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Entropy02:39

Entropy

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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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Entropy01:18

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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.
When an ideal gas expands isothermally, the disorder in the gas increases. From the molecular perspective, the gas molecules have more volume to move around in.
Consider an infinitesimal step in the expansion, which...
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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.
The statement can be further generalized to prove that entropy is a state function. Take a cyclic process between any two points on a p-V diagram.
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Second Law of Thermodynamics02:49

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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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Second Law of Thermodynamics00:53

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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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Entropy and the Second Law of Thermodynamics01:20

Entropy and the Second Law of Thermodynamics

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The second law of thermodynamics can be stated quantitatively using the concept of entropy. Entropy is the measure of disorder of the system.
The relation  between entropy and disorder can be illustrated with the example of the phase change of ice to water. In ice, the molecules are located at specific sites giving a solid state, whereas, in a liquid form, these molecules are much freer to move. The molecular arrangement has therefore become more randomized. Although the change in average...
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Transfer entropies within dynamical effects framework.

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Summary
This summary is machine-generated.

Transfer entropy (TE) quantifies causal couplings in time-series data. This study links TE to dynamical causal effects (DCEs), revealing how different TE versions predict distinct long-term system responses.

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Area of Science:

  • Complex Systems Analysis
  • Information Theory
  • Dynamical Systems Theory

Background:

  • Transfer entropy (TE) is a key metric for inferring causal relationships in time-series data.
  • The interpretation of TE solely as a coupling strength measure is often insufficient for understanding long-term dynamics.
  • Existing methods lack a clear link between TE values and the long-term consequences of inferred causal links.

Purpose of the Study:

  • To establish a theoretical relationship between Transfer Entropy (TE) and Dynamical Causal Effects (DCEs).
  • To investigate how different TE calculation methods impact the prediction of long-term system behaviors.
  • To analyze these relationships within a paradigmatic system of bidirectionally coupled linear overdamped oscillators.

Main Methods:

  • Analytical derivation of relationships between TE and DCEs.
  • Modeling of a stochastic dynamical system with bidirectionally coupled linear overdamped oscillators.
  • Comparison of original and infinite-history TE versions in predicting long-term DCEs.

Main Results:

  • Demonstrated a direct link between TE and DCEs, providing a richer interpretation of TE.
  • Showcased that different TE calculation methods (original vs. infinite-history) can yield qualitatively distinct DCE predictions.
  • Identified specific conditions under which TE versions diverge in their long-term dynamical implications.

Conclusions:

  • TE can be interpreted not only as a coupling strength but also as a predictor of dynamical causal effects.
  • The choice of TE calculation method is critical and can lead to fundamentally different conclusions about long-term system dynamics.
  • This work provides a framework for a more nuanced understanding and application of TE in complex systems analysis.