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Entropies for severely contracted configuration space.
G Cigdem Yalcin1, Carlos Velarde2, Alberto Robledo3
1Department of Physics, Istanbul University, 34134, Vezneciler, Istanbul, Turkey.
Heliyon
|July 22, 2016
Summary
Dual entropy expressions naturally apply to systems with contracting configuration spaces. The entropic index quantifies contraction, while the dual index defines the dimension where extensivity is restored, observed in nonlinear maps and chaos transitions.
Area of Science:
- Statistical mechanics
- Nonlinear dynamics
- Complex systems
Background:
- Systems with contracting configuration spaces exhibit unique entropic properties.
- Understanding phase-space contraction is crucial for characterizing transitions to chaos.
- Tsallis entropy provides a framework for non-extensive statistical mechanics.
Purpose of the Study:
- To demonstrate the applicability of dual Tsallis-type entropy expressions to systems with exceptional configuration space contraction.
- To investigate the role of entropic indices in characterizing phase-space contraction and extensivity restoration.
- To explore these concepts in low-dimensional nonlinear maps exhibiting transitions to chaos.
Main Methods:
- Applying dual entropy expressions of the Tsallis type.
- Analyzing phase-space contraction in ensembles of trajectories within nonlinear maps.
- Studying systems along the three routes to chaos.
Main Results:
- Dual entropy expressions naturally describe statistical-mechanical systems with contracting configuration spaces.
- The entropic index ([Formula: see text]) characterizes the contraction process.
- The dual index ([Formula: see text]) identifies the dimension for extensivity restoration.
- Observed in nonlinear maps at transitions between regular and chaotic behavior.
Conclusions:
- Dual Tsallis-type entropy is a powerful tool for analyzing systems with contracting phase space.
- The entropic and dual indices provide quantitative measures of contraction and extensivity.
- The findings are applicable to diverse phenomena including size-rank functions and urbanization processes.
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