- Email: [email protected]
Phase transition and scaling in the generalized traffic flow model
PHYSICA ELSEVIER
Physica A 246 (1997) 460470
Phase transition and scaling in the generalized traffic flow model Takashi Nagatani
*
Division of Thermal Science, College of Engineering, Shizuoka University, Hamamatsu 432, Japan
Received 9 June 1997
Abstract The optimal velocity traffic model is generalized to the power-law type of the optimal velocity. The phase transition among the freely moving phase, the coexisting phase, and the homogeneous congested phase is investigated by computer simulation and the linear stability theory. Phase diagrams are obtained for various values of the power. It is shown that the phase boundary and the critical point depend strongly upon the power. The scaling behaviors of the headway, the car velocity, and the propagation velocity of jams are studied near the critical point. It is found that the headway and the car velocity scale as (ac - a) ~ and (ac - a) B with ~ = ~ = n/(n + 1) and the propagation velocity of jams scales as (ac - a ) ; ' with 7 = 2n/(n + 1) where a is the sensitivity, ac is its critical value, and n is the power of the model. PACS: 05.70.Fh; 05.70.jk; 89.40.+k Keywords: Traffic flow; Phase transition; Critical phenomena; Scaling
1. Introduction Recently, traffic problems have attracted considerable attention [1]. Three different approaches have been used to describe the collective properties o f traffic flow: a microscopic approach [ 2 - 1 1 ] , a gas kinetic approach [ 12-17], and a macroscopic approach [ 18,19]. The traffic flow models are classified into the deterministic and stochastic models [2,6]. A stochastic cellular automaton (CA) model was introduced by Nagel and Schreckenberg [2]. They showed that the start-stop waves (traffic jams) appear in the congested traffic region as observed in real freeway traffic. Bando et al. [6] proposed the deterministic optimal velocity model in which a car accelerates or decelerates according to the dynamical equation o f car motion with the optimal velocity function. They also found that the density waves appear in the congested traffic. Lately, Komatsu * E-mail: [email protected]. 0378-4371/97/$17.00 Copyright @ 1997 Elsevier Science B.V. All rights reserved PH S0378-4371 (97)00376-2
T. Nayatani/Physica A 246 (1997) 460~t70
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and Sasa [20] derived the modified KdV equation from the optimal velocity model to describe the traffic jam. They showed that there is a critical point ac for the sensitivity a, and the characteristic length and time scale as (ac - a ) -J/2 and (ac - a ) -3/2 just below the critical point ac. Kerner and Konhauser [18] have also derived the KdV equation from the hydrodynamic equation of the traffic. In this paper, we generalize the optimal velocity model to study the phase transition and critical phenomena in the traffic flow. We investigate the phase transition among the freely moving phase, the coexisting phase, and the homogeneous congested phase by using computer simulation and the linear stability theory. We show that the phase boundary and the critical point depend on the power of the model. Furthermore, we study the scaling behavior of the traffic near the critical point. We show the dependence of the scaling exponents upon the power.
2. Generalized model and simulation
We generalize the optimal velocity model [6]. The optimal velocity model is described as follows. A car has the optimal velocity depending only upon the headway. A car is controlled in such a way that its car velocity adjusts the optimal velocity. Then, the equation of motion of car j is given by
d2xj/dt 2 = a { V ( A x j ) - dxj/dt}
with U ( A x j ) = tanh(Axj - Xc) + tanhxc,
(1)
where Axj(= Xi+l - xj) is the headway, a is the sensitivity of the car, and Xc is the safety distance. V(Axj) represents the optimal velocity. In this model, the maximal velocity and the minimal velocity are two and zero, respectively. We extend the optimal velocity model as follows. We assume that the acceleration of a car is proportional to the difference between the power of the optimal velocity and the power of the car velocity. The equation of motion of car j is given by
dZxj/dt 2 = a { V ( Axj )n
_
_
( dxi/d t ). }.
(2)
When n = 2, m(dxj/dt)2/2 represents the energy which car j has at time t, where m is the mass of the car. mV(Axj)2/2 represents the energy which a car moving with the optimal velocity has. A car accelerates or decelerates according to the difference between the energies. We study the phase transition and the scaling behavior in the generalized optimal velocity model described by Eq. (2). We perform a numerical simulation for the generalized optimal velocity model. Initially, cars are randomly distributed on the one-dimensional space with car density P0 and velocity v0. The boundary is periodic. In order to form a single jam, a hindrance is put at a point on the one-dimensional space. We assume that when a car reaches the hindrance it slows down instantly to low velocity vh. In time, a single jam is formed just behind the hindrance. After the jam is formed, the hindrance is removed. The jam propagates backward with constant propagation velocity Vp. Once the single jam is formed for a special range of the
T. Naoatani/Physica A 246 (1997) 460-470
462
1.0
n=2, xc=5 /~itical
point
0.9 a
0.8 0.7 0.6 0.5 3.0
4.0
5.0
6.0
7.0
zkx Fig. 1. The phase diagram in the (dx, a) plane for power n = 2 and safety distance xc = 5 where Ax is the headway and a is the sensitivity. The regions I, II, and IIl indicate the freely moving phase, the homogeneous congested phase, and the coexisting phase, respectively. Above the critical point, no phase transition occurs. The headways within and out of the jam are plotted by the circles for various values of sensitivity a, where car density is P0 = 0.2.
density, the jam is stable and does not break up. The jam has the form of the kinkantikink. The stable jam occurs at the intermediate density. For low density, a jam disappears in time, and all cars move freely with nearly maximal velocity. For high density, a jam also disappears in time, and a congested homogeneous traffic flow occurs. Thus, we can make a stable jam on the one-dimensional space for special values of density. We take the number N of cars as N = 100-400. The safety distance is set a s x c ~ 5.
Firstly, we calculate the traffic flow for power n = 2. Fig. 1 shows the plot of the headways within and out of the jam for various values of sensitivity a, where P0 = 0.2 and Vh = 1.0. When sensitivity a is less than critical value ac = 1.0, a single jam occurs. The headways within and out of the jam are plotted. When sensitivity a is larger than ac = 1.0, no jam appears. We calculate the density occurring at the jam. For low densities, the jam does not occur. The boundary between a jam and no jam is consistent with the curve on the right-hand side for low densities. Also, for high densities, the jam does not occur. The boundary is consistent with the curve on the left-hand side. In Fig. 1, the region I indicates the freely moving phase in which a car moves with about maximal velocity. The region II indicates the homogeneous congested traffic flow without jams. In the region III, the coexisting phase with the jam appears. In the coexisting phase, freely moving phase and the congested phase coexist. Thus, Fig. 1 shows the phase diagram among the freely moving phase, the coexisting phase, and the homogeneous congested phase for n = 2 and Xc = 5. The critical point is given by a = 1.0 and
72 Nagatani/Physica A 246 (1997) 460~170
463
n=2, xc=5 lcritical point
_
1.O 0.9 II
a
-/.
0.8
p
0.7
!
\
\
! !
0.6
0.5 0
\ I
I
I
0.5
1.0
1.5
2.0
V Fig. 2. The phase diagram in the (v,a) plane for the same values of parameters in Fig. 1. Regions 1, lI, and 11I indicate the freely moving phase, the homogeneous congested phase, and the coexisting phase, respectively. Xc = 5.0. The phase boundaries correspond to the saturation lines in the liquid-gas transition. Similarly, Fig. 2 shows the plot of the velocities within and out of the jam for various values of sensitivity a, where P0 = 0.2 and vh = 1.0. The density P0, safety distance Xc and the power n in Fig. 2 are same as the values in Fig. 1. Regions I, II and III indicate the freely moving phase, the homogeneous congested phase and the coexisting phase, respectively. In region III, the system separates into the two coexisting phases. The following quantities are considered as the order parameter S: S = ( A x f - Axj)
or
S = (vy - ~).),
(3)
where A x f is the headway out of the jam, Axj is the headway within the jam, vf is the car velocity out of the jam, and 4i is the car velocity within the jam. The order parameter S is different from zero below the critical point ac. Fig. 3 shows the phase boundaries and the critical points in the (Ax, a) plane for powers n = 1,2, 3, and 4. The critical point and the phase boundaries decrease with the increase of power n. Fig. 4 shows the phase boundaries and the critical points in the (v,a) plane for powers n = 1,2,3, and 4. These critical points and the phase boundaries show similar behaviors as in Fig. 3. Fig. 5 shows the plot of the critical point ac against power n. The circles indicate the simulation results. The critical point ac decreases largely with the increase of power n. The solid curve indicates the critical point ac calculated by the linear stability theory described lately. The simulation results are consistent with the values predicted by the linear stability theory. We study the scaling behavior of the coexisting phase just below the critical point. In the coexisting phase, we find the maximal headway Axmaxwithin the freely moving
464
T. Nagatani/Physica A 246 (1997) 460-470
n=l 2.0
1.5 a
1.0
0.5
0.0 3.0
4.0
Ax
5.0
6.0
7.0
Fig. 3. The phase boundaries and the critical points in the (Ax, a) plane for powers n = 1,2,3, and 4, where Ax is the headway and a is the sensitivity.
n=l 2.0
1.5 a
1.0
0.5
0.0 0
I
I
I
0.5
1.0
1.5
2.0
V
Fig. 4. The phase boundaries and the critical points in the (v,a) plane for powers n = 1,2,3, and 4, where v is the car velocity and a is the sensitivity.
phase and the m i n i m a l headway Axmin within the congested phase. In Fig. 6, the difference Axmax -Axrnin b e t w e e n the m a x i m a l h e a d w a y and the m i n i m a l headway is l o g - l o g plotted against the difference a c - a o f the sensitivity from the critical value ac for power n = 1,2, 3, 4, and 5. The difference Axmax - Axmin o f the headway
T. Nagatani/ Physica 4 246 (1997) 460-470
465
2.0 1.5 ac 1.0
0.5
0.0 0
I
I
I
I
I
l
2
3
4
5
n Fig. 5. The plot of the critical point ac against power n. The circles indicate the simulation results. The solid curve indicates the critical point ac calculated by the linear stability theory. .9 n=3/~//~ E
~Z/'X 0.1
,,ff~;/',
n=5 ,
,,,,,,
0.01
,
,
, , , ,
O. I
ac-a Fig. 6. The log-log plot of A x m a x - - Axmin against ac - a for power n = l, 2, 3, 4, and 5, where Axmax and Axmin are, respectively, the maximal and minimal headways and ac is the critical value of the sensitivity. scales as Axmax - AXmin ~ (ac - a) ~ with ~ = 0.51(n = 1), 0.79(n = 4)
and
0.64(n = 2),
0.83(n = 5 ) .
0.76(n = 3), (4)
Here we repaired part o f the systematic errors due to the finite system size b y w o r k i n g with an effective critical point which differs from the true ac but approaches the true critical point if the system size goes to infinity. Thus, we shift the critical point b y the value o f the inverse o f the system size for the effective critical point. Similarly, we find the m a x i m a l velocity /;max within the freely m o v i n g phase and the m i n i m a l velocity Vmin within the congested phase in the coexisting phase. Fig. 7
12 Nagatanil Physica A 246 (1997) 460-470
466
._= E
n=3t ~_~.o,¢
~>"
.=2
>
~.~o~'~-/~-3-'-~ n=4
E
/i'/"
0.1
,
0.01
,,,,,,I
,
,
,,,,,,
0.1
ac-a
Fig. 7. The l o g - l o g plot o f Pmax - Vmin against a¢ - a for power n = 1,2, 3, 4, and 5, where Vmax and /)min are, respectively, the maximal and minimal car velocities and ac is the critical value o f the sensitivity.
1.0 (X
0.5 o:0~
t
0
I 0
1
I 2
~
I
I
I
3
4
5
n Fig. 8. The plot of scaling exponents ~ a n d / 3 against power n. The data points are on the curve n/(n + 1 ).
shows the plot of difference /)max car velocity scales as
-
-
/)min against ac - a. Difference /)max
-
-
/)min of the
Pmax 1 /)min "~ ( a c - - a)/~
with [3 = 0.51(n = 1), 0.74(n = 3),
0.64(n = 2),
0.81(n = 4)
and
0.83(n = 5 ) .
(5)
Exponent ~ is consistent with exponent [3 within numerical accuracy. Fig. 8 shows the plot of exponents ~ and [3 against power n. The data points are on the curve n / ( n + 1). Thus, we find that the scaling exponents c~ and [3 of the headway and velocity are given by ct = [3 = n / ( n ÷ 1).
(6)
T. NagataniIPhysiea A 246 (1997) 460 470
467
n=4 /Y
.
~2
<
~o/O/n=l
- ~~y
0.1
0.01 "?' 0.01
,
, ,,,,,,I
ac-a
0.1
,
, ,,,,,,
1
Fig. 9. The log-log plot of Vp- Vp,c against ac - a for power n = l, 2, 3, and 4, where Vp is the propagation velocity of a jam and Vp,c is the propagation velocity at the critical point. W e study the scaling b e h a v i o r o f propagation velocity Vp o f the j a m (density wave). In the m o v i n g frame with propagation velocity Vp, the following conservation law holds (Vmin
--
Vp)/Axmin
=
(1)max
--
Vp)/AXmax.
(7)
Propagation velocity vp is obtained Vp = - ( Vmax/ Axma x -- Vmin/ Axmin )/( l / Axmin - 1/dXmax ).
(8)
B y taking the limit o f Eq. (8), propagation velocity Vp,c at the critical point is obtained vp, c = xc(dV(Ax)/dAX)Ax_xc - V(xc) = 4.0.
(9)
Critical p r o p a g a t i o n velocity Vp,c does not d e p e n d on p o w e r n. Fig. 9 shows the plot o f difference Vp - Vp,c against ac - a for n = 1,2, 3, and 4. The propagation velocity scales as Vp - Vp,c ~ (ac - a ) ~' with 7 = 1.0(n = 1),
1.27(n = 2),
1.46(n = 3)
and
1.56(n = 4 ) . (lO)
Fig. 10 shows the plot o f exponent Y against p o w e r n. The data points are on the curve 2n/(n + 1). Thus, we find that scaling exponent 7 o f the propagation velocity is given b y Y = 2n/(n + 1).
(11)
468
T. Nagatani/Physica A 246 (1997) 460-470
1.5
O
O
1.0
o.5 0
I
I
I
I
l
2
3
4
5
n Fig. 10. The plot of scaling exponent 7 against power n. The data points are on the curve 2n/(n + I).
Near the critical point, the characteristic length and time diverge as Lc~(ac-a)
-~
and
Tc~(ac-a)
-6.
(12)
The propagation velocity is given by /)p
-
-
Vp,c ~ Lc/Tc ,-~ (ac - a ) -~+6 .
(13)
Thus, exponent 6 o f the characteristic time is obtained 6 = 7 + ct = 3 n / ( n + 1).
(14)
When n = 1, exponent 6 agrees with value 6 = 3 obtained by the modified KdV [20].
3. Linear stability theory We apply the linear stability theory to our model. We consider the stability of a homogeneous traffic flow. The solution Xj, o of a homogeneous steady state is obtained by taking d 2 x j / d t 2 = 0: xj, o = b j + V ( b ) t
with b = L / N ,
(15)
where N is the number o f cars and b is the car spacing (identical headway). Let y j be a small deviation from the steady-state flow xj,0 : x j = xj, o + y j . Then, the linearized equation is obtained d 2y j / d t 2 = n a V ( b ) " - l ( f A y j - d y j / d t ) ,
(16)
32 Nagatani/Physica A 246 (1997) 460-470
469
where f = V~(b) is the derivative of the optimal velocity V ( A x ) at Ax = b. By expanding y j = e x p { i k j + (zl + iz2t}, the following equations of Zl and z2 are derived: z~ - z ~ +CZl - c f ( c o s k - 1) = 0,
(17)
2zlz2 + cz2 - c f s i n k = 0,
(18)
where c = n a V ( b ) "
1. By setting zt = 0, the neutral stability line is obtained
V ' ( b ) = n a V ( b ) n - 1 / ( c o s k + 1).
(19)
At k = 0 mode, the homogeneous traffic flow is unstable for a < 2 V ' ( b ) / n V ( b ) n-l .
(20)
When b = Xc, V~(xc) = 1 and V(xc) = 1. Then, if a > 2/n, a homogeneous traffic is always stable. The critical point is given by ac : 2/n .
(21)
The dependence of the critical point upon power n is shown by the solid curve in Fig. 5. The analytical result is consistent with the simulation results.
4. Summary We have generalized the optimal velocity traffic model to describe the spontaneous traffic j a m on a highway. We have investigated the phase transition and the scaling behavior near the critical point. We have shown the phase diagram among the freely moving phase, the coexisting phase, and the homogeneous congested phase for various values of power n in the generalized traffic model. It has been found that the critical point depends strongly upon power n. We have shown that the headway scales as (ac - a) ~, the car velocity scales as (ac - a)/~, and the jam propagation velocity scales as (ac - a) ~', where ~ = fl = n/(n + 1) and 7 = 2n/(n + 1). Also, we have applied the linear stability theory to the generalized traffic model. It has been found that the critical point predicted by the linear stability theory is consistent with that calculated by simulation.
References [1] D.E. Woll; M. Schreckenberg, A. Bachem (Eds.), Traffic and Grannular Flow, World Scientific, Singapore, 1996. [2] K. Nagel, M. Schreckenberg, J. Phys. ! 2 (1992) 2221. [3] O. Biham, A.A. Middleton, D.A. Levine, Phys. Rev. A 46 (1992) R6124. [4] T. Nagatani, Phys. Rev. E 48 (1993) 3290. [5] J.A. Cuesta, F.C. Matinez, J.M. Molera, A. Sanchez, Phys. Rev. E 48 (1993) 4175. [6] M. Bando, K. Hasebe, A. Nakayama, A. Shibata, Y. Sugiyama, Phys. Rev. E 51 (1995) 1035. [7] K.H. Chung, P.M. Hui, G.Q. Gu, Phys. Rev. E 51 (1995) 772.
470 [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
T. Naoatani/Physica A 246 (1997) 460-470
M. Schreckenberg, A. Schadschneider, K. Nagel, N. Ito, Phys. Rev. E 51 (1995) 2329. K. Nagel, H.J. Herrmann, Physica A 199 (1993) 254. G. Csanyi, J. Kertestz, J. Phys. A 28 (1995) 427. T. Nagatani, Phys. Rev. E 51 (1995) 922. I. Prigogine, R. Herman, Kinetic Theory of Vehicular Traffic, Elsevier, New York, 1971. S.L. Paveri-Fontana, Transport. Res. 9 (1975) 225. E. Ben-Naim, P.L. Krapivsky, S. Redner, Phys. Rev. E 50 (1994) 822. D. Helbing, Phys. Rev. E 53 (1996) 2366. D. Helbing, Physica A 233 (1996) 253. T. Nagatani, Physica A 237 (1997) 67. B.S. Kemer, P. Konhauser, Phys. Rev. E 48 (1993) 2335. B.S. Kemer, P. Konhauser, M. Schilke, Phys. Rev. E 51 (1995) 6243. K. Komatsu, S. Sasa, Phys. Rev. E 52 (1992) 5574.
Physica A 246 (1997) 460470
Phase transition and scaling in the generalized traffic flow model Takashi Nagatani
*
Division of Thermal Science, College of Engineering, Shizuoka University, Hamamatsu 432, Japan
Received 9 June 1997
Abstract The optimal velocity traffic model is generalized to the power-law type of the optimal velocity. The phase transition among the freely moving phase, the coexisting phase, and the homogeneous congested phase is investigated by computer simulation and the linear stability theory. Phase diagrams are obtained for various values of the power. It is shown that the phase boundary and the critical point depend strongly upon the power. The scaling behaviors of the headway, the car velocity, and the propagation velocity of jams are studied near the critical point. It is found that the headway and the car velocity scale as (ac - a) ~ and (ac - a) B with ~ = ~ = n/(n + 1) and the propagation velocity of jams scales as (ac - a ) ; ' with 7 = 2n/(n + 1) where a is the sensitivity, ac is its critical value, and n is the power of the model. PACS: 05.70.Fh; 05.70.jk; 89.40.+k Keywords: Traffic flow; Phase transition; Critical phenomena; Scaling
1. Introduction Recently, traffic problems have attracted considerable attention [1]. Three different approaches have been used to describe the collective properties o f traffic flow: a microscopic approach [ 2 - 1 1 ] , a gas kinetic approach [ 12-17], and a macroscopic approach [ 18,19]. The traffic flow models are classified into the deterministic and stochastic models [2,6]. A stochastic cellular automaton (CA) model was introduced by Nagel and Schreckenberg [2]. They showed that the start-stop waves (traffic jams) appear in the congested traffic region as observed in real freeway traffic. Bando et al. [6] proposed the deterministic optimal velocity model in which a car accelerates or decelerates according to the dynamical equation o f car motion with the optimal velocity function. They also found that the density waves appear in the congested traffic. Lately, Komatsu * E-mail: [email protected]. 0378-4371/97/$17.00 Copyright @ 1997 Elsevier Science B.V. All rights reserved PH S0378-4371 (97)00376-2
T. Nayatani/Physica A 246 (1997) 460~t70
461
and Sasa [20] derived the modified KdV equation from the optimal velocity model to describe the traffic jam. They showed that there is a critical point ac for the sensitivity a, and the characteristic length and time scale as (ac - a ) -J/2 and (ac - a ) -3/2 just below the critical point ac. Kerner and Konhauser [18] have also derived the KdV equation from the hydrodynamic equation of the traffic. In this paper, we generalize the optimal velocity model to study the phase transition and critical phenomena in the traffic flow. We investigate the phase transition among the freely moving phase, the coexisting phase, and the homogeneous congested phase by using computer simulation and the linear stability theory. We show that the phase boundary and the critical point depend on the power of the model. Furthermore, we study the scaling behavior of the traffic near the critical point. We show the dependence of the scaling exponents upon the power.
2. Generalized model and simulation
We generalize the optimal velocity model [6]. The optimal velocity model is described as follows. A car has the optimal velocity depending only upon the headway. A car is controlled in such a way that its car velocity adjusts the optimal velocity. Then, the equation of motion of car j is given by
d2xj/dt 2 = a { V ( A x j ) - dxj/dt}
with U ( A x j ) = tanh(Axj - Xc) + tanhxc,
(1)
where Axj(= Xi+l - xj) is the headway, a is the sensitivity of the car, and Xc is the safety distance. V(Axj) represents the optimal velocity. In this model, the maximal velocity and the minimal velocity are two and zero, respectively. We extend the optimal velocity model as follows. We assume that the acceleration of a car is proportional to the difference between the power of the optimal velocity and the power of the car velocity. The equation of motion of car j is given by
dZxj/dt 2 = a { V ( Axj )n
_
_
( dxi/d t ). }.
(2)
When n = 2, m(dxj/dt)2/2 represents the energy which car j has at time t, where m is the mass of the car. mV(Axj)2/2 represents the energy which a car moving with the optimal velocity has. A car accelerates or decelerates according to the difference between the energies. We study the phase transition and the scaling behavior in the generalized optimal velocity model described by Eq. (2). We perform a numerical simulation for the generalized optimal velocity model. Initially, cars are randomly distributed on the one-dimensional space with car density P0 and velocity v0. The boundary is periodic. In order to form a single jam, a hindrance is put at a point on the one-dimensional space. We assume that when a car reaches the hindrance it slows down instantly to low velocity vh. In time, a single jam is formed just behind the hindrance. After the jam is formed, the hindrance is removed. The jam propagates backward with constant propagation velocity Vp. Once the single jam is formed for a special range of the
T. Naoatani/Physica A 246 (1997) 460-470
462
1.0
n=2, xc=5 /~itical
point
0.9 a
0.8 0.7 0.6 0.5 3.0
4.0
5.0
6.0
7.0
zkx Fig. 1. The phase diagram in the (dx, a) plane for power n = 2 and safety distance xc = 5 where Ax is the headway and a is the sensitivity. The regions I, II, and IIl indicate the freely moving phase, the homogeneous congested phase, and the coexisting phase, respectively. Above the critical point, no phase transition occurs. The headways within and out of the jam are plotted by the circles for various values of sensitivity a, where car density is P0 = 0.2.
density, the jam is stable and does not break up. The jam has the form of the kinkantikink. The stable jam occurs at the intermediate density. For low density, a jam disappears in time, and all cars move freely with nearly maximal velocity. For high density, a jam also disappears in time, and a congested homogeneous traffic flow occurs. Thus, we can make a stable jam on the one-dimensional space for special values of density. We take the number N of cars as N = 100-400. The safety distance is set a s x c ~ 5.
Firstly, we calculate the traffic flow for power n = 2. Fig. 1 shows the plot of the headways within and out of the jam for various values of sensitivity a, where P0 = 0.2 and Vh = 1.0. When sensitivity a is less than critical value ac = 1.0, a single jam occurs. The headways within and out of the jam are plotted. When sensitivity a is larger than ac = 1.0, no jam appears. We calculate the density occurring at the jam. For low densities, the jam does not occur. The boundary between a jam and no jam is consistent with the curve on the right-hand side for low densities. Also, for high densities, the jam does not occur. The boundary is consistent with the curve on the left-hand side. In Fig. 1, the region I indicates the freely moving phase in which a car moves with about maximal velocity. The region II indicates the homogeneous congested traffic flow without jams. In the region III, the coexisting phase with the jam appears. In the coexisting phase, freely moving phase and the congested phase coexist. Thus, Fig. 1 shows the phase diagram among the freely moving phase, the coexisting phase, and the homogeneous congested phase for n = 2 and Xc = 5. The critical point is given by a = 1.0 and
72 Nagatani/Physica A 246 (1997) 460~170
463
n=2, xc=5 lcritical point
_
1.O 0.9 II
a
-/.
0.8
p
0.7
!
\
\
! !
0.6
0.5 0
\ I
I
I
0.5
1.0
1.5
2.0
V Fig. 2. The phase diagram in the (v,a) plane for the same values of parameters in Fig. 1. Regions 1, lI, and 11I indicate the freely moving phase, the homogeneous congested phase, and the coexisting phase, respectively. Xc = 5.0. The phase boundaries correspond to the saturation lines in the liquid-gas transition. Similarly, Fig. 2 shows the plot of the velocities within and out of the jam for various values of sensitivity a, where P0 = 0.2 and vh = 1.0. The density P0, safety distance Xc and the power n in Fig. 2 are same as the values in Fig. 1. Regions I, II and III indicate the freely moving phase, the homogeneous congested phase and the coexisting phase, respectively. In region III, the system separates into the two coexisting phases. The following quantities are considered as the order parameter S: S = ( A x f - Axj)
or
S = (vy - ~).),
(3)
where A x f is the headway out of the jam, Axj is the headway within the jam, vf is the car velocity out of the jam, and 4i is the car velocity within the jam. The order parameter S is different from zero below the critical point ac. Fig. 3 shows the phase boundaries and the critical points in the (Ax, a) plane for powers n = 1,2, 3, and 4. The critical point and the phase boundaries decrease with the increase of power n. Fig. 4 shows the phase boundaries and the critical points in the (v,a) plane for powers n = 1,2,3, and 4. These critical points and the phase boundaries show similar behaviors as in Fig. 3. Fig. 5 shows the plot of the critical point ac against power n. The circles indicate the simulation results. The critical point ac decreases largely with the increase of power n. The solid curve indicates the critical point ac calculated by the linear stability theory described lately. The simulation results are consistent with the values predicted by the linear stability theory. We study the scaling behavior of the coexisting phase just below the critical point. In the coexisting phase, we find the maximal headway Axmaxwithin the freely moving
464
T. Nagatani/Physica A 246 (1997) 460-470
n=l 2.0
1.5 a
1.0
0.5
0.0 3.0
4.0
Ax
5.0
6.0
7.0
Fig. 3. The phase boundaries and the critical points in the (Ax, a) plane for powers n = 1,2,3, and 4, where Ax is the headway and a is the sensitivity.
n=l 2.0
1.5 a
1.0
0.5
0.0 0
I
I
I
0.5
1.0
1.5
2.0
V
Fig. 4. The phase boundaries and the critical points in the (v,a) plane for powers n = 1,2,3, and 4, where v is the car velocity and a is the sensitivity.
phase and the m i n i m a l headway Axmin within the congested phase. In Fig. 6, the difference Axmax -Axrnin b e t w e e n the m a x i m a l h e a d w a y and the m i n i m a l headway is l o g - l o g plotted against the difference a c - a o f the sensitivity from the critical value ac for power n = 1,2, 3, 4, and 5. The difference Axmax - Axmin o f the headway
T. Nagatani/ Physica 4 246 (1997) 460-470
465
2.0 1.5 ac 1.0
0.5
0.0 0
I
I
I
I
I
l
2
3
4
5
n Fig. 5. The plot of the critical point ac against power n. The circles indicate the simulation results. The solid curve indicates the critical point ac calculated by the linear stability theory. .9 n=3/~//~ E
~Z/'X 0.1
,,ff~;/',
n=5 ,
,,,,,,
0.01
,
,
, , , ,
O. I
ac-a Fig. 6. The log-log plot of A x m a x - - Axmin against ac - a for power n = l, 2, 3, 4, and 5, where Axmax and Axmin are, respectively, the maximal and minimal headways and ac is the critical value of the sensitivity. scales as Axmax - AXmin ~ (ac - a) ~ with ~ = 0.51(n = 1), 0.79(n = 4)
and
0.64(n = 2),
0.83(n = 5 ) .
0.76(n = 3), (4)
Here we repaired part o f the systematic errors due to the finite system size b y w o r k i n g with an effective critical point which differs from the true ac but approaches the true critical point if the system size goes to infinity. Thus, we shift the critical point b y the value o f the inverse o f the system size for the effective critical point. Similarly, we find the m a x i m a l velocity /;max within the freely m o v i n g phase and the m i n i m a l velocity Vmin within the congested phase in the coexisting phase. Fig. 7
12 Nagatanil Physica A 246 (1997) 460-470
466
._= E
n=3t ~_~.o,¢
~>"
.=2
>
~.~o~'~-/~-3-'-~ n=4
E
/i'/"
0.1
,
0.01
,,,,,,I
,
,
,,,,,,
0.1
ac-a
Fig. 7. The l o g - l o g plot o f Pmax - Vmin against a¢ - a for power n = 1,2, 3, 4, and 5, where Vmax and /)min are, respectively, the maximal and minimal car velocities and ac is the critical value o f the sensitivity.
1.0 (X
0.5 o:0~
t
0
I 0
1
I 2
~
I
I
I
3
4
5
n Fig. 8. The plot of scaling exponents ~ a n d / 3 against power n. The data points are on the curve n/(n + 1 ).
shows the plot of difference /)max car velocity scales as
-
-
/)min against ac - a. Difference /)max
-
-
/)min of the
Pmax 1 /)min "~ ( a c - - a)/~
with [3 = 0.51(n = 1), 0.74(n = 3),
0.64(n = 2),
0.81(n = 4)
and
0.83(n = 5 ) .
(5)
Exponent ~ is consistent with exponent [3 within numerical accuracy. Fig. 8 shows the plot of exponents ~ and [3 against power n. The data points are on the curve n / ( n + 1). Thus, we find that the scaling exponents c~ and [3 of the headway and velocity are given by ct = [3 = n / ( n ÷ 1).
(6)
T. NagataniIPhysiea A 246 (1997) 460 470
467
n=4 /Y
.
~2
<
~o/O/n=l
- ~~y
0.1
0.01 "?' 0.01
,
, ,,,,,,I
ac-a
0.1
,
, ,,,,,,
1
Fig. 9. The log-log plot of Vp- Vp,c against ac - a for power n = l, 2, 3, and 4, where Vp is the propagation velocity of a jam and Vp,c is the propagation velocity at the critical point. W e study the scaling b e h a v i o r o f propagation velocity Vp o f the j a m (density wave). In the m o v i n g frame with propagation velocity Vp, the following conservation law holds (Vmin
--
Vp)/Axmin
=
(1)max
--
Vp)/AXmax.
(7)
Propagation velocity vp is obtained Vp = - ( Vmax/ Axma x -- Vmin/ Axmin )/( l / Axmin - 1/dXmax ).
(8)
B y taking the limit o f Eq. (8), propagation velocity Vp,c at the critical point is obtained vp, c = xc(dV(Ax)/dAX)Ax_xc - V(xc) = 4.0.
(9)
Critical p r o p a g a t i o n velocity Vp,c does not d e p e n d on p o w e r n. Fig. 9 shows the plot o f difference Vp - Vp,c against ac - a for n = 1,2, 3, and 4. The propagation velocity scales as Vp - Vp,c ~ (ac - a ) ~' with 7 = 1.0(n = 1),
1.27(n = 2),
1.46(n = 3)
and
1.56(n = 4 ) . (lO)
Fig. 10 shows the plot o f exponent Y against p o w e r n. The data points are on the curve 2n/(n + 1). Thus, we find that scaling exponent 7 o f the propagation velocity is given b y Y = 2n/(n + 1).
(11)
468
T. Nagatani/Physica A 246 (1997) 460-470
1.5
O
O
1.0
o.5 0
I
I
I
I
l
2
3
4
5
n Fig. 10. The plot of scaling exponent 7 against power n. The data points are on the curve 2n/(n + I).
Near the critical point, the characteristic length and time diverge as Lc~(ac-a)
-~
and
Tc~(ac-a)
-6.
(12)
The propagation velocity is given by /)p
-
-
Vp,c ~ Lc/Tc ,-~ (ac - a ) -~+6 .
(13)
Thus, exponent 6 o f the characteristic time is obtained 6 = 7 + ct = 3 n / ( n + 1).
(14)
When n = 1, exponent 6 agrees with value 6 = 3 obtained by the modified KdV [20].
3. Linear stability theory We apply the linear stability theory to our model. We consider the stability of a homogeneous traffic flow. The solution Xj, o of a homogeneous steady state is obtained by taking d 2 x j / d t 2 = 0: xj, o = b j + V ( b ) t
with b = L / N ,
(15)
where N is the number o f cars and b is the car spacing (identical headway). Let y j be a small deviation from the steady-state flow xj,0 : x j = xj, o + y j . Then, the linearized equation is obtained d 2y j / d t 2 = n a V ( b ) " - l ( f A y j - d y j / d t ) ,
(16)
32 Nagatani/Physica A 246 (1997) 460-470
469
where f = V~(b) is the derivative of the optimal velocity V ( A x ) at Ax = b. By expanding y j = e x p { i k j + (zl + iz2t}, the following equations of Zl and z2 are derived: z~ - z ~ +CZl - c f ( c o s k - 1) = 0,
(17)
2zlz2 + cz2 - c f s i n k = 0,
(18)
where c = n a V ( b ) "
1. By setting zt = 0, the neutral stability line is obtained
V ' ( b ) = n a V ( b ) n - 1 / ( c o s k + 1).
(19)
At k = 0 mode, the homogeneous traffic flow is unstable for a < 2 V ' ( b ) / n V ( b ) n-l .
(20)
When b = Xc, V~(xc) = 1 and V(xc) = 1. Then, if a > 2/n, a homogeneous traffic is always stable. The critical point is given by ac : 2/n .
(21)
The dependence of the critical point upon power n is shown by the solid curve in Fig. 5. The analytical result is consistent with the simulation results.
4. Summary We have generalized the optimal velocity traffic model to describe the spontaneous traffic j a m on a highway. We have investigated the phase transition and the scaling behavior near the critical point. We have shown the phase diagram among the freely moving phase, the coexisting phase, and the homogeneous congested phase for various values of power n in the generalized traffic model. It has been found that the critical point depends strongly upon power n. We have shown that the headway scales as (ac - a) ~, the car velocity scales as (ac - a)/~, and the jam propagation velocity scales as (ac - a) ~', where ~ = fl = n/(n + 1) and 7 = 2n/(n + 1). Also, we have applied the linear stability theory to the generalized traffic model. It has been found that the critical point predicted by the linear stability theory is consistent with that calculated by simulation.
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M. Schreckenberg, A. Schadschneider, K. Nagel, N. Ito, Phys. Rev. E 51 (1995) 2329. K. Nagel, H.J. Herrmann, Physica A 199 (1993) 254. G. Csanyi, J. Kertestz, J. Phys. A 28 (1995) 427. T. Nagatani, Phys. Rev. E 51 (1995) 922. I. Prigogine, R. Herman, Kinetic Theory of Vehicular Traffic, Elsevier, New York, 1971. S.L. Paveri-Fontana, Transport. Res. 9 (1975) 225. E. Ben-Naim, P.L. Krapivsky, S. Redner, Phys. Rev. E 50 (1994) 822. D. Helbing, Phys. Rev. E 53 (1996) 2366. D. Helbing, Physica A 233 (1996) 253. T. Nagatani, Physica A 237 (1997) 67. B.S. Kemer, P. Konhauser, Phys. Rev. E 48 (1993) 2335. B.S. Kemer, P. Konhauser, M. Schilke, Phys. Rev. E 51 (1995) 6243. K. Komatsu, S. Sasa, Phys. Rev. E 52 (1992) 5574.