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Related Concept Videos

Entropy Change in Reversible Processes01:10

Entropy Change in Reversible Processes

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.
Entropy Changes Accompanying Specific Processes01:21

Entropy Changes Accompanying Specific Processes

Entropy, a measure of disorder in a system, changes during phase transitions like freezing or boiling. At the transition temperature Ttrs, where two phases are in equilibrium, the phase transition is a reversible process. The entropy change can be calculated from a substance's enthalpy of transition using the equation ΔStrs = ΔtrsH /Ttrs.When a perfect gas expands isothermally from one volume to another, entropy increases logarithmically with volume. Conversely, isothermal compression results...
The Entropy as a State Function01:14

The Entropy as a State Function

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...
BIBO stability of continuous and discrete -time systems01:24

BIBO stability of continuous and discrete -time systems

System stability is a fundamental concept in signal processing, often assessed using convolution. For a system to be considered bounded-input bounded-output (BIBO) stable, any bounded input signal must produce a bounded output signal. A bounded input signal is one where the modulus does not exceed a certain constant at any point in time.
To determine the BIBO stability, the convolution integral is utilized when a bounded continuous-time input is applied to a Linear Time-Invariant (LTI) system.
Entropy and the Second Law of Thermodynamics01:20

Entropy and the Second Law of Thermodynamics

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...
Entropy and the Second Law of Thermodynamics01:26

Entropy and the Second Law of Thermodynamics

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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Characteristic Functions and Process Identification by Neural Networks.

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Microstate and Omega Complexity Analyses of the Resting-state Electroencephalography
06:40

Microstate and Omega Complexity Analyses of the Resting-state Electroencephalography

Published on: June 15, 2018

Ergodic parameters and dynamical complexity.

Rui Vilela Mendes1

  • 1CMAF - Instituto de Investigação Interdisciplinar, UL, Av. Gama Pinto 2, 1649-003 Lisboa, Portugal. vilela@cii.fc.ul.pt

Chaos (Woodbury, N.Y.)
|October 7, 2011
PubMed
Summary
This summary is machine-generated.

This study rigorously characterizes ergodic parameters beyond Lyapunov exponents using a cocycle formulation. It links dynamical Renyi entropies to local expansion rate fluctuations, offering new insights into complex dynamical systems.

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Area of Science:

  • Dynamical Systems Theory
  • Statistical Mechanics
  • Network Science

Background:

  • Lyapunov exponents are standard measures of chaos.
  • Characterizing complexity in dynamical systems requires advanced ergodic parameters.
  • Existing formulas for entropy and expansion rates have limitations.

Purpose of the Study:

  • To rigorously characterize ergodic parameters beyond the Lyapunov exponent.
  • To establish a relationship between dynamical Renyi entropies and local expansion rate fluctuations.
  • To demonstrate the application of these parameters in understanding complex systems.

Main Methods:

  • Utilizing a cocycle formulation for rigorous mathematical analysis.
  • Generalizing the Pesin formula to connect Renyi entropies and expansion rate fluctuations.
  • Applying the developed framework to diverse examples.

Main Results:

  • Introduction of new ergodic parameters alongside established ones.
  • A generalized Pesin formula relating dynamical Renyi entropies and local expansion rate fluctuations.
  • Demonstration of how these parameters characterize system complexity.

Conclusions:

  • The cocycle formulation provides a powerful tool for analyzing ergodic parameters.
  • The generalized Pesin formula offers deeper insights into the complexity of dynamical systems.
  • Ergodic parameters are crucial for understanding phenomena like synchronization and self-organized criticality.