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Entropy02:39

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Salt particles that have dissolved in water never spontaneously come back together in solution to reform solid particles. Moreover, a gas that has expanded in a vacuum remains dispersed and never spontaneously reassembles. The unidirectional nature of these phenomena is the result of a thermodynamic state function called entropy (S). Entropy is the measure of the extent to which the energy is dispersed throughout a system, or in other words, it is proportional to the degree of disorder of a...
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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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In the quest to identify a property that may reliably predict the spontaneity of a process, a promising candidate has been identified: entropy. Scientists refer to the measure of randomness or disorder within a system as entropy. High entropy means high disorder and low energy. To better understand entropy, think of a student’s bedroom. If no energy or work were put into it, the room would quickly become messy. It would exist in a very disordered state, one of high entropy. Energy must be...
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In the quest to identify a property that may reliably predict the spontaneity of a process, a promising candidate has been identified: entropy. Processes that involve an increase in entropy of the system (ΔS > 0) are very often spontaneous; however, examples to the contrary are plentiful. By expanding consideration of entropy changes to include the surroundings, a significant conclusion regarding the relation between this property and spontaneity may be reached. In thermodynamic models, the...
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The Second Law of Thermodynamics states that entropy, or the amount of disorder in a system, increases each time energy is transferred or transformed. Each energy transfer results in a certain amount of energy that is lost—usually in the form of heat—that increases the disorder of the surroundings. This can also be demonstrated in a classic food web. Herbivores harvest chemical energy from plants and release heat and carbon dioxide into the environment. Carnivores harvest the...
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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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Generation and Coherent Control of Pulsed Quantum Frequency Combs
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Defining chaos.

Brian R Hunt1, Edward Ott2

  • 1Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742, USA.

Chaos (Woodbury, N.Y.)
|October 3, 2015
PubMed
Summary
This summary is machine-generated.

We introduce a new, computationally feasible definition of chaos based on positive "expansion entropy." This method applies broadly to various dynamical systems, offering a practical approach to identifying chaotic behavior.

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Area of Science:

  • Dynamical Systems and Chaos Theory
  • Information Theory and Entropy Measures

Background:

  • Defining chaos computationally is challenging for complex systems.
  • Existing methods may lack general applicability or computational feasibility.

Purpose of the Study:

  • To propose a computationally feasible definition of chaos.
  • To introduce and define
  • expansion entropy
  • as a measure of chaos.
  • To compare this new definition with topological entropy.

Main Methods:

  • Development of a novel entropy-like quantity termed
  • expansion entropy
  • .
  • Application of this quantity to diverse systems including attractors, repellers, and non-periodically forced systems.
  • Comparison with established topological entropy under specific conditions.

Main Results:

  • Chaos is defined as occurring when expansion entropy is positive.
  • Demonstration of the general applicability of expansion entropy.
  • Equivalence between expansion entropy and topological entropy is established under certain conditions.

Conclusions:

  • The proposed definition of chaos using expansion entropy is computationally feasible and broadly applicable.
  • Expansion entropy offers a practical and robust method for identifying chaotic dynamics.
  • This work highlights the importance of entropy-based measures in defining and understanding chaos.