Andreas Schadschneider
Institute for Theoretical Physics
University of Cologne
Germany
www.thp.uni-koeln.de/~as www.thp.uni-koeln.de/ant-traffic
Modelling of Traffic Flow and
Related Transport Problems
Overview
• Highway traffic• Traffic on ant trails• Pedestrian dynamics• Intracellular transport
•basic phenomena
•modelling approaches
•theoretical analysis
•physics
Topics:
Aspects:
General topic: Application of nonequilibrium physics to various transport processes/phenomena
Introduction
Traffic = macroscopic system of interacting particles
Nonequilibrium physics:Driven systems far from equilibrium
Various approaches:• hydrodynamic• gas-kinetic• car-following• cellular automata
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
Asymmetric Simple Exclusion Process (ASEP):
1. directed motion2. exclusion (1 particle per site)
3. stochastic dynamics
Caricature of traffic:
“Mother of all traffic models”
For applications: different modifications necessary
Update scheme
• random-sequential: site or particles are picked randomly at each step (= standard update for ASEP; continuous time dynamics)
• parallel (synchronous): all particles or sites are updated at the same time
• ordered-sequential: update in a fixed order (e.g. from left to right)
• shuffled: at each timestep all particles are updated in random order
In which order are the sites or particles updated ?
ASEP
• simple• exactly solvable• many applications
Applications:
• Protein synthesis
• Surface growth
• Traffic
• Boundary induced phase transitions
ASEP = “Ising” model of nonequilibrium physics
Periodic boundary conditions
no or short-range correlations
fundamental diagram
Influence of Boundary Conditions
open boundaries: density not conserved!
exactly solvable for all parameter values!
Derrida, Evans, Hakim, Pasquier 1993
Schütz, Domany 1993
Phase Diagram
Low-density phase
J=J(p,)
High-density phase
J=J(p,)
Maximal current phase
J=J(p)
1.order
transition
2.order
transitions
Highway
Traffic
Spontaneous Jam Formation
Phantom jams, start-stop-waves
interesting collective phenomena
space
time
jam velocity:
-15 km/h
(universal!)
Experiment
Relation: current (flow) $ density
Fundamental Diagram
free flow
congested flow (jams)
more detailed features?
Cellular Automata Models
Discrete in • Space • Time• State variables (velocity)
velocity ),...,1,0( maxvv
dynamics: Nagel – Schreckenberg (1992)
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):
Interpretation of the Rules
1) Acceleration: Drivers want to move as fast as possible (or allowed)
2) Braking: no accidents
3) Randomization: a) overreactions at braking b) delayed acceleration c) psychological effects (fluctuations in driving) d) road conditions
4) Driving: Motion of cars
Realistic Parameter Values
Standard choice: vmax=5, p=0.5
Free velocity: 120 km/h 4.5 cells/timestep
Space discretization: 1 cell 7.5 m
1 timestep 1 sec
Reasonable: order of reaction time (smallest relevant timescale)
Discrete vs. Continuum Models
Simulation of continuum models:
Discretisation (x, t) of space and time necessary
Accurate results: x, t ! 0
Cellular automata: discreteness already taken into account in definition of model
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
Simulation
Analytical Methods
Mean-field: P(1,…,L)¼ P(1) P(L)
Cluster approximation:
P(1,…,L)¼ P(1,2) P(2,3) P(L)
Car-oriented mean-field (COMF):
P(d1,…,dL)¼ P(d1) P(dL) with dj = headway of car j (gap to car ahead)
1 2 3 4
1 2 3 4
1 2 3 4
d1=1 d2=0 d3=2
Particle-hole symmetry
Mean-field theory underestimates flow: particle-hole attraction
Fundamental Diagram (vmax=1)
vmax=1: NaSch = ASEP with parallel dynamics
ASEP with random-sequential update: no correlations (mean-field exact!)
ASEP with parallel update: correlations, mean-field not exact, but 2-cluster approximation and COMF
Origin of correlations?
Paradisical States
Garden of Eden state (GoE)
in reduced configuration space without GoE states: Mean-field exact!
=> correlations in parallel update due to GoE states
not true for vmax>1 !!!
(AS/Schreckenberg 1998)
(can not be reached by dynamics!)
Fundamental Diagram (vmax>1)
No particle-hole symmetry
Phase Transition?
Are free-flow and jammed branch in the NaSch model separated by a phase transition?
No! Only crossover!!
Exception: deterministic limit (p=0)
2nd order transition at 1
1
max vc
Andreas Schadschneider
Institute for Theoretical Physics
University of Cologne
Germany
www.thp.uni-koeln.de/~as www.thp.uni-koeln.de/ant-traffic
Modelling of Traffic Flow and
Related Transport Problems
Lecture II
Nagel-Schreckenberg Model
velocity ),...,1,0( maxvv
1. Acceleration
2. Braking
3. Randomization
4. Motion
vmax=1: NaSch = ASEP with parallel dynamics
vmax>1: realistic behaviour (spontaneous jams, fundamental diagram)
Fundamental Diagram II
free flow
congested flow (jams)
more detailed features?
high-flow states
Metastable States
Empirical results: Existence of
• metastable high-flow states
• hysteresis
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 ruleSimulation
NaSch model
VDR-model: phase separation
Jam stabilized by Jout < Jmax
VDR model
Jam Structure
Fundamental Diagram III
Even more detailed features?
non-unique flow-density relation
Synchronized Flow
New phase of traffic flow (Kerner – Rehborn 1996)
States of• high density and relatively large flow• velocity smaller than in free flow• small variance of velocity (bunching)• similar velocities on different lanes (synchronization)• time series of flow looks „irregular“• no functional relation between flow and density• typically observed close to ramps
3-Phase Theory
free flow
(wide) jams
synchronized traffic
3 phases
Cross-Correlations
free flow
jam
synchro
free flow, jam:
synchronized traffic:
1)(, Jcc
0)(, Jcc
Cross-correlation function:
cc J() / h (t) J(t+) i - h (t) i h J(t+)i
Objective criterion for classification of traffic
phases
Time Headway
free flow synchronized traffic
many short headways!!! density-dependent
Brake-light model
Nagel-Schreckenberg model
1. acceleration (up to maximal velocity)
2. braking (avoidance of accidents)
3. randomization (“dawdle”)
4. motion
plus:
slow-to-start rule
velocity anticipation
brake lights
interaction horizon
smaller cells
…
Brake-light model
(Knospe-Santen-Schadschneider-Schreckenberg 2000)
good agreement with single-vehicle data
Fundamental Diagram IV
a) Empirical results
b) Monte Carlo simulations
Test: „Tunneling of Jams“
Highway Networks
Autobahn networkof North-Rhine-Westfalia
(18 million inhabitants)
length: 2500 km67 intersections (“nodes”)830 on-/off-ramps (“sources/sinks”)
Data Collection
online-data from
3500 inductive loops
only main highways are densely equipped with detectors
almost no data directly
from on-/off-ramps
Online Simulation
State of full network through simulation based on available data “interpolation” based on online data: online simulation
classification into 4 states (available at www.autobahn.nrw.de)
Traffic Forecasting
state at 13:51forecast for 14:56actual state at 14:54
2-Lane Traffic
Rules for lane changes (symmetrical or asymmetrical)• Incentive Criterion: Situation on other lane is better• Safety Criterion: Avoid accidents due to lane changes
Defects
Locally increased randomization: pdef > p
Ramps have similar effect!
shock
Defect position
City Traffic
BML model: only crossings
Even timesteps: " moveOdd timesteps: ! move
Motion deterministic !
2 phases:
Low densities: hvi > 0
High densities: hvi = 0
Phase transition due to gridlocks
More realistic model
Combination of BML and NaSch models
Influence of signal periods,
Signal strategy (red wave etc), …
Chowdhury, Schadschneider 1999
Summary
Cellular automata are able to reproduce many aspects of
highway traffic (despite their simplicity):
• Spontaneous jam formation• Metastability, hysteresis• Existence of 3 phases (novel correlations)
Simulations of networks faster than real-time possible
• Online simulation• Forecasting
Finally!
Sometimes „spontaneous jam formation“ has a rather simple explanation!
Bernd Pfarr, Die ZEIT
Intracellular
Transport
Transport in Cells
(long-range transport)
(short-range transport)
• microtubule = highway• molecular motor
(proteins) = trucks• ATP = fuel
Molecular Motors
DNA, RNA polymerases: move along DNA; duplicate and transcribe DNA into RNA
Membrane pumps: transport ions and small molecules across membranes
Myosin: work collectively in muscles
Kinesin, Dynein: processive enzyms, walk along filaments (directed); important for intracellular transport, cell division, cell locomotion
Microtubule
24 nm
m10~
8 nm
- +
Mechanism of Motion
inchworm: leading and trailing head fixed
hand-over-hand: leading and trailing head change Movie
• 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
ASEP-like Model of Molecular Motor-Traffic
(Lipowsky, Klumpp, Nieuwenhuizen, 2001 Parmeggiani, Franosch, Frey, 2003 Evans, Juhasz, Santen, 2003)
q
D A
ASEP + Langmuir-like adsorption-desorption
Competition bulk – boundary dynamics
Phase diagram
3/1/ ad
0
SL H
Position of Shock is x=1 when SH x=0 when LS
wx0 1
L
HS
cf. ASEP
General belief: Coordination of two heads is required for processivity (i.e., long-distance travel along the track) of conventional TWO-headed kinesin.
KIF1A is a single-headed processive motor.
Then, why is single-headed KIF1A processive?
Single-headed kinesin KIF1A
Movie
2-State Model for KIF1A
state 1: “strongly bound”
state 2: “weakly bound”
Hydrolysis cycle of KIF1A
K KT
KDPKD
ATP
P
ADP
d
Bound on MT
Brownian& Ratchetmotion on MT
hydrolysis12
New model for KIF1A
ー +1 0 0 2 0 1 21 1 2 10
Brownian, ratchet
Attachment
2,1
2,1 Detachment
t
t +1
1
2
h2
1
s
1
f2
2 2
b
2
BrownianRelease ADP( Ratchet )
Hydrolysis
0
1
1
0
a d
Att. Det.
]ms[ -1h
]ms[ -1a
0.01 (0.0094)
0.1 (0.15)
0.00001 (1) 0.001 (100)
0.2 (0.9)
Blue: state_1Red: state_2
])[mMol(ATP
])nMol[A1(KIF0.00005 (5)
x
t
Phase diagram
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
Andreas Schadschneider
Institute for Theoretical Physics
University of Cologne
Germany
www.thp.uni-koeln.de/~as www.thp.uni-koeln.de/ant-traffic
Modelling of Traffic Flow and
Related Transport Problems
Lecture III
Dynamics on
Ant Trails
Ant trails
ants build “road” networks: trail system
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.
Chemotaxis
chemical trace: pheromones
Ant trail model
Basic ant trail model: ASEP + pheromone dynamics
• hopping probability depends on density of pheromones• distinguish only presence/absence of pheromones• ants create pheromones• ‘free’ pheromones evaporate
q q Q
1. motion of ants
2. pheromone update (creation + evaporation)Dynamics:
f f fparameters: q < Q, f
Ant trail model
q q Q
equivalent to bus-route model (O’Loan, Evans Cates 1998)
(Chowdhury, Guttal, Nishinari, A.S. 2002)
Limiting cases
f=0: pheromones never evaporate
=> hopping rate always Q in stationary state
f=1: pheromone evaporates immediately
=> hopping rate always q in stationary state
for f=0 and f=1: ant trail model = ASEP (with Q, q, resp.)
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!!
Experimental result
(Burd et al., 2002)
Problem: mixture of unidirectional and counterflow
Spatio-temporal organization
formation of “loose clusters”
early times steady state
coarsening dynamics:
cluster velocity ~ gap to preceding cluster
Traffic on Ant Trails
Formation of clusters
Analytical Description
Mapping on Zero-Range Process
ant trail model:
})1(1{)1()( // vxvx fqfQxu (v = average velocity)
phase transition for f ! 0 at 2qQ
qQc
Counterflow
hindrance effect through interactions (e.g. for communication)
plateau
Pedestrian
Dynamics
Collective Effects
• jamming/clogging at exits• lane formation • flow oscillations at bottlenecks• structures in intersecting flows
Lane Formation
Lane Formation
Oscillations of Flow Direction
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“
Long-ranged Interactions
Problems for complex
geometries:
Walls ’’screen“ interactions
Models with local interactions ???
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
Static Floor Field
• Not influenced by pedestrians• no dynamics (constant in time)• modelling of influence of infrastructure
Example: Ballroom with one exit
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
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)
Lane Formation
velocity profile
Friction
Friction: not all conflicts are resolved! (Kirchner, Nishinari, Schadschneider 2003)
friction constant = probability that no one moves
Conflict: 2 or more pedestrians choose the same target cell
Herding Behaviour vs. Individualism
Minimal evacuation times for optimal combination of herding and individual behaviour
Evacuation time as function of coupling strength to dynamical floor field
(Kirchner, Schadschneider 2002)
Large kD: strong herding
Evacuation Scenario With Friction Effects
Faster-is-slower effect
evacuation time
effective velocity
(Kirchner, Nishinari, A.S. 2003)
Competitive vs. Cooperative Behaviour
Experiment: egress from aircraft (Muir et al. 1996)
Evacuation times as function of 2 parameters:
• motivation level
- competitive (Tcomp)
- cooperative (Tcoop )
• exit width w
Empirical Egress Times
Tcomp > Tcoop for w < wc
Tcomp < Tcoop for w > wc
Model Approach
Competitive behaviour:
large kS + large friction
Cooperative behaviour:
small kS + no friction =0
(Kirchner, Klüpfel, Nishinari, A. S., Schreckenberg 2003)
Summary
Variants of the Asymmetric Simple Exclusion Process
• Highway traffic: larger velocities
• Ant trails: state-dependent hopping rates
• Pedestrian dynamics: 2-dimensional motion
• Intracellular transport: adsorption + desorption
Various very different transport and traffic problems can be described by similar models
Applications
Highway traffic:• Traffic forecasting• Traffic planning and optimization
Ant trails:• Optimization of traffic• Pedestrian dynamics (virtual chemotaxis)
Pedestrian dynamics:• safety analysis (planes, ships, football stadiums,…)
Intracellular transport:• relation with diseases (ALS, Alzheimer,…)
Collaborators
Cologne:Ludger SantenAnsgar KirchnerAlireza NamaziKai KlauckFrank ZielenCarsten BursteddeAlexander John Philip Greulich
Thanx to:
Rest of the world:
Debashish Chowdhury (Kanpur)
Ambarish Kunwar (Kanpur)
Vishwesha Guttal (Kanpur)
Katsuhiro Nishinari (Tokyo)
Yasushi Okada (Tokyo)
Gunter Schütz (Jülich)
Vladislav Popkov (now Cologne)
Kai Nagel (Berlin)
Janos Kertesz (Budapest)
Duisburg:
Michael Schreckenberg
Robert Barlovic
Wolfgang Knospe
Hubert Klüpfel
Torsten Huisinga
Andreas Pottmeier
Lutz Neubert
Bernd Eisenblätter
Marko Woelki