Traffic human-ca
-
Upload
zainuddin-kurnia -
Category
Education
-
view
159 -
download
0
Transcript of Traffic human-ca
![Page 1: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/1.jpg)
Andreas Schadschneider
Institute for Theoretical Physics
University of Cologne
www.thp.uni-koeln.de/~as www.thp.uni-koeln.de/ant-traffic
Cellular Automata Modelling of Traffic in Human and
Biological Systems
![Page 2: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/2.jpg)
Introduction
Modelling of transport problems:
space, time, states can be discrete or continuous
various model classes
![Page 3: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/3.jpg)
Overview
1. Highway traffic
3. Traffic on ant trails
5. Pedestrian dynamics
7. Intracellular transport
Unified description!?!
![Page 4: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/4.jpg)
Cellular Automata
Cellular automata (CA) are discrete in• space• time• state variable (e.g. occupancy, velocity)
Advantage: very efficient implementation for large-scale computer simulations
often: stochastic dynamics
![Page 5: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/5.jpg)
Asymmetric Simple
Exclusion Process
![Page 6: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/6.jpg)
Asymmetric Simple Exclusion Process
Asymmetric Simple Exclusion Process (ASEP):
• directed motion• exclusion (1 particle per site)
Caricature of traffic:
For applications: different modifications necessary
![Page 7: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/7.jpg)
Influence of Boundary Conditions
open boundaries:
Applications: Protein synthesis
Surface growth
Boundary induced phase transitions
exactly solvable!
![Page 8: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/8.jpg)
Phase Diagram
Low-density phase
J=J(p,α)
High-density phase
J=J(p,β)
Maximal current phase
J=J(p)
![Page 9: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/9.jpg)
Highway
Traffic
![Page 10: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/10.jpg)
Cellular Automata Models
Discrete in • Space • Time• State variables (velocity)
velocity ),...,1,0( maxvv =
![Page 11: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/11.jpg)
Update Rules
Rules (Nagel-Schreckenberg 1992)
• Acceleration: vj ! min (vj + 1, vmax)
• Braking: vj ! min ( vj , dj)
• Randomization: vj ! vj – 1 (with probability p)
• Motion: xj ! xj + vj
(dj = # empty cells in front of car j)
![Page 12: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/12.jpg)
Example
Configuration at time t:
Acceleration (vmax = 2):
Braking:
Randomization (p = 1/3):
Motion (state at time t+1):
![Page 13: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/13.jpg)
Interpretation of the Rules
• Acceleration: Drivers want to move as fast as possible (or allowed)
• Braking: no accidents
• Randomization: a) overreactions at braking b) delayed acceleration c) psychological effects (fluctuations in driving) d) road conditions
4) Driving: Motion of cars
![Page 14: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/14.jpg)
Simulation of NaSch Model
• Reproduces structure of traffic on highways - Fundamental diagram - Spontaneous jam formation
• Minimal model: all 4 rules are needed
• Order of rules important
• Simple as traffic model, but rather complex as stochastic model
![Page 15: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/15.jpg)
Fundamental Diagram
Relation: current (flow) $ density
![Page 16: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/16.jpg)
Metastable States
Empirical results: Existence of
• metastable high-flow states
• hysteresis
![Page 17: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/17.jpg)
VDR Model
Modified NaSch model: VDR model (velocity-dependent randomization)
Step 0: determine randomization p=p(v(t))
p0 if v = 0
p(v) = with p0 > p
p if v > 0
Slow-to-start rule
![Page 18: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/18.jpg)
NaSch model
VDR-model: phase separation
Jam stabilized by Jout < Jmax
VDR model
Simulation of VDR Model
![Page 19: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/19.jpg)
Dynamics on
Ant Trails
![Page 20: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/20.jpg)
Ant trails
ants build “road” networks: trail system
![Page 21: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/21.jpg)
Chemotaxis
Ants can communicate on a chemical basis:
chemotaxis
Ants create a chemical trace of pheromones
trace can be “smelled” by otherants follow trace to food source etc.
![Page 22: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/22.jpg)
q q Q
1. motion of ants
2. pheromone update (creation + evaporation)Dynamics:
f f f
parameters: q < Q, f
Ant trail model
q q Q
![Page 23: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/23.jpg)
Fundamental diagram of ant trails
different from highway traffic: no egoism
velocity vs. density
Experiments:
Burd et al. (2002, 2005)
non-monotonicity at small
evaporation rates!!
![Page 24: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/24.jpg)
Spatio-temporal organization
formation of “loose clusters”
early times steady state
coarsening dynamics
![Page 25: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/25.jpg)
Pedestrian
Dynamics
![Page 26: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/26.jpg)
Collective Effects
• jamming/clogging at exits• lane formation • flow oscillations at bottlenecks• structures in intersecting flows ( D. Helbing)
![Page 27: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/27.jpg)
Pedestrian Dynamics
More complex than highway traffic
• motion is 2-dimensional• counterflow • interaction “longer-ranged” (not only nearest neighbours)
![Page 28: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/28.jpg)
Pedestrian model
Modifications of ant trail model necessary sincemotion 2-dimensional:• diffusion of pheromones• strength of trace
idea: Virtual chemotaxis
chemical trace: long-ranged interactions are translated into local interactions with ‘‘memory“
![Page 29: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/29.jpg)
Floor field cellular automaton
Floor field CA: stochastic model, defined by transition probabilities, only local interactions
reproduces known collective effects (e.g. lane formation)
Interaction: virtual chemotaxis (not measurable!)
dynamic + static floor fields
interaction with pedestrians and infrastructure
![Page 30: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/30.jpg)
Transition Probabilities
Stochastic motion, defined by transition probabilities
3 contributions:• Desired direction of motion • Reaction to motion of other pedestrians• Reaction to geometry (walls, exits etc.)
Unified description of these 3 components
![Page 31: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/31.jpg)
Transition Probabilities
Total transition probability pij in direction (i,j):
pij = N¢ Mij exp(kDDij) exp(kSSij)(1-nij)
Mij = matrix of preferences (preferred direction)
Dij = dynamic floor field (interaction between pedestrians)
Sij = static floor field (interaction with geometry)
kD, kS = coupling strength
N = normalization (∑ pij = 1)
![Page 32: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/32.jpg)
Lane Formation
velocity profile
![Page 33: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/33.jpg)
Intracellular
Transport
![Page 34: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/34.jpg)
Intracellular Transport
Transport in cells:
• microtubule = highway• molecular motor (proteins) = trucks• ATP = fuel
![Page 35: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/35.jpg)
• Several motors running on same track simultaneously
• Size of the cargo >> Size of the motor
• Collective spatio-temporal organization ?
Fuel: ATP
ATP ADP + P Kinesin
Dynein
Kinesin and Dynein: Cytoskeletal motors
![Page 36: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/36.jpg)
Practical importance in bio-medical research
BlindnessKIF3A kinesinRetinitis pigmentosa
Sinus and Lung disease, male infertility
DyneinPrimary ciliary diskenesia/
Kartageners’ syndrome
Pigmentation defectMyosin VGriscelli disease
Hearing lossMyosin VIIUsher’s syndrome
Neurological disease; sensory loss
KIF1B kinesinCharcot-Marie tooth disease
SymptomMotor/TrackDisease
Goldstein, Aridor, Hannan, Hirokawa, Takemura,…………….
![Page 37: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/37.jpg)
ASEP-like Model of Molecular Motor-Traffic
α βq
D A
Parmeggiani, Franosch and Frey, Phys. Rev. Lett. 90, 086601 (2003)
ASEP + Langmuir-like adsorption-desorption
Also, Evans, Juhasz and Santen, Phys. Rev.E. 68, 026117 (2003)
![Page 38: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/38.jpg)
position of domain wall can be measured as a function of controllable parameters.
Nishinari, Okada, Schadschneider, Chowdhury, Phys. Rev. Lett. (2005)
KIF1A (Red)
MT (Green)10 pM
100 pM
1000pM
2 mM of ATP2 µm
Spatial organization of KIF1A motors: experiment
![Page 39: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/39.jpg)
Summary
Various very different transport and traffic problems can be described by similar models
Variants of the Asymmetric Simple Exclusion Process
• Highway traffic: larger velocities• Ant trails: state-dependent hopping rates• Pedestrian dynamics: 2d motion, virtual chemotaxis• Intracellular transport: adsorption + desorption
![Page 40: Traffic human-ca](https://reader036.fdocuments.us/reader036/viewer/2022081404/559772991a28ab420e8b4880/html5/thumbnails/40.jpg)
Collaborators
Cologne:
Ludger SantenAlireza NamaziAlexander JohnPhilip Greulich
Duisburg:
Michael Schreckenberg Robert BarlovicWolfgang KnospeHubert Klüpfel
Thanx to:
Rest of the World:
Debashish Chowdhury (Kanpur)
Ambarish Kunwar (Kanpur)
Katsuhiro Nishinari (Tokyo)
T. Okada (Tokyo)
+ many others