Abstract
We investigate a microscopical structure in a chain of cars waiting at a red signal on signal-controlled crossroads. Presented is an one-dimensional space-continuous thermodynamical model leading to an excellent agreement with the data measured. Moreover, we demonstrate that an inter-vehicle spacing distribution disclosed in relevant traffic data agrees with the thermal-balance distribution of particles in the thermodynamical traffic gas (discussed in [1]) with a high inverse temperature (corresponding to a strong traffic congestion). Therefore, as we affirm, such a system of stationary cars can be understood as a specific state of the traffic sample operating inside a congested traffic stream.
Introduction
Investigation of various transport systems is recently one of the prominent subjects of physics. Intention of relevant researches is to describe such systems (or phenomena) quantitatively, create their appropriate models (theoretical or numerical), and finally obtain the exact or numerical outputs comparable to real situations. Higher aspiration of those scientific researches can be found in finding a certain connection among the different phenomena and revealing a possible universality.
Currently, one of the strongly accented fields is an investigation of queuing systems. Under the terms of that field it has been discussed many various topics, for example, wide-ranging spectrum of vehicular traffic problems [2] , pedestrian dynamics [3] , escape panic [4] , longitudinal parking of cars on a street [5] , [6] , parallel parking [7] , [8] , or public transport in some Latin America countries [9] , [10] . All those subjects are in close connection with the Random Matrix Theory, theory of chaos, or theory of particle gases (see the references cited above). The main goal of this paper is to extend the set of queuing systems mentioned above by a stationary ensemble of cars waiting at a red signal on signal-controlled crossroads (see also [11] ).
Moreover, we are aiming to create an one-dimensional model of point-like vehicles producing the same inter-vehicle gap distributions as those detected among cars standing on signal-controlled intersections. In the second part of this article we demonstrate that such a model can be interpreted (on a microscopical level) as a thermodynamical gas of dimensionless particles exposed to a thermal bath. This analogy allows, as we assert, to find an exact form of relevant spacing distribution which can be consequently compared to the realistic gap-statistics.
Describing the system
The traffic data analyzed in this work were measured during a few days on a multi-lane intersection located near center of Prague. This intersection is a constituent of an extensive network of roads and crossroads inside the internal metropolis subplot depicts the initial state of the numerical scheme described in text whereas the bottom subplot demonstrates the final stationary state of the traffic sample, i.e the state when the cars are waiting for green signal. We note that the squares represent the model particles with the leading car being picked out.
and is therefore strongly saturated during the whole daytime practically. Furthermore, the time interval between two green signals (on one crossroad) is very short, which causes that some cars are not able to reach the threshold of following intersection (during one green phase) and have to wait therefore for another green light. This fact finally leads to a substantial decrease of average velocity of vehicles moving between crossroads, i.e one can observe the effects usually detected in congested traffic regime (see Ref. [2] ). Measured were bumper-to-bumper distances r i between subsequent cars ((i + 1)th and ith ones) waiting at a red signal (in one direction only). Data file contains 5022 digitally gauged events showing the mean inter-vehicle gap equal approximately to 149 centimeters. We add that the clearances were measured directly using the laser technology.
More detailed statistical analysis uncovers that a probability density p(r) for distance r between neighboring cars shows a similar behavior as that investigated between the eigenvalues of random matrices (see [12] ), zeros of Riemann zeta function (see [13] ), or vehicles moving inside the traffic stream on the freeways (see [1] ). Such a behavior (follow Fig. 2 ) demonstrates the presence of repulsive interactions among the elements in question. As well known, a spacing distribution of non-interacting elements shows a different distribution, in concrete: Poisson probability density
(r ≥ 0).
Since the traffic interaction (in local sense, of course) is usually quantified as power-law repulsion among the successive vehicles (see Ref. [1] and [14] ) let us suppose that a potential energy of the ensemble investigated reads as
Herein we assume that the stationary traffic state analyzed in this paper (i.e. the queue of waiting cars) is determined by the preceding process -traffic flow towards the intersection. Evidently, moving in traffic sample the driver is interacting with other cars and optimizing his/her motion to reach the threshold of the crossroad as soon as possible and, at the same time, avoid a crash with the preceding vehicle. Such a behavior corresponds to the thermodynamic effects governing the ensemble into a local thermal equilibrium (see [1] for details).
Modified Metropolis algorithm
Accepting the above-mentioned assumptions on thermodynamical aspects of the issue we formulate the following onedimensional traffic model based on principles of statistical physics. Consider N + 1 point-like particles (cars) located randomly (or equidistantly if advantageous) on a line (or on a circle) so that the mean gap among them is one, i.e.
where r i represents the gap between (i + 1)th and ith particles. Thus, the ordered positions
constitute the initial state for our simulation (see Fig. 1 ). The particles move along the line (or along the circle) accepting the undermentioned rules until the leading car reaches a fixed point (the threshold of new crossroads). In accord to a . Bars represent the probability density for bumper-to-bumper distance among the cars waiting at a red signal on intersections (measured in Prague). Data were re-scaled so that the mean spacing is equal to one. Points display the optimized result of the numerical scheme (Metropolis algorithm) for N = 100 and β model = 1.45. Finally, the curve displays the distribution (3) for the fitted value β f it ≈ 1.2488 (obtained by the number variance test.) realistic situation the overtaking cars are not permitted, i.e. the particles can not change their order. Let β model ≥ 0 denote the inverse temperature specifying the measure of chaos inside the ensemble simulated. We assume β model to be the only significant parameter of the model. The car positions are repeatedly updated (we use 20000 steps in our version) according to the following rules:
1. Calculated is the potential energy U 0 (using formula (1)) for the actual set of locations {x 1 , x 2 , . . . , x N +1 }.
2. We pick an index j ∈ {1, 2, . . . , N + thomastran@example.com.
3. We draw a random number δ equally distributed in the interval (0, 1). 
We calculate a value of potential energy
6. If U ′ ≤ U 0 the jth particle position take on a new value x
where ∆U = U ′ − U 0 , should be compared with a random number r equally distributed in (0, 1). Provided that the inequality w > r is fulfilled the jth particle position takes on the new value x ′ j too. Otherwise, the original configuration {x 1 , x 2 , . . . , x N +1 } remains unchanged.
The sketched procedure represents a modified Metropolis algorithm originally developed for chemistry purposes (in Ref. [15] ). This algorithm belongs to the category of Monte Carlo simulations (see Ref. [17] ) which are recently used for numerical modelling of statistical systems (as demonstrated in Ref. [16] , for example). The elaborated scheme of Metropolis ensures a relaxation of ensemble into a thermal-balance state when the energy fluctuates around a constant value being independent of initial configuration of particles (see Fig. 3 ). After reaching the thermal equilibrium (i.e. after approximately 5000 updates of configuration (Monte Carlo steps), as visible in Fig. 3 ) the ensemble lingers in this state until the simulation is interrupted. Then, as observed, corresponding probability density for inter-particle gaps depends on the inverse temperature β model only.
Our aim is to find the optimal value of inverse temperature β model so that the gap distribution p(r) corresponds to that measured among the cars on crossroads. Using a χ 2 −method (i.e. minimizing the sum of squares-deviations between two distributions in question) one can find that optimal value β model is approximately 1.45. Concretely, for fixed value β model the distribution p(r) is obtained. Then the χ 2 −test between empirical data and p(r) could be evaluated. The optimal value of β model is the one for which the corresponding sum of squares-deviations is minimal. To conclude, for value β model = 1.45 both processes (traffic and Metropolis procedure) generate practically the same gap distributions (see Fig.  2 ). Thus, the introduced procedure could represent a realistic model for behavior of the cars in the vicinity of the chosen intersection. (1) for N = 100 and β model = 1.45 during the run of Metropolis procedure (having 20000 steps) for random (or equidistant) initial locations of elements, respectively. Plotted is the average value (calculated for 100 repeated realizations of Metropolis). Grey curve represents the energy value (1) for one realization of Metropolis (when initial particle positions were chosen equidistantly).
Terminal state of thermodynamical traffic gas
As explored in articles [14] , [1] , and [18] , the traffic flow can be understood (on a microscopical level) as a thermodynamical gas of interacting cars exposed to a heat bath of inverse temperature β. Besides, the latter has an immediate relation to the traffic density. If accepting such a approach we describe the traffic ensemble (on the move) as a circular gas of point-like particles whose hamiltonian reads as
where v i and r i represent a ith car velocity and gap to the previous car, respectively. Quantity v denotes the desired velocity of the ensemble. Then (see the exact calculation in [1] ) the derived probability density p β (r) for a gap r among the successive vehicles is p β (r) = A exp βr
where the constants A and B are calculated via two normalization equations
According to Ref. [1] the following relations hold true:
Herein K 1 stands for a Mac-Donald's function (modified Bessel's function of the second kind) of the first order.
Since the situation investigated in this article is without any doubt the result of a preceding traffic flow (see [11] ) it is meaningful to expect that the clearance distribution among the cars waiting at the red-light-signal will be of the form (3). Indeed, as confirmed by an appropriate statistical analysis of the collected data (discussed later) the measured gap statistics (see Fig. 2 ) corresponds to the probability density (3) if the inverse temperature β of the thermodynamical model is β f it ≈ 1.2488.
We denote that this value has been determined by a more sophisticated method presented in next section. In addition, a positive comparison between the corresponding gap distributions supports the hypothesis that traffic stream can be locally understood as a stochastic gas whose elements are repulsed by the forwardly-directed nearest-neighbor power-law potential depending on a reciprocal distance between successive gas-elements. This correspondence, however, does not mean that traffic is a thermodynamical system, of course. 
Testing the statistical variance of data
If trying to find a more robust argumentation for an assertion on statistical similarities between the process investigated and the traffic model we can apply some of the techniques originally developed for purposes of the Random Matrix Theory (see the book [12] ). Usual way how to quantify the behavior of variances among the statistical data is in applying so-called number-variance test. Such a test is defined as follows.
Consider N spacings r 1 , r 2 , . . . , r N between the successive vehicles (or particles of model) and suppose that the mean distance taken over the complete ensemble is re-scaled to one, i.e. 
where the integer part ⌊N/L⌋ stands for the number of all subintervals included in the interval
and represents (in a traffic instance) the statistical variance in the number of vehicles operating at the same time inside a fixed part of the road of a length L.
As well known from Random Matrix Theory the number variance can be explicitly derived from the relevant spacing density p β (r). The significant advantage is remarkable sensitivity of the number variance ∆ n (L) to any change in the probability density p β (r) -i.e. to any change in the potential U (r 1 , r 2 , . . . , r n ) also. Whereas the number variance of independent events (or non-interacting elements) is the identity ∆ n (L) = L, for a thermodynamical traffic gas with nonzero inverse temperature β there has been numerically calculated (in Ref. [18] ) a different behavior, concretely: a linear dependence
with a slope χ ≈ 1 2.4360 β 0.8207 + 1 ≤ 1
and a shift γ ≈ β 5.1926 β + 2.3929 ≥ 0.
As understandable now, the comparison between the number variance of the collected data and the function (5) can be then used (together with the comparison of the relevant gap distributions) as a robust fitting procedure which is capable of revealing more detailed nuances among the distributions compared. If applied to our topic, such a procedure generates the optimal value (4) for which the exactly determined number variance (5) corresponds to the measured data (see Fig.  4 ). Note that both of curves ∆ n (L) are rapidly deflected from the line visualizing the number variance of non-interacting particles. It implies the presence of a strong repulsion among the vehicles. However, a small deviation is detected for larger L between the traffic data (plus signs in Fig. 4 ) and Metropolis data (points in the same figure) . Such a discrepancy can be explained by the simple fact that the respective temperatures (i.e. β model and β f it ) differ each from other.
Summary and discussion
Investigated was the traffic ensemble of vehicles waiting at a red-light-signal on signal-controlled crossroads. We have introduced the thermal space-continuous time-discrete traffic model of repulsing point-like elements based on the Metropolis algorithm. By the suitable choice of the inverse temperature parameter there were obtained the same statistical distributions as those produced by the real traffic process. Above that, we show that the investigated state of the realistic traffic sample can be predicted with the help of the thermal-equilibrium state for local thermodynamical gas whose point-like particles are repulsed by the short-range power-law potential (1). As demonstrated above, for the fitted value β ≈ 1.25 of reciprocal temperature the corresponding spacing distributions are practically the same. The correspondence between the traffic samples and presented theory is, moreover, supported by the robust test of number variance which reveals 1. the thermal feature of the topic -on microscopic scale, 2. the presence of strong interactions among the cars, 3. a deep connection between the stationary state of waiting cars and the preceding move of sample towards the intersection threshold.
To conclude, we assert that the configuration of vehicles waiting at a red-light-signal on signal-controlled crossroads is a product of local thermodynamics-like processes acting among the cars. All the accessible statistical analyses strongly support this fact. Therefore, the observed phenomenon can be understood as a traffic in especial super-congested state.

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