Phase transition behavior in a cellular automaton model with different initial configurations

Abstract

We investigate the dynamical transition from free-flow to jammed traffic,
which is related to the divergence of the relaxation time and susceptibility of
the energy dissipation rate $E_d$, in the Nagel-Schreckenberg (NS) model with
two different initial configurations. Different initial configurations give
rise to distinct phase transition. We argue that the phase transition of the
deterministic NS model with megajam and random initial configuration is first-
and second-order phase transition, respectively. The energy dissipation rate
$E_d$ and relaxation time follow power-law behavior in some cases. The
associated dynamic exponents have also been presented.

Full text

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arXiv:1102.5704v1 [nlin.CG] 28 Feb 2011
Phase transition behavior in a cellular automaton model with different initial
configurations
Wei Zhang1,2,∗Wei Zhang2,†and Wei Chen1‡
1Department of Physics, Fudan University, Shanghai 200433, China
2Department of Physics, Jinan University, Guangzhou 510632, China
(Dated: March 1, 2011)
We investigate the dynamical transition from free-flow to jammed traffic, which is related to the
divergence of the relaxation time and susceptibility of the energy dissipation rate Ed, in the Nagel-
Schreckenberg (NS) model with two different initial configurations. Different initial configurations
give rise to distinct phase transition. We argue that the phase transition of the deterministic NS
model with megajam and random initial configuration is first- and second-order phase transition,
respectively. The energy dissipation rate Edand relaxation time follow power-law behavior in some
cases. The associated dynamic exponents have also been presented.
PACS numbers: 05.65.+b, 45.70.Vn, 89.40.Bb
I.INTRODUCTION
Nonequilibrium phase transitions and various nonlin-
ear dynamical phenomena in traffic system have at-
tracted much attention of a community of physicists in
recent years. There are many theories to describe traf-
fic phenomenon such as fluid-dynamical theories, kinetic
theories, car-following theories and cellular automata
(CA) models[1–3]. The advantages of the cellular au-
tomata approaches show the flexibility to adapt compli-
cated features observed in real traffic [1, 4]. In addition
, CA theory is also a simple and useful approach for the
study of nonequilibrium steady states and their transition
mechanisms. So CA theory has been extensively applied
and investigated. The Nagel-Schreckenberg (NS) model
is a basic CA models describing one-lane traffic flow and
phase transition[5].
The question of the dynamical transition from free-flow
phase to jammed phase in NS model has been investi-
gated by several scholars[6–14]. However, to our knowl-
edge, the effects of the initial configuration on phase tran-
sition behavior have not been explored in detail so far,
and should be further investigated
The energy dissipation rate Ed proposed by us in
Refs.[15] is related to traffic phase transition, and can
be viewed as an order parameter. In the deterministic
NS model, there is a critical density below which the
parameter is zero which is associated with a free-flow
phase, but over which the order parameter is not zero
anymore representing the jammed phase. The study of
energy dissipation in traffic system has important realis-
tic significance. According to Refs.[16], more than 20%
fuel consumption and air pollution is caused by impeded
and ”go and stop” traffic. Due to the relevance of this
parameter for realistic case it is important to understand
∗Electronic address: tzwphys@jnu.edu.cn; wzhang2007065@gmail.com
†Electronic address: twzhang@jnu.edu.cn
‡Electronic address: phchenwei@gmail.com
it’s phase transition behavior thoroughly.
In this paper, using the order parameter Ed, we study
the nonequilibrium phase transition in the NS model with
random and megajam initial configurations. We argue
that the phase transition in the deterministic NS model
with megajam and random initial configuration is first-
and second-order phase transition, respectively. The re-
laxation time of Edin deterministic NS model with the
two initial configuration are distinct and will be analyzed.
And an associated susceptibility is numerically studied.
Some critical exponents will be presented in the following
section.
The paper is organized as follows. Section II is devoted
to the description of the NS model and the definition
of energy dissipation rate. In section III, the numeri-
cal studies of relaxation time and susceptibility of energy
dissipation rate in NS models are given, and the influ-
ences of the initial configuration on phase transition are
considered. The results are summarized in section IV.
II.DESCRIPTION OF THE MODEL AND
ORDER PARAMETER
The model is defined on a single lane road consisting
of L cells of equal size numbered by i = 1, 2, ··· , L
and the time is discrete. Each site can be either empty
or occupied by a car with the speed v = 0, 1, 2,··· ,
vmax, where vmaxis the speed limit. Let x(i,t) and v(i,t)
denote the position and the velocity of the ith car at time
t, respectively. The number of empty cells in front of the
ith car is denoted by d(i,t) = x(i+1,t)−x(i,t)−1. The
following four steps for all cars update in parallel with
periodic boundary.
(1) Acceleration:
v(i,t + 1/3) → min[v(i,t) + 1,vmax];
(2) Slowing down:
v(i,t + 2/3) → min[v(i,t + 1/3),d(i,t)];
(3) Stochastic braking:
v(i,t+1) → max[v(i,t+2/3)−1,0]with the probability
p;

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(4) Movement: x(i,t + 1) → x(i,t) + v(i,t + 1).
Iteration over these four update rules already gives re-
alistic results such as the spontaneous formation of traffic
jams, ”go and stop” wave. With increasing vehicle den-
sity dynamitic transition from free-flow phase to jammed
state occurs. There are two types of initial configura-
tion: random condition where all vehicles’ positions are
distributed randomly, and megajam configuration where
all vehicles stand in one big cluster.
The kinetic energy of the car moving with the velocity
v is mv2/2, where m is the mass of the vehicle. When
braking the kinetic energy reduces. Let Ed denotes en-
ergy dissipation rate per time step per vehicle. For sim-
ple, we neglect rolling and air drag dissipation and other
dissipation such as the energy needed to keep the motor
running while the vehicle is standing and moving in our
analysis, i.e. we only consider the energy lost caused by
speed-down. The dissipated energy of ith car from time
t − 1 to t is defined by
e(i,t) =
?m
2
0
?v2(i,t − 1) − v2(i,t)?
for v(i,t) < v(i,t − 1)
for v(i,t) ? v(i,t − 1).
(1)
Thus, the energy dissipation rate
Ed=1
T
1
N
t0+T
?
t=t0+1
N
?
i=1
e(i,t), (2)
where N is the number of vehicles in the system and t0
is the relaxation time, taken as t0 = 10L unless stated
otherwise. Assuming that energy dissipation per vehicle
at time t is e(t) =
N
ation time τ of e(t) need to get to the steady state as the
time obtained when |Ed− e(t)| < 10−4.
In this model, the particles are ”self-driven” and the
kinetic energy increases in the acceleration step. In the
stationary state, the value of the increased energy while
accelerating is equivalent to that of the dissipated energy
caused by speed-down, and the kinetic energy is constant
in the system. Generally, the mean density is denoted by
ρ = N/L.
1
?N
i=1e(i,t). We consider the relax-
III. NUMERICAL RESULTS
First, we investigate the influences of the initial con-
figuration on the relaxation time τ of energy dissipation
e(t) in the deterministic NS model. In the deterministic
case, the stochastic braking is not considered, i.e. p = 0.
Figure 1 shows the relaxation time τ as a function of the
vehicle density ρ with megajam and random initial con-
figuration in the case of vmax= 3,L = 1000. As shown
in Fig. 1, there is a critical slowing down; and the relax-
ation to the steady state becomes quite slow close to the
critical density ρc= 1/(1 + vmax). The relaxation time
τ diverges at the critical density ρc in the model with
FIG. 1: The relaxation time τ as a function of the vehicle
density ρ in the deterministic NS model with different initial
configuration for the case of vmax = 3, L = 1000.
random initial configuration, which are consistent with
a second-order phase transition. In the case of megajam
initial configuration, however, the relaxation time τ is
invalid below the critical density ρc for there is no en-
ergy dissipation occurs at any time. Above the critical
density, τ decreases linearly with increasing the vehicle
density. As reported in Refs.[15 ], above the critical den-
sity ρc, energy dissipation rate Ed occurs abruptly and
reaches the maximum value in the system with megajam
initial condition, which is different from that with ran-
dom initial configuration. The phase transition is not
continuous. Thus, we argue that the dynamical transi-
tion from free-flow traffic to jammed state in the system
with megajam initial condition can be viewed as first-
order phase transition. In the jammed phase, the relax-
ation to the steady state in the system with megajam
initial condition is slower than that with random initial
configuration, as shown in Fig. 1.
Figure 2 shows the relaxation time τ as a function of
the vehicle density ρ with different values of the speed
limit vmax in the case of random initial configuration.
From figure 2, we see that the maximal value of relaxation
time τmoccurs at the critical density ρc. And the time
τmdecreases with the increase of speed limit vmax. The
relaxation time τmis given as
τm=
L
vmax+ 1, (3)
which is compatible with the results obtained with an-
other order parameter[14]. Figure 3 shows the relaxation
time τm as a function of the system size L with differ-
ent values of the speed limit vmax. As shown in figure
3, symbol data are obtained from computer simulations,
and solid lines correspond to analytic results of the for-
mula (3).
Figure 4 shows the relaxation time τ as a function of
the vehicle density ρ with different values of the speed

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FIG. 2: The relaxation time τ as a function of the vehicle
density ρ for various values of the speed limit vmax in the
deterministic NS model with random initial configuration for
the case of L = 1000.
FIG. 3: The relaxation time τm as a function of the lattice
size L for various values of the speed limit vmax in the deter-
ministic NS model with random initial configuration. Symbol
data are obtained from computer simulations, and solid line
corresponds to analytic results of the formula (3).
limit vmax in the case of megajam initial configuration.
From figure 4, we see that the relaxation time τ decreases
linearly with the increase of the vehicle density over the
critical density ρc. The relaxation time τ can be written
as
τ =
L
vmax(1 − ρ). (4)
In figure 4, symbol data are obtained from computer sim-
ulations, and solid lines correspond to analytic results
of the formula (4). Compared with the theoretical re-
sults and simulation data, excellent agreement can be
obtained.
In the deterministic case, i.e. p = 0, the order param-
eter Edis zero when ρ ≤ ρcand Ed> 0 in the density
FIG. 4: The relaxation time τ as a function of the vehicle
density ρ for various values of the speed limit vmax in the
deterministic NS model with megajam initial configuration
for the case of L = 1000.. Symbol data are obtained from
computer simulations, and solid line corresponds to analytic
results of the formula (4).
FIG. 5: The susceptibility χp as a function of the vehicle
density ρ in NS model with random initial configuration for
various values of the speed limit vmax, for the case of L =
10000 and p = 0.01.
interval 1 > ρ > ρc. However, in the case of p ?= 0, the
parameter Ed is not zero anymore. The phase transi-
tion observed in the deterministic case is destroyed by
the stochastic braking probability p. The probability p
is the conjugated parameter of the order parameter of
NS model. In order to analyze the relationship between
the probability p, energy dissipation rate Ed and phase
behavior, we define the associated susceptibility
χp=∂Ed
∂p
????
p=0
.(5)
Figure 5 exhibits the relation of χpto the vehicle den-
sity ρ with various values of the speed limit and random

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FIG. 6: The log-log plot of energy dissipation rate Ed(scaled
by m ) at the critical density ρcas a function of the stochastic
braking probability p in NS model with random initial con-
figuration for the case of L = 10000, for various values of the
speed limit vmax.
initial condition in the case of p = 0.01,L = 10000. As
shown in figure 5, the susceptibility χpfirst increases with
the density ρ, then it decreases with ρ above the critical
density ρcwhere a maximum value is reached. At high
density region, the susceptibility χpis negative, i.e. en-
ergy dissipation rate Ed decreases with increasing the
probability p. When ρ > 0.8, the curves converges into
one curve and vmaxhas no influence on χp. The peak’s
values increase with the increase of the speed limit vmax.
The peak of χptends to diverge at the critical density ρc
in the case of p → 0. The divergence of χpis relevant to
the traffic phase transition.
From figure 5, we can observe that χp ∝ vmax when
ρ → 0. At the critical density ρc, for p → 0, we find that
the order parameter Edfollows a power-law behavior of
the form
Ed∝ pγ. (6)
Figure 6 shows that the critical exponent γ ≈ 0.483 ±
0.005 and the speed limit has no influences on γ. In
the NS model with megajam initial configuration, the
behavior of χp is similar with that with random initial
condition. The dynamic exponent γ remains unchanged
when varying the initial configuration. However, the ini-
tial condition has effects on the relaxation time τ near
the limit p → 0. Figure 7 exhibits time evolutions of
e(t) starting from the megajam and random initial con-
figuration in the case of vmax = 3,p = 0.005,L = 1000
and ρ = 0.3. As shown in figure 7, the relaxation to
steady state in the system with megajam initial condi-
tion is slower than that with random configuration, even
if the values of e(t) at the steady state are unique. Fig-
ure 8 exhibits the relation of relaxation time τ to the
stochastic braking probability p near the limit p → 0
FIG. 7: Time evolutions of energy dissipation e(t) (scaled by
m ) starting from the random and megajam initial config-
uration for the case of vmax = 3, p = 0.005, ρ = 0.3 and
L = 1000.
FIG. 8: The relaxation time τ as a function of stochastic
braking probability p near the limit p → 0 for several fixed
density in NS model with megajam initial configuration for
the case of vmax = 3 and L = 1000. Lines are guides for the
eyes.
with various values of vehicle density. From figure 8, we
see that
τ ∝ p−δ,(7)
and the dynamic exponent δ ≈ 1.04 ± 0.02.
Figure 9 shows the relaxation time τ as a function of
the system size L with megajam initial configuration in
the case of vmax= 3,ρ = 0.3 and p = 0.005. As shown in
figure 9, the relaxation time τ has a scaling form
τ ∝ Lν, (8)
and the exponent ν ≈ 1, which is compatible with the
results presented in Refs.[17,18].

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FIG. 9: The relaxation time τ as a function of the system size
L with megajam initial configuration for the case of vmax =
3,ρ = 0.3 and p = 0.005.
IV. SUMMARY
In this paper, we investigate traffic phase transition
behavior in the NS model considering different initial
configurations. Different initial condition gives rise to
distinct phase transition. We argue that the phase tran-
sition in the system with megajam and random configu-
ration is first- and second-order phase transition, respec-
tively. Using the order parameter Ed, we numerically
studied the relaxation time τ and susceptibility χp. The
two quantities diverge at the critical density ρc. We ana-
lyzed the relaxation time τmand τ theoretically. Theo-
retical analyses give an excellent agreement with numer-
ical results. Near the limit p → 0, the parameter Edand
relaxation time τ follow a power-law behavior. The dy-
namic exponent γ ≈ 0.483 ± 0.005, δ ≈ 1.04 ± 0.02 and
ν ≈ 1.
When p ? 0, the phase transition behavior becomes
more complicated. We will make further investigation
using the order parameter Ed. The associated dynamic
exponents and scaling laws will be presented elsewhere.
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