arXiv:1001.2081v2 [cond-mat.stat-mech] 7 Jul 2010 Autocorrelations in the totally asymmetric simple exclusion process and Nagel-Schreckenberg model Jan de Gier ∗ Department of Mathematics and Statistics, The University of Melbourne, VIC 3010, Australia Timothy M. Garoni † ARC Centre of Excellence for Mathematics and Statistics of Complex Systems, Department of Mathematics and Statistics, The University of Melbourne, VIC 3010, Australia Zongzheng Zhou Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China (Dated: July 7, 2010) We study via Monte Carlo simulation the dynamics of the Nagel-Schreckenberg model on a finite system of length L with open boundary conditions and parallel updates. We find numerically that in both the high and low density regimes the autocorrelation function of the system density behaves like 1 −| t | /τ with a finite support [ − τ, τ ]. This is in contrast to the usual exponential decay typical of equilibrium systems. Furthermore, our results suggest that in fact τ = L/c , and in the special case of maximum velocity v max = 1 (corresponding to the totally asymmetric simple exclusion process) we can identify the exact dependence of c on the input, output and hopping rates. We also emphasize that the parameter τ corresponds to the integrated autocorrelation time, which plays a fundamental role in quantifying the statistical errors in Monte Carlo simulations of these models. PACS numbers: 05.40-a, 05.60cd, 05.70Ln The totally asymmetric simple exclusion process (TASEP) [1] is a simple transport model, of fundamental importance in nonequilibrium statistical mechanics. In addition to its mathematical richness, it has applications ranging from molecular biology to freeway traffic. A TASEP consists of a chain of length L , with each site being either occupied by a particle or not, on which particles hop from left to right. See Fig. 1. If site i = 1 is vacant a particle will enter the system with probability α . If site i = L is occupied the particle will leave the system with probability β . In the bulk of the system, a particle on site i will hop to site i +1 with probably 1 − p provided i + 1 is vacant, otherwise it remains at site i . TASEPs exhibit boundary-induced phase transitions, governed by the parameters α , β and p . In general, for a given p , there exist three possible phases, depending on α and β : a low-density phase, a high-density phase, and a maximum-current (or maximum-flow) phase. In the context of traffic models, it is most appro- priate to update all sites in parallel at each time-step. The stationary distribution of the TASEP with fully- parallel updates [2, 3] is known exactly. (For reviews of the stationary properties of TASEPs with random se- quential updates see [4, 5].) The Nagel-Schreckenberg (NaSch) model [6] is an important generalization of the parallel-update TASEP, in which particles can move up to v$_{max}$ ∈ N sites per time step. The NaSch model is generally considered to be the minimal model for traffic on freeways [7]. While many results are known rigorously for the TASEP, our understanding of the NaSch model and its further generalizations typically rely on numeri- cal simulation. This is particularly true of traffic network models, in which the NaSch model is often a component (see for example [8–10]). In the current article we focus on dynamic (auto)correlation functions. The autocorrelations of the TASEP with random sequential update have been stud- ied in [11, 12] and display a separation of time scales between relaxation of local density fluctuations and col- lective domain wall motion. In particular, it was re- cently observed [12] that the TASEP with random se- quential update exhibits non-trivial oscillations in the power spectrum of the system density, in the low and high density phases. In this article, we further elucidate the nature of these non-trivial oscillations, and demon- strate that they extend to the NaSch model generally. We emphasize that all the simulations performed in this work used fully-parallel updates, including our simula- tions of TASEP (which we view as the special case of the NaSch model with v$_{max}$ = 1). $^{∗}$Electronic address: jdgier@unimelb.edu.au $^{†}$Electronic address: t.garoni@ms.unimelb.edu.au 1. INTRODUCTION α 1 − p β FIG. 1: A TASEP with L = 10.