Bernoulli TASEP — Empirical Density Profile     TASEPs

Setup. State space $\{0,1\}^{\mathbb{Z}}$; write $\eta_t(x) \in \{0,1\}$ for the occupancy of site $x \in \mathbb{Z}$ at time $t \in \mathbb{Z}_{\ge 0}$. Fix $r \in \mathbb{Z}_{\ge 1}$ and $p \in (0,1]$. Step initial condition: $\eta_0(x) = \mathbf{1}_{-r < x \le 0}$. Let $(\xi_{t,x})_{t \ge 0,\, x \in \mathbb{Z}}$ be i.i.d. $\mathrm{Bernoulli}(p)$.

Definition 1 (parallel update). $\eta_{t+1}$ is obtained from $\eta_t$ by moving, simultaneously, every particle at $x$ with $\xi_{t,x} = 1$ and $\eta_t(x+1) = 0$.

Definition 2 (right-to-left sequential update). $\eta_{t+1}$ is obtained from $\eta_t$ by processing sites in strict decreasing order of $x$: apply $(\eta(x), \eta(x+1)) \mapsto (0,1)$ whenever the current configuration satisfies $\eta(x) = 1$, $\eta(x+1) = 0$, and $\xi_{t,x} = 1$.

Proposition 1 (Evans–Rajewsky–Speer [2]). The stationary current of Definition 1 on $\mathbb{Z}/N\mathbb{Z}$ at density $\rho \in [0,1]$ is $$j_{\mathrm{par}}(\rho) = \tfrac{1}{2}\!\left(1 - \sqrt{1 - 4p\rho(1-\rho)}\,\right).$$

Proposition 2 (Rajewsky–Santen–Schadschneider–Schreckenberg [1]). The stationary current of Definition 2 on $\mathbb{Z}/N\mathbb{Z}$ at density $\rho \in [0,1]$ is $$j_{\mathrm{seq}}(\rho) = \frac{p\,\rho(1-\rho)}{1 - p\rho}.$$

Both currents are strictly concave on $[0,1]$ for $p \in (0,1)$. At the deterministic boundary $p = 1$ one has $j_{\mathrm{par}}(\rho) = \min(\rho, 1-\rho)$ and $j_{\mathrm{seq}}(\rho) = \rho$.

Scaling limit under step initial condition. Under Eulerian scaling $\xi = x/T$, the empirical measure $T^{-1}\sum_k \delta_{x_k(T)/T}$ is expected to converge in probability to $\rho_\infty(\xi)\, d\xi$, where $\rho_\infty$ is the entropy solution of $\partial_t \rho + \partial_x j(\rho) = 0$ with initial data $\rho_0(\xi) = \mathbf{1}_{\xi \le 0}$; see Kipnis–Landim [4]. For a strictly concave flux this yields the rarefaction $$\rho_\infty(\xi) = \begin{cases} 1, & \xi \le j'(1), \\ (j')^{-1}(\xi), & j'(1) < \xi < j'(0), \\ 0, & \xi \ge j'(0). \end{cases}$$

Corollary (explicit fans).

Parallel. $j_{\mathrm{par}}'(0) = p$, $j_{\mathrm{par}}'(1) = -p$; on $\xi \in (-p, p)$, $$\rho_\infty^{\mathrm{par}}(\xi) = \tfrac{1}{2}\!\left(1 - \mathrm{sgn}(\xi)\sqrt{\tfrac{\xi^2(1-p)}{p\,(p - \xi^2)}}\right).$$

Sequential. $j_{\mathrm{seq}}'(0) = p$, $j_{\mathrm{seq}}'(1) = -p/(1-p)$; on $\xi \in \bigl(-\tfrac{p}{1-p},\, p\bigr)$, $$\rho_\infty^{\mathrm{seq}}(\xi) = \tfrac{1}{p}\!\left(1 - \sqrt{\tfrac{1-p}{1 - \xi}}\,\right).$$

References

1. N. Rajewsky, L. Santen, A. Schadschneider, M. Schreckenberg. The asymmetric exclusion process: Comparison of update procedures. J. Stat. Phys. 92 (1998), 151–194.
2. M. R. Evans, N. Rajewsky, E. R. Speer. Exact solution of a cellular automaton for traffic. J. Stat. Phys. 95 (1999), 45–96. arXiv:cond-mat/9903287.
3. A. M. Povolotsky, V. B. Priezzhev. Determinant solution for the totally asymmetric exclusion process with parallel update. J. Stat. Mech. (2006), P07002.
4. C. Kipnis, C. Landim. Scaling Limits of Interacting Particle Systems. Grundlehren der mathematischen Wissenschaften 320, Springer, 1999.
5. A. Borodin, P. L. Ferrari, M. Prähofer, T. Sasamoto. Fluctuation properties of the TASEP with periodic initial configuration. J. Stat. Phys. 129 (2007), 1055–1080. arXiv:math-ph/0608056.