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

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.

Download Full-text

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

10.20944/preprints201809.0247.v1 ◽

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.

Download Full-text

Special Relativity and Its Newtonian Limit from a Group Theoretical Perspective

Symmetry ◽

10.3390/sym13101925 ◽

2021 ◽

Vol 13 (10) ◽

pp. 1925

Author(s):

Otto C. W. Kong ◽

Jason Payne

Keyword(s):

Phase Space ◽

Special Relativity ◽

Theoretical Perspective ◽

Hamiltonian Formulation ◽

Minkowski Spacetime ◽

Newtonian Limit ◽

Newtonian Theory ◽

Starting Point ◽

Coset Representation ◽

Phase Space Geometry

In this pedagogical article, we explore a powerful language for describing the notion of spacetime and particle dynamics intrinsic to a given fundamental physical theory, focusing on special relativity and its Newtonian limit. The starting point of the formulation is the representations of the relativity symmetries. Moreover, that seriously furnishes—via the notion of symmetry contractions—a natural way in which one can understand how the Newtonian theory arises as an approximation to Einstein’s theory. We begin with the Poincaré symmetry underlying special relativity and the nature of Minkowski spacetime as a coset representation space of the algebra and the group. Then, we proceed to the parallel for the phase space of a spin zero particle, in relation to which we present the full scheme for its dynamics under the Hamiltonian formulation, illustrating that as essentially the symmetry feature of the phase space geometry. Lastly, the reduction of all that to the Newtonian theory as an approximation with its space-time, phase space, and dynamics under the appropriate relativity symmetry contraction is presented. While all notions involved are well established, the systematic presentation of that story as one coherent picture fills a gap in the literature on the subject matter.

Download Full-text

EDUCING THE VOLUME OUT OF THE PHASE-SPACE BOUNDARY

International Journal of Modern Physics A ◽

10.1142/s0217751x06031417 ◽

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.

Download Full-text

An Entropy for Groups of Intermediate Growth

Advances in Mathematical Physics ◽

10.1155/2017/2863614 ◽

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.

Download Full-text

Shape Dynamics and the Linking Theory

10.1093/oso/9780198789475.003.0012 ◽

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.

Download Full-text