The remarkable scaling theory for cities [
1,
2,
3,
4] proposed by West and coworkers indicates that urban transportation networks are analogous to biological networks inside organisms transporting the energy to survive. As such, urban networks exhibit economies of scale when it comes to infrastructure needs; e.g., doubling the population of a city requires only about 1.8 times the number of gas stations, lane-miles of road, etc. Unlike biological networks, however, the social interactions taking place in cities produces about 2.2 times the wages, number of patents, crime and, as we will see here, traffic congestion costs. This is not good news for traffic congestion going forward, but this paper gives new insights into these growth mechanisms to better understand how to curb it.
Urban networks are the perfect example of simple local rules producing what appears to be complicated, chaotic and unpredictable dynamics. Indeed, the rules of traffic flow are exceedingly simple: “Advance if you can", yet its consequences can be very frustrating for the average commuter. An explanation for these unpredictable dynamics has long been conjectured [
5] based on percolation phase transitions and self-organized criticality. They show that the lifetime of traffic jams can be arbitrarily large as they obey a power-law distribution near the critical density. Power laws are the hallmark of fractal objects and complex systems having phase transitions, and due to their infinite variance are responsible for the chaotic and unpredictable dynamics. Fig.
Figure 1 shows these power-law dynamics in action on a single lane of traffic. When the density of cars,
, is slightly larger than the critical density,
, the jam clusters, which contains only bumper-two-bumper vehicles, percolate upstream (in the opposite traffic direction) eventually spilling over to the rest of the network. But if the density is slightly smaller, it is the traffic voids that percolate downstream. This strong dependence on initial conditions is what makes prediction challenging. Notice the self-similarity of the different triangles in the figure, revealing the fractal nature of traffic. Recent studies reveal power-law distributions for the jam clusters in Fig.
Figure 1 [
6], the average platoon size and the average speed of vehicles [
7] and the relaxation to stationary state [
8].
But a traffic theory based on the analogy with phase transitions is still lacking [
9,
10,
11,
12]. Two decades ago in an excellent review of traffic flow models, Nagel
et al [
12] conjecture that such an analogy would set traffic flow theory on one of the firmer foundations in physics, offering a coherent description from the microscopic driver behavior to the macroscopic system dynamics. The only obstacle seems to be the interpretation of thermal energy. Quoting [
12]: “[...]
once one has identified the traffic model parameter that corresponds to temperature, then some good analogy between the gas-liquid transition and laminar-jammed transition seems to be possible. However, without identification of that parameter, characterization of models is difficult.”
The two-fluid model of Prigogine and Herman [
13], based on their kinetic theory of vehicular traffic [
14], was a first attempt in this direction. They interpreted the temperature as the speed of vehicles, but the analogy with phase transitions is only formal [
15], and it does not allow descriptions at different scales. They made the important observation, embraced in this paper, that at a fundamental level traffic flow has only two phases: a gas phase where vehicles are moving and a liquid phase where vehicles are stopped.
This paper shows that interpreting the thermal energy of the particle system as the flow of cars through the network makes it possible to formulate a theory in complete analogy with the gas-liquid equilibrium phase transitions. This analogy is described by a single parameter, the order-parameter critical exponent in the theory of phase transitions, which is found to have several important interpretations, each discussed in the following sections of this paper: as the maximum throughput (capacity) of the network, as the aggregation level, as the nonlinear-term constant in the KPZ equation, and as the normalizing constant for congestion delays.
1. The liquid-gas coexistence curve
Phase transitions are characterized by an order parameter, the number density
of molecules (cars) in our case, whose critical value,
, separates the ordered phase (liquid/congestion) and of the disordered phase (gas/free-flow). Near the critical density most quantities of interest obey a power-law relationship of the reduced temperature
, where the critical temperature
depends on the specific fluid. In thermal equilibrium, the coexistence curve:
bounds a compact region in the density-temperature plane where the phase transition occurs, i.e. where the proportion of matter in one phase increases gradually from 0 to 1. An example of this coexistence curve was observed empirically by Guggenheim in 1945 for 8 simple fluids [
16] as an example of the “principle of corresponding states” in chemistry. This principle can be seen as a precursor of the remarkable concept of universality classes in statistical mechanics, which groups physical systems sharing the same critical exponents. Simple fluids and ferromagnets have been known to obey
since the 1890’s [
17], and more recently they are said to belong to the Ising universality class [
18].
Fig.
Figure 2 illustrates the coexistence curve for 77 simple fluids. It exhibits a remarkable low scatter as a consequence of universality. Notice that (
1) is symmetric with respect to the critical density, but the coexistence curve is not. To circumvent this one can make the shape of coexistence curves mirror-symmetric with respect to the line
, by applying the shear mapping described in [
23] and summarized in the Appendix. This mapping is illustrated in Fig.
Figure 2, where it has been applied to the entire left panel of the figure to make the point. Notice the remarkable agreement with the data of (
1) using
over the
entire density range after the transformation (right panel). In practical terms this means that one may always assume the coexistence curve to be symmetric with
which is the assumption hereafter in the paper.
1.1. The fundamental diagram (FD) of traffic flow
The FD gives the equilibrium relationship between any pair of traffic variables, typically flow (or current)
f, density
and speed
v, describing the motion of cars in a particular one-dimensional link in the network. Here the normalized flow of traffic,
, is interpreted as being proportional to the temperature
in the coexistence curve (
1). The Appendix makes this interpretation more precise. To derive the equation of the FD, solve for
in (
1) to get
, or better still
to ensure zero flow for empty and jam-density conditions, i.e.
. Therefore:
where the prefactor
–which corresponds to the maximum flow or capacity of the facility–ensures that both the free-flow and congestion wave speeds are always unity regardless of the value of
, i.e.
; see Fig.
Figure 3. This makes
the only parameter needed to fully describe traffic flow in equilibrium.
1.2. The microscopic traffic model and different aggregation levels
When
the FD becomes an isosceles-triangle; see Fig.
Figure 3. Triangular diagrams embody the simplest
microscopic rules: “Travel at the free-flow speed
unless impeded by the downstream car”. These rules are the basis for the influential Nagel-Schreckenberg (NaSch) model [
25] which adds stochastic fluctuations by introducing the probability
of slowing down by one speed unit for moving vehicles. But because the shape of the triangular diagram is irrelevant thanks to the shear symmetry discussed above, one can choose
giving the isosceles triangle in Fig.
Figure 3 without loss of generality. This means that the celebrated Newell’s car-following model [
26], which can be formulated as a cellular automaton [
27], is equivalent to elementary cellular automata rule 184 [
28], which in turn is equivalent to the totally asymmetric simple exclusion process (TASEP), for which many exact results are known. All experimental results in the remainder of the paper use the NaSch model with
, and it will be simply referred to as the “traffic model”.
Vehicle trajectories, either real or generated by the traffic model, can be aggregated at different levels. [
24] derived the
lane-level FD based on results from TASEP for
, which gives the steady-state average flow on a single circular lane of length
L with a constant density of vehicles. They found that the capacity in this case is
, which means that at the link-level we have:
The dashed lines in Fig.
Figure 3 depict the lane-level FD assuming (
4). It can be seen that the agreement with the proposed FD is exact for
, and that it deteriorates for
except near the critical density. The Appendix shows that reasonable values for the slowdown probability should satisfy
, and we conclude from (
4) that
is a reasonable interval at the lane-level FD.
At the network-level, the macroscopic FD (MFD) [
29] gives the average flow versus average density, the average taken across all links in the network under steady-state conditions. It has received much attention during the last decade as an efficient way to control urban traffic [
30,
31,
32,
33]. The insets in Fig.
Figure 2(right) show empirical MFDs of vehicular traffic for different European cities, where it can be seen that the proposed FD (
3) provides a good fit with
, depending on the city. Keep in mind that this results have to be taken with a grain of salt because of unresolved challenges in the experimental observation of the MFD: traffic conditions are seldom in steady-state across the network, traffic counts are typically available from detectors only in a subset of the links, and detector location within the link can have an important impact in the flow-density estimations. The insets in the left panel of the figure shows empirical MFDs for pedestrian traffic, which are devoid of these challenges and therefore more reliable. Notice how the value of
is highly correlated with the directionality of pedestrian traffic.
2. Dynamic deterministic behavior
The description of traffic flow through the FD
is an equilibrium one, valid only when the system is in steady state. Dynamic behavior can be extracted by adding the conservation law of vehicles on the road:
, known as the kinematic wave or LWR model [
34,
35], an has played a central role in traffic science to this day. An alternative formulation of this model that turns out to scale nicely to the stochastic case in the next section, is achieved using Newell’s traffic flow surface [
36,
37],
, that gives the cumulative number of vehicles having crossed location
x by time
t, starting from the passage of a reference vehicle. Its gradients are given by densities and flows:
, which means that the very existence of the FD implies
a Hamilton-Jacoby partial differential equation (PDE), whose solutions can be found by solving the shortest path problem:
known as the Hopfs-Lax formula [
38]. In this variational formulation the FD corresponds to the Hamiltonian of the system, whose corresponding Lagrangian is denoted
[
39,
40]. The shortest path problem consists in finding a trajectory
from
to the initial (and/or boundary) data at
that minimizes the total cost along the path, given by the term in brackets. As shown next, the noisy version of this minimum path problem is at the heart of a wide variety of dynamical systems with the same scaling properties.
3. Stochastic nonequilibrium behavior
It is well-known that random fluctuations have important impacts on systems in one spatial dimension, such as the one studied here, and therefore mean-field approximations are bound to fail. Nonequilibrium physics has made important advances in the last two decades to understand these impacts by formalizing the existence of dynamical universality classes [
41,
42,
43]. The Gaussian universality class is the best-known due to the ubiquity of the central limit theorem. It is embodied in the Brownian motion stochastic process, which typically applies in non-interacting particle systems, where particle positions/states are independent of each other, and therefore normally distributed with fluctuations (i.e. standard deviation) growing as
. But most complex random systems do not follow fall into this universality class due to spatial correlations from the interactions among nearby particles. These systems are believed to be in the Kardar-Parisi-Zhang (KPZ) universality class [
44], where fluctuations grow only as
[
45], and obey, instead of the normal distribution and Brownian motions, a distribution/process that depends on the initial conditions; see
Table 1. This dependence comes from the number of possible candidate solutions to (a noisy version of) (
6), which varies depending on the initial condition, with the shortest path cost given by the Tracy-Widom distributions [
46]. These distributions [
47,
48] play a key role in this universality class, and appears in a wide variety of fields including random matrix theory, queuing theory, shortest path on random media, combinatorics, surface growth models, big data regression models and many others [
45,
49]. This speaks to the remarkable breath and depth of KPZ universality, with new members being added regularly. The KPZ equation is the stochastic PDE member of this class, which is typically formulated in terms of a surface,
, representing the height of the interface of growth processes:
where
is a surface tension responsible for the particle relaxation/diffusion at the surface,
controls the strength of the nonlinear term, and
is a Gaussian space-time white noise with zero mean and variance (or diffusion coefficient)
.
That traffic flow belongs to the KPZ universality class is a conjecture at the moment, and it is the purpose of this section to strengthen the case. This conjecture was proven in [
24] for the NaSch model with
and for all values of the slowdown probability
. They also showed this numerically for
, which we are in a position to prove here thanks to the shear mapping introduced in
Section 1, whereby the shape of a triangular diagram is irrelevant and so is the value of
. In addition, they showed by simulation that this conjecture also applies in multilane roads with lane changes, which gives a strong indication that this conjecture might be true in general.
An additional argument to support this conjecture is based on the existence of the FD,
, which implies (
5) in the deterministic case. As in the KPZ equation, one can add viscosity and random noise for generality:
A second-order Taylor series expansion of
f around a stationary density
does give the KPZ equation, but unfortunately this comes with the condition
; see [
24]. This condition might be too restrictive in the case of traffic flow since it is commonly found that
is either
or exhibits a singularity (as in our case (
3)) near the critical density, which is the main interest here.
It turns out that for the FD proposed here (
3) this restriction disappears if one defines
i.e. the fluctuations
are the deviations of Newell’s traffic surface with respect to the deterministic solution
in (
6). The Appendix shows these deterministic solutions for important initial conditions; for flat initial conditions, which represent cars flowing at the critical density uninterrupted by bottlenecks, we have
per (
A4). Substituting (
9) in (
8) gives:
which is exactly the KPZ equation (
7) with
for
. For other values of
we have a generalized nonlinearity
and
(, which reveals yet another interpretation of the critical exponent
). Fortunately, the exact type of nonlinearity is not crucial when it comes to KPZ universality [
44,
56]; see the Appendix. The Appendix also shows that (
10) follows for step initial conditions, which represent the flow out of bottlenecks, such as traffic signals. This means that both interrupted and uninterrupted traffic are part of the KPZ class.
To give a visual intuition why traffic flow belongs to the KPZ universality class, Fig.
Figure 4 shows the fluctuation surface
per (
9) produced by the NaSch model with Brownian motion initial conditions and different stopping probabilities
. Notice how the fluctuation magnitude decreases with time, as expected from KPZ where they behave as
instead of
in uncorrelated flows. In the stochastic case the surface resembles a natural landscape, in agreement with Newell [
37] who noted that the variational formulation of traffic flow is similar to soil erosion models, which also appear to be KPZ universal [
57].
4. Scaling of traffic congestion
An important consequence of KPZ universality is that, in the long run, important variables having units of time, scale with the total lane-miles of the network
as
, where the dynamical exponent
z equals 3/2 in KPZ. Therefore, one expects that for large times, large transportation networks will exhibit such scaling behavior for the travel delay as a consequence of congestion,
. In the case of our traffic model, the delay is simply the total size of jam clusters, see Fig.
Figure 1. Surprisingly, it appears that the scaling relationship
is valid in general, not only in the large-scale limit. It was recently shown in [
6] that even for a single one-lane road (where
) (
11) holds exactly for the deterministic traffic model with white noise initial conditions. This result is based on the correspondence between delay and the area under the Brownian motion corresponding to the initial conditions, which are known to scale as
and to obey the Airy distribution [
58,
59,
60]; see Fig.
Figure 4(top). In the case of the stochastic traffic model, Fig.
Figure 5 (left) shows that (
11) applies in the physically relevant case
. Notice that the different curves for different stopping probabilities
tend to overlap for
as shown in the inset, with
given by (
4), revealing yet another interpretation of this parameter as the natural units to measure delay.
Although empirical data on
and
to validate (
11) are not readily available, one can use the delay and population,
M, estimates from the Urban Mobility Report [
61] shown in Fig.
Figure 5 (right), by expressing (
11) in terms of the city population. The fractal properties of urban network topology imply
, where
is the fractal dimension of the transportation network and
A is the urbanized area of the city [
62,
63,
64]. Assuming that the urbanized area scales linearly with population, i.e.
as suggested by Fig.
Figure 5 (right), we get:
In a first-order approximation the value of
in this equation is the average network fractal dimension,
, across the 500 cities in the data set. This can be estimated using the relationship
found in [
63] for the Dallas Fort Worth area, which gives
and the slope
matches the data in Fig.
Figure 5 (right). It can be shown that the second-order approximation gives a slope of 1.27. Alternatively, we can use the network fractal dimensions reported in [
64] for the largest 95 metro areas in the United States, which gives first- and second-order approximations for the slope
of 1.11 and 1.14, respectively.
This last estimate is an excellent agreement with West’s scaling theory for cities mentioned in the introduction, which predicts a slope of 1.15 for most externalities of urban living including wages, city GDP, number of patents, and crime. Based on the results in this section, it appears that congestion delays also follow this superlinear scaling law possibly with a larger slope of around 1.2 as suggested by the data. Although not shown here for brevity, the same data [
61] reveals that the excess fuel consumed produces a chart almost identical to Fig.
Figure 5(right) with a slope also around 1.2. This is consistent with the definition of excess fuel being consumed during the delay component of travel time, and therefore should scale similarly.
5. Discussion and outlook
It would be interesting to see if the thermodynamics analogy proposed in this paper is sufficient to allow the theory of critical phenomena and phase transitions to be used in solving urban congestion problems, as conjectured by Nagel
et al [
12] two decades ago. This would be a major milestone not only for urban congestion control and planning, but for many related complex systems with power-law dynamics. Elsewhere we will show that heat corresponds to Newell’s traffic surface, temperature to network production, and entropy to travel time, and that the second law of thermodynamics in this case means that travel times tend to increase near the critical density. To complete the description, the potential energy should be taken as the remaining distance to the destination, where particles rest, possibly including departure and arrival time penalties.
While fluids in thermal equilibrium belong to the Ising universality class, traffic does not because of its anisotropic behavior. The directed percolation universality class appears as a strong candidate, considering that its value for
accords well with the empirical network-level FDs in Fig.
Figure 2. This is an agreement with recent empirical observations which also suggest direct percolation as the universality class of urban networks [
66]. Far from equilibrium, we concluded that traffic flow near the critical density belongs to the KPZ universality class, as a consequence of the nonlinearity of the FD. This result should also apply for supply chains given the mathematical similarities [
67], and extend naturally to other systems of conservation laws with power-law dynamics.
An immediate consequence of KPZ universality is the scaling of congestion costs of delays plus fuel consumption, revealing a superlinear scaling with respect to population that is in line with West’s scaling theory for cities. This is problematic as it indicates that doubling the population of a city will more-than-double congestion costs, i.e. by a factor of
according to West’s theory or
as shown here. The exponent
in (
11) with
for the KPZ class, indicates that more sustainable growth could be afforded by lowering the dynamical critical exponent
z and-or the network fractal dimension
. Research is needed to devise mechanisms to lower
z; other known universality classes arising from the interaction conservation laws is the Fibonacci family [
42], but their dynamical exponents are all
and tend to the Golden ratio
. Lowering the network fractal dimension
appears easier to achieve in practice, possibly at the expense of lower connectivity. This suggests that there may be an optimal level of connectivity that strikes a balance between mobility and sustainability, which remains to be explored. Traffic control is unlikely to alter the scaling relationship (
11), but it can help with network resilience: Because traffic flow is naturally driven to be in the critical state that maximizes throughput, it exhibits self-organized criticality [
68], where the system becomes more and more fragile in the process. As it reaches this state of maximum throughput, the entire system performance can collapse for no apparent reason due to the power-law dynamics taking place [
5,
6,
69]. Resiliency can be improved by designing control strategies that prevent reaching the critical state, but without significant impacts in delays. These are all important questions that will hopefully find answers soon to improve the livability of cities going forward.
Acknowledgments
This research was partially funded by NSF Awards #1932451, #1826003 and by the TOMNET University Transportation Center at Georgia Tech.
Appendix
Appendix A.1. Fluid data for Figure 2
The temperature-density data for the simple fluids in
Figure 2 were obtained using the function ThermodynamicData[] in Mathematica [
70]. The list of the 77 fluids in Fig. 2 is: Acetone, Ammonia, Argon, Benzene, Butane, Butene, Carbon Dioxide, Carbon Monoxide, Carbonyl Sulfide, Cis Butene, Cyclohexane, Cyclopropane, Decane, Dodecane, Ethane, Ethanol, Ethylene, Fluorine, Heptane, Hexane, Hydrogen Sulfide, Isobutene, Isohexane, Isopentane, Krypton, Methane, Methanol, Neon, Neopentane, Nitrogen, Nitrogen Trifluoride, Nitrous Oxide, Nonane, Octane, Oxygen, Pentane, Perfluoropentane, Propylene, R11, R113, R114, R116, R12, R123, R124, R125, R13, R134a, R14, R141b, R142b, R143a, R152a, R21, R218, R22, R227ea, R23, R236ea, R236fa, R245ca, R245fa, R32, R365mfc, R41, RC318, Sulfur Dioxide, Sulfur Hexafluoride, Toluene, Trans Butene, Trifluoroiodomethane, Water and Xenon.
Appendix A.2. The forgotten symmetry
The well-known Galilean transformation
expresses the invariance of the physical world to inertial frames of reference moving at a constant velocity
, and leaves a conservation law identical. A lesser-used linear symmetry that leaves conservation laws invariant is the shear (or skewing or transvection) mapping:
which can be interpreted as using asynchronous time, where clocks start running at each location according to the passage of an observer moving at a constant velocity
. In conjunction with
leaves also delays and total distance traveled unchanged; see [
23] for details. The significance of the shear mapping (
A1)-(
A2) is that one can make the shape of coexistence curves mirror-symmetric with respect to the line
by using
. This mirror-symmetry guarantees that the critical exponent is indeed the same at either sides of the critical point, an assumption typically made in the phase transitions literature, which now finds a solid justification. In physical chemistry the line of symmetry for coexistence curves (connecting the critical point
to
) is known as the diameter and has been used to improve the estimations of the critical point by assuming that it is in fact rectilinear [
71]. Although it appears that no connections with the shear mapping have been made, the benefits of having a symmetrical coexistence curve were clear to this community who devised different methods to approximate it [
72,
73].
Appendix A.3. Flow ∝ temperature ∝ heat flow in equilibrium
It is important to point out that interpreting the flow of vehicles as the temperature is only valid in equilibrium. More rigorously, the flow is interpreted here as the
heat flow, which gives the rate of heat transfer per unit time. By Fourier’s law of thermal conduction the heat flow is proportional to the temperature gradient in the system. But for simple homogenous systems in thermal equilibrium, the temperature gradient is, in turn, proportional to the temperature
alone [
74].
Appendix A.4. Why the slowdown probability should be p s <0.1
To see this, consider a single vehicle driving on an empty lane according to the NaSch model; i.e. at each time step it drives with a free-flow speed of 1 with probability or it stops with probability . It follows that the speed at each time step is a Bernoulli random variable with parameter , whose coefficient of variation is . Using the rule of thumb of keeping this coefficient under 30 percent gives and the claim follows.
Appendix A.5. Deterministic solutions for flat and step initial conditions
The fundamental diagram corresponds to the Hamiltonian of the system, whose Legendre-Fenchel transform gives the Lagrangian. For the proposed fundamental diagram (
3) the Lagrangian is given by:
where
is a wave speed and the sign % means ”same as in line above”. Since the Lagrangian is independent of space and time, the solution of (
6) is greatly simplified. The time integral in (
6) simplifies to
where
y is now a scalar satisfying
. Accordingly, to find the solution at point
amounts to finding the minimum straight-line path from
to the initial boundary at
, where the action (= cost of each path) is
.
Appendix Flat initial conditions
This case corresponds to constant initial density
, typically the critical density, so that
. The action is then given by
which is minimized at
where
if
and + otherwise. This gives
. With critical initial conditions
:
Step initial conditions
Here the initial density is a step function with discontinuity at
, with jam density to the left and zero density to the right, so that
. The action is then given by
which is minimized at
regardless of the value of
, and our solution becomes:
which is valid in
. Substituting (
9)-(
A8) in (
8) gives (
10) immediately for
, while for
one gets:
where
and
tend two zero as time tends to infinity, which gives (
10) as sought.
Appendix A.6. The type of KPZ nonlinearity is irrelevant
Here additional arguments are given to verify that KPZ universality is expected regardless of the value of
. Quoting the original paper [
44], “Higher-order terms can also be present, but they are irrelevant, and will not modify the universal scaling properties.” To see this in our particular case (
10), if
we saw in
Section 1.1 that there is no singularity at the critical density and therefore one may use a second-order Taylor series expansion and obtain the KPZ equation; see [
24]. The case
is more challenging due to the singularity at the critical density, but fortunately there are strong indications in the literature that KPZ behavior is to be expected. Recall that the NaSch model belongs to the KPZ universality class [
24]. As shown previously here the NaSch model implies
and therefore we conclude that this case also exhibits KPZ behavior. For
the literature on the viscous Hamilton–Jacobi equation
[
75,
76,
77] tells us that the large time behavior is dominated by the nonlinear term, which is a requirement of KPZ universality.
Appendix A.7. Data Availability
All empirical data used in this paper has been adequately referenced in the main text and is available through those references.
Appendix A.8. Code Availability
The Mathematica code to generate the data for
Figure 1, 3, 4 and 5 in this paper is available upon request.
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