Preisach模型的状态转移图和返回点记忆的作用
State transition graph of the Preisach model and the role of return point memory
摘要：
Preisach模型作为零模型对于理解周期性驱动的无序系统中的记忆形成是有用的。例如，在非晶态固体中，对剪切的非热响应是由于局部塑性事件（软点）。正如我们中的一位最近所表明的那样，对外加剪切的塑性响应可以用一个有向网络来严格描述，该网络的过渡对应于一个或多个软点的变化状态。该图的拓扑结构取决于软点之间的相互作用，当这种相互作用可以忽略不计时，所得到的描述就变成了Preisach模型的描述。因此，将转换图拓扑与底层软点交互联系起来的第一步是在没有交互的情况下确定这种图的结构。在这里，我们执行的Preisach模型的转移图的详细分析。我们强调的重要作用，返回点内存组织成层次结构的循环和子循环的图形。我们的分析表明，该图的大部分拓扑结构实际上不是由描述单个元件的单个滞后行为的开关场的值决定的，而是由一个更粗糙的参数决定的，该参数是一个排列$ρ$，它规定了单个滞后元件在主滞后环被遍历时改变其状态的顺序。这反过来又使我们能够得到组合属性，如转换图中的主要回路的数量以及构成主要磁滞回线及其嵌套子回路的状态$| \mathcal{R} |$的数量。我们发现，$| \mathcal{R} |$等于置换$ρ$中所含的递增序列的数目。
英文摘要：
The Preisach model has been useful as a null-model for understanding memory formation in periodically driven disordered systems. In amorphous solids for example, the athermal response to shear is due to localized plastic events (soft spots). As shown recently by one of us, the plastic response to applied shear can be rigorously described in terms of a directed network whose transitions correspond to one or more soft spots changing states. The topology of this graph depends on the interactions between soft-spots and when such interactions are negligible, the resulting description becomes that of the Preisach model. A first step in linking transition graph topology with the underlying soft-spot interactions is therefore to determine the structure of such graphs in the absence of interactions. Here we perform a detailed analysis of the transition graph of the Preisach model. We highlight the important role played by return point memory in organizing the graph into a hierarchy of loops and sub-loops. Our analysis reveals that the topology of a large portion of this graph is actually not governed by the values of the switching fields that describe the individual hysteretic behavior of the individual elements, but by a coarser parameter, a permutation $ρ$ which prescribes the sequence in which the individual hysteretic elements change their states as the main hysteresis loop is traversed. This in turn allows us to derive combinatorial properties, such as the number of major loops in the transition graph as well as the number of states $| \mathcal{R} |$ constituting the main hysteresis loop and its nested subloops. We find that $| \mathcal{R} |$ is equal to the number of increasing subsequences contained in the permutation $ρ$.
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