Activated Random Walkers: Facts, Conjectures and Challenges
Activated Random Walkers: Facts, Conjectures and Challenges
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激活随机游走者：事实、猜想和挑战
DOI：
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发表时间：
2009
期刊：
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通讯作者：
V. Sidoravicius
中科院分区：
文献类型：
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作者：
R. Dickman;L. Rolla;V. Sidoravicius
研究了整数格点上具有跳跃（随机行走）动力学的粒子系统。粒子可以以两种状态存在，活跃或不活跃（睡眠）;只有前者可以跳跃。动力学保持了粒子的数量;在给定的位置上，粒子的数量没有限制。孤立的活性粒子以λ>0的速率进入休眠状态，然后保持休眠状态，直到在同一位置被另一个粒子加入。所有粒子都不活跃的状态是吸收。活动是否长期持续取决于粒子密度λ和睡眠率λ之间的关系。我们讨论了一般的情况，然后，对于一维完全不对称的情况下，研究之间的相变的活性相（足够大的粒子密度和/或小的λ）和吸收一个。我们还提出了参数的渐近平均跳跃速度在活动阶段，在吸收阶段的固定率，和生存的无限系统在临界状态。利用平均场理论和蒙特卡罗模拟，我们定位相边界。在对称和非对称两种情况下，相变似乎都是连续的，但临界行为非常不同。前一种情况的特征是临界指数为简单的整数或有理数（例如β=1），其相图与平均场理论的预言雅阁。我们提出的证据表明，对称的版本属于保守的随机沙堆，也被称为保守的定向渗流的普遍性类。模拟还揭示了一个有趣的瞬态现象的阻尼振荡的活动密度。
We study a particle system with hopping (random walk) dynamics on the integer lattice ℤd. The particles can exist in two states, active or inactive (sleeping); only the former can hop. The dynamics conserves the number of particles; there is no limit on the number of particles at a given site. Isolated active particles fall asleep at rate λ>0, and then remain asleep until joined by another particle at the same site. The state in which all particles are inactive is absorbing. Whether activity continues at long times depends on the relation between the particle density ζ and the sleeping rate λ. We discuss the general case, and then, for the one-dimensional totally asymmetric case, study the phase transition between an active phase (for sufficiently large particle densities and/or small λ) and an absorbing one. We also present arguments regarding the asymptotic mean hopping velocity in the active phase, the rate of fixation in the absorbing phase, and survival of the infinite system at criticality. Using mean-field theory and Monte Carlo simulation, we locate the phase boundary. The phase transition appears to be continuous in both the symmetric and asymmetric versions of the process, but the critical behavior is very different. The former case is characterized by simple integer or rational values for critical exponents (β=1, for example), and the phase diagram is in accord with the prediction of mean-field theory. We present evidence that the symmetric version belongs to the universality class of conserved stochastic sandpiles, also known as conserved directed percolation. Simulations also reveal an interesting transient phenomenon of damped oscillations in the activity density.