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