Gruppo di Fisica dei Sistemi Complessi, Dipartimento di Fisica and Centro L.Galvani Università di Bologna and INFN sezione di Bologna

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1 COMPLEXCITY: MODELING MOBILITY Gruppo di Fisica dei Sistemi Complessi, Dipartimento di Fisica and Centro L.Galvani Università di Bologna and INFN sezione di Bologna Armando Bazzani, Bruno Giorgini, Sandro Rambaldi, Graziano Servizi, Giorgio Turchetti O.Bernardi, M.Capriotti, G.Melchiorre, F.Zanlungo internet site

2 Complexity: a point of view Complexity is a key feature of nature whenever the reductionist process does not apply. Our approach consists in developing a new thermodynamics of intelligent particles (Van Neumann s automata). The City is a paradigm in the emerging theory of complex systems and an ideal laboratory to study the thermodynamics of Van Neumann s automata.

3 The A.I.M. project The Artificial Intelligence Mobility project: a virtual reality and artificial intelligence laboratory of urban mobility. 1) The main characters are the individual citizens. 2) The citizens decisions depend on the propensities related to individual features. 3) The mobility is the realization of the citizens propensities using different transportation networks.

4 The Mobilis project requires the modeling of two spaces: a physical space where objective interactions take place; a cognitive space that involves subjective evaluations and individual behaviours. The physical interactions among citizens in the urban space have been extensively studied considering the different kinds of mobility (pedestrian, public or private transportation). A theory of propensities, of information based interactions and of subjective space representation in under development.

5 A theory is also required to connect the cognitive space with the physical space. The Decision Dynamics Theory faces the problem of how an individual makes objective his propensities. On our opinion this can be based on the Bayes-deFinetti interpretation of the Probability Theory. In this way one can associate a probability to all the possible choices according to the individual free wills.

6 We are developing the AIM project following three research lines: 1) the citizens mobility at a building scale level(campus); 2) the pedestrian dynamics in presence of a public transportation network at a quarter level (Mobilis); 3) the traffic dynamics on a urban road network (AutoMobilis).

7 Campus model Finite volume effects (uncompressibility) Confortable space: any mobber tries to avoid other mobbers entering in such space Local vision: it allows the mobbers to get local information from the environment Mobber is in silico automaton.

8 We assume a continuous space and discrete time steps t. We use a first order dynamics (without explicit interaction forces) to increase the simulation performances (the goal is mobbers). The automaton has a desired velocity v i. The automaton tries to change its actual velocity v i to the desired velocity value v i (t + t) = v i (t) v (v i v i ) v i v i v is a parameter that determines the transverse temperature. The automaton moves according to x i (t + t) = x i (t) + v i(t + t) + v i (t) t 2

9 Physical collisions Elastic and Anelastic collisions v i = v i + (λ x ij ± (1 λ) v ij ) v ij x ij x ij 2 where the parameter λ [1/2, 1] is a function of x ij and x ij v ij = 0 (molecular dynamics).

10 Visual collisions The automata rotate their velocities to protect the comfortable space (Non-Newtonian interaction).

11 Flocking Oscillations at a door F F + The different clusters of automata exhibit a collective behaviour and the pression fluctuations are amplified (scrum effect).

12 Self-organized behaviour in a corridor

13 Let us consider the dynamics of the populations a and b on the transverse space with respect the desired velocities We applied the detailed balance principle by assuming that the transition probabilities mainly depend on binary and triple collisions (i.e. a collision between a pedestrian with a cluster of two pedestrians of the other population). p (a) lr p (a) rl = σ a,b ρ b,l (1 + cρ b,l ) + σ a,a ρ a,l = σ a,b ρ b,r (1 + cρ b,r ) + σ a,a ρ a,r where σ a,b and σ a,a are total scattering cross sections and c measures the weight of the triple collisions in the transition from one part to another with respect to the binary collisions.

14 After some algebraic manipulations we get ( cσ a,b ρ a,l ρ a,r σ ) a,a (ρ a,r ρ a,l ) = 0 2 Therefore we have the entropic solution ρ a,l = ρ a,r that gives a uniform distribution of the pedestrians along the corridoir and a non-trivial solution that satisfies the equation 2cρ 2 a,l cρ a,l + σ a,a σ a,b = 0 The solution reads ρ a,l = c ± c 2 8cσ a,a /σ a,b 4c with the existence condition c > 8σ a,a /σ a,b. 1 ± ( 1 4σ ) a,a cσ a,b

15 The stability analysis gives a bifurcation phenomenon at the critical value of c b transverse densities a c The transverse temperature of the system < v i v i 2 > that depends on the parameter v controls σ a,b.

16 Global vision effects The global vision defines a decision mechanisms.

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18 Cognitive properties: path network representation of the space We introduce a path network representation that allows the automaton to choose the desired velocity v i.

19 A real application: the new Rimini railway station

20 The Mobilis model Campus is a chronotopic model of the integrated citizen mobility: the citizen propensities and the decisions are introduced in a probabilistic way w 2 p 2 r c chronotopos p 1 w 1 node p j = r w 4 p 4 w jp j i w ip i A Chronotopos is a macroscopic region of the urban space where time-scheduled activities are present. p 3 w 3

21 Each citizen has a internal genetic code that relates its propensities to the Chronotopoi. In the actual version the genetic code is (for example) work, social class, age, sex. The genetic code defines the daily activities of the automata and (as a consequence) their mobility requests. This is done according to probabilistic algorithms. The model takes into account the public network transportation U(t) 0.3 U(t) 0.3 U(t) percentage W(t) percentage W(t) percentage W(t) time time time Mean field theory results.

22 Learning and Memory The automata are endowed of an internal mechanism to process information. We assocate a real variable x to the decision state of an automaton that has two possible choices. The decision state changes according to a stochastic equation x(t + T) = x(t) V x T + k Tξ(t) T is the decision scale time (ex. 1 day)

23 V (x) (utility function) is a double-well potential V(x) x

24 The equilibrium distribution is ρ(x) x

25 Let us suppose to give to each automaton an information on the choices of other automata; then the utility function is modified by a weight p(ν) V (x, ν) = V 0 (x) + p(ν)x con ν =< x + x > cooperazione 0.1 p(ν) non cooperazione ν

26 Then we have a bifurcation phenomenon for the equilibrium distribution ρ(x) x

27 Experiment in the Milano Bicocca Campus Scientific Area Humanistic area Administration Train Station CNR Sociology Mathematics Statistics

28 A research group of the Sociology Department of Milano Bicocca University has performed an experiment on the pedestrian mobility in the university campus using the signals of GPS recorders (100 students were involved). 400 low speed point number mean speed 100 high speed elocity distribution istantaneus pedestrian speed (m/s) V

29 BUS STATIONS HUMANISTIC AREA TRAIN STATION VIA PIRELLI SCIENTIFIC AREA Integrated density in the morning.

30 Simulation

31 HUMANISTIC AREA TRAIN STATION BUS STATIONS VIA PIRELLI SCIENTIFIC AREA Integrated density at noon.

32 Simulation

33 Future project: Venezia Carnival

34 The AutoMobilis model This is a private traffic model on a urban network which takes into account: 1) the physical interactions among vehicles and the geometrical constraints of the streets; 2) the traffic rules along the streets and in the different crossing topologies; 3) the different drivers behaviours and their propensities to the chronotopoi.

35 Physical interactions The space is continuous and the drivers have a desided velocity v i (r(s)) that depends on the local road curvature. We have a second order dynamics: x i (t + t) = x i (t) + v i(t + t) + v i (t) t 2 v i (t + t) = v i (t) α (v i v i )Θ(d ij d s ) t βθ(d s d ij ) t The parameters α and β (the acceleration and the deceleration capacity) may vary among the drivers to define different classes (aggressive, slow drivers).

36 Urban road network The roads network is built from elementary bricks: the streets and the crossings with different topology The streets take into account the traffic rules, the geometry and the parking possibility. The crossing have a multiple dynamical rules according to their topology.

37 Citizen behaviour The citizens know the street networks and they have propensities towards the chronotopoi. Their behaviour depend on a recursive path finder algorithm that defines the probabilities of different choices at crossings. The dynamics is governed by few nodes.

38 An application to a real case: Senigallia Roads networks with chronotopoi (collaboration with Senigallia Municipality).

39 Senigallia is a real laboratory for Auto-Mobilis Saline-Centro working day Saline Centro week-end 1000/h 1000/h h h Measures of traffic fluxes.

40 Application to Bologna The Bologna Mobility driven by the University Chronotopos (Volvo Italia partnership).

41 Conclusions The City is for us a new laboratory for the Complex Systems Physics. Modeling Mobility is the challenge for the Physics of the City.