Theory of ω−4/3 law of the power spectrum in dissipative flows

It is demonstrated that ω−4/3 law of the power spectrum with the angular frequency ω in dissipative flows is produced by the emission of dispersive waves from the antikink of an congested domain. The analytic theory predicts the spectrum is proportional to ω for relatively low frequency and ω−4/3 for high frequency. 05.40.-a,45.70.Mg 05.60.-k Typeset using REVTEX 1 Recently, much attention has been attracted to collective dynamics of dissipative particles [1,2]. In particular, physics of granular flows [3–6] and traffic flows [7,8] are developing subjects. In such dissipative flows, we often observe the coexistence of congested regions and dilute regions. It is important to know the mechanism of the emergence of congestion of traffic and granular flows. Although we have some exact results on the formation of congested domains in one-dimensional traffic flow [9,10], we still do not understand the details of the fluctuation of dissipative flows. In experiments of dissipative flows, we usually measure the power spectrum which is the Fourier transform of the auto-correlation function. It is known that traffic flows and granular flows in a pipe have the power spectra obeying ω law with the angular frequency ω [11,12]. Several years ago, Moriyama et al. [13] have confirmed that granular flow in a pipe should have the spectrum with β = 4/3. We also expect that the power spectrum obeying ω law is universal for dissipative flows in the coexistence of congested-flow and sparse-flow. [3,13–16] This law is robust in the experiments of granular flows, which can be observed without tuning of a suitable set of parameters. [13,16] Although the previous papers [3,13] proposed the mechanism of ω law, their derivation might be incomplete. We can list several defects in their derivation: (i) They assumed that the system is in a weakly stable region of homogeneous state. However, ω can be commonly observed in the case of the coexistence between congestion and the sparse-flow. The power exponent β is drastically small when there are no definite domains in systems. [15,16] (ii) The experiments [13,16] suggest that ω law is robust without fine-tuning when phase separations take place, but the theory assumes that the system is in the vicinity of the neutral curve of the linear stability analysis. (iii) The theoretical spectrum depends on the wave number but there is no wave-number dependence in the actual observation in experiments. [3,13] (iv) Although the theory assumes that the relaxation process of internal structures, it is not clear what the relevant relaxation process is. Therefore, one is skeptical of the validity of the previous theory to explain ω law. Recently, Takesue et al. [17] have solved a kink-diffusion problem in the totally asym2 metric simple exclusion process (TASEP) [18] and derived ω law of the power spectrum. Although TASEP contains only a kink which connects one congested domain with a dilute region, their analysis is suggestive to understand more realistic situations in traffic and granular flows. In this Communication, we thus try to re-derive ω law in the case of coexistence between congestion and sparse-flow. In order to proceed the analysis we should recall that all of one-dimensional models for traffic and granular flows in weakly unstable regions can be described by trains of quasisolitons stabilized by small dissipations. [3,19–21] In general, a dilute region is connected with a congested region by asymmetric interfaces [3,20,21] which may be characterized by the soliton equation. [19] We call a front interface the kink and a backward interface the antikink. The antikink is not stable in the actual situations and emits dispersive waves backward. The waves are caught by the next domain. In the simplest situation, we can ignore the widths of the kinks and antikinks which may be much smaller than the typical domain size. From the observation of experiments the power spectrum may not be related to the formation process of domains but be characterized by the emission of dispersive waves from an antikink. Thus, we ignore the formation of a congested domain but focus on the decay process of the domain. We also map the model onto a one-dimensional space, where the position fixed in an experimental system is denoted by x and the system size is L and the boundaries are located at x = ±L/2. For simplicity, we place a detector to measure the power spectrum at x = 0, i.e. the center of the system. Let us introduce the packing fraction φ(x, t) ≡ n(x, t)/n0 where n0 is the maximum density. If we assume that an idealistic congested domain exists in the system at time t = 0, the packing fraction is given by φ(x, t = 0) = 1 between x = x0 and x = x0 + l, and φ(x, 0) = 0 for otherwise, where l and x0 are the size of the domain and the position of an antikink at t = 0, respectively. The equivalent expression is