Nonequilibrium phase separation in traffic flows

Traffic jam in an optimal velocity model with the backward reference function is analyzed. An analytic scaling solution is presented near the critical point of the phase separation. The validity of the solution has been confirmed from the comparison with the simulation of the model. Typeset using REVTEX e-mail: [email protected] e-mail: [email protected] 1 Recently, the importance of cooperative behavior in dissipative systems consisting of discrete elements has been recognized among physists. As a result, granular materials have been studied extensively from physical point of views. [1]. Similarly, to know the properties of traffic jams in daily life is also an attractive subject not only for engineers but also physists [2]. There are some similarities between two phenomena in particular in the simplest situation where cars and particles are respectively confined in a highway and a long tube. Thus, it is interesting to clarify common and universal mathematical structure behind these phenomena. We propose here a model of the traffic flow ẍn = a[U(xn+1 − xn)V (xn − xn−1)− ẋn], (1) where xn and a are the positions of n th car, and the sensitivity, respectively. This model contains the psychological effect of drivers. Namely, the driver of xn takes care of not only the distance ahead xn+1−xn but also the backward distance xn−xn−1. The optimal velocity function U should be a monotonic increasing function of the distance of xn+1−xn and V −1 should be a monotonic decreasing function of xn − xn−1. Thus, we adopt U(h) = tanh(h− 2) + tanh(2); V (h) = 1 + f0(1− tanh(h− 2)) (2) for the later explicit calculation. We put these optimal velocity functions as the product form UV in (1), because the driver of xn cannot accelerate the car without enough the forward distance xn+1 − xn even when the distance xn − xn−1 becomes short. This model (1) with (2) is the generalization of the optimal velocity (OV) model proposed by Bando et al. [3] ẍn = a[U(xn+1 − xn)− ẋn]. (3) Our model is also similar to the model of granular flow in a one dimensional tube ẍn = ζ [Ũ(xn+1 − xn−1)− ẋn] + g(xn+1 − xn)− g(xn − xn−1) (4) 2 where the explicit forms of Ũ and the force g are not important in out argument. Although real systems contain variety of cars and particles and higher dimensional effects, we believe that the most essential parts of both traffic flows and granular pipe flows can be understood by pure one dimensional models (1) and (4). The reason is as follows: It is known [4] that model (4) supplemented by the white noise produces a power law in the frequency spectrum of the density correlation function S(q, ω) ∼ ω, whose exponent 4/3 is very close to the experimental value [4] and that by the lattice-gas automata simulation [5]. From this success the essential effects of randomness such as passing cars and variety of cars seem to be represented by the adding white noise to the models (1) and (4). Komatsu and Sasa [6] reveal that the original OV model can be reduced into the modified Korteweg-de Vries (MKdV) equation at the critical point (the averaged car distance h = 2) for the phase separation. They also show that symmetric kink solitons deformed by dissipative corrections describe a bistable phase separation. The exactly solvable models in which the essential characteristics of the optimal velocity model are included have been proposed [7]. However, as will be shown, the generalized optimal velocity model (1) and granular model (4) as well as the fluid model of traffic flows by Kerner and Konhäuser [8] and two fluid models in granular flows [9] are not reduced to MKdV equation but exhibit the phase separations between a linearly unstable phase and a stable phase [10]. Thus, there is a wider universality class of dissipative particle dynamics which contains (1), (4) and fluid models [8,9]. The aim of this Letter is to obtain an analytic scaled solution of (1). To demonstrate quantitative validity of our analysis we will compare it with the result of our simulation. After the completion of our analysis on (1), we will briefly discuss the relation of the result and the expected results in (4) and fluid models. Let us rewrite (1) as r̈n = a[U(h + rn+1)V (h+ rn)− U(h+ rn)V (h+ rn−1)− ṙn] (5) where h is the averaged distance of successive cars and rn is xn+1 − xn − h . 3 Now, let us consider the linear stability of (5). The linearized equation of (5) around rn(t) = 0 is given by r̈n = a[U (h)V (h)(rn+1 − rn) + U(h)V (h)(rn − rn−1)− ṙn] (6) where the prime refers to the differentiation with respect to the argument. With the aid of the Fourier transformation rq(t) = 1 N ∑N n=1 exp[−iqnh]rn(t) with q = 2πm/Nh and the total number of cars N we can rewrite (6) as (∂t − σ+(q))(∂t − σ−(q))rq(t) = 0 (7) with σ±(q) = − a 2 ± √ (a/2)2 − aDh[U, V ](1− cos(qh)) + ia(UV )′ sin(qh), (8) where we drop the argument h in U and V . Dh[U, V ] ≡ U (h)V (h) − U(h)V (h) denotes Hirota’s derivative. The solution of the initial value problem in (7) is the linear combination of terms in proportion to exp[σ+(q)t] and exp[σ−(q)t]. The mode in proportion to exp[σ−(q)t] can be interpreted as the fast decaying mode, while the term in proportion to exp[σ+(q)t] is the slow and more important mode. The violation of the linear stability of the uniform solution in (6) is equivalent to Re[σ+(q)] ≥ 0. Assuming qh 6= 0 (qh = 0 is the neutral mode), the instability condition is given by 2(UV ) cos( qh 2 ) ≥ aDh[U, V ]. Thus, the most unstable mode exists at qh → 0 and the neutral curve for long wave instability is given by a = an(h) ≡ 2(UV ) Dh[U, V ] . (9) The neutral curve in the parameter space (a, h) is shown in Fig.1 for f0 = 1/(1 + tanh(2)) in (2). For later convenience, we write the explicit form of the long wave expansion of σ+ in the vicinity of the neutral line σ+(q) = ic0qh− c0 a− an(h) an(h) (qh) − i 3 6 c0 − (qh) 4an(h) c0 +O((qh) ) (10)