Andreas Schadschneider Institute for Theoretical Physics University of Cologne as Cellular Automata.
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Transcript of Andreas Schadschneider Institute for Theoretical Physics University of Cologne as Cellular Automata.
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
Introduction
Modelling of transport problems:
space, time, states can be discrete or continuous
various model classes
Overview
1. Highway traffic
2. Traffic on ant trails
3. Pedestrian dynamics
4. Intracellular transport
Unified description!?!
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
Asymmetric Simple Exclusion Process
Asymmetric Simple Exclusion Process (ASEP):
1. directed motion2. exclusion (1 particle per site)
Caricature of traffic:
For applications: different modifications necessary
Cellular Automata Models
Discrete in • Space • Time• State variables (velocity)
velocity ),...,1,0( maxvv
Update Rules
Rules (Nagel-Schreckenberg 1992)
1) Acceleration: vj ! min (vj + 1, vmax)
2) Braking: vj ! min ( vj , dj)
3) Randomization: vj ! vj – 1 (with probability p)
4) Motion: xj ! xj + vj
(dj = # empty cells in front of car j)
Example
Configuration at time t:
Acceleration (vmax = 2):
Braking:
Randomization (p = 1/3):
Motion (state at time t+1):
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
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
NaSch model
VDR-model: phase separation
Jam stabilized by Jout < Jmax
VDR model
Simulation of VDR Model
Chemotaxis
Ants can communicate on a chemical basis:
chemotaxis
Ants create a chemical trace of pheromones
trace can be “smelled” by other
ants follow trace to food source etc.
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
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!!
Spatio-temporal organization
formation of “loose clusters”
early times steady state
coarsening dynamics
Collective Effects
• jamming/clogging at exits• lane formation • flow oscillations at bottlenecks• structures in intersecting flows ( D. Helbing)
Pedestrian Dynamics
More complex than highway traffic
• motion is 2-dimensional• counterflow • interaction “longer-ranged” (not only nearest neighbours)
Pedestrian model
Modifications of ant trail model necessary since
motion 2-dimensional:• diffusion of pheromones• strength of trace
idea: Virtual chemotaxis
chemical trace: long-ranged interactions are translated into local interactions with ‘‘memory“
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
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
Intracellular Transport
Transport in cells:
• microtubule = highway• molecular motor (proteins) = trucks• ATP = fuel
• 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
Practical importance in bio-medical research
Disease Motor/Track Symptom
Charcot-Marie tooth disease
KIF1B kinesin Neurological disease; sensory loss
Retinitis pigmentosa KIF3A kinesin Blindness
Usher’s syndrome Myosin VII Hearing loss
Griscelli disease Myosin V Pigmentation defect
Primary ciliary diskenesia/
Kartageners’ syndrome
Dynein Sinus and Lung disease, male infertility
Goldstein, Aridor, Hannan, Hirokawa, Takemura,…………….
ASEP-like Model of Molecular Motor-Traffic
q
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)
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 m
Spatial organization of KIF1A motors: experiment
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
Collaborators
Cologne:
Ludger Santen
Alireza Namazi
Alexander John
Philip Greulich
Duisburg:
Michael Schreckenberg
Robert Barlovic
Wolfgang Knospe
Hubert Klüpfel
Thanx to:
Rest of the World:
Debashish Chowdhury (Kanpur)
Ambarish Kunwar (Kanpur)
Katsuhiro Nishinari (Tokyo)
T. Okada (Tokyo)
+ many others