Sunday, November 29, 2009

A new solution to the statistics of hard elongated objects

A new solution to the statistics of hard elongated objects
M.Ansari, [0911.5312] (cond.mat/statistical Mechanics)

On: an analytical solution to the statistics of hard elongated objects (e.g. needles, rectangles, ellipses, etc)

Elasticity theory describes how a system under distortion is mechanically deformed. There are two approaches to quantitatively study this. The traditional approach is
to analyze the dislocation of fluid rigid boundaries. Depending on the properties of fluid (e.g. viscosity, compressibility, etc.) a variety of different cases appears.

The other approach, which is of our interest, replaces fluid with discrete objects and studies the short-range interaction between these objects. If the objects are spherically-symmetric, their alignments lead to translational ordered/disordered phases.

Frenkel in a series of papers developed a method that enables to analyze the stress and elasticity of hard spherically-symmetric objects.

A natural generalization is to replace the spherically-symmetric objects with elongated objects. These objects carry a coupling between their translational and rotational degrees of freedom and display orientational ordered/disordered phases; similar to liquid crystals.

Recently, in a series of papers a formalism for direct calculation of elastic properties of hard non-spherically symmetric objects was proposed by Murat, Kantor, and Farago. They considered hard stiffness for these objects in order to prevent the influence of orientational degrees of freedom into kinetic energy. Their method was
developed on the basis of different types of central and non-central short-range interactions; central potential depends only on the relative distance between particles, whereas noncentral potential depends on individual object orientations.

This formalism has been so far applied in different problems such as the wrapping of proteins in DNA, the ordering of complex liquids systems and percolation transitions, and the jamming transitions.

However, this formalism is hard to be solved analytically and a Monte-Carlo simulation should been used to extract its physical properties. Kantor and Kardar in
proposed a self-consistency check for this formalism in one dimension, where instead of non-spherically symmetric objects, needles are applied. The center of needles
are on a line and the angle of each needle orientation is randomly chosen. They solved this model by transfer matrix method numerically and reported an agreement
between the numerical and MC results.

The purpose of my recent publication is to reconsider elongated objects in one dimension and propose an analytical solution that, in our opinion, goes a step forward since it allows to obtain analytical formulation for some collective properties obtained so far only numerically.

To this end, we eliminate the dependence of free energy on the absolute value of an angle, thus make the orientation completely isotropic.

Interparticle distance and elasticity coefficients are derived into analytical formulations and verify the exact model results of Kardar and Kantor. We generalize our formalism to cover different types of elongated objects and repeat to derive them and verify recent results. This formalism allows to evaluate other properties of the same class, such as inverse distance between needles.

We verify the inverse distance law of sound pressure in high densities. As expected from a previous study on spherically-symmetric object, in needles model the expectation value of inverse distance below a critical pressure does not scale as the inverse of distance.

Saturday, September 26, 2009

Energy released by Volcano and earthquake

The energy released from a Volcano is 10^13 Joules. The chemical bond energy between carbon molecules to form 12 grams of carbon compound is typically 10^5 Joules. With the 10^13 Joules one can vaporize 10^3 Tonnes of Carbon.

In the case of an earthquake of magnitude 6, about 10^15 Joules of energy is released, some of these are released before the main shock. This amount is necessary to vaporize 10^5 Tonnes of carbon. Although most of the energy is released in the thermal form, but anyhow should it be unrealistic that one studies the formation of an earthquake cloud before an earthquake on top of the region gaining this energy.

P.S.> The above picture is taken from here.

Wednesday, April 29, 2009

Reports on NPB findings from M.H. Ansari and co-researchers provide new insights

Source: VerticalNews Physics

May 5th, 2008

"Without imposing the trapping boundary conditions and only from within the very definition of area it is shown that the loop quantization of area manifests an unexpected degeneracy in area eigenvalues," researchers in Waterloo, Canada report.
"This could lead to a deeper understanding of the microscopic description of a quantum black hole," wrote M.H. Ansari and colleagues.

The researchers concluded: "If a certain number of semi-classically expected properties of black holes are imposed on a quantum surface its entropy coincides with the Bekenstein-Hawking entropy."

Ansari and colleagues published their study in NPB...

Sunday, February 01, 2009

Classical and Quantum Geometry Conferences in 2009

Very High Energy Phenomena Feb 1-8 La Thuile

Dark Matter Feb 2-6 CERN

Dark Matter Feb 9-11 Florence

High-Freq Grav. Waves Feb 24-27 Huntsville

Dark Side of Gravity Mar 2-4 Florence

New Directions in Cosmology Mar 16-20 Beijing

Black Holes and LQG Mar 26-28 Valencia

Grishchuk Fest Apr 17 Cardiff

NAM/JENAM Meeting Apr 20-23 Hatfield
(Grav Waves session 21)

Zeldovich 95th Apr 20-23 Minsk 

Relativity in Astrometry Apr 27-May 1 Virginia Beach

Black Holes VII May 9-15 Banff

GW+HEN Workshop May 18-20 Paris

Relativistic Astrophysics May 19-21 Atlanta

Sobral Meeting May 26-29 Sobral

Chalonge Colloquoium May 28-29 Paris

Cosmological Magnetic Fields May 31-Jun 5 Ascona

Standard Model of Universe Jun 4-5 Paris

12th Eastern Gravity Meeting Jun 15-16 Rochester

Mathematical Relativity Jun 18-19 Lisbon

21st Rencontres de Blois Jun 21-26 Blois

Amaldi 8 Jun 21-26 New York

ICGA 9 Jun 28-Jul 1 Wuhan

Unity of the Universe Jun 29-Jul 1 Portsmouth

Invisible Universe Jun 29-Jul 3 Paris

Marcel Grossmann MG12 Jul 12-18 Paris

Chalonge: 13th Paris Cosmology Jul 23-25 Paris

CosmoSTATS09 JUl 26-31 Ascona

Loops '09 Aug 2-8 Beijing

IAU General Assembly Aug 3-14 Rio de Janeiro

Neutron Stars: IAU JD Aug 3-5 Rio de Janeiro

NIJMEGEN09 Aug 18-28 Nijmegen

Grassmannian conference Sept 14-19 Szczecin

Challenges in Cosmology Sept 2-5 Talloires

Space, Time and Beyond Oct 8-9 Golm

Galileo - Xu Guangqi Oct 26-30 Shanghai

Thanks to Malcolm MacCallum

Tuesday, January 13, 2009

Revewing a review on my work!

In Mathematical Reviews, all of my papers so far have been reviewed. One of the reviews is more extensive and after reading that I felt it could be a good idea if I have the chance to review this review in Gauge Invariance blog. I thank Maria Cristina Abbati for writing the original review.

Let me emphasize this review by no mean is to criticize the original review as it is fair in its real meaning. Moreover, this paper under discussion has been already published in NPB a year ago and the review was written recently. The review written for Math Rev is only a way to explain, publicize and partially simplify the contents of the paper briefly. Mathematical Reviews is a large archive of Physics and Math papers where these papers are reviewed again to be resorted.

The original review is in green color. The red words in between are not so to criticize something. The are marked red only to make my review easy to read and conclude. The red words do not oppose anything in the green lines, as they are mostly after one another.

In a previous paper [M. Ansari, NPB 783 (2007), no. 3, 179--212; MR2356347], the author gave an accurate description of the spectrum of the area operator as it is defined in loop quantum gravity.

He improved on previous results in the area, gave a new minimum value of the area in Planck's area unit, a_{min} = (3^0.5) * 2  \pi \gamma (where \gamma is the Immirzi parameter), and found a classification of the spectrum as a union of equidistant subsets which he called generations.

[I should correct the first part of the 2nd paragraph. In that article, I have not tried to give any new minimum value to the area.

In fact, this minimum value of area was reported by Abhay Ashtekar and Jerzy Lewandowski. They reported a spectrum of area in 1996. A few months later, Carlo Rovelli and his colleague reported the same spectrum of area using a completely different technique. Later on, Thomas Thiemann also studied this spectrum in more details and reported the same spectrum for a quantum of area.

And Yes!, in that paper I reported a new classification of the specrum of area, the so-called ladder symmetry. In fact there exists a universal formula for area that is way more compact that the one reported originally.]

This paper is complementary to the above paper and contains a detailed analysis of the degeneracy of the area's eigenvalues. The numerical results for the first hundred levels of the spectrum are summarized in scatterplots. Taking into account the generating properties, the degeneracy for a large area is estimated and results in \Omega(A) = g(a_{min}  ^N, where g(a_{min} is the degeneracy of the minimum value.

[Yes, I reported a hidden degeneracy as well as a ladder symmetry on area eigenvalues. 
However, one should not mistake this "numerical result" mentioned here as an approximation to an analytical result. In fact, there is no numerical result in this work. Everything has been reported in terms of theorems and they are exact result. Recently, Takashi Tamaki extended this classification of area eigenvalues in a different way. I will give it a brief review in later posts in gauge invariance.]

A kinematical entropy S is assigned to each area value, defined as the log of its degeneration. Assuming that A=Na_{min} for a large area, the formula S=A  log (  g(a_{min})  ) / a_{min} is obtained.

The formula is relevant when it is applied to the entropy of a non-rotating black hole and, compared with the Bekenstein-Hawking formula, determines the value of \gamma giving \gamma= log 3 / ( \pi 2^0.5  ) .

As the author remarks, he follows here the approach to black holes used by C. Rovelli (see the bibliography), where a surface in spacetime is initially considered and the expected properties of a black hole are imposed on its entropy at the semiclassical level. Another approach to black holes in loop quantum gravity was first studied by A. Ashtekar et al. [Phys. Rev. Lett. 80 (1998), no. 5, 904--907; MR1606489 (98k:83051)]. There, a boundary condition on spacetime was given at the classical level and the resulting Hamiltonian was quantized along the lines of loop quantization, giving a completely different kinematical space. In this approach the value for the Immirzi parameter is double Ansari's value.

[ The only notice to add here is that in the last sentence of above apargraph, one should note that this comparison is done in SO(3) group representations.]

A comparison of the two approaches is given by the author himself in the paper, but the question of the right black hole theory in loop quantum gravity and the value of the Immirzi parameter seems to be still open.

[ There is a main difference between the spin network black hole strategy, the one I studied in this paper, and the other opponent strategy that is called 'quantum isolated horizon picture.' 
The difference is that in the latter picture one needs to *believe* that different parts of a black hole horizon are distinguishable from one another. If this is not assumed, one does not get the entropy proportional to area, instead it will be proportional to A^0.5. This is a strange fact that honestly is true! This distinguish-ability is an additional assumption and since 1996 till now nobody showed why should we assume this when the states associated with the Chern-Simons horizon does not show that as a quantum fact to us. 
I neglectdiscussing the recent claims that the state of punctures are reported to be a sequence of punctures not a set of puncture. This assumption if is true generates an orderly difference between punctures, breaks a huge number of diffeomoerphism invariant class on the horizon and make each puncture distinguishable. However, there is no physical reason why should we prefer a sequence of punctures to a set of them. All of this seems to be passing a problem from distinguished states into an underlying diffeomorphism invariance breaking.

In the spin network black hole picture, which turns out to be a more quantum approach to understand a black hole, although the Thermodynamics of black hole and their evolutions have not been developed yet, but instead this picture does not suffer from the illness the other opponent suffers from. In other words, the entropy based on the degeneracy of spin network states is really proportional to the area A, and not A^0.5.

The reason is simple. The area operator when acts on the spin network states, it is in-sensitive to the completely tangential edges. The spin network states with different completely tangential edge states may belong to the same unification class of area eigenstates. However they may at the same time belong to completely different classes of other physical operators such as energy. As a consequence, any physical object that is described only by one parameter semi-classically, manifest area unification classes as a surface with distinguishable parts. In other words, two states with the same area that reside on a black hole horizon are distinguishable from one another only because their difference in their completely tangential edges, thus under the action of other operators they become recognizable. ]