Traffic Models and Traffic-Jam Transition in Quantum ($ N $+ 1)-Level Systems

Traﬃc models and traﬃc-jam transition in quantum (N+1)-level systems

Andrea Nava, Domenico Giuliano, Alessandro Papa, and Marco Rossi Dipartimento di Fisica, Universitàdella Calabria, Arcavacata di Rende I-87036, Cosenza, Italy INFN - Gruppo collegato di Cosenza, Arcavacata di Rende I-87036, Cosenza, Italy (Dated: April 14, 2021) We propose a model to implement and simulate different traffic-flow conditions in terms of quantum graphs hosting an (N+1)-level dot at each site, which allows us to keep track of the type and of the destination of each vehicle. By implementing proper Lindbladian local dissipators, we derive the master equations that describe the traffic flow in our system. To show the versatility and the reliability of our technique, we employ it to model different types of traffic flow (the symmetric three-way roundabout and the three-road intersection). Eventually, we successfully compare our predictions with results from classical models.

I. INTRODUCTION one implements the analogy with quantum mechanical systems by introducing an effective Hamiltonian Hˆ , as well as vehicles creation and annihilation operators, sim- In the last decades the traffic behavior and the as- ilar to creation and annihilation operators in quantum sociated traffic-jam transition have been widely studied mechanics. Yet, the expectation value of an observable within the master equation formalism for stochastic ex- Oˆ for a system in the state |ψi has not to be com- clusion processes of many-body systems, using models puted as hψ| Oˆ |ψi, but with respect to the “sum-vector” such as, for instance, the asymmetric exclusion process P (ASEP) as well as the total asymmetric exclusion process hs| = hn| with |ni being a basis of the Fock space, i.e., ˆ (TASEP)1–4. All these models rely on the possibility to it is defined as hs| O |ψi. simulate the dynamical behavior of the system at a coarse Despite its potentially wide applicability, the formal- graining scale with vehicles living on a lattice with many- ism based on the effective Hamiltonian Hˆ is not easy to body interactions, rather than considering a continuum implement, since it is not easy to compute the operator of possible states (representing vehicles position and ve- expectation values as described above. For this reason locity). Therefore, the evolution of the system is not in the last years a fully quantum formulation in terms expressed in terms of deterministic equations of motion, of a Markovian Lindblad master equation description for but by means of a set of stochastic rules that describe all density matrix has been introduced14,15, which allows for the different paths that a system can follow starting from expressing the stochastic classical rules in terms of Lind- a given initial state5,6. In fact, despite the traffic flow is a blad jump operators that ensure (vehicle) particle conser- classic problem and, therefore, it might seem appropriate vation exclusion processes by allowing, at the same time, to describe it by resorting to appropriate hydrodynamic for an easy implementation of open boundary conditions. models, the finite dimensions of vehicles introduce a nat- Furthermore, the approach allows to treat at the same ural spatial “quantization” of space. Indeed, this concept level classical incoherent evolution and pure quantum co- has been widely used in cellular-automaton descriptions herent evolution opening the possibility to use quantum of the traffic-flow problem, where both space and time are dot systems as experimental devices to simulate classical discrete5–7 (incidentally, it is worth stressing how a sim- traffic behavior16,17. ilar “quantization” of both space and time, followed by Remarkably, while a huge amount of work has been a systematic implementation of cellular-automaton ap- already done for traffic description in one-dimensional proach, has proved to be pretty effective in, e.g., the systems18–23, in the last years a wide interest is arising 8,9 analysis of the infection spread in real space ). around the possibility to describe different kinds of inter- In Ref.[10], the stochastic rules have been encoded sections24–26, searching for optimization procedures27–30. in the transition probabilities for elementary evolutions Along this line of research, in this paper we analyze from one state to another. Moreover, they have been graphs with multiple in/out sites in terms of multilevel shown to be equivalent to the consolidated cellular au- quantum dots. In particular, we are interested in the in- tomaton approach of Biham and Nagatani7,11,12. Of par- tersection between three possible routes where the exit arXiv:2104.06289v1 [cond-mat.stat-mech] 13 Apr 2021 ticular relevance is the fact that the rules can be con- direction of each vehicle is not stochastically chosen dur- verted into a set of linear operators acting on the vec- ing the hopping process, but it is rather an internal prop- tors of a proper Fock space, taking advantage of the for- erty of the vehicle. mal analogy between classical stochastic processes and Specifically, we discuss in detail three-road junctions, quantum mechanical formalism. The Fock space descrip- which we model in terms of a quantum system on a graph, tion has been developed in terms of both Bosonic and realized by means of multilevel quantum dots coupled to Fermionic operators on periodic and open systems (see, external reservoirs. Within our formalism, we use quan- for instance, Ref. [13] for a review on the subject). tum hopping operators between quantum dots to repre- Within the Fock space formulation of the problem, sent the flux of vehicles in the junction. At variance, 2 to account for vehicles entering the junction from out- the other N states describe different vehicle types and side/exiting from the junction, we introduce hopping op- destinations. Assuming m different vehicle types and n erators from/to external reservoirs to/from the internal different destinations, we have N = m × n ’occupied’ dots. Making use of quantum Lindblad master equation states, which we label as |ji = |(v − 1) n + di with inte- formalism we recover results in agreement with classical gers v ∈ {1, . . . , m} and d ∈ {1, . . . , n}. Within our nota- experiments. In doing so, we develop and discuss a three- tion, the levels |ji, with j = 1, ..., n, represent a vehicle of dot model, which encodes the general information about type 1 directed towards one of the n possible destinations, the intersection of three routes, and eventually generalize the levels |ji, with j = n + 1,..., 2n represent a vehicle it to a six-dot model. This allows us to implement differ- of type 2 directed towards one of the n possible destina- ent priority rules such as a roundabout, a right-hand pri- tions, and so on. On each site we introduce the operators: ority junction, and an intersection between a major and σj,0 = |ji h0| (σ0,j = |0i hj|), that create (destroy) a given 0 a minor road. Remarkably, despite the “minimal” setup vehicle-destination combination, and σj0,j = |j i hj|, with we employ, our model exhibits all the relevant features of (v −1)n+1 ≤ j, j0 ≤ vn, that describe a vehicle of type v a classical traffic-flow diagram, such as the existence of a changing its destination. These operators satisfy the con- N critical density of vehicles beyond which the traffic-jam † P 7,11,12 ditions σi,jσk,λ = δj,kσi,λ, σi,j = σj,i and σj,j = 1. phase transition sets in . j=0 Moving backwards along the correspondence with As a consequence the dot can only be empty or occu- quantum lattice systems, our model can be re- pied by a single vehicle/destination combination as it is garded as a “minimal”, pertinently adapted, version impossible to create more than a vehicle on the same site. 31,32 of the Y-junction of one-dimensional fermionic We can now build up the Hilbert space for a complex 33–35 and/or bosonic quantum systems, including spin network of L multi-level dots as the tensor product of the 36–39 chains , as well as junctions hosting the remarkable Hilbert spaces of each dot. It is more convenient, as we 40–42 “topological” realization of Kondo effect , which have will show later, to let N be site dependent. Therefore, we been largely addressed in the recent literature on corre- L Q lated quantum systems as simplest examples of devices have, in general, (N` +1) basis vectors in the enlarged exhibiting nontrivial phases/quantum phase transitions `=1 space, |j1, . . . , jLi. They are defined as the Kronecker in their phase diagram. tensor product of the basis vectors of each dot The paper is organized as follows:

|j1, . . . , jLi = |j1i ⊗ ... ⊗ |jLi . (1) • In Section II we present our multi-directional model, in which we represent vehicles as hardcore (`) In the same way, we label as σi,j the creation, annihi- bosons on a graph. Consistently with our identifi- lation and conversion operators acting on a given dot `: cation, we define the creation and annihilation op- these are realized as the tensor product of (L−1) identity erators for vehicles and write the Markovian Lind- matrices of dimension N` × N` and the σi,j operator at blad equation governing the time evolution. the site `, that is In Section III we present a minimal three-dot model • σ(`) = ⊗ ... ⊗ σ ⊗ ... ⊗ . (2) that allows us to address the main features of the i,j I i,j I three-route intersection. For example, the state |1, 0, 2i represents a graph of three • In Section IV we extend the analysis of Section III dots where we have an empty road on site two, while by introducing a six-dot model that allows us to sites one and three are filled by two different vehicles implement different priority rules. (or same vehicle type with different destinations); the (1) operator σ0,j = σ0,j ⊗ I ⊗ I destroys the vehicle j on site • In Section V we summarize our results and discuss 1, and so on. possible perspectives of our work. Having built an appropriate Fock space with intrin- sic exclusion and multi-directional vehicles, we only need to introduce the master equations that determine the II. THE MODEL stochastic time evolution of the traffic flow in terms of the incoherent classical hopping terms. To do so, we formu- We now provide our lattice Fock space description of late the incoherent (classical) dynamics of the open quan- the traffic flow in terms of multilevel quantum dots on an tum system in terms of a Lindblad equation for the time open quasi-one-dimensional, multiple way out network evolution of the density matrix ρ(t), describing the inter- by also accounting for the possibility of having differ- action between different dots and between the boundary ent vehicles types or driving styles. To do so, we di- dots and a set of external reservoirs in terms of the jump, 14 vide a generic road into sections of length a, where a or Lindblad, operators, Lk : is the mean dimension of a vehicle. Therefore, we de- X 1 n o scribe each section as a quantum (N + 1)-level dot. The ρ˙ (t) = L ρ (t) L† − L† L , ρ (t) . (3) k k 2 k k state |0i corresponds to an empty road section, while k 3

In our model we introduce two kinds of jump operators. depict our system, along with its minimal three-dot dis- Operators of the first kind act locally on the boundary cretized version (in the next Section we will use a more sites, creating and destroying vehicles; instead, opera- realistic six-dot discretization in order to properly imple- tors of the second kind describe the incoherent stochastic ment priority rules). The crossroad has three entry and transport of vehicles, thus playing a role similar to the exit roads, labeled by 1, 2, 3. Vehicles can enter and exit Hamiltonian governing the coherent transport in the Li- from each road. For the time being, we do not take into ouvillian equation. All these operators can be expressed account any priority rule: as the only over-all constraint, (`) we impose the exclusion principle, that is, the impossibil- in terms of the σi,j operators. The creation and annihilation Lindblad operators are ity for two vehicles to occupy the same dot at the same defined as time. We consider a single type of vehicles, so that each site is described as an (N+1)-level dot with N = 3, since q L(`) = Γ(`)σ(`) three is the number of possible destinations. in,j j j,0 In the minimal three-dot model each dot is labeled by q L(`) = γ(`)σ(`) (4) indices (1), (2) and (3), with the dot (`) representing out,j j 0,j the incoming road (` − 1) and the outgoing road (`). On (`) (`) each dot we label the four possible states as: |0i for the where Γ and γ , with ` a boundary site, are the cou- j j empty site, |1i for a vehicle that wants to exit at the first pling strengths to inject or remove a vehicle. The cou- available exit, |2i for a vehicle that wants to exit at the pling constants give the number of vehicles that would (`) second available exit and |3i for a vehicle that wants to like to enter or exit the system in the unit of time. Γj exit through the same road it entered from. is a complicated function of the properties of the road Assuming that there are no vehicles that go in and out (road surface, track width, and so on) and of the vehicle from the same road, at each dot we can totally remove flux that would enter the system. On the other hand, (`) from the Fock space the state |3i. As a result, the Lind- γj only depends on the properties of the road, and for blad equation describing the roundabout includes now 12 instance to get the flux of vehicles of type j exiting the jump operators: (`) system one should multiply γj by the number of vehi- (`) Three jump operators between consecutive dots, cles σ . We will come back to the details on the physical • j,j that convert the state |2i vehicle on site ` into the meaning of these constants in next Section, when we will state |1i vehicle on site `+1, as long as the destina- discuss specific models. tion site is empty. These are given by (we removed Within our notation, we can easily introduce operators inessential indices) that move a given vehicle from the site ` to the neigh- bouring site `0, given by (1,2) p (1) (2) (1) Lhop = t σ1,0σ0,2 0 q 0 (`,` ) (`,`0) (` ) (`) (2,3) p (3) (2) L = t 0 σ 0 σ , (5) (2) hop,j,j0 j,j j ,0 0,j Lhop = t σ1,0σ0,2 (6) (3,1) p (3) (1) (3) (`,`0) Lhop = t σ1,0σ0,2; with tj,j0 the coupling strength associated to the hopping process. The Lindblad equation should contain only • Six creating operators, two for each dot, injecting jump operators between neighboring sites in accordance vehicles from the reservoirs and for each available with the correct driving directions (a pedestrian can in destinations principle move forward and backward between two dots q q while a vehicle can in general only move forward). (1) (1) (1) (1) (1) (1) Having highlighted the general properties of the Lind- Lin,1 = Γ1 σ1,0 Lin,2 = Γ2 σ2,0 blad operators and of the equations that they should sat- q q L(2) = Γ(2)σ(2) L(2) = Γ(2)σ(2) (7) isfy, in the next Sections we provide their explicit expres- in,1 1 1,0 in,2 2 2,0 sion for specific models, i.e. a three-dot model in Section q q L(3) = Γ(3)σ(3) L(3) = Γ(3)σ(3); III and a six-dot model in Section IV. Specifically, we will in,1 1 1,0 in,2 2 2,0 be focusing onto the non-equilibrium steady state transport properties of these models, defined by the condition • Three annihilation operator, removing vehicles in ρ˙ = 0 and, in particular, on the expectation values of the state |1i from the system incoming and outgoing fluxes from and to the boundary (1) p (1) (1) reservoirs and the on site occupation levels. Lout = γ σ0,1 (2) p (2) (2) Lout = γ σ0,1 (8) (3) p (3) (3) III. THE THREE-DOT MINIMAL MODEL Lout = γ σ0,1 .

In this Section we study a three-way crossroad, where We refer to Fig. 1 for a clearer deﬁnition of the symbols. the roads meet within a small roundabout. In Fig. 1, we 4

FIG. 1: Sketch of the minimal conﬁguration for the open quantum crossroad. Each dot is associated to an entering road and to the subsequent exit road, including the link- ing roundabout segment. We assume that, to each dot, two monodirectional chains are coupled, injecting and removing (`) (`) vehicles with rates given by Γj and γ , respectively. The motion inside the roundabout is ruled by the hopping terms proportional to t(`).

A subtle, though extremely important point is that, if we restrict our operator set only to the 12 operators listed above, as t → ∞, the system would ultimately flow toward an “insulating” phase. Indeed, none of the operators above can change the state |2i⊗|2i⊗|2i, that would stop the flux of vehicles. This effect is known to arise in classical traffic flow when, at an intersection, vehicles FIG. 2: Total current across the junction and mean occupa- with the same priority meet at the same time. Clearly, in tion number as a function of the incoming, Γ, and outgoing, (`) real life this situation is released by a collective “smart” γ, couplings for α = 0.5, γC = 0.1, t = 1. behavior where drivers collaborate against a strict appli- cation of the driving rules (the so called “courtesy cross- the total incoming flux through dot `, given by ing”). To emulate this behavior, and therefore to recover D E a non-zero and physically sound current across the in- (`) (`) (`) (`) (`) (`) hIin,ì = Γ1 + Γ2 σ0,0 ≡ Tr Γ1 + Γ2 σ0,0ρ , tersection, we need to introduce a 13th collective jump (11) operator, that is, the and the outgoing flux through dot ` of vehicles of type 1, given by • Clockwise operator, that simultaneously move three vehicles one step forward and is given by D (`) (`)E (`) (`) hIout,ì = γ σ1,1 ≡ Tr γ σ1,1ρ . (12) √ (1) (2) (3) LC = γCσ1,2σ1,2σ1,2 , (9) In particular, from now on, we assume the coupling constants to be time-independent real numbers so that we with γC the coupling strength for such collective will focus on the non-equilibrium steady state values de- process. fined by settingρ ˙ = 0. In order to reduce the number of parameters, we now Despite being a really simple system, the three-dot mini- (`) consider a Z3-symmetric roundabout, by setting γ = γ, mal model allows us to derive useful traffic-flow informa- (`) (`) Γ1 = αΓ, and Γ2 = (1 − α)Γ, ∀` = 1, 2, 3, with α that tion. The observables we are interested in are the particle regulates the mismatch between vehicles that want to density on site `, given by exit at the first or second available exit.

D (`)E (`) Let Iin,` and Iout,` respectively denote the current (i.e. hn`i = 1 − σ0,0 ≡ 1 − Tr σ0,0ρ , (10) the ﬂux) of vehicles entering or exiting the roundabout 5

FIG. 3: Total current across the junction and mean occupa- (`) FIG. 4: Total density across the junction and mean occupation number as a function of Γ for γ = 3, γC = 0.1, t = 1 (`) tion number as a function of Γ for γ = 3, γC = 0.1, t = 1 and α = 0.25 (green curve), α = 0.5 (blue curve) and α = 0.75 and α = 0.25 (green curve), α = 0.5 (blue curve) and α = 0.75 (orange curve). (orange curve). at route `. In Fig. 2 we show the steady state current across the junction, I = (Iin,1 + Iin,2 + Iin,3) /3 = (Iout,1 + Iout,2 + Iout,3) /3, and the total mean density, n = (n1 + n2 + n3) /3. In drawing the plots, we have set a finite cooperative coupling γC = 0.1 and we have assumed an equal number of vehicles for both exit directions, that is, α = 0.5. We measure everything in units of t(`) = t ≡ 1. Two interesting behaviors emerge in both the current and the density. The current exhibits an optimal working point in the Γ direction, similar to the one that is observed in one-dimensional quantum fermionic, or spin, chains (see for instance, Fig. 3a of Ref. [43]); however, the quantum optimal working point in the γ direction is spoiled in the classical incoherent case we are FIG. 5: Fundamental traffic-flow diagram (traffic flux I as treating here. Furthermore, while in a quantum chain a function of the mean occupation n) for γ = 3, γC = 0.1, the optimal working point is in pair with a similar non- t(`) = 1 and α = 0.25 (green curve), α = 0.5 (blue curve) and monotonic behavior of the mean density (see Fig. 4 of α = 0.75 (orange curve). Ref. [43]), in the classical regime this is not the case and the density increases monotonically as a function of Γ, for any value of γ. The reason is that, in the coherent for γ = 3 and α = 0.25, 0.5, 0.75 recovering the so called quantum case, the optimal working point emerges as a fundamental traffic-flow diagram, that is the existence of consequence of the bulk-reservoir hybridization, while in a critical density value beyond which a traffic-jam phase the incoherent classical dynamics it is a consequence of occurs, with a consequent decrease in the current. In the exclusion statistic at the junction that generates a Fig. 5 we see a net increase of the current as a function competition between vehicles that want to occupy the of α. In fact, we expect a trend as such, as a consequence same dot from different sites. We also observe, as ex- of the fact that, the larger is α (the closer to 1), the less pected, that both current and density are independent the vehicles from different branches compete with each of γ for high enough values. This is due the absence other for the same site occupation, till they do not com- of backscattering terms: vehicles are allowed to move in pete at all for α = 1. only one direction, so, as soon as γ is strong enough in While the results we show in Figs. 3, 4 and 5 have order to sink all the incoming vehicles without delay, it been derived for the Z3 symmetric case, when generaliz- has no other effects on the junction. ing them to the asymmetric case, we readily infer that (`) The nonmonotonic behavior in the current and the mono- low/high values of the Γj are associated to a low/high- tonic one in the density are a footprint of the classical density regime, with the current that exhibits a max- traffic-jam transition. In Fig. 3 and in Fig. 4, we re- imum for a finite intermediate value of the coupling spectively show the dependence of the current and of the strengths. density on Γ. Combining the two figures, we draw Fig. 5, It is worth to note that, even if we do not explicitly where we show the current as a function of the density introduce in our model the delay time, which is a funda- 6 mental parameter of the car-following microscopic models based on continuum space-time description44, we are still able to recover a fundamental traffic-flow diagram describing the jamming transition from the free flow to the congested regime. This is not surprising as in stochastic space-discrete models, like the one we implemented, the role of the delay time is played by a combined effect of the exclusion principle (that does not allow vehicles to jump in occupied sites) and of the probabilistic na- ture of the hopping events (mimicking the vehicle veloc- ity changes)45–47. While these effects are already present in the case of a one dimensional lane43,48,49, in our model the traffic-jam transition emerges due the interaction between vehicles from different lanes and is then a direct consequence of the topology of the intersection. So far, we have not implemented any priority rule at the crossway, that is, we have assumed a sort of “wild” driving style, where any occupancy competition is stochastically resolved by the “fastest” vehicle. How- ever, in real-life experience, we know that the wild driving style can be effective in the low-density case, but becomes ineffective and even dangerous with increasing number of FIG. 6: Sketch of the six-dot configuration for the open quan- vehicles. For this reason, different kinds of priority rules tum crossroad. Compared to the minimal model in Fig. 1, we have to be considered and implemented: right-way pri- added three external sites. The baths that inject vehicles are ority, roundabout, presence of a traffic light, et cetera. now coupled to the external dots, with a flux proportional to Γ(`), ` = 4, 5, 6, while the outgoing baths are coupled to the Each of these rules has its own regime in which it works internal sites, with a flux proportional to γ(`), ` = 1, 2, 3. The better than the others. We can recover and predict such decision about the exit is made before entering into one of the regimes also in our quantum dot model, properly imple- internal dots of the roundabout: this decision is governed by menting priority rules into the jump operators. In order the coupling constants Γ(`), ` = 1, 2, 3, j = 1, 2. to deal with different priority rules on the same footing j and compare them with each other, in the next Section we move from a minimal three-dot model to a more ap- • The six creation operators, injecting vehicles from propriate six-dot model, which we describe below. the reservoirs, now act on the external sites: therefore, only three of them are left out and they create a vehicle without giving directional preferences: IV. THE SIX-DOT MODEL (4) p (4) (4) Lin = Γ σ1,0 The six-dot model is built out of the three-dot one (5) p (5) (5) with the addition of three two-level extra sites. The in- Lin = Γ σ1,0 (14) p ternal sites ` = 1, 2, 3 regulate the interaction between L(6) = Γ(6)σ(6); vehicles, exactly like in the three-dot model, while the in 1,0 external sites ` = 4, 5, 6 regulate the injection of vehi- In addition, we introduce six new hopping opera- cles into the system. In absence of priority rules, a case • tors, that allow a vehicle to jump from an external we have dubbed “wild case”, the Lindbladian dynamics to the corresponding internal dot. During this pro- is described by the following jump operators, pictorially cess the decision about the exit is made: a vehicle shown in Fig. 6: on the external dot ` + 3 can decide to become a type-|1i , or a type-|2i vehicle, on site `: • The three jump operators between consecutive dots that convert a |2i vehicle on site ` into a |1i vehicle q q L(4) = Γ(1)σ(1)σ(4) L(4) = Γ(1)σ(1)σ(4) on site `+1, with ` = 1, 2, 3, do not change, as long hop,1 1 1,0 0,1 hop,2 2 2,0 0,1 q q as the arrival site is empty: (5) (2) (2) (5) (5) (2) (2) (5) Lhop,1 = Γ1 σ1,0σ0,1 Lhop,2 = Γ2 σ2,0σ0,1 (15) q q (1) p (2) (1) (6) (3) (3) (6) (6) (3) (3) (6) (1) L = Γ1 σ1,0σ0,1 L = Γ2 σ2,0σ0,1; Lhop = t σ1,0σ0,2 hop,1 hop,2 (2) p (2) (3) (2) Lhop = t σ1,0σ0,2 (13) • The three annihilation operators removing vehicles in the state |1i from an internal site of the junction (3) p (3) (1) (3) Lhop = t σ1,0σ0,2; do not change: 7

can jump into the `-th roundabout dot from the external ` + 3 dot (` = 1, 2, 3) if and only if the (` − 1) [mod 3] site is not occupied by a type-|2i vehicle. On the other hand, when β = 0, an incoming vehicle on site ` + 3 does not know whether a vehicle in site ` − 1 [mod 3] wants to jump, so it has to give the priority if the site is occupied by either a type-|1i-type, or a type-|2i vehicle. We consider a three-way intersection with two major fluxes (` = 4, 5) and a minor flux (` = 6), setting Γ(4) = (5) (6) Γ = Γ> and Γ = Γ<. The other coupling constants are chosen consistently with the Z3 symmetry, that is we FIG. 7: Possible three-road intersection discussed within the set γ(`) = γ, t(`) = t, with unbalanced exit-intention flux, six-dot model. Both intersections share the same topology, (`) (`) (`) (`) Γ1 = α t,Γ2 = (1 − α )t with ` = 1, 2, 3. To build but they differ for the imposed priority rules. In the round- a phase diagram of the junction we consider two cases: about, vehicles inside the ring have the priority over vehicles the ‘wild case’, i.e. the case with hopping operators as from the external arms; in the symmetric intersection, right- in Eq.(15), and the ‘perfect-roundabout case’, in which hand priority rule applies; in the priority intersection, vehicles in the secondary arm must give way to the vehicles from both the hopping terms are given by Eq.(18) with β = 1. In major roads. both cases we compute the non-equilibrium steady state total current across the junction as a function of Γ> and Γ<, with Γ< < Γ>. (1) p (1) (1) Lout = γ σ0,1 In Fig. 8, panels a) and b), we show the regions in the plane Γ , Γ in which the wild or perfect roundabout (2) p (2) (2) < > Lout = γ σ0,1 (16) priority rules produce the greatest current in the junc- (`) (`) (3) p (3) (3) tion for α = 0.5 and α = 0.75, with ` = 1, 2, 3. We Lout = γ σ0,1. observe that, for low values of Γ< and Γ>, corresponding to a low-density regime, the wild case exhibits a greater • Clockwise operator, that simultaneously move three vehicles one step forward, continues to act current than the perfect roundabout one; in contrast, for on the internal sites and is given by higher values of Γ< and Γ>, that is in the high-density regimes, the perfect roundabout priority rule is more con- √ (`) L = γ σ(1)σ(2)σ(3) . (17) venient. We also note that increasing α enhances the C C 1,2 1,2 1,2 region in which the wild case shows a current greater than in the perfect roundabout case. In the following, we implement various priority rules, on the background of the sets of operators listed above, by All our results can be explained in terms of classical be- modifying the hopping operators in Eq.(13) or in Eq.(15). havior. Let us start from the high-density regime. First In addition, pertinently modifying the various hopping of all, let us note that in the configuration |1i ⊗ |1i ⊗ |1i operators, we will be able to describe and compare on vehicles do not compete with each other and can freely the same footing different road configurations, like the enter into-, and exit from-, the roundabout all together. ones shown in Fig. 7. This suggests that, getting rid of |2i-states from the internal sites should increase the current. Furthermore, the probability to have two consecutive dots both occupied A. The roundabout increases with Γ> and Γ<, as more vehicles are injected into the system from the reservoirs. In Fig. 9 we consider what happens to a specific configuration, involving If the intersection is realized by means of a small round- more than one vehicle, if the priority is given to the vehi- about, vehicles inside the roundabout must have the pri- cle inside the roundabout or to the vehicle from outside. ority over vehicles coming from outside. In terms of jump Clearly, in the roundabout configuration only the first operators, we implement this by modifying the operators case happens. At variance, in the wild case both possi- in Eq.(15) as follows bilities are realized, that is, vehicles can take the prior-

(4) q (1) (1) (4) (3) (3) ity regardless of whether they are inside, or outside the Lhop,j → Γj σj,0 σ0,1 σ0,0 + βσ1,1 junction. Now, as a general remark, we note that, if the vehicles inside the junctions have the priority, vehicles (5) q (2) (2) (5) (1) (1) Lhop,j → Γj σj,0 σ0,1 σ0,0 + βσ1,1 (18) leave the junction one turn earlier (this characteristic is quite general, as we have found the same behavior also (6) q (3) (3) (6) (2) (2) Lhop,j → Γj σj,0 σ0,1 σ0,0 + βσ1,1 , for other configurations with “many” vehicles). As such configurations are more and more likely at higher value with β measuring the promptness of the drivers to use of the density, we expect the roundabout configuration the turn signals. If β = 1, all the drivers correctly use the to win against the wild one at high values of Γ> and Γ<. lights to announce their direction. In this way a vehicle Instead, in the complementary low-density regime, that 8

While the wild setup is not realistic in real-life traﬃc conﬁgurations, due the high probability of accidents, the roundabout is not the only commonly used priority rule. Indeed, before its introduction, the right-hand (or left- hand) priority rule was the most commonly used one. It is then natural to compare in our model the roundabout with, say, the right-hand priority rule. In order to simulate the right-hand priority we only have to modify the operators in Eq.(13) as follows:

(1) p (1) (2) (1) (5) Lhop → t σ1,0σ0,2σ0,0 (2) p (2) (3) (2) (6) Lhop → t σ1,0σ0,2σ0,0 (19) (3) p (3) (1) (3) (4) Lhop → t σ1,0σ0,2σ0,0 . The hopping operators in Eq.(19) mean that a vehicle can jump forward only if there are no vehicles at its right. In this case, vehicles from outside have the priority on vehicles into the roundabout. As the “both vehicles tossing the coin” argument is not valid, we expect, on the basis of the arguments presented in Fig. 9, that the right-hand FIG. 8: Panel a): Phase diagram of a three-way roundabout rule always gives a current across the junction lower than (4) (5) with two major fluxes, Γ = Γ = Γ>, and a minor flux, the wild setup. Indeed, we find that this is the case. (6) Γ = Γ< and with constants γ and t Z3 symmetric. At each An indirect confirmation comes from panels c) and d) of point we plot which priority rule, between the wild one (blue Fig. 8, where we do not consider the unrealistic wild case region) and the perfect roundabout (red region), exhibits the and compare the “right-hand” and “roundabout” prior- higher current across the system. We assume a symmetry between incoming vehicles that want to exit at the first and ity rules for β = 1 and β = 0, respectively. We observe second available exit by setting α(`) = 0.5, with ` = 1, 2, 3. that the right-hand rule still survives as the best option (`) (`) in the small low-density corner of the phase diagram and The other parameters are set to t = 1, γC = 0.1, Γ = 0.5, j that this region increases with β. Also in this case, the γ(`) = 3. Panel b): Same as panel a) but with an asymme- try in the exit direction that favors the first available gate, results are in agreement with real-life traffic experience. α = 0.75. Panel c): Same as panel b) but now we compare Before getting to the actual discussion of our results, the right-hand priority rule (green region) and the perfect let us make two remarks: first, in the low-density regime, roundabout (β = 1 one (red region). Panel d): Same as panel configurations like the one discussed in Fig. 9 are unlikely b) but now we compare the right-hand priority rule (green re- and, therefore, the roundabout effectiveness is reduced gion) and the no-turn signal roundabout (β = 0 one (orange compared to the high-density regime, like in the previous region). case; second, the “double toss coin” argument cannot be applied to the right-hand rule, due the presence of the (`) σ0,0 operator in the Lhop operator. Now, let us assume is at low values of the Γ> and Γ< couplings, a config- that at the same time we have a vehicle on one of the uration like the one in Fig. 9 becomes unlikely, as only external sites, say ` = 4, and another vehicle on the few vehicles will be present into the system at the same internal site ` = 3. They both want to jump into ` = 1. time. On the other hand, we have to remember that If we allow the vehicle in ` = 4 to jump first (right- the hopping probability is a stochastic event. Clearly, if hand rule), we are able to free the site and allow another the priority rule allows only one of the vehicle to “toss vehicle to jump into it, thus increasing the current as the coin” while the other “must” wait, as in the round- vehicles can enter from both ` = 4 and ` = 6. On the about case, the risk is greater and, consequently the flux other side, letting the vehicle in ` = 3 jump first will not is smaller than in the case in which both vehicles are al- give the same benefit, as a vehicle from outside can jump lowed to “toss the coin”. For this reason we expect, as we into the system only from ` = 6 that, in the low-density explicitly show within our model, that in the low-density regime, is most likely empty, but not from ` = 4 that is regime the wild configuration is more convenient in terms occupied. At the end of the day, as far as the current of current flow with respect to a perfect roundabout. in the junction is concerned, the presence of vehicles on dots ` = 4, 5, 6 is less convenient than the presence in dots ` = 1, 2, 3, when few vehicles are around. This is B. Symmetric intersection: right-hand priority not the case in the high-density regime, when all the dots are expected to be occupied and the effect of setups like In the previous Subsection we have compared the wild the one in Fig. 9 dominates, making the roundabout more (no priority) setup with the roundabout priority rule. effective. Clearly the “polite” roundabout, β = 1, always 9

FIG. 9: Minimum number of turns, for a sample configuration, to allow all the vehicles to exit from the system for a given priority rule. On the first line, we assume right-hand priority, which requires 6 different turns. On the second line, we assume perfect roundabout priority and, in this case, five different turns are sufficient. Number 2 represent a vehicle that wants to exit at the second possible gate, number 1 a vehicle intentioned to exit at the first available gate. At each step we draw in red color the operator which is acting. wins against the “no-turn signal” roundabout, β = 0. For this reason in panel d) of Fig. 8 we compare the right- hand rule and the no-turn signal roundabout, observing a similar behavior as in panel c) of Fig. 8, but with an enlarged (green) region in which the right-hand priority rule produces a greater current.

C. The priority intersection

The priority intersection is a quite common road configuration. We have a two-arm main road, that carries most of the traffic, and a side road with a minor traf- (4) (5) fic flow. Consequently, we set Γ = Γ = Γ> and (6) Γ = Γ<. The other coupling constants are chosen (`) (`) as γ = γ, t = t, with unbalanced Z3-broken exit- intention flux: α(1) = α(2) = 0.75, α(3) = 0.5, being (`) (`) (`) (`) Γ1 = α t,Γ2 = (1 − α )t. In the priority intersec- FIG. 10: Phase diagram of a priority rule intersection with (4) (5) tion, vehicles from sites ` = 4, 5 have the priority over two major priority fluxes, Γ = Γ = Γ>, and a minor sec- (6) vehicles from ` = 6 and vehicles from ` = 4 that want to ondary flux, Γ = Γ<. At each point we plot which priority turn left have to give the priority to vehicles from ` = 5. rule, between the priority intersection one (yellow region), the In our model this is realized starting from the wild rules perfect roundabout (red region) (both with β = 1) and the defined in the first part of this Section and by making traffic-light (blue region), exhibits the higher current across the replacements: the system. We assume that the main roads (` = 4, 5) ac- comodate most of the flow, setting α(1) = α(2) = 0.75. The p L(1) → t(1)σ(2)σ(1)σ(5) flux through the secondary road is taken symmetric, setting hop 1,0 0,2 0,0 (3) (`) α = 0.5. The other parameters are t = 1, ΓC = 0.1, (3) p (3) (1) (3) (4) (`) (`) Lhop → t σ1,0σ0,2σ0,0 (20) Γj = 1, γ = 3. For the traffic-light case, we set T = 2 and t1 = 0.75T . (6) q (3) (3) (6) (2) (2) Lhop,j → Γj σj,0 σ0,1 σ0,0 + βσ1,1 , with β measuring the promptness of the drivers to an- traffic light that regulates the major and minor fluxes. nounce their direction (as in the roundabout case, β = 1 In fact, within our model the traffic light is easily in- means that drivers correctly use turn signals). In Fig. 10 troduced by setting time-dependent coupling constants we compare the priority intersection rule with β = 1 with that are alternatively zero, that is Γ(4) = Γ(5) = 0 if (6) the current that we would have by replacing the junction t ∈ [nT, nT + t1] and Γ = 0 if t ∈ [nT + t1, (n + 1)T ], with the “polite” (β = 1) roundabout or by setting a and then computing the mean current over the periodic- 10 ity of the traffic light T . In doing this, we assume that roundabout, a right-hand priority junction and an inter- both t1 and T are much larger than the characteristic section between a major and a minor road by means of time in which the current reaches its stationary value. a six-dot model, with three levels for each dot. For this We observe that priority intersection rules are more ef- model we could report in various cases (shown in panels ficient at low density, i.e. for small Γ> and Γ< but, at a), b), c), and d) of Fig. 8 and in Fig. 10) which config- high-density, the roundabout and traffic light compete uration shows the greater current (or, in other words, is as the best configuration, depending on the values of Γ> more effective in letting vehicles cross the junction) de- and Γ<. This result shows a remarkable agreement with pending on incoming fluxes of vehicles from the external real-life traffic data and classical traffic models, as one roads. Our results qualitatively agree with real data, as can for instance verify by comparing our Fig. 10 with reported for instance in Ref. [50]. Figs. 8-11 of Ref. [50]. An advantage of the model we proposed is that it pro- vides a microscopic simulation (by means of a system of quantum dots) of a macroscopic system, a junction be- V. CONCLUSIONS tween three roads. It would be interesting to compare other predictions of our model or generalizations of it to real data of traffic in junctions. If these further compar- We proposed to model a three-road junction by means isons, as the ones we made in this paper, will be suc- of a graph of multilevel quantum dots coupled to external cessful, one could propose dot models (and possibly, a reservoirs. Hopping between quantum dots represent the realization of them in a laboratory) as microscopic sim- flux of vehicles inside the junction. Jumps from/to ex- ulators of real traffic situations. ternal reservoir to/from internal dots represent vehicles Further extensions of our work are related to the pos- entering/exiting the junction. Each level of the quantum sibility to consider more complex graphs, as for instance dots represents a possible kind of vehicle and/or destina- four-leg intersections, car-pedestrian intersections, inter- tion. The vehicle destination is chosen from the begin- action between different kinds of vehicles. In addition one ning, being an inner property of the vehicle, giving rise could investigate the light-signalized intersection, trying to complex fluxes combinations that are not observed to predict the optimum value for the time dependence of in a junction of (two-level) spin or fermionic quantum the hopping parameters. chains, where a particle can stochastically be annihilated Finally, within the Lindblad master equation formal- from any possible exit point. We first studied a three-dot ism, it is possible to introduce a Liouvillian quantum model, which represents vehicles entering and exiting a term, describing quantum coherent evolution of the den- junction without any rule. Even if it looks like a mini- sity matrix. This would allow to investigate coherent and mal setup, such model already exhibits the main features stochastic evolution on the same footing, exploring the of a typical traffic-flow diagram, that is the existence of crossover between quantum and classical steady states. a critical density of vehicles beyond which a traffic-jam phase occurs and the current (or flux of vehicles) de- creases (Fig. 5). In addition, in analyzing the current Acknowledgements – We thank P. Pantano for in- through the intersection as a function of the incoming sightful comments on the manuscript and C. Paletta for flux, Γ, we found an optimal working point, in which the useful discussions at the early stage of this work. current (or the traffic-flow) is maximal (Fig. 2). A. N. was financially supported by POR Calabria However, to better describe a realistic system, we FESR-FSE 2014/2020 - Linea B) Azione 10.5.12, grant added to the minimal model above three external dots, no. A.5.1. D. G. and M. R. acknowledge financial support which allow to mimic various priority rules used in road from Italy’s MIUR PRIN projects TOP-SPIN (Grant No. junctions. In this way, we have been able to implement a PRIN 20177SL7HC).

1 M. Kaulke and S. Trimper, Journal of Physics A: Mathe- 5 S. Maerivoet and B. De Moor, Physics Re- matical and General 28, 5445 (1995), URL https://doi. ports 419, 1 (2005), ISSN 0370-1573, URL org/10.1088/0305-4470/28/19/002. https://www.sciencedirect.com/science/article/ 2 A. Lazarescu and K. Mallick, Journal of Physics A: Math- pii/S0370157305003315. ematical and Theoretical 44, 315001 (2011), URL https: 6 O. Biham, A. A. Middleton, and D. Levine, Phys. Rev. A //doi.org/10.1088/1751-8113/44/31/315001. 46, R6124 (1992), URL https://link.aps.org/doi/10. 3 A. Ayyer and D. Roy, Scientiﬁc Reports 7, 13555 1103/PhysRevA.46.R6124. (2017), ISSN 2045-2322, URL https://doi.org/10.1038/ 7 T. Nagatani, Journal of Physics A: Mathematical and Gen- s41598-017-12768-8. eral 26, L781 (1993), URL https://doi.org/10.1088/ 4 S. Sandow and S. Trimper, Europhysics Letters (EPL) 21, 0305-4470/26/17/005. 799 (1993), URL https://doi.org/10.1209/0295-5075/ 8 R. Pastor-Satorras, C. Castellano, P. Van Mieghem, 21/8/001. and A. Vespignani, Rev. Mod. Phys. 87, 925 (2015), 11

URL https://link.aps.org/doi/10.1103/RevModPhys. 30 M. E. Foulaadvand and P. Maass, Phys. Rev. E 87.925. 94, 012304 (2016), URL https://link.aps.org/doi/10. 9 A. Nava, A. Papa, M. Rossi, and D. Giuliano, Phys. Rev. 1103/PhysRevE.94.012304. Research 2, 043379 (2020), URL https://link.aps.org/ 31 C. Chamon, M. Oshikawa, and I. Affleck, Phys. Rev. Lett. doi/10.1103/PhysRevResearch.2.043379. 91, 206403 (2003), URL https://link.aps.org/doi/10. 10 M. Doi, Journal of Physics A: Mathematical and General 9, 1103/PhysRevLett.91.206403. 1465 (1976), URL https://doi.org/10.1088/0305-4470/ 32 M. Oshikawa, C. Chamon, and I. Affleck, Jour- 9/9/008. nal of Statistical Mechanics: Theory and Experiment 11 T. Nagatani, Phys. Rev. E 48, 3290 (1993), URL https: 2006, P02008 (2006), URL https://doi.org/10.1088/ //link.aps.org/doi/10.1103/PhysRevE.48.3290. 1742-5468/2006/02/p02008. 12 T. Nagatani, Journal of the Physical Society of Japan 63, 33 D. Giuliano and P. Sodano, New Journal of Physics 1228 (1994), https://doi.org/10.1143/JPSJ.63.1228, URL 10, 093023 (2008), URL https://doi.org/10.1088/ https://doi.org/10.1143/JPSJ.63.1228. 1367-2630/10/9/093023. 13 G. Schütz (Academic Press, 2001), vol. 19 of 34 D. Giuliano and P. Sodano, Nuclear Physics B 811, 395 Phase Transitions and Critical Phenomena, pp. 1– (2009), URL https://www.sciencedirect.com/science/ 251, URL https://www.sciencedirect.com/science/ article/pii/S0550321308006457. article/pii/S106279010180015X. 35 A. Cirillo, M. Mancini, D. Giuliano, and P. So- 14 K. Temme, M. M. Wolf, and F. Verstraete, New Journal dano, Nuclear Physics B 852, 235 (2011), URL of Physics 14, 075004 (2012), URL https://doi.org/10. https://www.sciencedirect.com/science/article/ 1088/1367-2630/14/7/075004. pii/S0550321311003543. 15 M. de Leeuw, C. Paletta, and B. Pozsgay, 36 A. M. Tsvelik, Phys. Rev. Lett. 110, 147202 (2013), URL Constructing integrable lindblad superoperators (2021), https://link.aps.org/doi/10.1103/PhysRevLett.110. 2101.08279. 147202. 16 G. G. Giusteri, F. Mattiotti, and G. L. Celardo, Phys. 37 N. Crampé and A. Trombettoni, Nuclear Physics B Rev. B 91, 094301 (2015), URL https://link.aps.org/ 871, 526 (2013), URL https://www.sciencedirect.com/ doi/10.1103/PhysRevB.91.094301. science/article/pii/S055032131300120X. 17 M. T. Maurer, J. König,and H. Schoeller, Phys. Rev. 38 D. Giuliano, P. Sodano, A. Tagliacozzo, and A. Trom- Research 2, 033440 (2020), URL https://link.aps.org/ bettoni, Nuclear Physics B 909, 135 (2016), URL doi/10.1103/PhysRevResearch.2.033440. https://www.sciencedirect.com/science/article/ 18 T. Reichenbach, E. Frey, and T. Franosch, New Journal of pii/S0550321316300876. Physics 9, 159 (2007), URL https://doi.org/10.1088/ 39 D. Giuliano, A. Nava, and P. Sodano, Nuclear Physics B 1367-2630/9/6/159. 960, 115192 (2020), URL https://www.sciencedirect. 19 K. Nagel and M. Paczuski, Phys. Rev. E 51, com/science/article/pii/S0550321320302777. 2909 (1995), URL https://link.aps.org/doi/10.1103/ 40 B. Béri and N. R. Cooper, Phys. Rev. Lett. 109, PhysRevE.51.2909. 156803 (2012), URL https://link.aps.org/doi/10. 20 D. Helbing, Phys. Rev. E 55, R25 (1997), URL https: 1103/PhysRevLett.109.156803. //link.aps.org/doi/10.1103/PhysRevE.55.R25. 41 A. Altland, B. Béri,R. Egger, and A. M. Tsvelik, Journal 21 B. S. Kerner, S. L. Klenov, and P. Konhäuser,Phys. Rev. of Physics A: Mathematical and Theoretical 47, 265001 E 56, 4200 (1997), URL https://link.aps.org/doi/10. (2014), URL https://doi.org/10.1088/1751-8113/47/ 1103/PhysRevE.56.4200. 26/265001. 22 G. Orosz, R. E. Wilson, R. Szalai, and G. Stépán,Phys. 42 E. Eriksson, A. Nava, C. Mora, and R. Egger, Phys. Rev. Rev. E 80, 046205 (2009), URL https://link.aps.org/ B 90, 245417 (2014), URL https://link.aps.org/doi/ doi/10.1103/PhysRevE.80.046205. 10.1103/PhysRevB.90.245417. 23 H. Ito and K. Nishinari, Phys. Rev. E 89, 042813 (2014), 43 A. Nava, M. Rossi, and D. Giuliano, Phys. Rev. B URL https://link.aps.org/doi/10.1103/PhysRevE. 103, 115139 (2021), URL https://link.aps.org/doi/ 89.042813. 10.1103/PhysRevB.103.115139. 24 M. Ebrahim Fouladvand, Z. Sadjadi, and M. Reza Shae- 44 M. Bando, K. Hasebe, A. Nakayama, A. Shibata, and bani, Phys. Rev. E 70, 046132 (2004), URL https:// Y. Sugiyama, Phys. Rev. E 51, 1035 (1995), URL https: link.aps.org/doi/10.1103/PhysRevE.70.046132. //link.aps.org/doi/10.1103/PhysRevE.51.1035. 25 B. A. Toledo, V. Muñoz,J. Rogan, C. Tenreiro, and J. A. 45 Kai Nagel and Michael Schreckenberg, J. Phys. I France Valdivia, Phys. Rev. E 70, 016107 (2004), URL https: 2, 2221 (1992), URL https://doi.org/10.1051/jp1: //link.aps.org/doi/10.1103/PhysRevE.70.016107. 1992277. 26 B. Ray and S. N. Bhattacharyya, Phys. Rev. E 46 P. Wagner, The European Physical Journal B 84, 713 73, 036101 (2006), URL https://link.aps.org/doi/10. (2011), ISSN 1434-6036, URL https://doi.org/10.1140/ 1103/PhysRevE.73.036101. epjb/e2011-20722-8. 27 D. Chowdhury and A. Schadschneider, Phys. Rev. E 47 A. F. Siqueira, C. J. Peixoto, C. Wu, and W.-L. 59, R1311 (1999), URL https://link.aps.org/doi/10. Qian, Transportation Research Part B: Method- 1103/PhysRevE.59.R1311. ological 87, 1 (2016), ISSN 0191-2615, URL 28 E. Brockfeld, R. Barlovic, A. Schadschneider, and https://www.sciencedirect.com/science/article/ M. Schreckenberg, Phys. Rev. E 64, 056132 (2001), pii/S019126151600028X. URL https://link.aps.org/doi/10.1103/PhysRevE. 48 B. Derrida, M. R. Evans, V. Hakim, and V. Pasquier, 64.056132. Journal of Physics A: Mathematical and General 26, 1493 29 Y. Yokoya, Phys. Rev. E 69, 016121 (2004), URL https: (1993), URL https://doi.org/10.1088/0305-4470/26/ //link.aps.org/doi/10.1103/PhysRevE.69.016121. 7/011. 12

49 G. Benenti, G. Casati, T. c. v. Prosen, D. Rossini, and M. Znidariˇc,Phys.ˇ Rev. B 80, 035110 (2009), URL https: //link.aps.org/doi/10.1103/PhysRevB.80.035110. 50 A. N. Arshi, W. K. Alhajyaseen, H. Nakamura, and X. Zhang, Transportation Research Part A: Policy and Practice 118, 52 (2018), ISSN 0965- 8564, URL https://www.sciencedirect.com/science/ article/pii/S0965856417312302.