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Entropy

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...

Entropy

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 Thermodynamics

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 Thermodynamics

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 Processes

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 Function

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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A maximum entropy framework for nonexponential distributions.

Jack Peterson¹, Purushottam D Dixit, Ken A Dill

¹Department of Mathematics, Oregon State University, Corvallis, OR 97331.

Proceedings of the National Academy of Sciences of the United States of America

|December 4, 2013

Summary

This summary is machine-generated.

A new framework explains power-law distributions in social, economic, and biological systems. Maximizing entropy with economies of scale predicts these power-law tails, fitting diverse real-world data.

Keywords:

fat tail heavy tail social physics statistical mechanics thermostatistics

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Area of Science:

Complexity Science
Statistical Physics
Network Science

Background:

Power-law distributions are prevalent across diverse scientific domains, including social networks, economics, and biology.
Existing models often struggle to provide a unified explanation for the emergence of these distributions.

Purpose of the Study:

To develop a common theoretical framework for understanding probability distributions with power-law tails.
To identify the underlying principles that lead to power-law behavior in various systems.

Main Methods:

Derived distribution functions by modeling a 'joiner particle' entering a community with economies of scale.
Employed maximization of Boltzmann-Gibbs-Shannon entropy subject to an energy-like constraint.
Tested the predicted function against 13 diverse empirical distribution datasets.

Main Results:

The derived framework predicts distributions with power-law tails, reducing to the Boltzmann distribution without economies of scale.
The model provides excellent fits to empirical data from social networks (friendship links), biological systems (protein interactions), and socio-economic events (terrorist attack severity).

Conclusions:

The proposed entropic framework offers a unified explanation for power-law distributions driven by economies of scale.
This approach provides valuable insights into the conditions favoring power-law behavior in natural and social sciences.
The model's success across disparate fields highlights its broad applicability.