Crowded Particles - From Ions to Humans

Transcription

1 Westfälische Wilhelms-Universität Münster Institut für Numerische und Angewandte Mathematik February 13, 2009

2 1 Ions Motivation One-Dimensional Model Entropy

3 1 Ions Motivation One-Dimensional Model Entropy 2 Social Force Model and Pedestrian Motion Investiagtions for the 1D Model Limiting Behaviour

4 Motivation One-Dimensional Model Entropy 1 Ions Motivation One-Dimensional Model Entropy 2 Social Force Model and Pedestrian Motion Investiagtions for the 1D Model Limiting Behaviour

5 Biological Background Ions Motivation One-Dimensional Model Entropy located in membrane cells proteins with a hole in their middle regulate movement of inorganic ions through impermeable cell membrane exact structure not known topic of interest in biophysics, medicine

6 One-Dimensional Hopping Model Motivation One-Dimensional Model Entropy Probabilities n(x, t) = P(negatively charged ion at position x at time t) p(x, t) = P(positively charged ion at position x at time t)

7 Motivation One-Dimensional Model Entropy transitionrate: Π n/p + (x, t) = P(jump of n/p from position x to x + h in (t, t + t)) P(x + h is empty) 1 t

8 Motivation One-Dimensional Model Entropy transitionrate: Π n/p + (x, t) = P(jump of n/p from position x to x + h in (t, t + t)) 1 t P(x + h is empty) potential V (x, t) given

9 Motivation One-Dimensional Model Entropy transitionrate: Π n/p + (x, t) = P(jump of n/p from position x to x + h in (t, t + t)) 1 t P(x + h is empty) potential V (x, t) given probability that a negative particle is located at position x at time t + t: n(x, t + t) = n(x, t)(1 Π + (x, t) Π (x, t)) +n(x + h, t)π (x + h, t) + n(x h, t)π + (x h, t)

10 Motivation One-Dimensional Model Entropy transitionrate: Π n/p + (x, t) = P(jump of n/p from position x to x + h in (t, t + t)) 1 t P(x + h is empty) potential V (x, t) given probability that a negative particle is located at position x at time t + t: n(x, t + t) = n(x, t)(1 Π + (x, t) Π (x, t)) +n(x + h, t)π (x + h, t) + n(x h, t)π + (x h, t) Resulting Model t n = (D(1 m) n Dn m n(1 m) V ) t p = (D(1 m) p Dp m + p(1 m) V ) D diffusion coefficient, mass m(x, t) = n(x, t) + p(x, t)

11 Motivation One-Dimensional Model Entropy V satisfies the Poisson equation: λ 2 V xx (x, t) = n(x, t) p(x, t) + f (x) f (x): Permanent charge on the membrane

12 Solutions for Equilibrium Ions Motivation One-Dimensional Model Entropy t n = (D(1 m) n Dn m n(1 m) V ) t p = (D(1 m) p Dp m + p(1 m) V ) Solutions for Equilibrium n(x, t) = p(x, t) = exp(v (x,t)/d) exp(v (x,t)/d)+β exp( V (x,t)/d)+α β exp( V (x,t)/d) exp(v (x,t)/d)+β exp( V (x,t)/d)+α α and β are constants, α, β > 0

13 Numerical Results Ions Motivation One-Dimensional Model Entropy

14 Motivation One-Dimensional Model Entropy Entropy E = (n log (n) + p log (p) + (1 m) log (1 m) nv + pv ) dx decreasing during the process minimal in equilibrium state

15 1 Ions Motivation One-Dimensional Model Entropy 2 Social Force Model and Pedestrian Motion Investiagtions for the 1D Model Limiting Behaviour

16 Under normal Condition Ions Fastest route, not the shortest one Individual speed: most comfortable one (dependent of age, sex, purpose of the trip...) Speeds are Gaussian distibuted (mean value: 1,34 m/s, St-d 0.26 m/s) Certain distance from other pedestrians and boundaries Resting pedestrians are unifomly distributed about the available space

17 In Case of Competitive Evacuation (Panic) Nervousness try to move faster than normal Interactions become physical in nature Uncoordinated passing of bottlenecks Jams build up. Arching and clogging at exits Physical interactions add up dangerous pressures up to 4,5 tons Escape is slowed down by injured or fallen people Herding behaviour

18 Social Force Model (Helbing) Equation of Motion m i dv i (t) dt dr i (t) dt = v i (t) vi 0 = m e0 i (t) v i(t) i + f ij + τ i j( i) W f iw r i position of pedestrian i v i velocity acceleration dv i (t) dt m i mass τ i acceleration time vi 0 desired velocity, e 0 i desired direction

19 Forces Ions f body ij fij interact f body ij fij slid f ij = f interact ij + f body ij + f slid ij = A i e [(R ij d ij)/b ij] nij = k(r ij d ij )n ij = κ(r ij d ij ) v t jit ij and fij slid only if pedestrians i and j touch each other (panic) A i, B i, k, κ constants d ij distance n ij = (nij 1, n2 ij ) = (r i r j )/d ij normalized vector from j to i t ij = ( nij 2, n1 ij ) tangential direction vji t = (v j v i ) t ij tangential velocity difference R ij = (R i + R j ) sum of radii

20 Additional Forces Ions Walls are treated analogously Anisotropy: f ij = A i exp[(r ij d ij )/B i ]n ij ( λ i + (1 λ i ) 1 + cosϕ ) ij 2 λ < 1: happenings in front more weighted than events behind ϕ ij : angle between e i and n ij Sights: forces of type (1) (B larger, A smaller and negative) Groups: fij att = C ij n ij Fluctuation term ξ i (1)

21 Self-Organization Phenomena Helbing s Model describes the following phenomena quite realistically: Segregation Lane Formation

22 Self-Organization Phenomena Helbing s Model describes the following phenomena quite realistically: Segregation Lane Formation Oscillations Bottlenecks: Passing of direction

23 Self-Organization Phenomena Helbing s Model describes the following phenomena quite realistically: Segregation Lane Formation Oscillations Bottlenecks: Passing of direction Intersections Unstable traffic

24 Behaviour in Competitive Evacuation (Panic) People are getting nervous Higher level of fluctuations Higher desired velocity Herding behaviour: (1 n i )e i + n i e 0 e 0 j i i = (1 n i )e i + n i e 0 j Freezing by heating: i n i nervousness

25 Behaviour in Competitive Evacuation (Panic) Faster is slower Effect : Phantom Panics : vi 0 (t) = [1 n i (t)] vi 0 (0) + n i (t)vi max n i (t) = 1 v i(t) v 0 i (0)

26 Optimization Ions Series of columns in the middle of a corridor Funnelshaped geometrie at bottlenecks Two small doors better than a large one Intersections: Guidance arrangements which lead to a roundabout, an obstacle in the centre Slim queues Zigzag shape Avoidance of staircases Escape routes should not have a constant width Column placed asymetrically in front of exits

27 Model for hard Bodies with Remote Action General Model (Helbing) vi 0 f i = m i v i τ i m i dv i dt = f i j i A i( r j r i d i ) A potential d i = a i + b i v i safety margin; a i and b i are constants

28 Seyfried s Model Ions The Model f i (t) = { G i (t) ; if v i (t) > 0 max(0, G i (t)) ; if v i (t) 0 G i (t) = v i 0 v i (t) τ i ( 1 e i r i+1 (t) r i (t) d i (t) e i and B i : strength and the range of the force ) Bi

29 Investigations for the Model Old Model dv i (t) dt = { G i (t), if v i (t) > 0 max(0, G i (t)), if v i (t) 0 G i (t) = v i 0 v i (t) τ i Problem: Do pedestrians go backwards? ( 1 e i r i+1 (t) r i (t) d i (t) ) gi

30 Acceleration Term Ions Seyfried s term: G i (t) = v i 0 v i (t) τ i ( 1 e i r i+1 (t) r i (t) d i (t) ) gi New Acceleration Term G i (t) = { v 0 i v i (t) τ i e i m i ( h r i+1 (t) r i (t) d i (t)) gi, r i+1 r i d i > 0, else

31 Improved Model Ions New Model H(v i (t), G i (t)) = dv i dt = G i (t) H(v i (t), G i (t)) { 0, if v i (t) 0 and G i 0 1, if G i (t) δ or v i (t) ɛ { Old model: dv i (t) G i (t), if v i (t) > 0 = dt max(0, G i (t)), if v i (t) 0 Main difference: Interpolation Interpolation decreases the deceleration in case the velocity comes close to zero.

32 Results Ions do not go backwards, if their initial safety margin is large enough, because The interpolation we used lessened the deceleration in case the velocity came close to zero start deceleration sufficiently long before they approach an obstacle The more a person decelerates, the less becomes the velocity and the required safety margin Model works well!

33 Existence and Uniqueness Lemma 1: Let r i+1 (0) r i (0) > d i (0) r Ṅ v N i. Then the ODE-System r v 1 v 1 1 G 1 (t, r 1, r 2, v 1 ) H(v 1 (t), G 1 (t)) r v 2 v 2 2 = G 2 (tr 2, r 3, v 2 ) H(v 2 (t), G 2 (t)) ẏ = f(t, y).. v N G N (t, r N, r 1, v N ) H(v N (t), G N (t)) has a local solution. The solution is unique.

34 Velocity Density Relation 1,4 1,2 1 0,8 Velocity 0,6 0,4 0, ,5 1 1,5 2 2,5 Density

35 Limiting Behaviour Ions What happens if the number of pedestrians N tends to? We investigate the general formulation: dv i dt = v i 0 v i + 1 τ i m A (N(x i+1 x i ) d i )

36 General Distribution of Particles Equation for particle distribution function f N = f N (x 1, x 2,..., x N, t): t f N + ( ) dxi xi dt f N = 0 i

37 General Distribution of Particles Equation for particle distribution function f N = f N (x 1, x 2,..., x N, t): t f N + ( ) dxi xi dt f N = 0 i In our case t f N + i ([ xi vi 0 1 ] ) m A (N(x i+1 x i ) d i ) f N = 0

38 General Distribution of Particles Equation for particle distribution function f N = f N (x 1, x 2,..., x N, t): t f N + ( ) dxi xi dt f N = 0 i In our case t f N + i ([ xi vi 0 1 ] ) m A (N(x i+1 x i ) d i ) f N = 0 Analogon of BBGKY hierarchy (classical kinetic theory): mk N (x k, x k 1,..., x 1, t) = f N (x 1,..., x N, t)dx k+1 dx k+2...dx N R N k

39 General Distribution of Particles Equation for k-th marginal: t mk N + i k R ([ xi vi 0 1 m A(N(x i+1 x i ) d i ) B i ] ) mk+1 N dx k+1 = 0

40 General Distribution of Particles Equation for k-th marginal: t mk N + i k R ([ xi vi 0 1 m A(N(x i+1 x i ) d i ) B i ] ) mk+1 N dx k+1 = 0 We assume mk N to be dependent of mn 1 x i x i 1 i k, furthermore and the distances ρ denotes density function. m N 1 (x 1 ) N ρ(x)

41 Results Ions Equation from k-th particle marginal in the limit N t ρ(x, t)+ ([ x v 0 1 ( )] ) 1 m A ρ(x, t) d ρ(x, t) = 0

42 Results Ions Equation from k-th particle marginal in the limit N t ρ(x, t)+ ([ x v 0 1 ( )] ) 1 m A ρ(x, t) d ρ(x, t) = 0 Continuity equation: t ρ + x (vρ) = 0 v = v 0 1 ( ) 1 m A ρ(x, t) d

43 Conclusions Ions In the limits N and τ 0 all stochastic fluctuations disappear Motion can be described by a deterministic equation! Motion does not depend on initial distribution

44 Ions Analyze existing data New experiments (train evacuation) Group dynamics Influence of audible signals (message, voice) Influence of fitness

45 Thank you for your Attention!