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.

SECOND-ORDER LAGRANGIAN DYNAMICS IN THE PHASE-SPACE: SOME EXAMPLES

2012 ◽  
Vol 27 (10) ◽  
pp. 1250062
Author(s):  
CONSTANTIN BIZDADEA ◽  
MARIA-MAGDALENA BÂRCAN ◽  
MIHAELA TINCA MIAUTĂ ◽  
SOLANGE-ODILE SALIU
Keyword(s):  
Phase Space ◽  
Scalar Field ◽  
Finite Number ◽  
Configuration Space ◽  
Degrees Of Freedom ◽  
Hamiltonian Formulation ◽  
Second Order ◽  
Lagrangian Formulation ◽  
Field Theories ◽  
Lagrangian Dynamics

By means of a class of nondegenerate models with a finite number of degrees of freedom, it is proved that given a Hamiltonian formulation of dynamics, there exists an equivalent second-order Lagrangian formulation whose configuration space coincides with the Hamiltonian phase-space. The above result is extended to scalar field theories in a Lorentz-covariant manner.

scholarly journals Poisson-Lie T-duality of WZW model via current algebra deformation

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Francesco Bascone ◽  
Franco Pezzella ◽  
Patrizia Vitale
Keyword(s):  
Phase Space ◽  
Configuration Space ◽  
Current Algebra ◽  
Degrees Of Freedom ◽  
Target Space ◽  
Moody Algebra ◽  
Hamiltonian Description ◽  
Direct Sum ◽  
Group Manifold ◽  
Two Parameter

Abstract Poisson-Lie T-duality of the Wess-Zumino-Witten (WZW) model having the group manifold of SU(2) as target space is investigated. The whole construction relies on the deformation of the affine current algebra of the model, the semi-direct sum $$ \mathfrak{su}(2)\left(\mathrm{\mathbb{R}}\right)\overset{\cdot }{\oplus}\mathfrak{a} $$ su 2 ℝ ⊕ ⋅ a , to the fully semisimple Kac-Moody algebra $$ \mathfrak{sl}\left(2,\mathrm{\mathbb{C}}\right)\left(\mathrm{\mathbb{R}}\right) $$ sl 2 ℂ ℝ . A two-parameter family of models with SL(2, ℂ) as target phase space is obtained so that Poisson-Lie T-duality is realised as an O(3, 3) rotation in the phase space. The dual family shares the same phase space but its configuration space is SB(2, ℂ), the Poisson-Lie dual of the group SU(2). A parent action with doubled degrees of freedom on SL(2, ℂ) is defined, together with its Hamiltonian description.

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 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.

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.

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 Semi-Classical Localisation Properties of Quantum Oscillators on a Noncommutative Configuration Space

2015 ◽  
Vol 22 (04) ◽  
pp. 1550021 ◽  
Author(s):  
Fabio Benatti ◽  
Laure Gouba
Keyword(s):  
Phase Space ◽  
Configuration Space ◽  
Classical Limit ◽  
Coherent States ◽  
Point Of View ◽  
Quantum Mechanical ◽  
Wigner Functions ◽  
Quantum Mechanical Oscillators ◽  
Mechanical Oscillators ◽  
Space Noncommutativity

When dealing with the classical limit of two quantum mechanical oscillators on a noncommutative configuration space, the limits corresponding to the removal of configuration-space noncommutativity and position-momentum noncommutativity do not commute. We address this behaviour from the point of view of the phase-space localisation properties of the Wigner functions of coherent states under the two limits.

A Quasi-three-dimensional Finite-volume Shallow Water Model for Green Water on Deck

2013 ◽  
Vol 57 (03) ◽  
pp. 125-140
Author(s):  
Daniel A. Liut ◽  
Kenneth M. Weems ◽  
Tin-Guen Yen
Keyword(s):  
Shallow Water ◽  
Finite Volume ◽  
Time Domain ◽  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Green Water ◽  
Water Model ◽  
Ship Motions ◽  
Shallow Water Model

A quasi-three-dimensional hydrodynamic model is presented to simulate shallow water phenomena. The method is based on a finite-volume approach designed to solve shallow water equations in the time domain. The nonlinearities of the governing equations are considered. The methodology can be used to compute green water effects on a variety of platforms with six-degrees-of-freedom motions. Different boundary and initial conditions can be applied for multiple types of moving platforms, like a ship's deck, tanks, etc. Comparisons with experimental data are discussed. The shallow water model has been integrated with the Large Amplitude Motions Program to compute the effects of green water flow over decks within a time-domain simulation of ship motions in waves. Results associated to this implementation are presented.

scholarly journals Non-Smooth Geodesic Flows and Classical Mechanics

1969 ◽  
Vol 12 (2) ◽  
pp. 209-212 ◽  
Author(s):  
J. E. Marsden
Keyword(s):  
Phase Space ◽  
Kinetic Energy ◽  
Hamiltonian Systems ◽  
Configuration Space ◽  
Smooth Function ◽  
Cotangent Bundle ◽  
Classical Mechanics ◽  
Riemannian Metric ◽  
Geodesic Flows ◽  
Image Position

As is well known, there is an intimate connection between geodesic flows and Hamiltonian systems. In fact, if g is a Riemannian, or pseudo-Riemannian metric on a manifold M (we think of M as q-space or the configuration space), we may define a smooth function Tg on the cotangent bundle T*M (q-p-space, or the phase space). This function is the kinetic energy of q, and locally is given by

Sign in / Sign up

Export Citation Format

Share Document