Related Experiment Video
Updated: Mar 10, 2026
08:15
Synthesis of Nine-atom Deltahedral Zintl Ions of Germanium and their Functionalization with Organic Groups
Published on: February 11, 2012
14.4K
Formal groups and Z-entropies
1Departamento de Física Teórica II (Métodos Matemáticos de la Física), Facultad de Físicas, Universidad Complutense de Madrid, 28040 Madrid, Spain; Instituto de Ciencias Matemáticas, C/ Nicolás Cabrera, No. 13-15, 28049 Madrid, Spain.
Proceedings. Mathematical, Physical, and Engineering Sciences
|December 14, 2016
Summary
We introduce Z-entropies, a new family of multi-parametric entropies generalizing Boltzmann and Rényi entropies. These Z-entropies are composable, a crucial property for information theory applications in classical and quantum systems.
Area of Science:
- Information Theory
- Statistical Mechanics
- Mathematical Physics
Background:
- Rényi entropy is a well-established measure in information theory.
- Boltzmann and Tsallis entropies are known composable entropies within the trace-form class.
- Existing entropy frameworks have limitations in generalizing composable properties.
Purpose of the Study:
- Introduce a new family of multi-parametric entropies, termed Z-entropies.
- Demonstrate that Z-entropies generalize Boltzmann and Rényi entropies.
- Explore the composability property of Z-entropies and its mathematical underpinnings.
Main Methods:
- Mathematical derivation of Z-entropies.
- Analysis of the composability axiom for generalized entropies.
- Investigation of group-theoretical structures related to entropy composition.
Main Results:
- Rényi entropy is identified as the first example of the new Z-entropy family.
- Z-entropies are shown to generalize both Boltzmann and Rényi entropies.
- Composability of Z-entropies is established, linking them to group theory.
Conclusions:
- Z-entropies represent a novel class of composable entropies with broad applicability.
- The group-theoretical structure is key to understanding the statistical properties of Z-entropies.
- These findings offer new mathematical tools for classical and quantum information theory.
More Related Videos
Related Concept Videos
02:39Entropy
37.0K
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...
37.0K
01:18Entropy
3.7K
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...
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...
3.7K
02:38Third Law of Thermodynamics
22.4K
A pure, perfectly crystalline solid possessing no kinetic energy (that is, at a temperature of absolute zero, 0 K) may be described by a single microstate, as its purity, perfect crystallinity,and complete lack of motion means there is but one possible location for each identical atom or molecule comprising the crystal (W = 1). According to the Boltzmann equation, the entropy of this system is zero.
22.4K
01:20Entropy and the Second Law of Thermodynamics
5.1K
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...
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...
5.1K
01:26Entropy and the Second Law of Thermodynamics
145
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...
145
01:10Entropy Change in Reversible Processes
3.3K
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 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.
3.3K