![]() ![]() Non-trivial joint distributions, however, result in non-trivial restrictions on configuration space and possibly non-exponential scaling of W( N). In the absence of interactions, W( N) grows exponentially as the joint distribution over microscopic variables factorize. This is also a necessary classification for system-size independent modeling. A first classification of all systems is given by the mere size of the configuration space W in the function of the number of microscopic variables N. (By first-order statistics we mean visiting probabilities, whereas higher-order statistics refer to spatial and temporal correlations). In many cases, the microscopic dynamical rules governing the system are not known instead, their effect on first-order and higher-order statistics over configuration space form the basis of understanding. A useful level of description is provided by (generalized) statistical mechanics, an effort to identify relations between relevant observable summary statistics of stochastic dynamics over configuration space. The hallmark of such systems is that their global behavior emerges out of a large number of stochastic variables interacting in a non-trivial way 6, 7, 8, 9. Today, witnessing the feedback loop of developing digital technologies and increasing amount of data collected, there has been an ever increasing need and opportunity to understand and control complex biological, social or technological systems 1, 2, 3, 4, 5. We believe that this unified macroscopic picture of emergent degrees of freedom constraining mechanisms provides a step towards finding order in the zoo of strongly interacting complex systems. We demonstrate that knowing any two strongly restricts the possibilities of the third. We present a macroscopic formalism describing this interplay between first-order statistics, higher-order statistics, and configuration space growth. Here we show that contrary to the generally held belief, not only strong correlations or history-dependence, but skewed-enough distribution of visiting probabilities, that is, first-order statistics, also play a role in determining the relation between configuration space size and system size, or, equivalently, the extensive form of generalized entropy. In systems with strongly interacting variables, or with variables driven by history-dependent dynamics, this is no longer true. Independent or weakly interacting variables render the number of configurations scale exponentially with the number of variables, making the Boltzmann–Gibbs–Shannon entropy extensive. The concept of entropy connects the number of possible configurations with the number of variables in large stochastic systems. ![]()
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