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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

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 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.

Entropy and the Second Law of Thermodynamics

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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.
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...

Second Law of Thermodynamics

Second Law of Thermodynamics

In the quest to identify a property that may reliably predict the spontaneity of a process, a promising candidate has been identified: entropy. Processes that involve an increase in entropy of the system (ΔS > 0) are very often spontaneous; however, examples to the contrary are plentiful. By expanding consideration of entropy changes to include the surroundings, a significant conclusion regarding the relation between this property and spontaneity may be reached. In thermodynamic models, the...

Second Law of Thermodynamics

Second Law of Thermodynamics

The Second Law of Thermodynamics states that entropy, or the amount of disorder in a system, increases each time energy is transferred or transformed. Each energy transfer results in a certain amount of energy that is lost—usually in the form of heat—that increases the disorder of the surroundings. This can also be demonstrated in a classic food web. Herbivores harvest chemical energy from plants and release heat and carbon dioxide into the environment. Carnivores harvest the...

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The Case for Shifting the Rényi Entropy.

Francisco J Valverde-Albacete¹, Carmen Peláez-Moreno¹

¹Department of Signal Theory and Communications, Universidad Carlos III de Madrid, 28911 Leganés, Spain.

Entropy (Basel, Switzerland)

|December 3, 2020

Summary

This summary is machine-generated.

We present a new Rényi entropy formulation linked to the Hölder mean, offering novel insights into information potential and statistical mechanics. This approach also enables calculations from Shannon cross-entropy and escort probabilities.

Keywords:

Hölder means Shannon-type relations escort distributions generalized weighted means shifted Rényi entropy

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Area of Science:

Information Theory
Statistical Mechanics
Mathematical Physics

Background:

Rényi entropy is a fundamental concept in information theory, generalizing Shannon entropy.
Existing formulations present challenges in relating Rényi entropy to other mathematical means and physical quantities.
Understanding these relationships is crucial for advancements in data analysis and theoretical physics.

Purpose of the Study:

To introduce a novel variant of the Rényi entropy definition.
To align the Rényi entropy with the Hölder mean for enhanced theoretical insights.
To explore new connections between Rényi entropy, information potential, and statistical mechanics.

Main Methods:

Reformulating the definition of the r-th order Rényi Entropy.
Expressing the r-th order Rényi Entropy as the logarithm of the inverse of the r-th order Hölder mean.
Deriving methods to compute Rényi entropies from Shannon cross-entropy and escort probabilities.

Main Results:

A new Rényi entropy definition is proposed, directly linked to the Hölder mean.
Novel insights into the relationship between Rényi entropy and concepts like information potential and partition functions are revealed.
Expressions for calculating Rényi entropies from Shannon cross-entropy and escort probabilities are derived.

Conclusions:

The proposed Rényi entropy formulation offers a more unified perspective within information theory and statistical mechanics.
This work facilitates new calculations and deeper understanding of entropy measures.
The utility of shifting Rényi entropy in specific applications is highlighted, suggesting avenues for future research.