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Related Concept Videos

Entropy Change in Reversible Processes01:10

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In the Carnot engine, which achieves the maximum efficiency between two reservoirs of fixed temperatures, the total change in entropy is zero. The observation can be generalized by considering any reversible cyclic process consisting of many Carnot cycles. Thus, it can be stated that the total entropy change of any ideal reversible cycle is zero.
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The first law of thermodynamics is quantitatively formulated via an equation relating the internal energy of a system, the heat exchanged by it, and the work done on it. A quantitative formulation of the second law of thermodynamics leads to defining a state function, the entropy.
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Consider an isolated system in which a hot object is placed in contact with a cold one. This is an irreversible process that eventually leads both objects to reach the same equilibrium temperature. It is crucial to note that the constituents of any substance exhibit increased disorder at higher temperatures. As a cold substance absorbs heat, its constituents become more disordered. The energy transfer from a hotter object to a cooler one increases the system's disorder or randomness. This...
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The Entropy as a State Function01:14

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Consider an arbitrary process that moves between two specific states (A and B) in a cyclic manner. This process is reversible and broken down into smaller parts that each follow a Carnot cycle. A Carnot cycle has two isothermal (constant temperature) processes. During these processes, the ratio of the amount of heat transferred to their respective temperature remains constant. The other two processes in the Carnot cycle are also reversible but adiabatic, which means they occur without any heat...
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Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section
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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.

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|July 22, 2016
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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.

Keywords:
Nonlinear dynamical systemsNonlinear physicsPhysicsStatistical physics

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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.