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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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Maximum entropy principle in recurrence plot analysis on stochastic and chaotic systems.

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Chaos (Woodbury, N.Y.)
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Summary
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We introduce maximum entropy (Smax), a new method for analyzing dynamic systems. This parameter-free approach accurately quantifies time series complexity and correlates well with the Lyapunov exponent.

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Area of Science:

  • Dynamical Systems Theory
  • Nonlinear Dynamics
  • Time Series Analysis

Background:

  • Recurrence analysis is crucial for understanding dynamic systems, with methods evolving since Poincaré.
  • Existing recurrence methods for nonlinear systems often require manual parameter tuning, limiting their applicability.
  • The entropy of recurrence microstates offers a novel perspective on system dynamics.

Purpose of the Study:

  • To introduce a new parameter-free recurrence quantifier, maximum entropy (Smax).
  • To demonstrate Smax's ability to automatically determine the optimal recurrence neighborhood (epsilon-vicinity).
  • To establish Smax and epsilon as valuable tools for analyzing deterministic and stochastic systems.

Main Methods:

  • Development of the maximum entropy (Smax) quantifier based on recurrence plot microstate diversity.
  • Automatic determination of the optimal recurrence neighborhood (epsilon-vicinity) using the Smax concept.
  • Application and validation of Smax and epsilon on both deterministic and stochastic dynamic systems.

Main Results:

  • Smax provides a parameter-free measure, eliminating the need for manual selection of the recurrence neighborhood.
  • The determined optimal epsilon serves as a novel quantifier of dynamical properties.
  • Smax exhibits a strong correlation with the Lyapunov exponent, indicating its effectiveness.

Conclusions:

  • Maximum entropy (Smax) is a robust and user-friendly recurrence quantifier for time series analysis.
  • Smax offers improved accuracy and applicability compared to traditional methods, especially in nonlinear dynamics.
  • This approach enhances the study of recurrence properties in complex dynamical systems.