Physics Institute, Federal University of Rio Grande do Sul, 91501-970 Porto Alegre, Brazil.
Phys Rev E. 2023 Mar;107(3-1):034127. doi: 10.1103/PhysRevE.107.034127.
The Ising model on networks plays a fundamental role as a testing ground for understanding cooperative phenomena in complex systems. Here we solve the synchronous dynamics of the Ising model on random graphs with an arbitrary degree distribution in the high-connectivity limit. Depending on the distribution of the threshold noise that governs the microscopic dynamics, the model evolves to nonequilibrium stationary states. We obtain an exact dynamical equation for the distribution of local magnetizations, from which we find the critical line that separates the paramagnetic from the ferromagnetic phase. For random graphs with a negative binomial degree distribution, we demonstrate that the stationary critical behavior as well as the long-time critical dynamics of the first two moments of the local magnetizations depend on the distribution of the threshold noise. In particular, for an algebraic threshold noise, these critical properties are determined by the power-law tails of the distribution of thresholds. We further show that the relaxation time of the average magnetization inside each phase exhibits the standard mean-field critical scaling. The values of all critical exponents considered here are independent of the variance of the negative binomial degree distribution. Our work highlights the importance of certain details of the microscopic dynamics for the critical behavior of nonequilibrium spin systems.
网络中的伊辛模型作为理解复杂系统中协同现象的一个基本模型发挥着重要作用。在这里，我们解决了具有任意度分布的随机图上伊辛模型的同步动力学在高连通极限下的问题。根据控制微观动力学的阈噪声分布，模型演化为非平衡定态。我们得到了局域磁化强度分布的精确动力学方程，从中我们找到了将顺磁相与铁磁相分开的临界线。对于具有负二项式度分布的随机图，我们证明了固定点临界行为以及局域磁化强度前两个矩的长时间临界动力学取决于阈噪声的分布。特别是对于代数阈噪声，这些临界性质由阈分布的幂律尾部决定。我们进一步表明，每个相中平均磁化强度的弛豫时间表现出标准的平均场临界标度。这里考虑的所有临界指数的值都与负二项式度分布的方差无关。我们的工作强调了微观动力学的某些细节对非平衡自旋系统临界行为的重要性。
