=10^{6}$
particles and $\Delta{x}=10^{-2},$ and of the ARZ model (28) (solid line) at
the computational time $t=1$, $\Delta{x}=10^{-3}$ for $\eta=10^{-2}$ (small
support of the interaction kernel) and $\lambda=0.5$. Left column:
$\varepsilon=10^{-2}$; right column: $\varepsilon=10^{-3}$

Figure 6 shows that for a sufficiently small support of the interaction
kernel, in particular $\eta=10^{-2}$, both the traffic density $\rho$ and the
mean speed $u$ computed out of the particle model tend to be well reproduced
by the ARZ model when the scaling parameter $\varepsilon$ decreases from
$10^{-2}$ to $10^{-3}$. This is in agreement with the hydrodynamic limit
predicted by the theory.

Figure 7: Solution of the particle model (17)-(18) (markers) with $N=10^{6}$
particles and $\Delta{x}=10^{-2},$ and of the ARZ model (28) (solid line) at
the computational time $t=1$, $\Delta{x}=10^{-3}$ for $\eta=10^{-1}$ (large
support of the interaction kernel) and $\lambda=0.5$. Left column:
$\varepsilon=10^{-3}$; right column: $\varepsilon=10^{-4}$

Figure 7 shows instead that if the support of the interaction kernel is not
small, in this case $\eta=10^{-1}$, then there is no agreement between the
solutions of the particle and the ARZ models even for values of the scaling
parameter much smaller than before, in this case
$\varepsilon=10^{-3},\,10^{-4}$. In particular, we notice that the particle
solutions obtained with the two tested values of $\varepsilon$ are virtually
the same, thereby suggesting that the hydrodynamic limit has been
substantially reached numerically. The fact that such solutions do not
coincide with the one produced by the ARZ model proves that, in the regime of
large $\eta$, the ARZ model is not the macroscopic counterpart of the particle
system. Indeed, that the approximation (21), which the hydrodynamic limit
leading to the ARZ model is based on, is not valid in the considered regime.

Figure 8: Solution of the particle model (29)-(31) (markers) with $N=10^{6}$
particles and $\Delta{x}=10^{-2},$ and of the ARZ-like model (40) (solid line)
at the computational time $t=1$, $\Delta{x}=10^{-3}$ for $\eta=10^{-2}.$ Left
column: $\varepsilon=10^{-1},\,\lambda=10^{-1}$; middle column:
$\varepsilon=10^{-2},\,\lambda=10^{-1};$ right column:
$\varepsilon=10^{-2},\,\lambda=10$

Finally, Figure 8 shows a comparison between the solutions produced by the
generalised FTL particle model (29)-(31) with

$\pazocal{V}(s)=\frac{s}{1+s},\qquad\Psi_{\lambda}(s)=\frac{\lambda
s}{1+\lambda s}$ (42)

and the generalised ARZ model (40) in the regime $\eta\ll 1$. We solve the
particle model by an algorithm analogous to Algorithm 2 with due modifications
in the interaction rules, cf. (29)-(30), and the interaction kernel, cf. (31).
Furthermore, as initial condition for the headway $h$ we prescribe the
function

$h_{0}(x)=\begin{cases}0.5\quad x<0\\\ 0.6\quad x>0.\end{cases}$

Also in this case we observe (Figure 8 – left and middle columns) that,
consistently with the theory developed in Section 4, the particle solution
approaches the macroscopic solution as $\varepsilon$ decreases from $10^{-1}$
to $10^{-2}$, provided $\lambda$ is sufficiently small ($\lambda=10^{-1}$ in
this case). If instead $\lambda$ is not small enough (e.g., $\lambda=10$,
Figure 8 – right column) the particle solution may be affected consistently by
both the function $\Psi_{\lambda}$ used in the interactions (29) and the
cutoff (31), in such a way that the hydrodynamic limit (40) does not represent
accurately the actual macroscopic dynamics.

Figure 9: Comparison of the speed functions (41), (42) based on the
relationship $s=\frac{\ell_{0}}{\rho}$

######  Remark (Speed function comparison).

It may be instructive to compare qualitatively the speed function (42), say
$\pazocal{V}_{2}$, used in this numerical test with the one introduced in
(41), say $\pazocal{V}_{1}$. We observe that the two functions have a slightly
different meaning: (41) is an optimal speed towards which vehicle speeds relax
depending on the traffic congestion, whereas (42) is the actual speed of
vehicles expressed in terms of their headway. Nevertheless, invoking the
heuristic relationship $s\propto\frac{1}{\rho}$ we may rewrite (42) as

$\pazocal{V}_{2}(\rho)=\frac{\ell_{0}}{\rho+\ell_{0}},$

$\ell_{0}>0$ being the proportionality constant between $s$ and
$\frac{1}{\rho}$, which is commonly understood as the characteristic length of
a vehicle. Based on this expression of $\pazocal{V}_{2}$, Figure 9 shows that
the trend of the two speed functions is qualitatively the same, hence that
they are consistent with one another.

###  5.3 Speeds approaching zero

Figure 10: Left: initial condition. Right: solution of the particle model
(17)-(18) (markers) with $N=10^{6}$ particles, $\Delta{x}=10^{-2}$ and of the
ARZ model (28) (solid line) at the computational time $t=1$ with
$\Delta{x}=10^{-3}$ for $\eta=10^{-2}$, $\lambda=0.5$ and
$\varepsilon=10^{-2}$

Figure 11: Left: initial condition. Right: solution of the particle model
(29)-(30) (markers) with $N=10^{6}$ particles, $\Delta{x}=10^{-2}$ and of the
ARZ-like model (40) (solid line) at the computational time $t=1$ with
$\Delta{x}=10^{-3}$ for $\eta=10^{-2}$, $\lambda=0.5$ and
$\varepsilon=10^{-2}$

Finally, we reconsider the two second order models in the case of speeds
approaching zero, which for the generalised ARZ-like model means a mean
headway approaching zero. The precise initial conditions are plotted in the
left panels of Figures 10, 11. The right panels of the same figures show that,
at successive times, the traffic density increases rapidly close to the
initial discontinuity, because the speed/headway of vehicles switches suddenly
from a high to a low value. We observe that both models do not predict the
formation of a queue, i.e. a backward travelling density wave, but rather a
pointwise accumulation of the density. At the macroscopic level, this is due
to the well-known lack of maximum principle for second order models, which
implies no a priori upper bound on the vehicle density. However, from the
numerical simulations we infer that the macroscopic models are closely
reproducing, also in this case, the original particle trends in the proper
hydrodynamic regimes. Therefore, we conclude that such a pointwise
accumulation of the density is consistent with the (possibly generalised)
follow-the-leader dynamics in the regime of small non-locality of the
interactions, hence that it is not just an “analytical drawback” of the
macroscopic description.

##  6 Conclusions

In this paper, we have derived first and second order non-local traffic models
as physical limits of fundamental particle dynamics, such as optimal speed and
follow-the-leader dynamics. This is, in our view, a first result establishing
a structural link between microscopic vehicle dynamics widely recognised as
“first principles” of traffic and non-local macroscopic descriptions of the
flow of vehicles. It is worth stressing that our approach differs from other
approaches in the literature, which consider instead the convergence of ad-hoc
particle discretisations to macroscopic models in the many-particle limit.
What these approaches actually prove is, as a matter of fact, that selected
particle discretisations may be “numerically” consistent with the desired
macroscopic models, which remain postulated a priori.

The techniques we have used have their roots in the classical collisional
kinetic theory revisited in the light of the application to interacting multi-
agent systems. More specifically, we have considered hydrodynamic limits of
Povzner-type kinetic equations (possibly with cutoff), which are a natural
setting to describe non-local vehicle interactions at the mesoscopic scale.

On one hand, we have shown that vehicle interactions based on non-local
optimal speed dynamics are well described at the macroscopic scale by a scalar
conservation law with non-local flux, whose form however differs slightly from
the usual ones assumed in first order models postulated heuristically. On the
other hand, we have shown that the macroscopic Aw-Rascle-Zhang traffic model,
and possible generalisations of it, emerges as the prototypical hydrodynamic
limit of non-local follow-the-leader dynamics with arbitrary (but compactly
supported) interaction kernel, as long as the measure of the support of the
latter is sufficiently small. These results have also been fully supported by
numerical comparisons between the solutions of the particle models and those
of the corresponding hydrodynamic limits.

Further research in this direction may concern the rigorous statement of the
formal limits proposed in this paper, as well as the study of the hydrodynamic
limit of non-local follow-the-leader vehicle interactions in the case of an
interaction kernel with non-small support.

## Acknowledgements

This work was partially supported by the Italian Ministry for University and
Research (MUR) through the “Dipartimenti di Eccellenza” Programme (2018-2022),
Department of Mathematical Sciences “G. L. Lagrange”, Politecnico di Torino
(CUP: E11G18000350001) and through the PRIN 2017 project (No. 2017KKJP4X)
“Innovative numerical methods for evolutionary partial differential equations
and applications”.

F.A.C. and A.T. are members of GNFM (Gruppo Nazionale per la Fisica
Matematica) of INdAM (Istituto Nazionale di Alta Matematica), Italy.

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# Question Generation for Evaluating Cross-Dataset Shifts in Multi-modal
Grounding

Arjun R Akula
University of California, Los Angeles
<EMAIL_ADDRESS>
###### Abstract

Visual question answering (VQA) is the multi-modal task of answering natural
language questions about an input image. Through cross-dataset adaptation
methods, it is possible to transfer knowledge from a source dataset with
larger train samples to a target dataset where training set is limited.
Suppose a VQA model trained on one dataset train set fails in adapting to
another, it is hard to identify the underlying cause of domain mismatch as
there could exists a multitude of reasons such as image distribution mismatch
and question distribution mismatch. At UCLA, we are working on a VQG module
that facilitate in automatically generating OOD shifts that facilitates in
systematically evaluating cross-dataset adaptation capabilities of VQA models.

##  1 Background & Motivation

VQA is the challenging AI task of answering natural language questions about
an input image Antol et al. (2015); Anderson et al. (2018). Several datasets
have been proposed to measure the progress on this task such as VQA 2.0 Goyal
et al. (2017), VizWiz Gurari et al. (2018), Visual7w Zhu et al. (2016), GQA
Hudson and Manning (2019), to name a few. Recently, self-attention and multi-
modal pre-training based methods Tan and Bansal (2019); Lu et al. (2019); Cao
et al. (2020) demonstrated superior performance on these datasets. Despite
great progress, the state-of-the-art methods are found to be less effective
when the distribution of $\langle image,question\rangle$ pairs in testing set
are different from the training set Chao et al. (2018); Akula and Zhu (2022b);
Gardner et al. (2020), necessitating the importance of developing cross-
dataset adaptation methods Akula and Zhu (2019a); Carlson et al. (2003);
Soricut and Marcu (2003); LeThanh et al. (2004).

Through cross-dataset adaptation methods, it is possible to transfer knowledge
from a source dataset with larger train samples to a target dataset where
training set is limited. However, VQA datasets differ in the way they are
collected, making them significantly different in the distribution of input
visual and language features Chao et al. (2018). These multi-modal
distribution shifts make it difficult to measure adaptation capabilities and
has yet to be well-studied. For example, consider a domain adaptation setting
between VQA 2.0 Goyal et al. (2017) vs. VizWiz Gurari et al. (2018) datasets.

Suppose a VQA model trained on VQA 2.0 train set fails in adapting (e.g.
through fine-tuning) to Vizwiz. From this observation, it is difficult to
identify the real cause of domain mismatch as there could exists a multitude
of reasons such as (a) image distribution mismatch (e.g. VQA 2.0 consists of
high quality images compared to Vizwiz); (b) question distribution mismatch
(e.g. VQA 2.0 questions are less conversational than Vizwiz questions); (c)
insufficient sample size (e.g. VQA 2.0 consists of relatively large number of
training samples); and (d) a combination of image and question distribution
mismatches Akula and Zhu (2019a); Akula et al. (2020a); Akula and Zhu (2019b);
Akula et al. (2021c, d, b, 2020c); R Akula et al. (2019); Pulijala et al.
(2013); Gupta et al. (2012).

Figure 1: Examples from VQA 2.0 and VizWiz datasets. Existing works on domain
adaptation compare the VQA model’s performance on the test splits $\langle
I_{vqa},Q_{vqa}\rangle$ and $\langle I_{vzwz},Q_{vzwz}\rangle$. In this work,
we propose a Visual Question Generation module to generate additional tests
sets $\langle I_{vqa},Q_{vzwz}\rangle$ and $\langle I_{vzwz},Q_{vqa}\rangle$
that facilitate in disentangled, incremental, and fine-grained evaluation of
cross-dataset generalization.

##  2 Problem Definition

There exists two methods to generate cross-datasets shifts:

  1. 1.

Human Annotations: We can ask human annotators (eg. AMT turkers) to write
VizWiz style queries for VQA 2.0 images and VQA 2.0 style questions for VizWiz
images. Although, the quality of annotations will be high, this approach is
costly and cannot scale to multiple datasets Akula et al. (2013, 2018, 2021a);
Gupta et al. (2016); Akula et al. (2019b); Akula (2021); Akula et al. (2019a,
2020b).

  2. 2.

Visual Question Generation (VQG): In this work, instead of using human
annotators, we propose a VQG module that facilitate in automatically
generating OOD shifts for VQA datasets. This facilitates in systematically
evaluating cross-dataset adaptation capabilities of VQA models. Specifically,
using our VQG module, we generate additional test sets for source and target
datasets by controlling and disentangling distribution shifts in vision and
language features Akula and Zhu (2022a); Agarwal et al. (2018); Akula et al.
(2019c); Akula (2015); Palakurthi et al. (2015); Agarwal et al. (2017);
Dasgupta et al. (2014).

For example, as shown in Figure 1, using $\langle image,question\rangle$ pairs
from source and target test sets involve shift in both visual and language
features such as $\langle I_{vqa},Q_{vqa}\rangle$ to $\langle
I_{vzwz},Q_{vzwz}\rangle$. We augment these test sets to facilitate measuring
adaptation capabilities of VQA models on incremental (systematic) shifts such
as $\langle I_{vqa},Q_{vqa}\rangle$ to $\langle I_{vqa},Q_{vzwz}\rangle$ to
$\langle I_{vzwz},Q_{vzwz}\rangle$.

##  3 Summary of Contributions

Below we summarize our key contributions:

  1. 1.

Proposing and Implementing a Visual Question Generation (VQG) module for
generating questions and answers from the images.

  2. 2.

Using our proposed VQG, we generate OOD test splits for VQA 2.0 and VizWiz
datasets

  3. 3.

We show that our generated OOD splits help in quantifying the systematic
cross-dataset shifts in VQA models.

##  4 Approach

We will leverage state-of-the-art implementations to train an end-to-end VQG
model. Specifically, we use train sets of VQA 2.0 and VizWiz datasets and
train VQG model mapping input image to questions with an additional dataset
source indicator specifying the source of the sample. After we train our VQG,
during inference, we change the dataset indicator to generate cross-dataset
image and question pairs. For example, we pick VQA 2.0 image and provide the
dataset indicator as VizWiz, for generating VizWiz style questions on VQA 2.0
images. We will experiment with several contextual features (such as adding
bounding box annotations, pre-training on image captioning datasets, etc) to
control the quality of the generated questions.

##  5 Datasets and VQA Models

We experiment with VQA 2.0 and VizWiz datasets. We use ViLBERT Lu et al.
(2019), a pretrain-then-transfer approach, as the state-of-the-art VQA model
for our adaptation experiments.

##  6 Initial Experiments

We have first started selecting a state-of-the-art VQG model. We choose an
existing implementation based on mutual information maximization Krishna et
al. (2017). We incorporated the following additional cues to the input to
generate the cross-dataset splits:
(1) Source of the dataset (eg: VQA, VizWiz)
(2) First Three Words of Question: We found that VizWiz questions start with
unique words such as “Can you please” and “Please tell me”. So, we believe
providing the first three words of the question as additional guidance would
further help the model to understand the distribution style of questions that
we like to generate.
(3) We are currently working on integrating answer categories as additional
inputs to the module. The work by Krishna et al. (2017) proposed16 categories
on VQA dataset such as spatial, binary and count. We are leveraging these
categories to make the generated questions more diverse.

Training VQA models on ViLBERT: We have completed training VQA models using
ViLBERT architecture. This step took us more time as this takes up to 5 days
to train the model. Once we get decent questions generated using VQG, we will
immediately start our adaptation experiments using the VQA models trained
using ViLBERT.

##  7 Conclusion

We performed cross-dataset evaluation with VQA 2.0 and VizWiz datasets. To do
this, we proposed a Visual Question Generation (VQG) module for generating
questions and answers from the images.Our experiments demonstrate that our
generated OOD splits help in quantifying the systematic cross-dataset shifts
in VQA models.

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