Since the development of recurrence plots (RP) and recurrence quantification analysis (RQA), there has been a growing interest in many areas in studying physical systems using recursion techniques. In particular, as part of the RQAs, we observed the development of the concept of recurrence microstates, defined as small blocks obtained from a recurrence graph. It can be shown that some other RQAs can be calculated as a function of recurrence microstates, and the probabilities of occurrences of these microstates can define an information entropy, the so-called entropy of recurrence microstates. It was also observed that recurrence microstates and recurrence entropy can distinguish between correlated and uncorrelated stochastic and deterministic states, due to their symmetry properties. In this paper we propose analytical expressions for calculating the entropy of recurrence microstates, avoiding the need to sample a large set of recurrence microstates. The results can be particularly important in quantifying small amounts of data, where significant sampling of microstates may not be possible. In this paper we propose analytical expressions to compute the entropy of recurrence microstates avoiding the need to sample a large set of recurrence microstates. We show that our results are accurate for cases where the probability distribution function is known. For other situations, the results can be calculated approximately. Another important fact is that the approximate results can be generalized to any size of microstate, making them a powerful tool for calculating the entropy of recurrence. Our approximate methods allow us to know what these properties are and how to exploit this quantifier in the best possible way, with minimal memory usage. Finally, we show that our analytical results are in remarkable agreement with numerical simulations.

中文翻译：

---

计算重现微观状态熵的解析结果

自从递归图（RP）和递归量化分析（RQA）的发展以来，许多领域对使用递归技术研究物理系统越来越感兴趣。特别是，作为 RQA 的一部分，我们观察了循环微观状态概念的发展，定义为从循环图中获得的小块。可以证明，其他一些 RQA 可以计算为循环微观状态的函数，并且这些微观状态出现的概率可以定义信息熵，即所谓的循环微观状态熵。还观察到，由于其对称性，循环微观状态和循环熵可以区分相关和不相关的随机和确定性状态。在本文中，我们提出了用于计算循环微观状态熵的解析表达式，避免了对大量循环微观状态进行采样的需要。结果对于量化少量数据尤其重要，因为在这种情况下不可能对微观状态进行大量采样。在本文中，我们提出了解析表达式来计算循环微观状态的熵，从而避免了对大量循环微观状态进行采样的需要。我们证明，对于已知概率分布函数的情况，我们的结果是准确的。对于其他情况，可近似计算结果。另一个重要的事实是，近似结果可以推广到任何大小的微观状态，这使得它们成为计算递归熵的强大工具。我们的近似方法使我们能够知道这些属性是什么，以及如何以尽可能最好的方式利用这个量词，同时使用最少的内存。最后，我们表明我们的分析结果与数值模拟非常一致。