Wilkinson Joseph W P, Prosen Tomaž, Garrahan Juan P
School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, United Kingdom.
Centre for the Mathematics and Theoretical Physics of Quantum Non-equilibrium Systems, University of Nottingham, Nottingham, NG7 2RD, United Kingdom.
Phys Rev E. 2022 Mar;105(3-1):034124. doi: 10.1103/PhysRevE.105.034124.
We study the dynamics and statistics of the Rule 150 reversible cellular automaton (RCA). This is a one-dimensional lattice system of binary variables with synchronous (Floquet) dynamics that corresponds to a bulk deterministic and reversible discretized version of the kinetically constrained "exclusive one-spin facilitated" (XOR) Fredrickson-Andersen (FA) model, where the local dynamics is restricted: A site flips if and only if its adjacent sites are in different states from each other. Similar to other RCA that have been recently studied, such as Rule 54 and Rule 201, the Rule 150 RCA is integrable, however, in contrast is noninteracting: The emergent quasiparticles, which are identified by the domain walls, behave as free fermions. This property allows us to solve the model by means of matrix product ansatz. In particular, we find the exact equilibrium and nonequilibrium stationary states for systems with closed (periodic) and open (stochastic) boundaries, respectively, resolve the full spectrum of the time evolution operator and, therefore, gain access to the relaxation dynamics, and obtain the exact large deviation statistics of dynamical observables in the long-time limit.
我们研究了150号规则可逆细胞自动机（RCA）的动力学和统计学性质。这是一个由二元变量组成的一维晶格系统，具有同步（弗洛凯）动力学，它对应于动力学受限的“异或单自旋促进”（XOR）弗雷德里克森 - 安德森（FA）模型的体确定性和可逆离散版本，其中局部动力学受到限制：当且仅当一个位点的相邻位点处于彼此不同的状态时，该位点才会翻转。与最近研究的其他RCA（如54号规则和201号规则）类似，150号规则RCA是可积的，然而，与之不同的是它是非相互作用的：由畴壁识别的涌现准粒子表现为自由费米子。这一性质使我们能够通过矩阵乘积假设来求解该模型。具体而言，我们分别找到了具有封闭（周期性）和开放（随机）边界的系统的精确平衡态和非平衡稳态，解析了时间演化算符的全谱，从而得以研究弛豫动力学，并获得了长时间极限下动力学可观测量的精确大偏差统计。
