scholarly journals Group entropies: from phase space geometry to entropy functionals via group theory

2018 ◽  
Author(s):  
Henrik Jeldtoft Jensen ◽  
Piergiulio Tempesta
Keyword(s):  
Group Theory ◽  
Phase Space ◽  
Degrees Of Freedom ◽  
Formal Group ◽  
Exponential Dependence ◽  
Information Theoretic ◽  
Entropy Functionals ◽  
Phase Space Geometry ◽  
Generalized Entropies ◽  
Space Volume

The entropy of Boltzmann-Gibbs, as proved by Shannon and Khinchin, is based on four axioms, where the fourth one concerns additivity. The group theoretic entropies make use of formal group theory to replace this axiom with a more general composability axiom. As has been pointed out before, generalized entropies crucially depend on the number of allowed number degrees of freedom $N$. The functional form of group entropies is restricted (though not uniquely determined) by assuming extensivity on the equal probability ensemble, which leads to classes of functionals corresponding to sub-exponential, exponential or super-exponential dependence of the phase space volume $W$ on $N$. We review the ensuing entropies, discuss the composability axiom, relate to the Gibbs' paradox discussion and explain why group entropies may be particularly relevant from an information theoretic perspective.

scholarly journals Group Entropies: From Phase Space Geometry to Entropy Functionals via Group Theory

Entropy ◽  
2018 ◽  
Vol 20 (10) ◽  
pp. 804 ◽  
Author(s):  
Henrik Jeldtoft Jensen ◽  
Piergiulio Tempesta
Keyword(s):  
Group Theory ◽  
Phase Space ◽  
Degrees Of Freedom ◽  
Functional Form ◽  
Theoretical Perspective ◽  
Formal Group ◽  
Exponential Dependence ◽  
Entropy Functionals ◽  
Phase Space Geometry ◽  
Space Volume

The entropy of Boltzmann-Gibbs, as proved by Shannon and Khinchin, is based on four axioms, where the fourth one concerns additivity. The group theoretic entropies make use of formal group theory to replace this axiom with a more general composability axiom. As has been pointed out before, generalised entropies crucially depend on the number of allowed degrees of freedom N. The functional form of group entropies is restricted (though not uniquely determined) by assuming extensivity on the equal probability ensemble, which leads to classes of functionals corresponding to sub-exponential, exponential or super-exponential dependence of the phase space volume W on N. We review the ensuing entropies, discuss the composability axiom and explain why group entropies may be particularly relevant from an information-theoretical perspective.

scholarly journals EDUCING THE VOLUME OUT OF THE PHASE-SPACE BOUNDARY

2006 ◽  
Vol 21 (19n20) ◽  
pp. 3967-3988
Author(s):  
MANUEL A. COBAS ◽  
M. A. R. OSORIO ◽  
MARÍA SUÁREZ
Keyword(s):  
Phase Space ◽  
Finite Volume ◽  
Configuration Space ◽  
Degrees Of Freedom ◽  
Surface Boundary ◽  
Duality Symmetry ◽  
Volume Dependence ◽  
Volume Limit ◽  
Space Boundary ◽  
Space Volume

We explicitly show that, in a system with T-duality symmetry, the configuration space volume degrees of freedom may hide on the surface boundary of the region of accessible states with energy lower than a fixed value. This means that, when taking the decompactification limit (big volume limit), a number of accessible states proportional to the volume is recovered even if no volume dependence appears when energy is high enough. All this behavior is contained in the exact way of computing sums by making integrals. We will also show how the decompactification limit for the gas of strings can be defined from a microcanonical description at finite volume.

scholarly journals A new class of entropic information measures, formal group theory and information geometry

2019 ◽  
Vol 475 (2222) ◽  
pp. 20180633 ◽  
Author(s):  
Miguel Á. Rodríguez ◽  
Álvaro Romaniega ◽  
Piergiulio Tempesta
Keyword(s):  
Group Theory ◽  
General Class ◽  
Formal Group ◽  
Information Geometry ◽  
Riemannian Structure ◽  
Natural Properties ◽  
Information Measures ◽  
New Class ◽  
Theoretical Structure ◽  
Generalized Entropies

In this work, we study generalized entropies and information geometry in a group-theoretical framework. We explore the conditions that ensure the existence of some natural properties and at the same time of a group-theoretical structure for a large class of entropies. In addition, a method for defining new entropies, using previously known ones with some desired group-theoretical properties is proposed. In the second part of this work, the information geometrical counterpart of the previous construction is examined and a general class of divergences are proposed and studied. Finally, a method of constructing new divergences from known ones is discussed; in particular, some results concerning the Riemannian structure associated with the class of divergences under investigation are formulated.

scholarly journals An Entropy for Groups of Intermediate Growth

2017 ◽  
Vol 2017 ◽  
pp. 1-8
Author(s):  
Nikolaos Kalogeropoulos
Keyword(s):  
Phase Space ◽  
Power Law ◽  
Metric Space ◽  
Degrees Of Freedom ◽  
Parameter Variation ◽  
Nonextensive Statistical Mechanics ◽  
Phase Space Volume ◽  
Entropy Functional ◽  
Ergodic Behavior ◽  
Space Volume

One of the few accepted dynamical foundations of nonadditive (“nonextensive”) statistical mechanics is that the choice of the appropriate entropy functional describing a system with many degrees of freedom should reflect the rate of growth of its configuration or phase space volume. We present an example of a group, as a metric space, that may be used as the phase space of a system whose ergodic behavior is statistically described by the recently proposed δ-entropy. This entropy is a one-parameter variation of the Boltzmann/Gibbs/Shannon functional and is quite different, in form, from the power-law entropies that have been recently studied. We use the first Grigorchuk group for our purposes. We comment on the connections of the above construction with the conjectured evolution of the underlying system in phase space.

Shape Dynamics and the Linking Theory

2018 ◽  
Author(s):  
Flavio Mercati
Keyword(s):  
Phase Space ◽  
Degrees Of Freedom ◽  
Line Element ◽  
Hamiltonian Formulation ◽  
Operational Definition ◽  
Extended Phase Space ◽  
Extended Phase ◽  
Problem Of Time ◽  
Definition Of ◽  
Matter Fields

This chapter explains in detail the current Hamiltonian formulation of SD, and the concept of Linking Theory of which (GR) and SD are two complementary gauge-fixings. The physical degrees of freedom of SD are identified, the simple way in which it solves the problem of time and the problem of observables in quantum gravity are explained, and the solution to the problem of constructing a spacetime slab from a solution of SD (and the related definition of physical rods and clocks) is described. Furthermore, the canonical way of coupling matter to SD is introduced, together with the operational definition of four-dimensional line element as an effective background for matter fields. The chapter concludes with two ‘structural’ results obtained in the attempt of finding a construction principle for SD: the concept of ‘symmetry doubling’, related to the BRST formulation of the theory, and the idea of ‘conformogeometrodynamics regained’, that is, to derive the theory as the unique one in the extended phase space of GR that realizes the symmetry doubling idea.

scholarly journals Quantum information probes of charge fractionalization in large-N gauge theories

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Brandon S. DiNunno ◽  
Niko Jokela ◽  
Juan F. Pedraza ◽  
Arttu Pönni
Keyword(s):  
Degrees Of Freedom ◽  
Gauge Theories ◽  
Entanglement Entropy ◽  
Coarse Grained ◽  
Strongly Coupled ◽  
Information Theoretic ◽  
Large N ◽  
Wide Range ◽  
Charge Fractionalization ◽  
Electric Flux

Abstract We study in detail various information theoretic quantities with the intent of distinguishing between different charged sectors in fractionalized states of large-N gauge theories. For concreteness, we focus on a simple holographic (2 + 1)-dimensional strongly coupled electron fluid whose charged states organize themselves into fractionalized and coherent patterns at sufficiently low temperatures. However, we expect that our results are quite generic and applicable to a wide range of systems, including non-holographic. The probes we consider include the entanglement entropy, mutual information, entanglement of purification and the butterfly velocity. The latter turns out to be particularly useful, given the universal connection between momentum and charge diffusion in the vicinity of a black hole horizon. The RT surfaces used to compute the above quantities, though, are largely insensitive to the electric flux in the bulk. To address this deficiency, we propose a generalized entanglement functional that is motivated through the Iyer-Wald formalism, applied to a gravity theory coupled to a U(1) gauge field. We argue that this functional gives rise to a coarse grained measure of entanglement in the boundary theory which is obtained by tracing over (part) of the fractionalized and cohesive charge degrees of freedom. Based on the above, we construct a candidate for an entropic c-function that accounts for the existence of bulk charges. We explore some of its general properties and their significance, and discuss how it can be used to efficiently account for charged degrees of freedom across different energy scales.

Phase space geometry of constrained systems

1995 ◽  
Vol 105 (3) ◽  
pp. 1539-1545 ◽  
Author(s):  
V. P. Pavlov ◽  
A. O. Starinetz
Keyword(s):  
Phase Space ◽  
Constrained Systems ◽  
Space Geometry ◽  
Phase Space Geometry
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