A lattice model for active-passive pedestrian dynamics: a quest for drafting effects Thoa Thieu 1 Joint work with: Matteo Colangeli 2 , Emilio N. M. Cirillo 3 and Adrian Muntean 4 1 Gran Sasso Science Institute (GSSI), L’Aquila, Italy 2 University of L’Aquila, Italy 3 Sapienza University of Rome, Italy 4 Karlstad University, Sweden Crowds models and control, Marseille, France, 3-7/6/2019 Thoa Thieu A lattice model for active-passive pedestrian dynamics: a quest for drafting effects 1 / 22
Outline 1 Introduction 2 Empty corridor model Lattice gas model Numerical results on empty corridor model 3 The corridor model in presence of an obstacle Numerical results on the corridor with an obstacle 4 Conclusion and open questions 5 References Thoa Thieu A lattice model for active-passive pedestrian dynamics: a quest for drafting effects 2 / 22
Lattice gas model Consider a lattice gas dynamics: particles jump on nearest neighbor sites obeying an exclusion principle. • The occupation number η ( x ) is � 1 if there is a particle at x ∈ Λ , η ( x ) = 0 otherwise . In most cases Λ = Z d . A point η is called a configuration. • The generator of Markov jump process acts on a function f : Ω − → R is given by � c ( x , y , η ) [ f ( η x , y ) − f ( η )] , Lf ( η ) = x , y ∈ Λ where c ( x , y , η ) is the jump rates. Thoa Thieu A lattice model for active-passive pedestrian dynamics: a quest for drafting effects 3 / 22
Drafting effect in aerodynamics Thoa Thieu A lattice model for active-passive pedestrian dynamics: a quest for drafting effects 4 / 22
Drafting effect in lattice gas model We consider a particle model of two species of particles [1], called ”active” (A) and ”passive” (P): Thoa Thieu A lattice model for active-passive pedestrian dynamics: a quest for drafting effects 5 / 22
Drafting effect in lattice gas model We consider a particle model of two species of particles [1], called ”active” (A) and ”passive” (P): • active species, knowing where to go (they are aware of the location of the exit door on the lattice) Thoa Thieu A lattice model for active-passive pedestrian dynamics: a quest for drafting effects 5 / 22
Drafting effect in lattice gas model We consider a particle model of two species of particles [1], called ”active” (A) and ”passive” (P): • active species, knowing where to go (they are aware of the location of the exit door on the lattice) • passive species, randomly exploring the environment Thoa Thieu A lattice model for active-passive pedestrian dynamics: a quest for drafting effects 5 / 22
Drafting effect in lattice gas model We consider a particle model of two species of particles [1], called ”active” (A) and ”passive” (P): • active species, knowing where to go (they are aware of the location of the exit door on the lattice) • passive species, randomly exploring the environment These two species perform a simple exclusion process on a 2D lattice. Thoa Thieu A lattice model for active-passive pedestrian dynamics: a quest for drafting effects 5 / 22
Drafting effect in lattice gas model We consider a particle model of two species of particles [1], called ”active” (A) and ”passive” (P): • active species, knowing where to go (they are aware of the location of the exit door on the lattice) • passive species, randomly exploring the environment These two species perform a simple exclusion process on a 2D lattice. Thoa Thieu A lattice model for active-passive pedestrian dynamics: a quest for drafting effects 5 / 22
Gedankenexperiment Add N more active A box with N passive particles particles in the box Will the evacuation be faster or slower? Thoa Thieu A lattice model for active-passive pedestrian dynamics: a quest for drafting effects 6 / 22
Gedankenexperiment Add N more active A box with N passive particles particles in the box Will the evacuation be faster or slower? Our results show that the motion of the active particles enhances the mobility of the passive ones. Drafting effect on a discrete lattice . Thoa Thieu A lattice model for active-passive pedestrian dynamics: a quest for drafting effects 6 / 22
Outline 1 Introduction 2 Empty corridor model Lattice gas model Numerical results on empty corridor model 3 The corridor model in presence of an obstacle Numerical results on the corridor with an obstacle 4 Conclusion and open questions 5 References Thoa Thieu A lattice model for active-passive pedestrian dynamics: a quest for drafting effects 7 / 22
Empty corridor model Consider a SEP on a 2 D square lattice Λ = { 1 , . . . , L x } × { 1 , . . . , L y } , ∂ Λ is closed, except a small set D = { ( x , y ) ∈ Λ : y = 1 , x ∈ [ x ex , x ex + w ex − 1] } , located on the upper horizontal row of Λ. The visibility region located in the top of the lattice domain V = { 1 , . . . , L x } × { 1 , . . . , L v } with L v ∈ [1 , L y ]. Thoa Thieu A lattice model for active-passive pedestrian dynamics: a quest for drafting effects 8 / 22
Empty corridor model Hopping rates definition: • The hopping rate of the species P from w to z as: � � c ( P ) ( w , z ) = η ( P ) ( w ) 1 − η ( P ) ( z ) − η ( A ) ( z ) . • The hopping rate of the species A from w to z as: � � c ( A ) ( w , z ) = (1 + ε ( v )) η ( A ) ( w ) 1 − η ( P ) ( z ) − η ( A ) ( z ) , where ε ( v ) = ε > 0 inside the visibility region, otherwise ε ( v ) = 0. Thoa Thieu A lattice model for active-passive pedestrian dynamics: a quest for drafting effects 9 / 22
Numerical scheme The model has been simulated by using the following scheme • Define all of the rates c ( A ) ( w , z ) and c ( P ) ( w , z ) of all possible probabilities that all particles can leave the system • Calculate the cumulative function λ equal to the total rates of c ( A ) ( w , z ) and c ( P ) ( w , z ) • Pick up a number τ at random with exponential distribution of parameter λ • Time then update to t + τ and a bond is chosen with probability equal to the corresponding rate divided by λ • A particle hops from the occupied site to the empty site of the chosen bond Thoa Thieu A lattice model for active-passive pedestrian dynamics: a quest for drafting effects 10 / 22
Numerical results on empty corridor model Figure 1: Microscopic configurations of the lattice model sampled at different times (time increases from top to bottom, left to right) by running the sets of Monte Carlo simulations. Thoa Thieu A lattice model for active-passive pedestrian dynamics: a quest for drafting effects 11 / 22
Numerical results on empty corridor model 760 740 average evacuation time 720 700 680 660 640 620 0 0.2 0.4 0.6 0.8 1 ε Figure 2: Behavior of average evacuation time of particles in presence of both species A and P (empty triangles for L v = 2, empty circles for L v = 5, empty pentagons for L v = 7, empty squares for L v = 15), and only P (filled circles) as a function of ε . The values of parameters in the simulations are L = 15, w ex = 7, N A = N P = 70. Thoa Thieu A lattice model for active-passive pedestrian dynamics: a quest for drafting effects 12 / 22
Numerical results on empty corridor model 760 740 average evacuation time 720 700 680 660 640 620 2 4 6 8 10 12 14 L v Figure 3: Behavior of average evacuation time of particles in presence of both species A and P (empty triangles for ε = 0 . 1, empty circles for ε = 0 . 3, empty squares for ε = 0 . 5), and only P (filled circles) as a function of L v . The values of parameters in the simulations are L = 15, w ex = 7, N A = N P = 70. Thoa Thieu A lattice model for active-passive pedestrian dynamics: a quest for drafting effects 13 / 22
Outline 1 Introduction 2 Empty corridor model Lattice gas model Numerical results on empty corridor model 3 The corridor model in presence of an obstacle Numerical results on the corridor with an obstacle 4 Conclusion and open questions 5 References Thoa Thieu A lattice model for active-passive pedestrian dynamics: a quest for drafting effects 14 / 22
Numerical results on the corridor with an obstacle Figure 4: Microscopic configurations of the lattice model sampled at different times (time increases from top to bottom, left to right) by running the sets of Monte Carlo simulations. Thoa Thieu A lattice model for active-passive pedestrian dynamics: a quest for drafting effects 15 / 22
Numerical results on the corridor with an obstacle 820 800 average evacuation time 780 760 740 720 700 680 0 0.2 0.4 0.6 0.8 1 ε Figure 5: Behavior of average evacuation time of particles in presence of both species A and P (empty triangles for L v = 2, empty circles for L v = 5, empty pentagons for L v = 7, empty squares for L v = 15), and only P (filled circles) as a function of ε . The values of parameters in the simulations are L = 15, w ex = 7, N A = N P = 70. Thoa Thieu A lattice model for active-passive pedestrian dynamics: a quest for drafting effects 16 / 22
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