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Related Concept Videos

Entropy02:39

Entropy

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...
Entropy01:18

Entropy

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...
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...
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.
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...

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Correction: Bianconi, G. The Quantum Relative Entropy of the Schwarzschild Black Hole and the Area Law. <i>Entropy</i> 2025, <i>27</i>, 266.

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Large Scale Energy Efficient Sensor Network Routing Using a Quantum Processor Unit
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Large Scale Energy Efficient Sensor Network Routing Using a Quantum Processor Unit

Published on: September 8, 2023

Entropy of network ensembles.

Ginestra Bianconi1

  • 1The Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, 34014 Trieste, Italy.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|April 28, 2009
PubMed
Summary

This study introduces a statistical mechanics framework for analyzing complex network ensembles. It resolves the paradox of scale-free networks having low structural entropy by showing they are the most probable distribution.

Area of Science:

  • Statistical mechanics applied to network science.
  • Analysis of complex network ensembles.
  • Characterization of network structures.

Background:

  • Traditional random network models often fail to capture complex systems' nontrivial features.
  • Understanding the statistical properties of diverse network types (undirected, directed, weighted) is crucial.
  • The concept of entropy is used to quantify the diversity of networks within an ensemble.

Purpose of the Study:

  • To generalize random network concepts using statistical mechanics for diverse network ensembles.
  • To define and evaluate structural entropy for networks with specific degree sequences.
  • To resolve the paradox between scale-free networks' prevalence and their low structural entropy.

Main Methods:

  • Developed a statistical mechanics framework to model network ensembles.

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  • Defined and calculated structural entropy for undirected, uncorrelated, simple networks.
  • Proved that scale-free degree distributions are the most likely for a given structural entropy.

    • Main Results:

    • The framework successfully describes various network ensembles, including those with community structure or distance-dependent link probabilities.
    • Structural entropy was defined and evaluated for networks with a given degree sequence.
    • Scale-free degree distributions were shown to be statistically favored, resolving a key paradox.

    Conclusions:

    • The generalized statistical mechanics approach provides a powerful tool for network ensemble analysis.
    • The findings explain the prevalence of scale-free networks by their high probability under specific entropy constraints.
    • The framework has implications for developing advanced network-construction algorithms.