View
1
Download
0
Category
Preview:
Citation preview
Traffic systems and random Traffic systems and random matrix theorymatrix theory
phenomenologphenomenologicalical descriptiondescription
Petr SebaDepartment of Physics, University Hradec Kralove,
Czech Republic
in collaboration with
Milan KrbalekTechnical University Prague
Traffic systems arefar from equilibrium
strongly interacting
non linear
described usually with the help of cellular automata (Nagel Schreckenberg model)
the traffic density is strongly changing with space and time – traffic jams etc.
Typical example (satellite observation)
time
space
Basic assumption: (without external perturbations like building jobs etc.) the actual traffic situation depends mainly on the traffic density
– fundamental diagram of the transport theory
Random matrix theorythe eigenvalue distribution of random matrices is
identical with the spacing distribution of certain gases of interacting particles in equilibrium with prescribed temperature (Dyson gas, Pechukas gas etc.)
spectral unfolding - distinguishing between the internal and external dynamics - between processes taking place on different time scales
Traffic systems and random matrices
Of particular interest for the traffic theory is the level spacing distribution (time headway, distance clearing) and the
number variance (traffic density fluctuations).
The main clue is the unfolding of the traffic data
The Dutch and German highways
Data collected by induction loops
Measured was the time of passage, velocity and length of the vehicle
Lanes were separated, overtaking cars eliminated
Free traffic – low density
0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
1.2
1.4
car distance
prob
abili
ty d
ensi
tyKOELN − free regime
Psychological gap
Synchronized traffic – alert drivers
0 0.5 1 1.5 2 2.5 3 3.5 40
0.2
0.4
0.6
0.8
1
1.2
1.4
The Loevenisch data compared with GUE band matrix(band width= 5)
The GUE distribution for full matrix
Psychological gap
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.5
1
1.5
2
2.5
3
Re−scaled time interval
Number variance
0 1 2 3 4 50
1
2
3
4
5
Interaction more complex: bus transport in Cuernavaca (Mexico)
bus owned by the driver
there is not time table
there are not particular stops
the traffic is quite turbulent
not interacting busses – Poisson distribution of arrivals
information about the preceding bus has its value - market
Bus arrives
Arrival time is notified
Next bus arrivesThe information is sold
to the driver
Interaction between buses appears and the Poisson distribution is diminished
0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
1.2bus spacing
P(s
)
s
What comes out?
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
1.2Integrated spacing density
s
I(s)
0 0.2 0.4 0.60
0.05
0.1
0.15
0.2
0 1 2 3 4 5 60
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Number of vehicles
vari
anceNumber variance
Mathematical explanation
JINHO BAIK, ALEXEI BORODIN, PERCY DEIFT, AND TOUFIC SUIDANJ. Phys. A: Math. Gen. 39 (2006) 8965-8975
GUE spacing statistics comes out when the particles undergo random Poisson walk that is conditioned not to intersect. This may explain the time headway distributions of the buses as well as that observed on the highways. Various numerical simulations of the traffic (cellular automata) lead also to similar results.
But: do we need the “random walk” ?
Car parking - another system with interacting comG:\DCIM\102_PANA\P1020477.JPGponets
NOT
S. Rawal and G. J. Rodgers, Physica A 346 (2005) 621.AY Abul-Magd - arXiv Physics 0510136
results from London – 500 cars
Data from Hradec Kralove (Czech republic)
bidirectional traffic - right side parking. One way traffic – left side parking.
GUE GOE
Different parking technique
Do we need cars? A small excursion to linguistic
Instead of interacting cars let us investigate interacting phonems
Not all random combinations of phonems can be really pronounced – there is an interaction between them.
Take a language where graphems (letters) = phonems, i.e. you code for all phonems and you speak what you write. Moreover – neglect interaction between phonems belonging to different words
P r o n o u n c e1
2
Assume that all phonems have equal weight
What comes out?
0 0.5 1 1.5 2 2.5 3 3.5 4 4.50
0.2
0.4
0.6
0.8
1
1.2
normalized distance between repeating letters in a word
prob
abili
ty d
ensi
ty
Ovidius, Methamorphoseon
0 0.5 1 1.5 2 2.5 3 3.5 4 4.50
0.2
0.4
0.6
0.8
1
1.2
normalized distance between repeating letters in a word
pro
babili
ty d
ensi
ty
Genesis, Vulgata
0 0.5 1 1.5 2 2.5 3 3.5 4 4.50
0.2
0.4
0.6
0.8
1
1.2
normalized distance between repeating letters in a word
pro
babili
ty d
ensi
ty
Genesis, English translation
0 0.5 1 1.5 2 2.5 3 3.5 4 4.50
0.2
0.4
0.6
0.8
1
1.2
normalized distance between repeating letters in a word
prob
abili
ty d
ensi
ty
Japan text in the ’roomaji’ transliteration
0 0.5 1 1.5 2 2.5 3 3.5 4 4.50
0.2
0.4
0.6
0.8
1
1.2
normalized distance between repeating letters in a word
prob
abili
ty d
ensi
ty
Esperanto, Andersen story
probability increase for small distances
Recommended