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The authors have declared that no competing interests exist.

Car traffic in urban systems has been studied intensely in past decades but models are either limited to a specific aspect of traffic or applied to a specific region. Despite the importance and urgency of the problem we have a poor theoretical understanding of the parameters controlling urban car use and congestion. Here, we combine economical and transport ingredients into a statistical physics approach and propose a generic model that predicts for different cities the share of car drivers, the _{2} emitted by cars and the average commuting time. We confirm these analytical predictions on 25 major urban areas in the world, and our results suggest that urban density is not the most relevant variable controlling car-related quantities but rather are the city’s area size and the density of public transport. Mitigating the traffic (and its effect such as _{2} emissions) can then be obtained by reducing the urbanized area size or, more realistically, by improving either the public transport density or its access. In particular, increasing the population density is a good idea only if it also increases the fraction of individuals having access to public transport.

As most humans now live in urban areas and two-third of the world population will live in cities by 2050 [

On the other hand, car traffic has been studied at various granularities—and mostly at the theoretical level of a single lane—with various tools ranging from agent-based modelling (see for example [

According to the classical urban ecomomics model of Fujita and Ogawa [_{R}(

In order to discuss commuting costs, we assume that a proportion _{car} and _{MRT} in order to choose the less costly transportation mode.

(a) For a given agent located at a random location _{0} with probability 1 − _{0} = 1km). In this case the journey to work located at the central business district (CBD) is made by car (dashed line). (b) With probability _{car} of car (dashed line) and the cost _{MRT} of MRT (the trip is depicted by the red line) in order to choose the less costly transportation mode to go to the CBD.

The existence of a single central business district, the location of homes, and the density of MRT are considered as exogenous variables. The important endogenous variable is here the share of car users and the time spent in traffic jams (allowing then to estimate CO2 emissions). All the assumptions used in this model are of course approximations to the reality but we claim here that our model captures the essence of the traffic in large urban areas phenomenon. Starting with a model containing all these various parameters would actually be not tractable and would hide the critical ingredients.

In order to define the model completely we have to specify the expressions for the generalized costs. We will omit all other forms of commuting (walking, cycling, etc.), and we neglect spatial correlations between the densities of public transport and residence or population, which is an important assumption that certainly needs to be refined in future studies. The fraction

For cars, we include congestion described by the Bureau of Public Roads function (see [_{c} is the daily cost of a car, _{c} and _{m} are respectively the car and MRT velocities, _{c} > _{m}) but is more expensive. Once an individual has chosen a mode he sticks to it and will not reconsider his choice even if the traffic evolves. In other words, we assume here that individual habits have a longer time scale than traffic dynamics.

Individual mobility is then governed by comparing these costs _{car} and _{MRT} and will depend on exogenous parameters such as car and subway velocities, car costs, etc. In the general mode choice theory (see for example [_{car} and _{MRT} there is a probability _{C} = _{car} − _{MRT}) to choose the car. The function _{car} < _{MRT} which implies a condition on the value of time of the form _{m}(_{m}(_{car} = _{MRT} between generalized costs of car and MRT leads to the critical distance given by

The limit between the two areas evolves with congestion: the larger the traffic (curves from blue to green) and the larger the area in which rapid transit is beneficial compared to car driving. The grey solid vertical line corresponds to the size of the urban area and indicates the critical value of time (dashed red line) below which rapid transit is advantageous in the whole agglomeration whatever the value of congestion. The values of the parameters are chosen here as: _{c} = 15 $, _{c} = 40 km/h, _{m} = 30 km/h,

Writing

When population increases, car traffic has essentially two sources: first, individuals may not have access to the MRT and second, if they have access to it, they might be too far and will prefer to take the car. This can be summarized by the following differential equation for the variation of car traffic T when P varies

We have to plug in this expression together with _{0}exp(−_{0}) and we then obtain at dominant order in _{c}/_{m} − 1/_{c}). We note here that in the uniform density case _{0}, we obtain ^{2}/

This result

Compiling data from 25 megacities in the world (see

The red line is the prediction of our model (^{2} = 0.69). Given the absence of any tunable parameter the agreement is satisfactorily, and discrepancies are probably mostly due to the existence of other modes of transport (walking or cycling), lower car ownership rates, or a higher cost of the MRT, etc.

Our model also provides a prediction for the transport-related gas emissions and we will focus on the _{2} case for which we obtained data. We make the simplest assumption where these emissions are proportional to the total time spent on roads. The quantity of _{2} emitted for a driver residing at _{i} _{i} is of the form _{2} emitted by car and per capita is given by
_{2} emissions on _{2} emissions and areas such as New York which appears to be one of the largest transport CO_{2} emitter in the world [_{2} emissions. Most importantly, we identify urban sprawl (_{2} emissions. Mitigating the traffic is therefore not obtained by increasing the density but by reducing the area size and improving the public transport density. Increasing the population at fixed area would increase the emission of _{2} (due to an increase of traffic congestion leading to an increase of _{2} emissions.

The red line is the linear fit of the predicted form _{2}tons/km/hab/year (the Pearson coefficient is 0.79). We had no congestion estimate for Seoul and Tokyo and we used an average congestion rate

We also note that

Finally, we can also estime the average commuting time (details are given in _{c} averaged over the population the following expression
_{0}(1 −

We presented a parsimonious and generic model for the car traffic and its consequences in large cities, and tested it against empirical data for 25 large cities in the world. This approach illustrates how a combination of statistical physics, economical ingredients and empirical validation can lead to a robust understanding of systems as complex as cities. In particular, this approach is in contrast with the commonly accepted idea that urban density is pivotal, and our aim here was to capture the essence of the urban mobility phenomenon and to obtain analytical results for the car traffic and the quantity of emitted _{2}. Our analysis shows that traffic related quantities are governed by three factors: access to mass rapid transit, congestion effects and the urban area size. In order to reduce _{2} emissions for example, our model suggests to increase public transport access either by increasing the density around MRT stations or to increase the density of public transport (in contrast with the conclusions of an econometric study in the US [_{2} emissions. Finally, we insist on the fact that in this model we voluntarily left out a number of parameters such as other transportation modes (buses, tramway), polycentrism, the transport network structure, fuel price and tax, dynamic road pricing, etc. but our main point was to fill a gap for understanding traffic in urban areas by proposing a parsimonious model with the smallest number of parameters and the largest number of predictions in agreement with data. Given the simplicity of this model we cannot expect a perfect agreement with data for various and different cities, but it seems that this approach captures correctly all the trends and identifies correctly the critical factors for car traffic. This seems to be a basic requirement before adding other factors and increasing the complexity of the model. Also, it seems at this stage necessary to first encourage the measure and sharing of data such as the density of public transport in order to propose further tests of our theoretical framework.

We studied 25 metropolitan areas from Europe, America, Asia and Australia. The number of cities was limited by the availability of data on MRT accessibility and reliable modal share estimates. All the data is freely available and we list here their sources. We also provide a file together with this submission with all the data used in this study (see

The definition of a metropolitan area varies from one country to another but we aimed at assembling a statistically coherent set of agglomerations. Populations and areas were collated from Wikipedia data on metropolitan statistical areas in accordance with national definitions.

Various indicators were compiled from diverse sources: the quantity _{2} emissions from [

Values of time were assessed by taking half of the hourly wage after tax for each city [

Congestion delays were taken from the TomTom index [

The road capacity was computed from the congestion delay ^{1/μ} with the value

For the velocities _{m} and _{c} and the costs _{c} and _{m} wikipedia data [_{m} ≈ 30km/h. The free flow car velocity _{c} also depends a bit on the city and varies from 30 km/h (without congestion effect) in European cities such as Paris to 56 km/h in some american cities [_{c} ≈ 40 km/h.

For _{c}, we used a cost simulator [_{c} ≈ 15 USD per trip.

For

The commuting time for MRT users is given by _{c} averaged over the population the following expression
_{0}(1 − _{eff}’ for each city and compare it to the average value _{eff} (and the ratio _{eff}/

We perform the fit on the single parameter

This file contains all data used in this paper (in a format.xlsx).

(XLSX)

VV thanks the IPhT for support during his internship.