具有返回点记忆迟滞模型的状态转移图的结构：I.一般理论
The structure of state transition graphs in hysteresis models with return point memory: I. General Theory
摘要：
本文研究了慢变外场驱动下无序系统的非热准静态动力学。我们的兴趣是在一个自动机描述（AQS-A），表示AQS动态的驱动字段触发的状态转换的图形。这些系统的一个特殊特征是返回点记忆（RPM），即系统在磁场循环时返回到相同微观状态的趋势。众所周知，存在三个条件，（1）一个偏序的设置配置;（2）一个无通过的性质;（3）绝热响应单调驱动场，意味着RPM。当周期性地驱动时，这样的系统在至多一个周期的瞬态之后进入循环响应。然而，条件（1）-（3）仅是充分的而不是必要的。事实上，我们表明，AQS动态自然提供了一个更具选择性的偏序，由于其连接到磁滞回线，是一个自然的选择，建立RPM。这使我们能够考虑AQS-A表现出RPM，而不一定具有不通过属性。我们称这种自动机为$\ell$ AQS-A，并给出了它们的状态转移图的结构。我们的中心发现是，RPM约束的{ \em内循环}结构的磁滞回线，即其层次组织成子循环，但不是{ \em间循环}结构。我们证明了内部循环结构的拓扑结构具有一个自然的表示在有序树和相应的状态转移图是平面的。另一方面，RPM属性不会显著限制循环间的转换。因此，一个表现出RPM并受到周期性强迫的系统在进入周期性响应之前可能会经历大量的瞬态循环。这样的系统甚至可以表现出次谐波响应。
英文摘要：
We consider the athermal quasi-static dynamics (AQS) of disordered systems driven by a slowly varying external field. Our interest is in an automaton description (AQS-A) that represents the AQS dynamics in terms of the graph of state transitions triggered by the driving field. A particular feature of these systems is return point memory (RPM), a tendency for the system to return to the same microstate upon cycling the field. It is known that the existence of three conditions, (1) a partial order on the set of configuration; (2) a no-passing property; and (3) an adiabatic response to monotonous driving fields, implies RPM. When periodically driven, such systems settle into a cyclic response after a transient of at most one period. However conditions (1)-(3) are only sufficient but not necessary. In fact, we show that the AQS dynamics naturally provides a more selective partial order which, due to its connection to hysteresis loops, is a natural choice for establishing RPM. This enables us to consider AQS-A exhibiting RPM without necessarily possessing the no-passing property. We call such automata $\ell$AQS-A and work out the structure of their state transition graphs. Our central finding is that RPM constrains the {\em intra-loop} structure of hysteresis loops, namely its hierarchical organization into sub loops, but not the {\em inter-loop} structure. We prove that the topology of the intra-loop structure has a natural representation in terms of an ordered tree and that the corresponding state transition graph is planar. On the other hand, the RPM property does not significantly restrict inter-loop transitions. A system exhibiting RPM and subject to periodic forcing can thus undergo a large number of transient cycles before settling into a periodic response. Such systems can even exhibit subharmonic response.
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