Abstract
We study via computer simulations (using various serial and parallel updating techniques) the time evolution of shocks, particularly the shock widthσ(t), in several versions of the two-dimensional asymmetric simple exclusion process (ASEP). The basic dynamics of this process consists of particles jumping independently to empty neighboring lattice sites with ratesp up=p down=p ⊥ andp left<p right. If the system is initially divided into two regions with densitiesρ left<ρ right, the boundary between the two regions corresponds to a shock front. Macroscopically the shock remains sharp and moves with a constant velocityv shock=(p right−ρ left)(1−p left−p right). We find that microscopic fluctuations causeσ to grow ast β, β≈1/4. This is consistent with theoretical expectations. We also study the nonequilibrium stationary states of the ASEP on a periodic lattice, where we break translation invariance by reducing the jump rates across the bonds between two neighboring columns of the system by a factorr. We find that for fixed overall density ρavg and reduction factorr sufficiently small (depending onρ avg and the jump rates) the system segregates into two regions with densitiesρ 1 andρ 2=1−ρ 1, where these densities do not depend on the overall densityρ avg. The boundary between the two regions is again macroscopically sharp. We examine the shock width and the variance in the shock position in the stationary state, paying particular attention to the scaling of these quantities with system size. This scaling behavior shows many of the same features as the time-dependent scaling discussed above, providing an alternate determination of the resultΒ≈1/4.
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Alexander, F.J., Cheng, Z., Janowsky, S.A. et al. Shock fluctuations in the two-dimensional asymmetric simple exclusion process. J Stat Phys 68, 761–785 (1992). https://doi.org/10.1007/BF01048875
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DOI: https://doi.org/10.1007/BF01048875