Abstract
The concept of entropy connects the number of possible configurations with the number of variables in large stochastic systems. Independent or weakly interacting variables render the number of configurations scale exponentially with the number of variables, making the Boltzmann–Gibbs–Shannon entropy extensive. In systems with strongly interacting variables, or with variables driven by history-dependent dynamics, this is no longer true. 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. We present a macroscopic formalism describing this interplay between first-order statistics, higher-order statistics, and configuration space growth. We demonstrate that knowing any two strongly restricts the possibilities of the third. 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.
Highlights
The concept of entropy connects the number of possible configurations with the number of variables in large stochastic systems
Note that based on observing the dynamics over configuration space, this is a meaningful definition of system size, whereas the mere number of variables is not: think of a configuration space defined by many copies of the same variable
System size independent models can be formulated in terms of homogeneous functions of N, without paying much attention to the fact that dynamics takes place on configuration space
Summary
The concept of entropy connects the number of possible configurations with the number of variables in large stochastic systems. 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] 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]. Non-trivial joint distributions, result in non-trivial restrictions on configuration space and possibly non-exponential scaling of W(N) Assuming that most of the relevant generalized entropic forms can be written as a sum of a pointwise function g over probabilities, W
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