Faster is Slower Effect B. Maury DMA, Ecole Normale Suprieure & - PowerPoint PPT Presentation

Faster is Slower Effect B. Maury DMA, Ecole Normale Supérieure & Laboratoire de Mathématiques d’Orsay With F. Al Reda, S. Faure, A. Roudneff-Chupin, F. Santambrogio, J. Venel Marseille, June 5 th 2019

Experimental evidence (and counter-evidence…) Garcimartin et al. 14’ Zuriguel et al. 16’ Nicolas et al. 16’ Parisi et al. 15’ Observed phenomena : Capacity Drop and Faster is Slower effects

Experimental evidence (and counter-evidence…) Complementary CDF for time lapses

Related phenomenon : role of an obstacle

Faster is Slower Effect in other contexts (or: the best is the enemy of the good ) For general systems : counter-effective increase of the forcing term (possibly above a certain threshold) Examples Expiratory Flow Limitation Hurtling Droplet Gouttes, bulles, perles et ondes David Quéré, Françoise Brochard-Wyart et Pierre-Gilles de Gennes

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Faster is Slower Effect in other contexts (or: the best is the enemy of the good )

Faster is Slower Effect in crowd motion Helbing’s S ocial Force Model : du i dt = m i � � f w τ ( U i − u i ) + f ij + m i ik , j ̸ = i k Reproduces the FiS effect with an additional friction term Cellular Automata (Schadschneider, Seyfried, Klüpfel … ) Von Neumann Again, friction makes it work: in case of a conflict, there is a non-zero probability that no competitor moves

Faster is Slower Effect in crowd motion N.B. all computations performed with Python Package c romosim pypi.org/project/cromosim/ (S. Faure & B.M.) Mathematical aspects described in Crowds in Equations, B.M. & S. Faure, World Scientific

Faster is Slower Effect Alternative standpoint, an underlying bizarre Laplace operator

Macroscopic model (with A. Roudneff Chupin & F. Santambrogio) ty U = U ( x ) Spontaneous velocity Feasible densities K = { ρ ∈ P ( Ω ) , 0 ≤ ρ ≤ 1 a.e. } � ∂ρ � ∂ t + ∇ · ( ρ u ) = 0 � � � u = P C ρ U, � re ρ = 1 Cone of feasible velocities : nonnegative divergence wherever The projection can be formulated as a unilateral Darcy problem � u + ∇ p = � U � � 0 −∇ · u ≤ � � � 0 p ≥ � � � � = 0 , u · ∇ p � ω

Macroscopic setting : mathematical issues The problem takes the form of a conservation law ∂ρ ∂ t + ∇ · F ( ρ ) = 0 , where F ( ρ ) = ρ P C ρ ( U ) is non-local, non-smooth in both ways : the velocity field u is simply L 2 , and its dependence upon ρ is highly non-smooth. − → Standard theory is not applicable Idea : extend Moreau’s catching up algorithm to measures ρ k +1 = ( id + τ U ) # ρ k transport (prediction), � ˜ � � ρ k +1 = P K � ρ k +1 � � (in the Wasserstein sense) ˜ projection (correction), � Th. (with A. Roudneff-Chupin & F. Santanbrogio) : the evolution problems admits a solution

Evacuation of a room = u P C ρ U . � � u + ∇ p = U � p = 0 � 0 −∇ · u ≤ � � � 0 p ≥ � � � � − ∆ p = −∇ · U > 0 , = 0 , u · ∇ p � ω p = 0 At the exit : u · n = U · n − ∂ p ∂ n ≥ U · n People exit faster as they would if they were alone : no capacity drop, no clogging.

x Faster is Slower effect ? p = 0 speed correction factor β = β ( x ) − ∆ p β = −∇ · β U > p = 0 ∂ p β Z Z Z J ( β ) = u β · n = β U β · n − ∂ n Γ out Γ out Γ out � − ∆ q = 0 in Ω , The gradient of J is is U · ∇ q � � q = 1 on Γ out � � where q solves the adjoint problem : q = 0 on Γ up � � � ∂ q � ∂ n = 0 on Γ w � for « reasonable » choices of U : � U · r q � 0 Faster is Faster effect …

Microscopic level : A granular (purely selfish) model (with J. Venel) at q 1 , q 2 , . . ., q N ∈ R 2 . N individuals, centered at K = { q ∈ R 2 N , D ij ( q ) = | q j − q i | − 2 r ≥ 0 ∀ i ̸ = j } . Set of feasible configurations U = ( U 1 , . . . , U N ) Spontaneous velocities r r e ij − e ij q i D ij q j u = dq dt = P C q U , C q = { v , D ij ( q ) = 0 ⇒ e ij · ( v j − v i ) ≥ 0 }

Individual dissatisfaction (distance to the exit) Gradient flow framework � Ψ ( q ) = D ( q i ) + I K ( q ) . Dissatisfaction functional i dq dt ∈ − ∂ Ψ ( q ) ∂ Ψ ( q ) = { v, Ψ ( q ) + v · h ≤ Ψ ( q + h ) ∀ h } Non overlapping constraint � 0 if q ∈ K � I K ( q ) = � + ∞ if q / ∈ K �

Micro-macro similarities r r U = ( U 1 , . . . , U N ) . spontaneous velocities e ij − e ij x i D ij x j Constraint when i and j are in contact G ij · u ≥ 0 G ij = ∇ D ij ( x ) = (0 , . . . , 0 , − e ij , 0 , . . . , 0 , e ij , 0 , . . . , 0) ∈ R 2 N Macro Micro � � � � u + B ⋆ p = u − = U, p ij G ij U, � u + ∇ p = U � � � � � i ∼ j � � � 0 −∇ · u ≤ 0 , Bu ≤ � � � � � 0 − G ij · u ≤ ∀ i ∼ j, � � � 0 p ≥ � � � ≥ 0 , p � � � � p ≥ 0 , � � � � = 0 , � u · ∇ p � = 0 . p · Bu � � G ij · u > 0 = ⇒ p ij = 0 . � ω

Remark : « Standard » laplacian on the primal network p j p i Macroscopic u ij = − c ij ( p j − p i ) Ohm / Poiseuille / Fick law u = � k r p X u ij = 0 r · u = 0 Kirchhoff law j ∼ i X c ij ( p i − p j ) = 0 Discrete harmonicity �r · k r p = 0 j ∼ i Maximum principle

Discrete Poisson problem Discrete Continuous BB ? p = BU − ∆ p = −∇ · U > 0 , In 1d : .... ×  2 − 1 0 0  · · ⎛ 1 − 1 0 · · 0 ⎞   − 1 2 − 1 0   · ·   0 1 − 1 · · ·   0 − 1   BB ? = ⎜ ⎟ · · · B =     ⎜ ⎟ · · · · · 0   · · · · · ⎝ ⎠      2 − 1  · · · · 1 − 1 · ·     0 0 − 1 2 · · -1 In higher dimensions : a bit different -1 2 -1 2 -1 2 -1 -1 2 -1 2 -1

Here : non-standard laplacian on the dual network Pressures defined at edges (i.e. contact points)

Here : non-standard laplacian on the dual network Pressures defined at edges (i.e. contact points)

j i k The matrix is not diagonally dominant in general + some extra diagonal terms are positive

Consequence : no maximum principle BB ? p = BU > 0 p > 0 does not imply 4 -disk system, glued together + + - + +

Faster is slower effect in the microscopic situation ? speed correction factors β = ( β i ) i - - + + BB ? p � = B ( β � U ) n i J ( β ) = − B ? p · n i The gradient of J is U � B ? q BB ? q = Bn i where q solves the adjoint problem : No reason for J to be positive: some individuals may accelerate the egress of the blue guy by reducing their speed