It is worth noting that the two expressions (1) and (17) lead to the same leading-order expression (13) for τ$_{int}$$_{,n}$ . Indeed, inserting (17) into (9) we obtain Since ( e − 1) = O (1) for τ$_{exp}$ = O (1), the terms arising from the exponential decay of ρ$_{n}$ ( t ) are O ( ε ) in the low and high density phases. Some remarks are in order. Firstly, for the determin- istic ( p = 0) parallel-update TASEP, the static variance var( n ) can be computed analytically from the known re- sults for the two-point function [3]. In the low density phase it is given by and for the high density region α is replaced by β . We expect that var( n ) = O (1 /L ) would remain true when p > 0, and indeed for v$_{max}$ > 1 as well. In general, therefore, we expect the prefactor in (20) to be O (1) in L . We begin this section by comparing the power spec- trum found in [12] with the Fourier transform of (1). We then present the exact result for the collective velocity for TASEP, before presenting the results of our simulations. Finally, we note that [12] fit (19) to their data with a very small value of A/D . This small value follows from the fact that the numerical simulations in [12] were performed along the mean field line of the TASEP with random sequential update, where, theoretically, var( n ) is identically zero. It is surprising that [12] were still able to extract a meaningful signal on this line. Let N denote the number of occupied sites in a TASEP system, and let ̂ N ( ω ) denote the discrete Fourier trans- form of a particular time series N$_{1}$, N$_{2}$, . . . , N$_{T}$ , as de- fined in (7). The quantity I ( ω ) := T 〈 ∣ ∣ ̂ N ( ω ) ∣ $^{2}$〉 is what [12] refer to as the power spectrum of N . They find that for the continuous-time TASEP in the low-density phase where A, D and v are parameters, which [12] set empiri- cally to v ≈ 0 . 4, D ≈ 20 and A ≈ 1 / 500. We now attempt to compare (19) with the corre- sponding result derived from (1). From (8) we see that f$_{N}$ ( ω ) ∼ 〈 ∣ ∣ ̂ N ( ω ) ∣ $^{2}$〉 as T → ∞ , hence we should compare (19) with f$_{N}$ ( ω ) = L $^{2}$f$_{n}$ ( ω ), where f$_{n}$ ( ω ) is computed via (1). Although our empirical observations of the be- havior (1) were made in the discrete time case of fully- parallel updates, (1) can be interpreted as a well defined continuous function on R . In fact, the fully-parallel up- date rule becomes equivalent to the random sequential update in the limit ε → 0 of rescaled variables 1 − p = ε , α = ˜ αε and β = ˜ βε . Here, ˜ α and ˜ β are the usual injection and extraction rates of the TASEP in continuous time. To compare with the continuous time result (19), we compute f$_{n}$ ( ω ) via the continuous-time Fourier trans- form, so that (1) and (3) predict Now, since ω = 2 πm/T , for sufficiently large T we have exp( Dω $^{2}$L/v$^{3}$) ≈ 1. This is exactly the regime used by [12] in their Fig. 3 ( L = 1000 or L = 32000 and T = 10 $^{6}$). Therefore, in this regime we can identify (19) with (20) if v = | v$_{c}$ | and The negativity of the collective velocity in the high- density phase is simply due to the fact that it is the prop- agation of holes from right to left, rather than of particles from left to right, that is important in this phase. Using these exact expressions for v$_{c}$ the expression (3) now becomes The collective velocity [13] is the drift of the center of mass of a momentary local fluctuation of the stationary state, and is related to the current (flow) J and bulk density ρ$_{b}$ via v$_{c}$ = ∂J ( ρ$_{b}$ ) /∂ρ$_{b}$ . An exact expression for v$_{c}$ ( α, β, p ) is available [2] for the case of parallel-update TASEP. If we define, for convenience, the function The stationary distribution of the TASEP with fully- parallel updates [2, 3] is known exactly. In particular, if α < β, 1 − √ p such TASEPs reside in a low-density phase, while for β < α, 1 − √ p a high-density phase results, with α = β < 1 − √ p defining a coexistence line of the two phases (corresponding to a first order phase transition). For α, β > 1 − √ p by contrast, the system resides in a maximum-current phase, in which the density is precisely 1 / 2. 2 τ$_{int}$$_{,n}$ = τ + ( ε (1 − ε ) + 2 ε e 1 /τ$_{exp}$ − 1 ) 1 τ . (18) 1 /τ$_{exp}$ − 1 var( n ) = α (1 − α ) (1 + α ) 3 1 L + O ( L − $^{2}$) , (22) I ( ω ) T ≈ 2 v ω 2 A D [ 1 − e − Dω $^{2}$L/v3 cos ( L ω v )] , (19) g ( x, p ) = (1 − p )((1 − x ) 2 − p ) (1 x ) 2 + p (2 x 1) , (23) v$_{c}$ ( α, β, p ) = { g ( α, p ) , low density phase , − g ( β, p ) , high density phase . (24) then f$_{N}$ ( ω ) = 2 | v$_{c}$ | ω 2 var( n ) L [ 1 − cos ( L ω | v$_{c}$ | )] . (20) A D = var( n ) L. (21) τ = { L/g ( α, p ) , α < β, 1 − √ p, L/g ( β, p ) , β < α, 1 p. (25) 3.2. Collective velocity 3.1. Power spectrum 3. TASEP 4