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Related Concept Videos

The Entropy as a State Function01:14

The Entropy as a State Function

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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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Entropy and the Second Law of Thermodynamics01:20

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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.
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Entropy and the Second Law of Thermodynamics01:26

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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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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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Riemannian-geometric entropy for measuring network complexity.

Roberto Franzosi1, Domenico Felice2,3, Stefano Mancini2,3

  • 1QSTAR and INO-CNR, largo Enrico Fermi 2, I-50125 Firenze, Italy.

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Summary
This summary is machine-generated.

This study introduces information geometry to quantify network complexity. A novel method associates networks with Riemannian manifolds, using their volume to define entropy for effective complexity measurement.

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Area of Science:

  • Complex Systems Science
  • Information Geometry
  • Network Science

Background:

  • Quantifying complexity is a central challenge in complex systems science.
  • Existing methods may not fully capture the intricate nature of complex networks.
  • Information geometry offers a novel mathematical framework for analyzing complex structures.

Purpose of the Study:

  • To develop a quantitative measure of network complexity using information geometry.
  • To associate any given network with a differentiable geometric object (Riemannian manifold).
  • To utilize the volume of this manifold to define a new entropy measure for complexity assessment.

Main Methods:

  • Employing information geometry to construct a Riemannian manifold for any network.
  • Defining network entropy based on the volume of the associated manifold.
  • Applying the developed entropy measure to analyze various network types.

Main Results:

  • Successfully detected a classical phase transition in random and scale-free networks.
  • Effectively characterized small exponential random graphs and configuration models.
  • Demonstrated the capability of the proposed entropy in measuring complexity for real-world networks.

Conclusions:

  • The proposed information geometry-based entropy provides a robust measure of network complexity.
  • This approach offers a powerful tool for analyzing phase transitions and structural properties in diverse networks.
  • The method is applicable to both theoretical network models and empirical network data.