Abstract
A characteristic feature of complex systems is their deep structure, meaning that the definition of their states and observables depends on the level, or the scale, at which the system is considered. This scale dependence is reflected in the distinction of micro- and macro-states, referring to lower and higher levels of description. There are several conceptual and formal frameworks to address the relation between them. Here, we focus on an approach in which macrostates are contextually emergent from (rather than fully reducible to) microstates and can be constructed by contextual partitions of the space of microstates. We discuss criteria for the stability of such partitions, in particular under the microstate dynamics, and outline some examples. Finally, we address the question of how macrostates arising from stable partitions can be identified as relevant or meaningful.
Highlights
The study of complex systems includes a whole series of other interdisciplinary approaches: system theory (Bertalanffy [10]), cybernetics (Wiener [11]), self-organization (Foerster [12]), fractals (Mandelbrot [13]), synergetics (Haken [14]), dissipative (Nicolis and Prigogine [15]) and autopoietic systems (Maturana and Varela [16]), automata theory (Hopcroft and Ullmann [17], Wolfram [18]), network theory (Albert and Barabási [19], Boccaletti et al [20], Newman et al [21]), information geometry (Ali et al [22], Cafaro and Mancini [23]), and more
Entropy 2016, 18, 426 the study of complex systems as far as it has evolved from nonlinear dynamics are deterministic chaos (Stewart [31], Lasota and Mackey [32]), coupled map lattices (Kaneko [33], Kaneko and Tsuda [34]), symbolic dynamics (Lind and Marcus [35]), self-organized criticality (Bak [36]) and computational mechanics (Shalizi and Crutchfield [37])
The KMS condition induces a contextual topology, which is basically a coarse-graining of the microstate space (Li ) that serves the definition of the states of statistical mechanics (Ls )
Summary
The concepts of complexity and the study of complex systems represent some of the most important challenges for research in contemporary science. Entropy 2016, 18, 426 the study of complex systems as far as it has evolved from nonlinear dynamics are deterministic chaos (Stewart [31], Lasota and Mackey [32]), coupled map lattices (Kaneko [33], Kaneko and Tsuda [34]), symbolic dynamics (Lind and Marcus [35]), self-organized criticality (Bak [36]) and computational mechanics (Shalizi and Crutchfield [37]) This ample (and incomplete) list notwithstanding, it is fair to say that one important open question is the question for a fundamental theory with a universal range of applicability, e.g., in the sense of an axiomatic basis, of nonlinear dynamical systems. Corresponding microstates and macrostates and their associated observables are in general non-trivially related to one another
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