Information geometry of scaling expansions of non-exponentially growing configuration spaces

Abstract

Many stochastic complex systems are characterized by the fact that their configuration space doesn't grow exponentially as a function of the degrees of freedom. The use of scaling expansions is a natural way to measure the asymptotic growth of the configuration space volume in terms of the scaling exponents of the system. These scaling exponents can, in turn, be used to define universality classes that uniquely determine the statistics of a system. Every system belongs to one of these classes. Here we derive the information geometry of scaling expansions of sample spaces. In particular, we present the deformed logarithms and the metric in a systematic and coherent way. We observe a phase transition for the curvature. The phase transition can be well measured by the characteristic length r, corresponding to a ball with radius 2r having the same curvature as the statistical manifold. Increasing characteristic length with respect to the size of the system is associated with sub-exponential sample space growth is associated with strongly constrained and correlated complex systems. Decreasing of the characteristic length corresponds to super-exponential sample space growth that occurs for example in systems that develop structure as they evolve. Constant curvature means exponential sample space growth that is associated with multinomial statistics, and traditional Boltzmann-Gibbs, or Shannon statistics applies. This allows us to characterize transitions between statistical manifolds corresponding to different families of probability distributions.

arXiv comment20 pages, 3 figures, 3 appendices

References in corpus (28)

• Superstatistics
• Black hole thermodynamical entropy
• Statistical mechanics in the context of special relativity
• Measuring thermodynamic length
• Asymptotically scale-invariant occupancy of phase space makes the entropy Sq extensive
• Axioms and uniqueness theorem for Tsallis entropy
• Constructing a statistical mechanics for Beck-Cohen superstatistics
• Stochastic thermodynamic interpretation of information geometry
• Deformed exponentials and logarithms in generalized thermostatistics
• When do generalized entropies apply? How phase space volume determines entropy
• Group entropies, correlation laws and zeta functions
• Understanding scaling through history-dependent processes with collapsing sample space
• Maximum Entropy Principle in statistical inference: case for non-Shannonian entropies
• Information Geometry and Phase Transitions
• Geometry of escort distributions
• The three faces of entropy for complex systems -- information, thermodynamics and the maxent principle
• Generalized entropies and logarithms and their duality relations
• Generalized Shannon-Khinchin Axioms and Uniqueness Theorem for Pseudo-additive Entropies
• Generalized isotropic Lipkin-Meshkov-Glick models: ground state entanglement and quantum entropies
• Classification of complex systems by their sample-space scaling exponents
• Sample space reducing cascading processes produce the full spectrum of scaling exponents
• Extreme robustness of scaling in sample space reducing processes explains Zipf's law in diffusion on directed networks
• On the Derivation of power-law distributions within standard statistical mechanics
• The world according to Renyi: Thermodynamics of multifractal systems
• Information geometric duality of $\varphi$-deformed exponential families
• Black hole horizons can hide positive heat capacity
• Towards an Information Geometric characterization/classification of Complex Systems. I. Use of Generalized Entropies
• Maximum configuration principle for driven systems with arbitrary driving

Cited by in corpus (2)

• An overview of generalized entropic forms
• Thermodynamics of exponential Kolmogorov-Nagumo averages