We address the question of the time needed by $N$ particles, initially located on the first sites of a finite one-dimensional lattice of size $L$, to exit that lattice when they move according to a TASEP transport model. Using analytical calculations and numerical simulations, we show that when $N\ll L$, the mean exit time of the particles is asymptotically given by $T_N\left(L\right)\sim L+{\beta }_N\sqrt{L}$ for large lattices. Building upon exact results obtained for two particles, we devise an approximate continuous space and time description of the random motion of the particles that provides an analytical recursive relation for the coefficients ${\beta }_N$. The results are shown to be in very good agreement with numerical results. This approach sheds some light on the exit dynamics of $N$ particles in the regime where $N$ is finite while the lattice size $L\to \infty$. This complements previous asymptotic results obtained by Johansson [Commun. Math. Phys. 209, 437 (2000)] in the limit where both $N$ and $L$ tend to infinity while keeping the particle density $N/L$ finite.

中文翻译：

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完全不对称简单排除过程的退出时间

我们解决所需时间的问题$氮$粒子，最初位于尺寸有限的一维晶格的第一个位置$L$，当它们根据 TASEP 传输模型移动时退出该晶格。通过分析计算和数值模拟，我们表明当$氮\ll L$，粒子的平均退出时间渐近由下式给出${\mathrm{时间}}_{氮}（L）～L+{\beta }_{氮}\sqrt{L}$对于大格子。基于两个粒子获得的精确结果，我们设计了粒子随机运动的近似连续空间和时间描述，为系数提供了解析递归关系${\beta }_{氮}$。结果表明，结果与数值结果非常吻合。这种方法为我们的退出动态提供了一些线索$氮$粒子在其中$氮$是有限的，而晶格尺寸$L\to \mathrm{无穷大}$。这补充了 Johansson [ Commun. 数学。物理。 209 , 437 (2000)] 在两者的限制下$氮$和$L$趋于无穷大，同时保持粒子密度$氮/L$有限。