Could Zipf's Law and Dark Matter be linked in the cosmos?

[Justice Hunter]

Watched a video recently by Vsauce explaining elegantly how Zipf's law is so influential in many, normally overlooked processes in nature.

I'm curious to know if there could be any sort of Zipf distribution associated with galaxy rotation or dark matter. I know that Information Entropy follows a power law, so i don't think it would be surprising to find one lurking around in the cosmos somewhere's.

I also find it curious that Darkmatter takes up 84% of the matter in the universe, which is a notable property of a Zipf distribution.

Anyway, I'm no scientist or mathematician, I'm just a curious guy.

 

[wolram]

Do you have a link?

 

[Dale]

It is actually not very informative to simply go around fitting distributions to things like the amount of dark matter in the universe. Since we have only N=1 universe to study, such fits are little more than numerology.

What is of more interest is if there is a specific physical theory which would predict a given distribution. So, as wolram mentioned, do you have a scientific reference regarding a specific theory which predicts some particular Zipf distribution?

 

[Justice Hunter]

Do you have a link?

If you mean a link to the Vsauce video :

What is of more interest is if there is a specific physical theory which would predict a given distribution. So, as wolram mentioned, do you have a scientific reference regarding a specific theory which predicts some particular Zipf distribution?

This is actually what I'm curious to find out, if there have been any findings of a ziph distribution in dark matter specifically.

It is actually not very informative to simply go around fitting distributions to things like the amount of dark matter in the universe. Since we have only N=1 universe to study, such fits are little more than numerology.

Ya, i mean i don't think at face-value it would be very useful, but i could imagine that if dark-matter fits a pattern similar to other processes in nature, it could be a small part of a bigger picture as to what dark matter actually is.

 

[Dale]

Sorry this is not a valid reference. If you (or anyone else interested) find a scientific reference about this then please PM me and i can reopen the thread so that we can continue the discussion on that basis.

Edit: the OP has a related reference that he or she would like to discuss. The thread is reopened for discussing that reference.

 

[Justice Hunter]

So after some searching, i found this particular paper on the subject.
http://pubs.sciepub.com/ijp/3/2/2/

I'll admit i don't fully understand the paper, but from what i can gather, when Dark Matter and Dark Energy Quanta collide, their paths follow a Pareto Distribution, which accounts for the obscene ratio of Dark Matter to Dark energy, and ultimately Baryonic matter.

To quote directly from the paper :
"Instead of trying to speculate on what particles might be involved in the mysterious dark matter and dark energy observed in the universe, the present study starts in the other end and asks what kind of natural requirements on the world around us could possibly result in the creation of something that mass-wise dwarfs everything else in the universe. Once a reasonable candidate for such a mechanism has been identified, then we might perhaps be in a better position to consider the question of whatever particles might be involved.

The present study suggests that Lorentz covariance of the particle propagation, and the resulting, forced, successive particle doubling, might perhaps be the strongest (and possibly only) candidate for the mechanism causing the huge and dominating amounts of dark energy and dark matter that has been observed in the universe."

Thoughts?

Edit: follow up question: Could this possibly rule out all theories related to revisions of gravity to fit Dark Energy/Matter?

 

[Jimster41]

Pretty interesting. Paper below is same author laying the groundwork.

If I am following - a hypothetical wave propagating without dispersion and subject to Lorenz co-variance in an expanding continuous medium must create "quanta".
I'm not sure I really get how equation13 jumps to the Pareto distribution for particle creation along propagation length.

It seems like he is depicting the expanding vacuum as a Self-Organized-Critical system like a Bak–Tang–Wiesenfeld model?

Equations 5-12 in the paper below specify how the requirement of Lorenz co-variance and no dispersion result in a "multiplication" of quanta (v=2) in the context of the Boltzmann Transport Equation. There also appears to be a FAQ section at the end.

[Edit] I realize after reading more carefully expansion is not critical to his model - just propagation from a source. But would an observer co-moving with a "propagating wave" see particle creation?http://arxiv.org/abs/1003.3870

Lorentz-covariant quantum transport and the origin of dark energy
Arne Bergstrom
(Submitted on 19 Mar 2010 (v1), last revised 4 May 2011 (this version, v7))
A possible explanation for the enigma of dark energy, responsible for about 76 % of the mass-energy of the universe, is obtained by requiring only that the rigorous continuity equation (the Boltzmann transport equation) for quanta propagating through space should have the form of a Lorentz-covariant and dispersion-free wave equation. This requirement implies (i) properties of space-time which an observer would describe as uniform expansion in agreement with Hubble's law, and (ii) that the quantum transport behaves like in a multiplicative medium with multiplication factor = 2. This inherent, essentially explosive multiplicity of vacuum, thus caused by the requirement of Lorentz-covariance, is suggested as a potential origin of dark energy. In addition, it is shown (iii) that this requirement of Lorentz-covariant quantum transport leads to an apparent accelerated expansion of the universe.

 

[Justice Hunter]

It seems like he is depicting the expanding vacuum as a Self-Organized-Critical system like a Bak–Tang–Wiesenfeld model?

I believe this is exactly the implications of the paper, that really, dark energy is more or less a side effect of how quantum stuff follows paths through space-time after interactions (or rather duplication?), and that the quantum stuff is, instead of regular matter, dark matter that undergo this duplication.

Again i am not fully versed to understand exactly what the paper means, but perhaps if someone could correct me if I'm mistaken.

 

[Jimster41]

It is actually not very informative to simply go around fitting distributions to things like the amount of dark matter in the universe. Since we have only N=1 universe to study, such fits are little more than numerology.

What is of more interest is if there is a specific physical theory which would predict a given distribution. So, as wolram mentioned, do you have a scientific reference regarding a specific theory which predicts some particular Zipf distribution?

Are you saying that models of Self Organized Critical systems are - numerology?

If you enter "Sneppen Bak" or "Self Organizing Criticality" into arxiv you will get a huge list of papers looking at the properties of models and games that exhibit 1/f and power-law behavior...
Are you are saying these have no grounding in fundamental physics?

I'm hoping to see someone talk about SOC in the context of Lie Group governed LQG spin-foam evolution. Here are a couple I just downloaded.
http://arxiv.org/abs/cond-mat/0611201
Self-organized network evolution coupled to extremal dynamics
Diego Garlaschelli, Andrea Capocci, Guido Caldarelli
(Submitted on 8 Nov 2006 (v1), last revised 10 Jun 2008 (this version, v2))
The interplay between topology and dynamics in complex networks is a fundamental but widely unexplored problem. Here, we study this phenomenon on a prototype model in which the network is shaped by a dynamical variable. We couple the dynamics of the Bak-Sneppen evolution model with the rules of the so-called fitness network model for establishing the topology of a network; each vertex is assigned a fitness, and the vertex with minimum fitness and its neighbours are updated in each iteration. At the same time, the links between the updated vertices and all other vertices are drawn anew with a fitness-dependent connection probability. We show analytically and numerically that the system self-organizes to a non-trivial state that differs from what is obtained when the two processes are decoupled. A power-law decay of dynamical and topological quantities above a threshold emerges spontaneously, as well as a feedback between different dynamical regimes and the underlying correlation and percolation properties of the network.

http://arxiv.org/abs/nlin/0512050
Dynamic Critical approach to Self-Organized Criticality
Karina Laneri (1), Alejandro F. Rozenfeld (1), Ezequiel V. Albano (1) ((1) INIFTA, La Plata, Argentina)
(Submitted on 19 Dec 2005)
A dynamic scaling Ansatz for the approach to the Self-Organized Critical (SOC) regime is proposed and tested by means of extensive simulations applied to the Bak-Sneppen model (BS), which exhibits robust SOC behavior. Considering the short-time scaling behavior of the density of sites (ρ(t)) below the critical value, it is shown that i) starting the dynamics with configurations such that ρ(t=0)→0 one observes an {\it initial increase} of the density with exponent θ=0.12(2); ii) using initial configurations with ρ(t=0)→1, the density decays with exponent δ=0.47(2). It is also shown that he temporal autocorrelation decays with exponent Ca=0.35(2). Using these, dynamically determined, critical exponents and suitable scaling relationships, all known exponents of the BS model can be obtained, e.g. the dynamical exponent z=2.10(5), the mass dimension exponent D=2.42(5), and the exponent of all returns of the activity τALL=0.39(2), in excellent agreement with values already accepted and obtained within the SOC regime.

 

[Jimster41]

@Justice Hunter

Here's a pretty recent review of the study of Self Organizing Critical phenomenon. My understanding is that Zipf's law is considered a hallmark of SOC by folks who study it. But overall SOC is a somewhat controversial subject.

http://arxiv.org/abs/1504.04991
25 Years of Self-Organized Criticality: Concepts and Controversies
Nicholas Watkins, Gunnar Pruessner, Sandra Chapman, Norma Bock Crosby, Henrik Jensen
(Submitted on 20 Apr 2015)
Introduced by the late Per Bak and his colleagues, self-organized criticality (SOC) has been one of the most stimulating concepts to come out of statistical mechanics and condensed matter theory in the last few decades, and has played a significant role in the development of complexity science. SOC, and more generally fractals and power laws, have attacted much comment, ranging from the very positive to the polemical. The other papers in this special issue (Aschwanden et al, 2014; McAteer et al, 2014; Sharma et al, 2015) showcase the considerable body of observations in solar, magnetospheric and fusion plasma inspired by the SOC idea, and expose the fertile role the new paradigm has played in approaches to modeling and understanding multiscale plasma instabilities. This very broad impact, and the necessary process of adapting a scientific hypothesis to the conditions of a given physical system, has meant that SOC as studied in these fields has sometimes differed significantly from the definition originally given by its creators. In Bak's own field of theoretical physics there are significant observational and theoretical open questions, even 25 years on (Pruessner, 2012). One aim of the present review is to address the dichotomy between the great reception SOC has received in some areas, and its shortcomings, as they became manifest in the controversies it triggered. Our article tries to clear up what we think are misunderstandings of SOC in fields more remote from its origins in statistical mechanics, condensed matter and dynamical systems by revisiting Bak, Tang and Wiesenfeld's original papers.

 

[Jimster41]

I just saw this over on the "Loop and Allied QG" bibliography thread. I'm don't want to stray too far from your original post but the paper you found does seem to me to point to the unresolved relationship between QM and GR and invoke "self organizing criticality".

This paper is very recent but it appears to make some connection between "discrete quantum geometry" (Quantized GR) and fractals, which are related to SOC.

It appears to be 203 pages long though!

http://arxiv.org/abs/1510.08706
Discrete quantum geometries and their effective dimension
Johannes Thürigen
(Submitted on 29 Oct 2015)
In several approaches towards a quantum theory of gravity, such as group field theory and loop quantum gravity, quantum states and histories of the geometric degrees of freedom turn out to be based on discrete spacetime. The most pressing issue is then how the smooth geometries of general relativity, expressed in terms of suitable geometric observables, arise from such discrete quantum geometries in some semiclassical and continuum limit. In this thesis I tackle the question of suitable observables focusing on the effective dimension of discrete quantum geometries. For this purpose I give a purely combinatorial description of the discrete structures which these geometries have support on. As a side topic, this allows to present an extension of group field theory to cover the combinatorially larger kinematical state space of loop quantum gravity. Moreover, I introduce a discrete calculus for fields on such fundamentally discrete geometries with a particular focus on the Laplacian. This permits to define the effective-dimension observables for quantum geometries. Analysing various classes of quantum geometries, I find as a general result that the spectral dimension is more sensitive to the underlying combinatorial structure than to the details of the additional geometric data thereon. Semiclassical states in loop quantum gravity approximate the classical geometries they are peaking on rather well and there are no indications for stronger quantum effects. On the other hand, in the context of a more general model of states which are superposition over a large number of complexes, based on analytic solutions, there is a flow of the spectral dimension from the topological dimension d on low energy scales to a real number 0<α<d on high energy scales. In the particular case of α=1 these results allow to understand the quantum geometry as effectively fractal.

 

[Justice Hunter]

I just saw this over on the "Loop and Allied QG" bibliography thread. I'm don't want to stray too far from your original post but the paper you found does seem to me to point to the unresolved relationship between QM and GR and invoke "self organizing criticality".

This paper is very recent but it appears to make some connection between "discrete quantum geometry" (Quantized GR) and fractals, which are related to SOC.

It appears to be 203 pages long though!

http://arxiv.org/abs/1510.08706
Discrete quantum geometries and their effective dimension
Johannes Thürigen
(Submitted on 29 Oct 2015)
In several approaches towards a quantum theory of gravity, such as group field theory and loop quantum gravity, quantum states and histories of the geometric degrees of freedom turn out to be based on discrete spacetime. The most pressing issue is then how the smooth geometries of general relativity, expressed in terms of suitable geometric observables, arise from such discrete quantum geometries in some semiclassical and continuum limit. In this thesis I tackle the question of suitable observables focusing on the effective dimension of discrete quantum geometries. For this purpose I give a purely combinatorial description of the discrete structures which these geometries have support on. As a side topic, this allows to present an extension of group field theory to cover the combinatorially larger kinematical state space of loop quantum gravity. Moreover, I introduce a discrete calculus for fields on such fundamentally discrete geometries with a particular focus on the Laplacian. This permits to define the effective-dimension observables for quantum geometries. Analysing various classes of quantum geometries, I find as a general result that the spectral dimension is more sensitive to the underlying combinatorial structure than to the details of the additional geometric data thereon. Semiclassical states in loop quantum gravity approximate the classical geometries they are peaking on rather well and there are no indications for stronger quantum effects. On the other hand, in the context of a more general model of states which are superposition over a large number of complexes, based on analytic solutions, there is a flow of the spectral dimension from the topological dimension d on low energy scales to a real number 0<α<d on high energy scales. In the particular case of α=1 these results allow to understand the quantum geometry as effectively fractal.

This is a really great find, i can barely wrap my head around it since i don't know how to do all the equations, but I'm going to take the time to read through this carefully. care to maybe summarize what's happening in the paper that makes it fractal? might make it easier for me to understand.

 

[Jimster41]

This is a really great find, i can barely wrap my head around it since i don't know how to do all the equations, but I'm going to take the time to read through this carefully. care to maybe summarize what's happening in the paper that makes it fractal? might make it easier for me to understand.

I wish I had the ability to do that!

I do think he's saying something a bit radical. Pretty much everything I can get from the intro is tantalizing though.

 

[Jimster41]

I realize now that paper is pulling together a set of building blocks the author and others have laid. Some links below.http://arxiv.org/abs/1208.0354
Laplacians on discrete and quantum geometries
Gianluca Calcagni, Daniele Oriti, Johannes Thürigen
(Submitted on 1 Aug 2012 (v1), last revised 17 May 2013 (this version, v2))
We extend discrete calculus for arbitrary (p-form) fields on embedded lattices to abstract discrete geometries based on combinatorial complexes. We then provide a general definition of discrete Laplacian using both the primal cellular complex and its combinatorial dual. The precise implementation of geometric volume factors is not unique and, comparing the definition with a circumcentric and a barycentric dual, we argue that the latter is, in general, more appropriate because it induces a Laplacian with more desirable properties. We give the expression of the discrete Laplacian in several different sets of geometric variables, suitable for computations in different quantum gravity formalisms. Furthermore, we investigate the possibility of transforming from position to momentum space for scalar fields, thus setting the stage for the calculation of heat kernel and spectral dimension in discrete quantum geometries.http://arxiv.org/abs/1311.3340
Spectral dimension of quantum geometries
Gianluca Calcagni, Daniele Oriti, Johannes Thürigen
(Submitted on 13 Nov 2013 (v1), last revised 18 Jun 2014 (this version, v2))
The spectral dimension is an indicator of geometry and topology of spacetime and a tool to compare the description of quantum geometry in various approaches to quantum gravity. This is possible because it can be defined not only on smooth geometries but also on discrete (e.g., simplicial) ones. In this paper, we consider the spectral dimension of quantum states of spatial geometry defined on combinatorial complexes endowed with additional algebraic data: the kinematical quantum states of loop quantum gravity (LQG). Preliminarily, the effects of topology and discreteness of classical discrete geometries are studied in a systematic manner. We look for states reproducing the spectral dimension of a classical space in the appropriate regime. We also test the hypothesis that in LQG, as in other approaches, there is a scale dependence of the spectral dimension, which runs from the topological dimension at large scales to a smaller one at short distances. While our results do not give any strong support to this hypothesis, we can however pinpoint when the topological dimension is reproduced by LQG quantum states. Overall, by exploring the interplay of combinatorial, topological and geometrical effects, and by considering various kinds of quantum states such as coherent states and their superpositions, we find that the spectral dimension of discrete quantum geometries is more sensitive to the underlying combinatorial structures than to the details of the additional data associated with them.

http://arxiv.org/abs/1412.8390
Dimensional flow in discrete quantum geometries
Gianluca Calcagni, Daniele Oriti, Johannes Thürigen
(Submitted on 29 Dec 2014 (v1), last revised 21 Apr 2015 (this version, v2))
In various theories of quantum gravity, one observes a change in the spectral dimension from the topological spatial dimension d at large length scales to some smaller value at small, Planckian scales. While the origin of such a flow is well understood in continuum approaches, in theories built on discrete structures a firm control of the underlying mechanism is still missing. We shed some light on the issue by presenting a particular class of quantum geometries with a flow in the spectral dimension, given by superpositions of states defined on regular complexes. For particular superposition coefficients parametrized by a real number 0<α<d, we find that the spatial spectral dimension reduces to ds≃α at small scales. The spatial Hausdorff dimension of such class of states varies between 1 and d, while the walk dimension takes the usual value dw=2. Therefore, these quantum geometries may be considered as fractal only when α=1, where the "magic number" dsspacetime≃2 for the spectral dimension of space\emph{time}, appearing so often in quantum gravity, is reproduced as well. These results apply, in particular, to special superpositions of spin-network states in loop quantum gravity, and they provide more solid indications of dimensional flow in this approach.

http://arxiv.org/abs/1409.3150
Group field theories for all loop quantum gravity
Daniele Oriti, James P. Ryan, Johannes Thürigen
(Submitted on 10 Sep 2014 (v1), last revised 14 Feb 2015 (this version, v2))
Group field theories represent a 2nd quantized reformulation of the loop quantum gravity state space and a completion of the spin foam formalism. States of the canonical theory, in the traditional continuum setting, have support on graphs of arbitrary valence. On the other hand, group field theories have usually been defined in a simplicial context, thus dealing with a restricted set of graphs. In this paper, we generalize the combinatorics of group field theories to cover all the loop quantum gravity state space. As an explicit example, we describe the GFT formulation of the KKL spin foam model, as well as a particular modified version. We show that the use of tensor model tools allows for the most effective construction. In order to clarify the mathematical basis of our construction and of the formalisms with which we deal, we also give an exhaustive description of the combinatorial structures entering spin foam models and group field theories, both at the level of the boundary states and of the quantum amplitudes.

 

[Jimster41]

I did try to read the basis papers and part of the big one. Actually I think they are trying to qualify some things others have said about space time geometry being "fractal". To do so they survey the candidate Loop Quantum Gravity theories, further develop the tools for calculating properties of space time that would result from choices of parameters of those theories, and explore via numerical simulations what happens to the Spectral Dimension (which is related to the "heat kernel") when some of those parameters are varied. Their development of the discrete Laplacian is in support of this but also supports other explorations of LQG theories. I do think that the spectral dimension and "heat kernel" could be related conceptually to the thesis of the paper you found re Zipf's Law and Dark Matter but they also supersede it in terms of precision and currency. I had not known that others had claimed space time might be fractal. These authors confirm that under very specific and potentially interesting choices of LQG parameters it might be but theirs appears more to be a negative result re that claim. To which I say... So then why all the darn fractals everywhere? Gotta be a reason. But then as they point out it isn't clear what a "fractal" is in the first place.