Many aspects of natural phenomena cannot be addressed efficiently if strict domain boundaries based on length and time scales are assumed. On the contrary, most complex systems are characterized by having highly nontrivial interactions between dynamics on different scales. In many cases, hierarchical organization still exists but the separation of different levels is defined by functionality rather than on traditional physical scales. From this perspective, satisfactory descriptions of dynamical hierarchies should use functionality to define levels, rather than e.g., length-scale.
In general, defining separation between levels in a hierarchical structure is nontrivial. In statistical data analysis, for example, detecting structures such as hierarchies of clusters in a data set is an area which still attracts much attention from the research community. If no additional apriori information is provided to guide the definition of clusters and differentiation between levels, e.g., as a parametrized model in terms of a set of probabilistic distribution functions, the task is not well defined. The same is true for hierarchical structures in dynamical systems. Traditional definitions of separation are based on physical scales, defined with respect to physical constants, e.g., the Planck constant or some natural system defined dimensional units such as lattice distance, or in statistical physics in terms of converging averages from a large number of degrees of freedom.
A general definition of a level of description in a dynamical system is based on closure. The central idea is that a level in the hierarchy should be causally self-contained, i.e. the information to make optimal predictions of its future time evolution is contained in the system itself. For a wide class of deterministic and stochastic dynamical systems this means that a level of description should have a Markovian dynamics, i.e. given the current state of the system the history contains no additional information about the future time evolution. A hierarchy consists of levels of description as well as projective maps that define the upward causality in the hierarchy. More coarse levels of description must be defined as projective maps from more detailed levels. The situation can be summarized in the following diagram:
where ∑1 denotes the detailed state space and ∑2 denotes the coarse grained state space, the condition for ∏ to be a proper reduction of the detailed dynamics v(t) is that there exist a closed dynamics w(t) in the coarser level. As mentioned above the central requirement on w(t) is that it is a Markovian dynamics.
