Density fluctuations and phase separation in a traffic flow model

6 Pages·0.52 MB·English

by  S. Lubeck

1
8 Density fluctuations and phase separation in a
9
traffic flow model
9
1

n
a S. Lu¨beck, M. Schreckenberg,and K.D. Usadel
J

1 Theoretische Physik,Gerhard-Mercator-Universit¨at, 47048 Duisburg,Deutschland
2

]
h
c Abstract. Within the Nagel-Schreckenberg traffic flow model we con-
e
siderthetransition from thefreeflowregimetothejammed regime.We
m
introduce a method of analyzing the data which is based on the local
- densitydistribution.Thisanalyzesallowsustodeterminethephasedia-
t
a gram andtoexaminetheseparation ofthesystem intoacoexisting free
t flow phase and a jammed phase above the transition. The investigation
s
. ofthesteadystatestructurefactoryieldsthatthedecomposition inthis
t
a phasecoexistenceregimeisdrivenbydensityfluctuations,providedthey
m exceed a critical wavelength.
-
d
n
1 Introduction
o
c
[
Over the past few yearsmuch attention has been devotedto the study of traffic

1 flow.Since the seminalworkof Lighthilland Whitham in the middle ofthe 50’s
v [1] many attempts have been made to construct more and more sophisticated
0
models which incorporate various phenomena occurring in real traffic (for an
2
overview see [2]). Recently, a new class of models, based on the idea of cellular
2
1 automata, has been proven to describe traffic dynamics in a very efficient way
0 [3]. Especially the transition from free flow to jammed traffic with increasing
8 car density could be investigated very accurately. Nevertheless, besides various
9
indications [4], no unique description for a dynamical transition has been found
/
t (seeforinstance[5,6]andreferencestherein).Inthisarticleweconsideramethod
a
m of analysis which allows us to identify the different phases of the system and to
describe the phase transition, i.e., considering the fluctuations which drive the
-
d transition, and determining the phase diagram.
n We consider a one-dimensional cellular automaton of linear size L and N
o
c particles. Each particle is associated the integer values vi ∈ {0,1,2,...,vmax}
: and di 0,1,2,3,... , representing the velocity and the distance to the next
v ∈ { }
forward particle [3]. For each particle, the following update steps representing
i
X the acceleration, the slowing down, the noise, and the motion of the particles
r are done in parallel: (1) if vi < di then vi Min vi +1,vmax , (2) if vi > di
a → { }
then v d , (3) with probability P v Max v 1,0 , and (4) r r +v ,
i i i i i i i
→ → { − } →
where r denotes the position of the i-th particle.
i
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inTrafficandGranularFlow97,editedbyD.E.WolfandM.Schreckenberg,Springer,Singapore(1998)
2 Simulation and Results
Figure 1 shows a space-time plot of the system. Each dot corresponds to a
particle at a given time step. The global density ρ = N/L exceeds the critical
g
density andjams occur.Traffic jams are characterizedby a highlocaldensity of
the particles andbya backwardmovementofshockwaves[1].One canseefrom
Fig.1 that in the jammed regimethe systemis inhomogeneous,i.e.,traffic jams
with a high local density and free flow regions with a low local density coexist.
In order to investigate this transition one has to take this inhomogeneity into
account.
Traditionallyone determines the so-calledfundamental diagram,i.e., the di-
agramofthe flow vs the density.The globalflowis givenby, Φ = ρ v , where
g
h i
v denotes the averagedvelocity of the particles. This non-local measurements
h i
are not sensitive to the inhomogeneous character of the system, i.e., the infor-
mation about the two different coexisting phases is lost. In the following we
consider a method of analysis which is based on the measurement of the local
density distribution p(ρ) [6]. The localdensity ρ is measuredon a section of the
system of size δ according to
N
1
ρ = θ(δ r ). (1)
i
ρ δ −
g i=1
X
The local density distribution p(ρ) is plotted for various values of the global
density ρ in Fig. 2. In the case of small values of ρ , see Fig. 2a, the particles
g g
can be considered as independent (see below) and the local density distribution
e
m
ti
←
space →
Fig.1. Space-time plot for vmax =5, P =0.5,andρg >ρc. Note the separation
of the system in high and low density regions.
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inTrafficandGranularFlow97,editedbyD.E.WolfandM.Schreckenberg,Springer,Singapore(1998)
0.25 0.15
a) b)
ρ<ρ ρ≈ρ
0.20 g c g c
0.10
ρ) 0.15 ρ)
(p 0.10 (p
0.05
0.05
0.00 0.00
0.00 0.10 0.20 0.30 0.00 0.10 0.20 0.30
ρ ρ
0.06 0.03
c) ρg>ρc d) ρg>>ρc
0.04 0.02
ρ) ρ)
(p (p
0.02 0.01
0.00 0.00
0.00 0.10 0.20 0.30 0.40 0.00 0.20 0.40 0.60
ρ ρ
Fig.2.Thelocaldensitydistributionp(ρ)forvariousvaluesoftheglobaldensity,
vmax =5,P =0.5andδ =256.Thedashedlinecorrespondstothecharacteristic
density of the free flow phase.
Fig.3. The local density distribution p(ρ ,ρ) as a function of the global den-
g
sity (horizontal axis) and local density (vertical axis), respectively. The colors
correspondtothevaluesoftheprobabilityp(ρ ,ρ),increasingfromblacktored.
g
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inTrafficandGranularFlow97,editedbyD.E.WolfandM.Schreckenberg,Springer,Singapore(1998)
101 101
ρ<ρ ρ≈ρ
g c g c
)k 100 )k 100
(S (S
a) b)
10-1 10-1
0.0 1.0 2.0 3.0 0.0 1.0 2.0 3.0
k k
101 102
ρ>>ρ
ρ>ρ g c
g c 101
)k 100 )k
(S (S
100
c) d)
10-1 10-1
0.0 1.0 2.0 3.0 10-3 10-2 10-1 100 101
k k
Fig.4. The structure factor S(k) for P =0.5, vmax =5 and for various values
of the density ρ. The dashed line marks the characteristic wavelength λ0 of the
free flow phase.
is simply Gaussian with the mean values ρ and a width which scales with √δ.
g
Increasing the global density, jams occur and the distribution displays two dif-
ferentpeaks(Fig. 2c).The firstpeak correspondsto the density offree particles
andinthephasecoexistenceregimethepositionofthispeakdoesnotdependon
the globaldensity (seethe dashedlines inFig.2).The secondpeak is locatedat
largerdensitiesandcharacterizesthejammedphase.Withincreasingdensitythe
second peak occurs in the vicinity of the critical density ρ (Fig. 2b) and grows
c
further (Fig. 2c) until it dominates the distribution in the sense that the first
peak disappears (Fig. 2d). The two peak structure of the local density distribu-
tionclearlyreflectsthe coexistenceofthe freeflowandjammedphaseabovethe
critical value ρ . In Fig. 3 we present the probability distribution as function of
c
the global and local density. Above a certain value of the global density ρ the
g
two peak structure occurs. The behavior of the first peak yields a criterion to
determine the transitionpoint [6] andone getsρ =0.0695 0.0007for P =0.5
c
±
and vmax, respectively.
In order to describe the spatial decomposition of the coexisting phases we
measured the steady state structure factor [7]
2
L
1
S(k) = η(r)eikr , (2)
L*(cid:12)(cid:12)Xr=1 (cid:12)(cid:12) +
(cid:12) (cid:12)
(cid:12) (cid:12)
where η(r)=1 if the lattice site r is o(cid:12)ccupied and(cid:12)η(r)=0 otherwise. In Fig. 4
we plot the structure factor S(k) for the same values of the global density as in
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inTrafficandGranularFlow97,editedbyD.E.WolfandM.Schreckenberg,Springer,Singapore(1998)
1.0
ρ(Φ )
max
0.8
) 0.6
1
+ f+j
max
v v =2
ρ (g 0.4 vmmaaxx=3
v =4
max
v =5
0.2 vmmaaxx=8 f
from [8]
0.0
0.0 0.2 0.4 0.6 0.8 1.0
P
Fig.5. The phase diagram of the Nagel-Schreckenbergmodel. Note that in the
non-deterministic region 0 < P < 1 the density of the maximum flow exceeds
the density of the transition point.
Fig.2,i.e.,below,inthevicinity,aboveandfarawayofthetransitionpoint.Itis
remarkablethatS(k)exhibits amaximumforallconsideredvaluesofthe global
density at k0 0.72 (dashed lines in Fig. 4). This value correspondence to the
characteristic≈wave length λ0 = 2π of the density fluctuations in the free flow
k0
phase.ThesteadystatestructurefactorisrelatedtotheFouriertransformofthe
real space density-density correlationfunction. The wavelength λ0 corresponds
to a maximum of the correlation function, i.e., λ0 describes the most likely
distance of two particles in the free flow phase. For low densities the structure
factor is almost independent of the density and displays a minimum for small
k values indicating the lack of long-range correlations. Crossing the transition
point the smallest mode S(k = 2π) increases quickly. This suggests that the
L
jammed phase is characterized by long-range correlations which decay in the
limit ρ ρ algebraically as one can see from the log-log plot in Fig. 4d.
g c
≫
UptonowweonlyconsideredthecaseP =0.5.The phasediagraminFig.5
shows the P dependence of the transition density ρ . f denotes the free flow
c
phase andf+j correspondsto the coexistenceregionwherethe systemseparates
in the free flow and jammed phase. The dashed line displays the P dependence
ofthe maximumflowobtainedfromananalysisofthe fundamentaldiagram[8].
The critical densities ρ , where the phase transition takes place, are lower than
c
the density values of the maximum flow. Measurements of the relaxation time,
which is expected to diverge ata transitionpoint [4], confirmthis result [9] (see
Fig. 5). But one has to mention that the determination of the critical density
via relaxationtimes leads in the coexistence regime f+j to unphysicalresults,in
the sense that the relaxation time becomes negative [8,9].
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inTrafficandGranularFlow97,editedbyD.E.WolfandM.Schreckenberg,Springer,Singapore(1998)
3 Conclusions
In conclusion we have studied numerically the Nagel-Schreckenberg traffic flow
model using a local density analysis. Crossing the critical line of the system
a phase transition takes place from a homogeneous regime (free flow phase)
to an inhomogeneous regime which is characterized by a coexistence of two
phases (free flow traffic and jammed traffic). The decomposition in the phase
coexistence regime is driven by density fluctuations, provided they exceed a
critical wavelength λ .
c
References
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2. TrafficandGranular Flow,editedbyD.E.Wolfetal.,(WorldScientific,Singapore,
1996).
3. K. Nagel and M. Schreckenberg,J. Phys. I 2, 2221 (1992).
4. G. Cs´anyi and J. Kert´esz, J. Phys. A 28, L427 (1995).
5. M. Sasvari and J. Kert´esz, Phys.Rev.E 56, 4104 (1997).
6. S. Lu¨beck, M. Schreckenberg,and K.D. Usadel, Phys. Rev. E 57, 1171 (1998).
7. M. Schmittmann and R. K. P. Zia, in Phase Transition and Critical Phenomena,
edited byC. Domb and J. L. Lebowitz, (AcademicPress, London, 1995), Vol. 17.
8. B. Eisenbl¨atter, diploma thesis, Universit¨at Duisburg,1995 (unpublished).
9. B.Eisenbl¨atter,L.Santen,A.Schadschneider,andM.Schreckenberg,Phys.Rev.E
57, (1998).