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Necessary Condition of Self-Organisation in Nonextensive Open Systems
1Department of Physics, Faculty of Science, Ege University, Izmir 35100, Turkey.
Entropy (Basel, Switzerland)
|March 29, 2023
Summary
This study introduces new q-Gibbsian equalities for self-organization in nonextensive systems, connecting q-renormalized and q-relative entropies. Numerical analysis of the logistic map demonstrates their effectiveness in quantifying complexity and self-organization.
Area of Science:
- Statistical Mechanics
- Nonextensive Thermodynamics
- Chaos Theory
- Complex Systems
Background:
- Investigates the evolution of systems from equilibrium to arbitrary states using power-law forms and q-exponentials.
- Highlights the necessity of q-Gibbsian equalities for self-organization in nonextensive open systems.
- Addresses the challenge of deriving connections between different entropy forms without predefined effective Hamiltonians.
Purpose of the Study:
- To theoretically derive connections between q-renormalized entropies (ΔS˜q) and q-relative entropies (KLq) in Bregman and Csiszar forms.
- To explain the relationship between Klimantovich's renormalized entropy and Kullback-Leibler relative entropy.
- To numerically verify the application of these concepts using the logistic map as a model of complexity.
Main Methods:
- Introduced new q-Gibbsian equalities as a condition for self-organization.
- Theoretically derived connections between q-renormalized and q-relative entropies.
- Employed the logistic map (Xt+1=1-aXt2) to numerically measure self-organization using q-renormalized entropy through period doublings and chaotic band mergings.
Main Results:
- Established theoretical links between q-renormalized entropies and q-relative entropies in both Bregman and Csiszar forms.
- Demonstrated that the logistic map's complexity and self-organization levels correlate with q-renormalized entropy.
- Identified a unique q* value for system evolution, consistent across different entropy forms, and validated with literature values for a specific parameter range.
Conclusions:
- The study successfully connects different forms of entropy in nonextensive systems and quantifies self-organization.
- The logistic map analysis confirms the utility of q-renormalized entropy in understanding complex system dynamics.
- A unique q* value characterizes system evolution, providing a novel metric for analyzing state transitions in complex systems.
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