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=== Recent Work ===

=== Recent Work ===

Barnett et al. <ref name=":6">Barnett L, Seth AK. Dynamical independence: discovering emergent macroscopic processes in complex dynamical systems. Physical Review E. 2023 Jul;108(1):014304.</ref> proposed the concept of dynamical decoupling by judging the decoupling of macroscopic and microscopic dynamics based on transfer entropy to judge the occurrence of emergence. That is, emergence is characterized as the macroscopic variables and microscopic variables being independent of each other and having no causal relationship, which can also be regarded as a causal emergence phenomenon.  

Barnett et al. <ref name=":6">Barnett L, Seth AK. Dynamical independence: discovering emergent macroscopic processes in complex dynamical systems. Physical Review E. 2023 Jul;108(1):014304.</ref> proposed the concept of [[dynamical decoupling]] by judging the decoupling of macroscopic and microscopic dynamics based on [[transfer entropy]] to judge the occurrence of emergence. That is, emergence is characterized as the macroscopic variables and microscopic variables being independent of each other and having no causal relationship, which can also be regarded as a causal emergence phenomenon.

In 2024, Zhang Jiang et al. <ref name=":2">Zhang J, Tao R, Yuan B. Dynamical Reversibility and A New Theory of Causal Emergence. arXiv preprint arXiv:2402.15054. 2024 Feb 23.</ref> proposed a new causal emergence theory based on singular value decomposition. The core idea of this theory is to point out that the so-called causal emergence is actually equivalent to the emergence of dynamical reversibility. Given the Markov transition matrix of a system, by performing singular value decomposition on it, the sum of the <math>\alpha</math> power of the singular values is defined as the reversibility measure of Markov dynamics (<math>\Gamma_{\alpha}\equiv \sum_{i=1}^N\sigma_i^{\alpha}</math>), where [math]\sigma_i[/math] is the singular value. This index is highly correlated with effective information and can also be used to characterize the causal effect strength of dynamics. According to the spectrum of singular values, this method can directly define the concepts of '''clear emergence''' and '''vague emergence''' without explicitly defining a coarse-graining scheme.

In 2024, [[Zhang Jiang]] et al. <ref name=":2">Zhang J, Tao R, Yuan B. Dynamical Reversibility and A New Theory of Causal Emergence. arXiv preprint arXiv:2402.15054. 2024 Feb 23.</ref> proposed a new causal emergence theory based on [[singular value decomposition]]. The core idea of this theory is to point out that the so-called causal emergence is actually equivalent to the emergence of dynamical reversibility. Given the Markov transition matrix of a system, by performing singular value decomposition on it, the sum of the <math>\alpha</math> power of the singular values is defined as the [[reversibility measure]] of Markov dynamics (<math>\Gamma_{\alpha}\equiv \sum_{i=1}^N\sigma_i^{\alpha}</math>), where [math]\sigma_i[/math] is the singular value. This index is highly correlated with [[effective information]] and can also be used to characterize the causal effect strength of dynamics. According to the spectrum of singular values, this method can directly define the concepts of '''clear emergence''' and '''vague emergence''' without explicitly defining a coarse-graining scheme.

== Quantification of causal emergence ==

== Quantification of causal emergence ==