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The system density, n , which is simply the fraction of sites which are occupied, is an important quantity in many applications, including traffic modeling. The rela- tionship between density and flow is known as the fun- damental diagram in the traffic engineering literature. While the stationary-state expectation 〈 n 〉 of n is well understood for the NaSch model, and in fact known rig- orously for the TASEP, the dynamic behavior of n$_{t}$ is non-trivial. In this article we numerically study the au- tocorrelation function ρ$_{n}$ ( t ) := ( 〈 n$_{0}$n$_{t}$ 〉 − 〈 n 〉 $^{2}$) / var( n ) of the general NaSch model, and find a very simple form for its finite-size scaling. Up to very small corrections, our simulations show that in both the high and low density phases we simply have 

for some constant τ ∝ L . 

The linear decay in (1) is in sharp contrast to the usual exponential decay typical of equilibrium systems. In fact, as discussed in section 2.2, there are good theoretical rea- sons to believe that ρ$_{n}$ ( t ) must ultimately decay exponen- tially on sufficiently long time scales, rather than exhibit the strictly finite support suggested by (1). However, as demonstrated by the simulations in sections 3 and 4, any corrections to the finite-support behavior displayed in (1) are extremely weak, and in practice (1) provides a very accurate approximation to the behavior of ρ$_{n}$ ( t ) through- out the low and high density phases. In particular, (1) provides a very good approximation to ρ$_{n}$ ( t ) for values of p relevant for traffic modeling. 

The Fourier series of ρ$_{n}$ ( t ) gives the power spectrum of n , and we note that taking the Fourier series of (1) does indeed produce oscillations as reported in [12]. Indeed, we have 

The discussion in [12] focused on the case v$_{max}$ = 1, with random sequential updates. However, our simulations show that (1), and hence (2), hold more generally for the NaSch model with arbitrary v$_{max}$ . 

The specific form (1) of the autocorrelation function has some interesting consequences for the design of Monte Carlo simulations. In particular, as discussed in sec- tion 2, assuming the validity of (1) we immediately have τ = 2 τ$_{int}$$_{,n}$ where τ$_{int}$$_{,n}$ is the integrated autocorrelation time of n . The integrated autocorrelation time can be interpreted loosely as the number of time steps between “effectively independent” samples. It is therefore reason- able to conjecture that the parameter τ should equal the amount of time it takes a fluctuation of the stationary state to traverse the system. If we let v denote the speed of such a fluctuation then we might reasonably expect 

## 1.1. Density autocorrelations 

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that τ = L/v . In section 3 we present numerical results that strongly suggest that in fact, for TASEP, we have 

where v$_{c}$ ( α, β, p ), the collective velocity [2, 13], is known exactly. The results (1) and (3) are consistent with the suggestions in [12] that the physical origins of the ob- served oscillations in the power spectrum of n are related to the time needed for a fluctuation to traverse the entire system. 

Furthermore, while no exact expression for v$_{c}$ ( α, β, p, v$_{max}$ ) is known for the general NaSch model, the simulations presented in section 4 demonstrate that the scaling form (3) extends to general v$_{max}$ . In addi- tion, in the deterministic limit ( p = 0) simple physical arguments produce an exact relationship between v$_{c}$ and v$_{max}$ which is in excellent agreement with the numerical results. 

The remainder of this article is organized as follows. In section 2, we briefly review some pertinent general theory relating to autocorrelations and then discuss some general consequences of (1). In section 3, we present our numerical evidence supporting (1) and (3) for TASEP, and also describe the exact expression for v$_{c}$ ( α, β, p ) in this case. We also explain relationship between (1) and (3) and the results presented in [12]. In section 4, we briefly review the definition of the NaSch model before presenting our numerical results for ρ$_{n}$ ( t ) in this case. Finally, we conclude in section 5 with a discussion. 

We begin by briefly recalling some standard definitions and results. Consider a Monte Carlo simulation of an ergodic Markov chain, and assume that sufficient time has passed that the system has reached stationarity. If one now measures an observable X at each time step one obtains a stationary time series X$_{1}$, X$_{2}$, . . . whose autocovariance function is defined to be 

The expectation 〈·〉 here is with respect to the stationary distribution, and we note that C$_{X}$ (0) = var( X ). The corresponding autocorrelation function is then defined as 

Finally, assuming C$_{X}$ ( t ) to be absolutely summable, its Fourier transform defines the spectral density 

The spectral density is closely related to the Fourier transform of the time series. Specifically, given any sta- 

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## 2. AUTOCORRELATIONS 

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