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Absolute Entropies and the Third Law of Thermodynamics01:23

Absolute Entropies and the Third Law of Thermodynamics

Ludwig Edward Boltzmann developed a definition for entropy, which stated that absolute entropy is proportional to the natural logarithm of the number of possible combinations of particles. Entropy stands alone among state functions as the only one whose absolute values can be determined.Consider a gas sample confined to a container. As the container expands, the energy levels of gas molecules become more closely spaced. This increases the number of available energy states, thereby increasing...
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...
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...
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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Differential Scanning Calorimetry — A Method for Assessing the Thermal Stability and Conformation of Protein Antigen
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Generalized entropies and logarithms and their duality relations.

Rudolf Hanel1, Stefan Thurner, Murray Gell-Mann

  • 1Section for Science of Complex Systems, Medical University of Vienna, 1090 Vienna, Austria.

Proceedings of the National Academy of Sciences of the United States of America
|November 7, 2012
PubMed
Summary

Superstatistics offers generalized entropies for complex systems violating Shannon-Khinchin axioms. This study reveals a duality that uniquely defines escort probabilities, leading to a theory of generalized logarithms related to entropy scaling.

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Area of Science:

  • Statistical mechanics
  • Complex systems theory
  • Information theory

Background:

  • Standard statistical mechanics relies on Boltzmann-Gibbs entropy, based on Shannon-Khinchin axioms.
  • Complex systems often violate these axioms, necessitating generalized entropy forms.
  • Superstatistics provides a framework for maximum entropy principles with generalized entropies.

Purpose of the Study:

  • To explore generalized entropies in systems violating the composability axiom.
  • To investigate the duality between escort and non-escort probability implementations of the maximum entropy principle.
  • To derive a theory of generalized logarithms arising from axiom violation and their relation to entropy scaling.

Main Methods:

  • Utilizing the superstatistics framework to formulate a maximum entropy principle.
  • Analyzing the duality connecting escort and non-escort probability methods.
  • Deriving generalized logarithms and examining their functional forms.

Main Results:

  • The duality between escort and non-escort probabilities uniquely determines the escort probability.
  • A complete theory of generalized logarithms, arising from the violation of the composability axiom, is derived.
  • The functional forms of these generalized logarithms are shown to be related to the asymptotic scaling behavior of the entropy.

Conclusions:

  • The identified duality provides a unique escort probability for generalized statistical systems.
  • This leads to a comprehensive theory of generalized logarithms and their connection to entropy scaling.
  • The findings offer new insights into the statistical description of non-Markovian and nonergodic complex systems.