Tag Archives: polyhedra

A Helium Atom of Space: Dynamical Instability of the Isochoric Pentahedron by Coleman-Smith and Mullery

This week I have been reviewing a paper on the  Isochoric  Pentahedron. In this paper, the authors present an analysis of the dynamics of the equifacial pentahedron on the Kapovich-Millson phase space under a volume preserving Hamiltonian. The classical dynamics of polyhedra under this Hamiltonian may arise from the classical limit of the node volume operators in loop quantum gravity. The pentahedron is the simplest nontrivial polyhedron for which the dynamics may be chaotic.  Canonical and microcanonical estimates of the Kolmogorov-Sinai entropy suggest that the pentahedron is a strongly chaotic system. The presence of chaos is further suggested by calculations of intermediate time Lyapunov exponents which saturate to non-zero values.

Introduction

Black holes act as thermodynamic systems whose entropy is proportional to the area of their horizon  and a temperature that is inversely proportional to their mass. They may be fast scramblers and show deterministic chaos.  Einstein’s field equations suggest that dynamical chaos, and  the tendency to lose information  is a generic property of classical gravitation. For  microscopic black holes with masses near the Planck mass, which possess only a small number of degrees of freedom – we need to consider if there a smallest black hole that can act as a thermal system and what mechanism drives the thermal equilibration of black holes at the microscopic level. The pursuit of these questions requires a quantum theory of gravity.

The authors consider the problem of the microscopic origin of the thermal properties of space-time in the framework of Loop Quantum Gravity . In LQG  the structure of space-time emerges naturally from the dynamics of a graph of SU(2) spins. The nodes of this graph can be thought of as representing granules of space-time, the spins connecting these nodes can be thought of as the faces of these granules. The volume of these granules, along with the areas of the connected faces are quantized. A recent focus has been on finding a semi-classical description of the spectrum of the volume operator at one of these nodes. There have been several reasonable candidates for the quantum volume operator and a semi-classical limit may pick out a particular one of these forms. The volume preserving deformation of polyhedra has recently emerged as a candidate for this semi-classical limit. In this scheme the black hole thermodynamics can be derived in the limit of a large number N of polyhedral faces. Here the deformation dynamics of the polyhedron is a secondary contribution after the configuration entropy of the polyhedron, which can be readily developed from the statistical mechanics of polymers.

The dynamics of the elementary polyhedron, the tetrahedron, can be exactly solved and semi-classically quantized through the Bohr-Sommerfeld procedure. The volume spectrum arising from quantizing this classical system has shown agreement with full quantum calculations. If the tetrahedron is the hydrogen atom of space, the next complex polyhedron, the pentahedron (N = 5), can be considered as the analogue of the helium atom. The dynamical system corresponding to the isochoric pentahedron with fixed face areas has a four-dimensional phase space compared with two dimensional phase space of the tetrahedron. Non-integrable Hamiltonian systems exhibit behaviors including Hamiltonian chaos.
There are two distinct classes of polyhedra with five faces, the triangular prism and a pyramid with a quadrilateral base. The latter forms a measure zero subset of allowed configurations as its construction requires reducing one of the edges of the triangular prism to zero length.

This article reviews the symplectic Kapovich-Millson phase space of polyhedral configurations and  a method by which it is possible to uniquely construct a triangular prism or quadrilateral pyramid for each point in the four-dimensional phase space. It also reviews a method for computing the volume of any polyhedron from its face areas and their normals.

Polyhedra and Phase Space

A convex polyhedron is a collection of faces bounded with any number of vertices.  The areas Al and normals nl of each face are sufficient to uniquely characterize a polyhedron. The polyhedral closure relationship

helequ1

is a sufficient condition on Al to uniquely define a polyhedron with N faces. The space of shapes of polyhedra  is defined as the space of all
polyhedra modulo to their orientation in three-dimensional space:

helequ2

The shape space of convex polyhedra with N faces is  2(N – 􀀀3) dimensional; in particular, the shape space of the tetrahedron (N = 4) is two-dimensional and that of the pentahedron (N = 5) is four dimensional. This space admits a symplectic structure, which can be defined by introducing a Poisson bracket:

helequ3

Canonical variables with respect to this Poisson bracket are defined by setting firstly helequpk. Then the canonical momenta in the Kapovich-Millson space are defined as helequmodpk and the conjugate positions are given by the angle q given by   helequangle and we have:

helequpq

This may be visualized by representing the polyhedron as a polygon with edges given by the vectors helequvk  this generally gives a non-planar polygon. Now systematically triangulate this polygon, the inserted edges are the conjugate momenta p and q the angles between each of these edges are the conjugate positions. An illustration of the pentagon associated with a pentahedron in shown below:

helfig1

An example configuration of the system in the polygon representation, the phase space coordinates plotted here are z = {0.3, 0.4, 0.9, 0.91}. The normal vectors are plotted as the red solid arrows and the momentum vectors are plotted as the dashed blue arrows. The associated polyhedron is also shown. All polyhedral faces have area fixed to one, so all polygonal edges have unit length .

The shape of the phase space

The geometric structure of the polyhedron itself, particularly the fixed face areas, induces certain restrictions upon the phase space. The position space is 2π periodic by construction. The momentum space is restricted by the areas of the faces, from the triangle inequality

helequ14a

Heron’s formula for the area of a triangle can be used to simplify the above inequalities:

helequ15

where a, b,c are the edges of the triangle.

Considering the triangles Δ1  and Δ2 then inorder for the system to be in a reasonable configuration we require that the area of each of these triangles be non zero.

Hamiltonian and Polyhedral Reconstruction

We can use the volume of the pentahedron at a given point in the phase space as the Hamiltonian. This ensures that trajectories generated by Hamilton’s equations will deform the pentahedron while maintaining a constant volume. Consider a vector field F(x) = ⅓x, using the divergence theorem we can find the volume of a polyhedron

helequ18

We can compute the volume of a polyhedron specied as a set of normals and areas once we know the location of a point upon each face.

helfig5

A section in the q1, q2 plane through the Hamiltonian evaluated at p1 = p2 = 0:94, all face areas are fixed to 1. The contours are isochors, the color scheme is brighter at larger volumes.

In the investigation of the phase space of the unit area triangular prism the authors found a great deal of structure in the Hamiltonian and in the distribution of configurations. The phase space contains moderate regions of local stability and large regions of local dynamical instability.

The distribution of local Lyapunov exponents appears to be correlated with the boundaries in the configuration space. They calculated the average dynamical instability measures in the canonical and microcanonical ensembles and obtained values that are comparable to those found in well-known chaotic systems.

helfig10

The density of the positive real components of LLE’s plotted against the volume of the system

Conclusions
The large degree of dynamical instability found in the isochoric pentahedron with unit area faces provides a starting point for a bottom-up investigation of the origin of thermal behavior of gravitational field configurations in loop quantum gravity. That the dynamical instability occurs in the simplest polyhedron where
it can suggests that it will be a generic property of more complex polyhedra. Any coupling to other polyhedral configurations can be expected to enhance the degree of instability. At low energies, the pentahedron appears to be a fast scrambler of information.