Simulatin local mechanisms as a qualitative explanation to temporal fractality in galaxy formation

8/12/2019 Simulatin local mechanisms as a qualitative explanation to temporal fractality in galaxy formation

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Simulating local mechanisms as a qualitative

explanation of temporal fractality in galaxy formation

Guifre S anchez

[email protected]

under the direction of

Mr. Alec Resnick

Massachusetts Institute of Technology

Research Science Institute

July 30, 2014


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Abstract

Cens [1] recent, large scale simulations of galaxy formation have revealed a fractal distribu-

tion in time in addition to their known, fractal distribution in space. A qualitative, systems

explanation for why this behavior was observeda simple local/global interactionhas been

put forward but not tested. We generalize, simplify, and simulate this qualitative description

of a reservoir of density and an element formation process to discern how fundamental the

local/global effects are to the observed scaling behavior. We nd that even at its most gen-

eral Cens mechanism produces a scaling behavior robust to changes in dimensionality and

spatial resolution, strongly suggesting that it is core to the observed temporal behavior in

galaxy formation.

Summary

Renyue Cen recently documented that galaxy formation is not only fractal in space but is also

fractal in time. This is surprising because it means that observing galaxy formation events

at multiple lengths and time scales reveals similar behavior. Understanding the origin of this

phenomenon is core to understanding galaxy formation, an important problem in modern

astrophysics. However, Cens work suggests but does not corroborate a mechanism explaining

this behavior. We generalize, simplify and simulate Cens hypothesis to provide evidence that

the same qualitative behavior does not depend on the physical details of Cens simulation.

We nd strong evidence to support this and suggest some future directions for documenting

the various effects of increasing our simulations physical realism on this qualitative results.


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1 Introduction

A cloud is made of billows upon billows upon billows that look like clouds. As

you come closer to a cloud you dont get something smooth, but irregularities at

a smaller scale.

Benot B. Mandelbrot (20 November 1924 - 14 October 2010)

Fractals have been commonly associated with geometry and space. One of their main char-

acteristics is that they exhibit repeating patterns that display at any scale, so in fact, the

concept of self-similarity needs not only to apply to geometrical structures [2],[3]. Geometry

helps us to understand their nature, to be more clear about the meaning of being fractal,

but it is only a way to express a mathematical behavior. Naturally, fractals can be treated

only using mathematics, and this fact extends and generalizes their properties, which turn

out to acquire important complexity. Using time in order to nd fractal structures has been

hardly considered, although the existent possibility of it. Considering that, the following re-

search will report an investigation on a discovery in astrophysics, where fractal distributions

in time, regarding the process of galaxy formation, have been found.

In [1] a study of the properties of the distributions of star formation events in time through

a computer simulation of the universe is presented, eventually conrming the existence of a

power-law relation between them.

A power-law is a relation between two quantities that is expressed mathematically as follows

[4]: y x . The meaning of this formula is that, as we increase minimally the x variable,

y increases or decreases if the scaling parameter is positive or negative, respectively

considerably rapid, depending on how big is the value of .The author (Renyue Cen) analyzed the behavior of two numbers, n50 and n90 , dened as the

number of top star formation peaks that made up 50% and 90% of total amount of stellar

mass of galaxies appeared in the simulation.

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Star formation peaks are dened as time periods during the evolution of a galactic system

when the total stellar mass produced per unit time is relatively high. Cen catalogued and

ranked order a complete list of all peaks considering the total stellar mass they provided to

the galaxies. That is the reason for the top word appeared in their denition.The evolution of these numbers as we advance in time seems to follow a power-law with

roughly the same scaling parameter . In fact, the estimated value for the scaling parameter

is 1 , where is an approximation of the golden ratio .

The main result of Cens work is shown in Figure 1.

Figure 1: Top panel shows the probability density distribution of computed scaling param-eters 50 and 90 for all galaxies with stellar masses greater than 10 10 M (Solar masses) atz = 0.62. Bottom panel shows the median of n50 and n90 as a function of time for all galaxieswith same characteristics.

As time advances, n50 and n90 decrease. That is explained as follows: galaxy formation is

a process that covers large ranges in space and time. In space, star formation starts being

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very dispersed, with brief star formation peaks appearing at more or less constant rate in

time. Small star formation events present higher probability to be formed during the start of

the galaxy formation process. But there remains a chance for large star formation peaks to

appear, as the gas density distribution gets more and more dened in some regions becauseof the provided uctuations by small star formation events. Cen analyzed the top ones that

made up the 50% and 90% of the total stellar mass of the system. However, it would seem

that as we advance in time, as there is less gas to fuel star formation, the likelihood of big star

formation events to appear is reduced, and therefore, these top ranked peak events should

be increasing. Nevertheless, we need to emphasize that galaxy formation is a process that

covers huge ranges in space and time, so although there is a little probability, big star forma-

tion events can happen. The mass they produce is so high that they overcome the generated

stellar mass from previous star formation events, triggering that we need considerably less

amount of top star formation events, covering a determined percentage of the total stellar

mass of the studied galaxy. Now it is clear that those numbers need to decrease over time,

but in fact, is not obvious that they follow the simple scheme provided by a power-law That

is why Cens work is relevant, because despite the apparently chaotic fashion governing the

process of galaxy formation, he was able to nd an analytical function to describe the be-

havior of star formation distributions in time. Cens work could be the start to unveil the

complete understanding of galactic formation.

Although Cens nding still needs to be qualitatively understood, he suggested a hypothesis

that can be taken as a starting point to begin research: galaxies are normally embedded in

a gas reservoir, which provides fuel to form stars. We consider the appearance of a trigger,

when some gas is driven inwards to some point in the gas reservoir, to fuel star formation.Cen suggested that trigger events are not usually followed or preceded by larger triggers, but

by smaller ones. And supposing a larger trigger appearing after a large previous one, there

would be a signicant decay in terms of stellar mass produced by the last trigger, because

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of the amount of gas collected by the rst appeared one. This compensated behavior could

lead star formation peaks to present a fractal distribution.

In this work, we consider generalizing Cens hypothesized mechanism, to conrm the ex-

istence of fractal distributions in systems where physics are not implied. Thus, our mostgeneral description of Cens hypothesis is given by the following explanation: a system is

constituted by the distribution of some material, i.e gas, in space; trigger events of varying

sizes can create elements, i.e. stars, depending on the concentration of that material given

a region of the system, i.e. density. We establish, in accordance to Cens hypothesis that

the probability of same-sized or larger elements to be formed, relative to elements formed

by previous trigger events, turns to be reduced as we advance in time. According to that,

time causes elements to become smaller-sized, at a rate that is still to be conrmed as a

power-law.

The spatial distributions of galaxies have been extensively studied observationally, conrming

them being fractal over specic ranges in space. Power-laws are considered mathematically

as fractal objects because they also present self-similarity characteristics, fact that explains

why do we considered them to describe fractal distributions. Hence, in some way, Cens

work suggests a fundamental joint spatio-temporal self-organization governing the process

of galaxy formation, which could lead to nd a model that completely describes galactic

behavior.

We expect Cens hypothesis to be valid for a variety of different systems. According to that,

we construct a design of a model based on our general description of his hypothesis to perform

computer simulations, looking at event timing in order to characterize fractal distributions.

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2 Simulations

We designed a system modeling it in 1, 2 and 3 dimensions. In 1 dimension, we consider the

system to be a line, in 2, a square, and in 3, a cube. We provide some logical and fundamental

considerations that describe the main rules governing our simulations in the conceptual list

in page 6.

2.1 Methodology

We consider a system to be mainly dened by spatial and temporal resolution. First, we

dene spatial resolution S r and time resolution T r to be the ratio between trigger radius r

and side length of the eld H ; and the ratio between time advanced per time step d t and the

total simulation time T , respectively. As we get closer to higher resolutions (small S r and T r

values), our system approximates to continuous models approaching realism. The main pur-

pose of the characterization is to nd a resolution conguration leading us to observe similar

behavior as described in Cens result, i.e. fractal distributions in time of trigger events. After

running the simulation we will have a list of all triggers that formed an element with theircorresponding capacity values t , and a list of all elements with their corresponding charge

values q . Histograms will be used to represent our data considering bin size to be determined

by the Freedman-Diaconis rule, in order to ensure representative statistical analysis.

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Denitions

A field is dened as the region of space embedding the distribution of the consideredmaterial. The eld is also characterized by side length H ; corresponding to line length,square side length and cube side length respectively for each dimensional case.

The eld is spatially discretized in cells, forming a grid . The grid is dened by the numberof cells C n per side.

A cell is dened by position 1, 2, or 3 coordinate values are considered according to eachdimensional case ( x,y,z ), density value c, and side length d h, which is dened as in eldsdescription (considering cells to be segments, squares or cubes depending on 1D, 2D or3D cases).

Initial conditions are provided, giving a constant cell density value cI at the beginning of the simulation.

Triggers are dened by position distinguishing each dimensional case ( x,y,z ), absorption

behavior, action area and capacity value t . , describes the number of cells a trigger can cover to absorb density. is a segment,

a square or a cube, considering 1, 2, or 3 dimensions respectively, and is dened bytrigger radius r , which is the minimum distance between the position of the trigger(centered in ) and the extreme of the segment, the side of the square, or the sideof the cube.

Absorption behavior, characterize the way triggers acquire density from neighboringcells

t , indicates the amount of total density a trigger can absorb from each cell within .

Elements are formed by triggers and are dened by position considering differences ineach dimensional case ( x,y,z ) and charge value q .

Triggers are placed randomly in space and time. We consider a trigger to have 50% prob-ability of appearing every time step.

Triggers cannot absorb elements.

t and q are values within ranges (1 , max ) and (1 , q max ) respectively.

dt is time advanced per time step

T is total simulation time

T n = T / dt is total number of time steps.

Boundary problems are solved using periodic boundary conditions .

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2.2 Diffusion

The most intuitive approach we consider to model density uctuations in a eld is having

diffusion as the governing process to describe its evolution in time. Hence, the heat equation

is used because it describes the trend of temperature, i.e. density, to reach homogeneity:

ut

2 ux 2

+ 2 uy 2

+ 2 uz 2

= 0 (1)

ut

2 u = 0 (2)

Where u is an arbitrary scalar function dening a value for each point ( x,y,z ) within a

considered region of space given a time t. is a positive coefficient whose value is simplied tobe 1 when mathematical treatment is required, as we will consider. Equation 2 shows another

way to express 1, using the Laplacian operator [5] 2 . The discretized heat equations [6],[7]

used in the simulation to resolve thermal diffusion distinguishing 1, 2, and 3dimensional

cases are:

1D : t +1i = ti +

dtdh2

ti+1 + ti 1 2

ti

2D : t +1i,j = ti,j + dtdh2

ti +1 ,j + ti 1 ,j + ti,j +1 + ti,j 1 4ti,j

3D : t +1i,j,k = ti,j,k +

dtdh2

ti+1 ,j,k + ti 1 ,j,k +

ti,j +1 ,k +

ti,j 1 ,k +

ti,j,k +1 +

ti,j,k 1 6

ti,j,k

Where ti , ti,j and ti,j,k are the density functions of the eld, equivalent to u in the heat

equation. Indices i, j and k refer to cell positions relative to the grid, and t indicates time

step. Finite difference approximations have been applied to obtain the presented equations.

2.3 Model

Every trigger is initialized with a random value for t which falls in the interval (1 , max ).

Trigger capacity values t are used to dene the fraction of density t / max absorbed from

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cells within , covering a determined number of cells m N . Thus, every trigger is allowed

to form an element, with charge value S = mi=1

t max c,i , where c,i is considered the density

value of one of the m cells within . After absorption, cell density values are reduced from

ctk , representing density value of an arbitrary cell (within ) k at time step t, to ct +1k =

1 t max ctk . This fact, drives every cell in the system to reach density value ck 0 as the

total simulation time tends to innity, T , assuming triggers appearing indenitely. We

consider our system to be consistent because, as we advance in time, element charge values

q will be reduced because two facts: 1) triggers are not considered to evaluate diffusion and

2) they cannot absorb formed elements. Trigger absorption behavior is illustrated in Figure

3.

Figure 2: When a trigger is placed (1), the action area of the trigger is dened, as wellas a capacity value t , chosen randomly from the interval (1 , max ). After that, the trigger

absorbs proportionally to the ratio t / max a quantity of density from each cell c,k within .The sum of all these quantities S will be the charge value q of the element the trigger willform (3). New density values for each cell c,k are provided, subtracting the absorption valuethe element acquired as charge, i.e. ct +1k = 1

t max c

tk . Diffusion computation is evaluated

every time step, but it is not considered for elements nor triggers.

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3 Results

For each dimensional case, we evaluate 3 different spatial resolutions with a x value for time

resolution T r = 5 10 5 , which ensures the system to evolve enough as for recreating fractal

distributions in time. We rst consider histograms representing the distribution of trigger

events according to the charge value q of the elements they formed.

Then, we evaluate in some cases the likelihood of the distribution to t with a power-law

using linear regression in logarithmic scaled plots. Bins in histograms valued with 0 frequency

are simply not considered for the calculation of scaling parameters. We refer values for spatial

and temporal resolutions using number of cells per side C n , and total number of time steps

T n . Then S r = C 1n and T r = T 1n .

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3.1 Element charge value q distribution

3.1.1 1D

Figure 3: Trigger event distribution according to element charge value q in 1 dimension. We reducedranges in each histogram to provide a correct visualization of the distribution. Spatial resolutionvalues are, from left to right, 1000 1 , 5000 1 and 10000 1 .

Figure 4: Logarithmic plot of trigger distribution according to charge value q in 1 dimension atspatial resolution S r = 10000

1 . We estimated the scaling parameter to be 0.85 with correlationcoefficient r 2 = 0 .71.

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3.1.2 2D

Figure 5: Trigger event distribution according to element charge value q in 2 dimensions. Valuesfor spatial resolution are 100 1 , 500 1 and 1000 1 for left, middle and right panels respectively.

Figure 6: We evaluate if power-law ts for spatial resolutions 100 1 and 500 1 . We observe that atS r = 500

1 linear regression (right panel) provides a good adjustment, meaning that a power-lawcould be considered to describe the distribution. Left and middle log-log plots represent differentq ranges for same histogram in Figure 5 at S r = 100

1 . Left panel covers all q values computedin the simulation. Middle panel does not consider the rst interval in the histogram. Power-lawparameters obtained at S r = 100

1 have been = 1.24 with r 2 = 0 .85. At S r = 500 1 , = 1.10

and r 2 = 0 .96

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3.1.3 3D

Figure 7: Trigger event distribution ac-cording to element charge value q in 3 di-

mensions. Left and right upper corner his-tograms correspond to same spatial reso-lution S r = 50

1 , but different q ranges.Left histogram considers all q values gen-erated during the simulation while rightavoids the small ones in order to offera better view of the distribution. Leftand right bottom histograms correspondto spatial resolutions 10 1 and 100 1 re-spectively. At S r = 10

1 , values for bin-size according to Freedman Diaconis rulewere too small as for providing visual-izable data. We reduced the number of breaks then to 500, and modied the q range to (0 , 9500).

Figure 8: Logarithmic scale plots of histograms shown in Figure 7. Each plot correspond to sameS r as described in Figure 7. Power law parameters at spatial resolutions 10

1 , 50 1 and 100 1 havebeen: ( 1 = 1.28,r 21 = 0 .80), ( 2 = 1.08,r 22 = 0 .86) and ( 3 = 1.05,r 23 = 0 .91), respectively.

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3.2 Trigger capacity value t distribution

Figure 9: Distribution of trigger events according to capacity value t . 1, 2 and 3 dimensionsfrom left to right. Time resolution is T r = 5 10

5 . Spatial resolutions are 10 4 , 10 3 and 10 2

for 1, 2 and 3 dimensions, respectively. As we are selecting randomly t values, we observe

what could be expected, a uniform distribution.

4 Discussion and Conclusions

The results we obtained strongly suggest Cens hypothesis is reasonable. We observed power-

law behavior in the distribution of trigger by element charge q . Our investigation was not

physically realistic at all, demonstrating that the general scheme of Cens mechanism is all

that is required to create power law behavior. The non-specicity and general nature of our

system strongly suggest that Cens mechanism is part of if not the entire explanation for

his observed results.

Nonetheless, it is interesting to explore the dependence of the systems power law behavior

on various dimensions of increasing its physical realism. We considered increasing spatial

resolution as the rst of these, but many others ( e.g. a non-instantaneous absorption mech-

anism) merit further investigation.

In simulations with the lowest spatial resolutions, the smallest q valued interval comprise

the majority of the trigger distribution. We attribute this to the combination of three facts:

1) at some point, triggers are lling every cell in the grid, 2) diffusion is not applied, and 3)

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there are not free cells for triggers to absorb density.

In the 2D and 3D cases, the scaling parameter decreases as we increase spatial resolution

(Figures 5 and 7). This may be explained by the fact that as triggers have more space to

absorb from, the importance of diffusion decreases, meaning the charge distribution tendstoward uniformity for increasing spatial resolutions. If spatial resolution is reduced, given a

xed time resolution T r , triggers more rapidly ll the available space, meaning the diffusion

becomes more prominent in determining density distribution.

As the system advances, large triggers will be less likely to have access to the required den-

sity, making smaller elements increasingly likely, which may explain the spikes we observe in

the rst intervals in low spatial resolution histograms. However, the character of the trend

to uniformity as we increase spatial resolution is not clear, e.g. in Figure 5 (2D case), it is

not clear whether the distribution converges to a linear relation or to some power law.

This and other ways of increasing the physical realism of the system merit more investigation

to understand which aspects of Cens hypothesis are essential in which ways to the scaling

behavior of the system in time and space.

The fact that we observed the power-law behavior described herein is robust to changes

in dimensionality or physical realism ( e.g. instantaneous absorption) suggests that Cens

mechanism is fundamental to the fractal distribution of star formation events in time.

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References

[1] R. Cen. Temporal self-organization in galaxy formation. Available at http://arxiv.

org/abs/1403.5265 (2014/Mar/20).

[2] U. of Warwick. Lectures on fractals and dimension theory. Available at http:

//homepages.warwick.ac.uk/ ~ masdbl/dimension-total.pdf .

[3] B. A. Steinhurst. Notions of dimension. Available at http://math.cornell.edu/

~steinhurst/docs/dimension.pdf (2010).

[4] A. Clauset, C. R. Shalizi, and M. E. Newman. Power-law distributions in empirical data.

Available at http://arxiv.org/abs/0706.1062 (2007/Feb/7).

[5] A. S. U. Huei-Ping Huang. Numerical method for laplaces equation. Available at http:

//www.public.asu.edu/ ~hhuang38/pde\_slides\_numerical\_laplace.pdf (2014).

[6] G. W. Recktenwald. Finite-difference approximations to the heat equation. www.nada.

kth.se/ ~jjalap/numme/FDheat.pdf (2011/Mar/6).

[7] I. S. U. Ambar K. Mitra, Department of Aerospace Engineering. Finite difference method

for the solution of laplace equation. Available at http://akmitra.public.iastate.

edu/aero361/design\_web/Laplace.pdf (2014).

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A Visualization

Figure 10 illustrates a qualitative visualization of the temporal evolution of the designed

model for each dimensional case.

Figure 10: Temporal evolution is considered as we go down. Left, middle, and right side of the

gure represent the evolution of the system considering 1, 2 and 3 dimensions, respectively.

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Simulatin local mechanisms as a qualitative explanation to temporal fractality in galaxy formation