TASEP on a Ring in Sub-relaxation Time Scale

Abstract

Interacting particle systems in the KPZ universality class on a ring of size L with O(L) number of particles are expected to change from KPZ dynamics to equilibrium dynamics at the so-called relaxation time scale $t=O(L^{3/2})$. In particular the system size is expected to have little effect to the particle fluctuations in the sub-relaxation time scale $1\ll t\ll L^{3/2}$. We prove that this is indeed the case for the totally asymmetric simple exclusion process (TASEP) with two types of initial conditions. For flat initial condition, we show that the particle fluctuations are given by the Airy$_1$ process as in the infinite TASEP with flat initial condition. On the other hand, the TASEP on a ring with step initial condition is equivalent to the periodic TASEP with a certain shock initial condition. We compute the fluctuations explicitly both away from and near the shocks for the infinite TASEP with same initial condition, and then show that the periodic TASEP has same fluctuations in the sub-relaxation time scale.

Appendix: Density Profile of TASEP with Periodic Step Initial Condition

In this appendix, we summarize the macroscopic picture of the periodic TASEP and the infinite TASEP with periodic step initial condition (1.9) via solving the Burger’s equation. We state the density profile, and the locations of the shock and any given particle as time t without much details since the computation is standard. Furthermore we do not study the issue of the convergence in the hydrodynamic limit to the Burger’s solution; the computations in this Appendix are used only to provide intuitive ideas and are not used in the proofs of the theorems.

We assume that $0<\rho \le \frac{1}{2}$. Consider the Burger’s equation for the infinite TASEP

$$\begin{aligned} \frac{{\mathrm d}}{{\mathrm d}t}q(x;t) +\frac{{\mathrm d}}{{\mathrm d}x}\left( q(x,t)(1-q(x,t))\right) =0 \end{aligned}$$

(5.1)

with the periodic initial condition

$$\begin{aligned} q(x;0)={\left\{ \begin{array}{ll} 1,\qquad &{}-\rho \le x-[x]-1\le 0,\\ 0,\qquad &{}0< x-[x]< 1-\rho , \end{array}\right. } \end{aligned}$$

(5.2)

where [x] means the largest integer which is less than or equal to x. The entropy solution q(x; t) represents the local density profile at location xL and time tL. Note that the solution is also periodic, $q(x+1, t)=q(x,t)$, and hence q(x; t) also represents the local density profile for the periodic TASEP with the same initial condition.

We now solve the above Burger’s equation explicitly. Due to the periodicity, we state the formula of q(x; t) for x only in an interval of length 1.

For time $t\le \frac{1}{4\rho }$, there is no shock and the solution is given by the following: For $0\le t\le \rho$,

$$\begin{aligned} q(x;t)= {\left\{ \begin{array}{ll} 1,&{} -\rho \le x\le -t,\\ \frac{1}{2}-\frac{1}{2t}x,&{} -t<x<t,\\ 0,&{} t\le x<1-\rho . \end{array}\right. } \end{aligned}$$

(5.3)

For $\rho \le t\le \frac{1}{4\rho }$,

$$\begin{aligned} q(x,t)= {\left\{ \begin{array}{ll} \displaystyle \frac{1}{2}-\frac{1}{2t}x, &{}\quad -2\sqrt{\rho t} +t\le x\le t,\\ 0, &{}\quad t<x< -2\sqrt{\rho t} +t+1. \end{array}\right. } \end{aligned}$$

(5.4)

The shocks are generated at time $t=\frac{1}{4\rho }$ at the locations $\frac{1}{4\rho }+\mathbb {Z}$. (In terms of the TASEP, the above time corresponds to the time $\frac{1}{4\rho }L=\frac{1}{4\rho ^2}N$.) Let us denote by $x_\mathrm{s}(t)$ the location of the shock of the Burger’s equation at time t which was initially generated at the location $-1+\frac{1}{4\rho }$, i.e. $x_\mathrm{s}(\frac{1}{4\rho })=-1+\frac{1}{4\rho }$. One can find that the shock location is given by

$$\begin{aligned} x_\mathrm{s}(t)=-\frac{1}{2}+(1-2\rho )t \end{aligned}$$

(5.5)

and the density profile is given by

$$\begin{aligned} q(x;t)=\frac{1}{2}-\frac{1}{2t}x, \quad x_\mathrm{s}(t)\le x < x_\mathrm{s}(t)+1 \end{aligned}$$

(5.6)

for all $t\ge \frac{1}{4\rho }$. This shows that the density profile difference at the shock, $\Delta q_\mathrm{s}(t):= \lim _{x\rightarrow x_\mathrm{s}(t)^+} q(x; t)-\lim _{x\rightarrow x_\mathrm{s}(t)^-} q(x; t)$, is given by $\Delta q_\mathrm{s}(t)=\frac{1}{2t}$ at time $t\ge \frac{1}{4\rho }$. As $t\rightarrow \infty$, this gap tends to zero and $q(x;t)\rightarrow \rho$ for all $x\in \mathbb {R}$. However, the density profile is not yet “flat enough” when $t\ll L^{1/2}$ (which corresponds to the sub-relaxation time scale $t\ll L^{3/2}$ in TASEP). Indeed, note that that when $t\ll L^{1/2}$, the gap satisfies $\Delta q_\mathrm{s}(t)\gg \frac{1}{L^{1/2}}$ (and the absolute value of the slope of the density profile at continuous points is $\gg \frac{1}{L^{1/2}}$.) In terms of the TASEP scale of time and space, $\Delta q_\mathrm{s}(t) L \gg L^{1/2} \gg (tL)^{1/3}$ which means that the gap is greater than the KPZ height fluctuations.

Given the formula of the density profile, we can compute the expected location of the $[\alpha N]$-th particle (the one initially located at $-N+[\alpha N]$) heuristically. Here $\alpha$ is an arbitrary constant satisfying $0<\alpha \le 1$. This particle meets a shock at the discrete (rescaled by L) times

$$\begin{aligned} \frac{\left( \sqrt{k-\alpha } +\sqrt{k+1-\alpha }\right) ^2}{ 4\rho }, \qquad k=1,2,\ldots . \end{aligned}$$

(5.7)

The particles location (rescaled by L) is heuristically given by

$$\begin{aligned} \begin{aligned} X_{\alpha }(t)&=(\sqrt{t}-\sqrt{(k+1-\alpha )\rho })^2-(k+1-\alpha ) \rho +k,\\&=t(1-\rho )+(\sqrt{t\rho }-\sqrt{k+1-\alpha })^2-(1-\rho )(1-\alpha )+ X_{\alpha }(0) \end{aligned} \end{aligned}$$

(5.8)

for time satisfying

$$\begin{aligned} \frac{\left( \sqrt{k-\alpha } +\sqrt{k+1-\alpha }\right) ^2}{ 4\rho }\le t <\frac{\left( \sqrt{k+1-\alpha } + \sqrt{k+2-\alpha }\right) ^2}{ 4\rho }. \end{aligned}$$

(5.9)