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A Proposito del Principio di Massima Entropia

Carlo Sempi

Published Jan 1, 1986• 0 citations
OpenAlex

Abstract

The maximum-entropy principle has recently been discussed in this journal by R. Rosa. The present note stresses a few points made or touched upon there. First it is shown how to derive, by means of the principle under discussion, the quantal statistics, along with the intermediate ones. Then the point is made that the correct application of the principle and its ultimate acceptance by the scientific community require that one uses an appropriate representation for the entropy. Thus great importance is attributed to the characterizations of the various entropies.

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The maximum-entropy principle has recently been discussed in this journal by R. Rosa. The present note stresses a few points made or touched upon there. First it is shown how to derive, by means of the principle under discussion, the quantal statistics, along with the intermediate ones. Then the point is made that the correct application of the principle and its ultimate acceptance by the scientific community require that one uses an appropriate representation for the entropy. Thus great importance is attributed to the characterizations of the various entropies.
We show that the Jaynes principle is indeed a proper inference scheme when applied to compound systems and will correctly produce the entangled maximum entropy states compatible with appropriate data. This is shown in a particular example considered recently by Horodecki et al. [Phys. Rev. A 59, 1799 (1999)], by including the dispersion of the entanglement along with its mean value as constraints and an application of the uncertainty principle. We also construct a ``thermodynamiclike'' description of the entanglement arising out of the maximum entropy principle.
In this Letter, we show that the Shore-Johnson axioms for the maximum entropy principle in statistical estimation theory account for a considerably wider class of entropic functional than previously thought. Apart from a formal side of the proof where a one-parameter class of admissible entropies is identified, we substantiate our point by analyzing the effect of weak correlations and by discussing two pertinent examples: two-qubit quantum system and transverse-momentum behavior of hadrons in high-energy proton-proton collisions.
The maximum entropy principle discovered by Jaynes E T governs a search of the most reasonable probability assignment of stochastic variable,based on the measured data.One of the simplest applications of this principle lies in that all the thermodynamical relations are derived from the maximizing entropy of the system with energy as a single measured value.The maximum entropy principle is used then for the radiation field as a quantum system.Our results show that with an increase of the number of measurable quantities,the thermal field,the coherent states as well as the squeezed states are automatically introduced in turn;in particular,the dependence of all the properties of these radiation fields on the thermal noise is formulated quantitatively.The fundamental results obtained in this paper are expected to provide a basis for applications of the maximum entropy principle in some fields,such as quantum optics,quantum informatics,biophotonics,and related interdisciplinary science.
We maximize Shannon conditional entropy when the mean value of the random variables "state energy" and "number of particles in the state" are assigned. As particular cases, we derive the Bose-Einstein, Fermi-Dirac and intermediate statistics. Two assumptions have been introduced in order to guarantee the existence of the Lagrange multipliers; these latter are interpreted as sufficient conditions for the thermodynamical equilibrium. Neither Sstirling formula nor the the hypothesis of contninuous numbers are used.
The Heisenberg position-momentum uncertainty relation is a cornerstone of quantum mechanics. However, its standard formulation is not fully consistent with special relativity. While partial understanding has been achieved in the ultra-relativistic regime, a comprehensive description is still lacking, particularly in the intermediate velocity domain, where particle speeds remain well below the speed of light yet relativistic corrections are expected to become appreciable. This regime constitutes the most promising arena for experimentally probing relativistic modifications of quantum uncertainty. By adopting a variational approach, in this work we derive a relativistic extension of the Heisenberg algebra within the framework of κκκ-deformed Kaniadakis statistics. The latter emerges from the application of the Maximum Entropy Principle to Kaniadakis entropy, a one-parameter generalization of the Boltzmann-Gibbs-Shannon entropy naturally induced by Lorentz transformations. We investigate the physical implications of the resulting uncertainty relation, deriving constraints on the Kaniadakis parameter from precision measurements of the fine-structure constant and confronting our construction with other extensions discussed in the recent literature.