A lattice model for active-passive pedestrian dynamics: a quest for - - PowerPoint PPT Presentation

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

1Gran Sasso Science Institute (GSSI), L’Aquila, Italy 2University of L’Aquila, Italy 3Sapienza University of Rome, Italy 4Karlstad 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

• beying an exclusion principle.
• The occupation number η(x) is

η(x) =

• 1 if there is a particle at x ∈ Λ,

0 otherwise. In most cases Λ = Zd. A point η is called a configuration.

• The generator of Markov jump process acts on a function f : Ω −

→ R is given by Lf (η) =

• x,y∈Λ

c(x, y, η) [f (ηx,y) − f (η)] , 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

A box with N passive particles Add N more active 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

A box with N passive particles Add N more active 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 2D square lattice Λ = {1, . . . , Lx} × {1, . . . , Ly}, ∂Λ is closed, except a small set D = {(x, y) ∈ Λ : y = 1 , x ∈ [xex, xex + wex − 1]}, located on the upper horizontal row of Λ. The visibility region located in the top of the lattice domain V = {1, . . . , Lx} × {1, . . . , Lv} with Lv ∈ [1, Ly].

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

620 640 660 680 700 720 740 760 0.2 0.4 0.6 0.8 1

average evacuation time ε

Figure 2: Behavior of average evacuation time of particles in presence of both species A and P (empty triangles for Lv = 2, empty circles for Lv = 5, empty pentagons for Lv = 7, empty squares for Lv = 15), and only P (filled circles) as a function of ε. The values of parameters in the simulations are L = 15, wex = 7, NA = NP = 70.

Thoa Thieu A lattice model for active-passive pedestrian dynamics: a quest for drafting effects 12 / 22

Numerical results on empty corridor model

620 640 660 680 700 720 740 760 2 4 6 8 10 12 14

average evacuation time Lv

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 Lv. The values of parameters in the simulations are L = 15, wex = 7, NA = NP = 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

680 700 720 740 760 780 800 820 0.2 0.4 0.6 0.8 1

average evacuation time ε

Figure 5: Behavior of average evacuation time of particles in presence of both species A and P (empty triangles for Lv = 2, empty circles for Lv = 5, empty pentagons for Lv = 7, empty squares for Lv = 15), and only P (filled circles) as a function of ε. The values of parameters in the simulations are L = 15, wex = 7, NA = NP = 70.

Thoa Thieu A lattice model for active-passive pedestrian dynamics: a quest for drafting effects 16 / 22

Numerical results on the corridor with an obstacle

680 700 720 740 760 780 800 820 2 4 6 8 10 12 14

average evacuation time Lv

Figure 6: 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 Lv. The values of parameters in the simulations are L = 15, xex = 5, NA = NP = 70.

Thoa Thieu A lattice model for active-passive pedestrian dynamics: a quest for drafting effects 17 / 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 18 / 22

Conclusions and open questions

• Active particles reduce the evacuation time of all particles in the

system, even assuming that agents do not exchange any information

• Too smart active particles can limit the drafting effect
• The drafting effect is more important in the presence of an obstacle in

the geometry ֒ → Open questions: Study the drafting effect with different location of the exit doors and obstacles, different ratio between the number of active and passive particles and, hydrodynamic limits for two different populations.

Thoa Thieu A lattice model for active-passive pedestrian dynamics: a quest for drafting effects 19 / 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 20 / 22

References

Colangeli, M., Muntean, A., Richardson, O., Thieu, T.: Modelling interactions between active and passive agents moving through heterogeneous environments. (Chapter 8, Crowd Dynamics vol.1 - Theory, Models, and Safety Problems, MSSET series) (2018)

Thoa Thieu A lattice model for active-passive pedestrian dynamics: a quest for drafting effects 21 / 22

Thanks for your attention!

Thoa Thieu A lattice model for active-passive pedestrian dynamics: a quest for drafting effects 22 / 22