Popular in Course
Popular in Physics 2
This 17 page Class Notes was uploaded by Vernice Schuster on Wednesday September 23, 2015. The Class Notes belongs to PHYS750 at Drexel University taught by RobertGilmore in Fall. Since its upload, it has received 19 views. For similar materials see /class/212525/phys750-drexel-university in Physics 2 at Drexel University.
Reviews for STQuantumFieldTheory
Report this Material
What is Karma?
Karma is the currency of StudySoup.
You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!
Date Created: 09/23/15
Chapter 3 Fractals 31 Examples of Fractals A fractal is a geometric object which is selfsimilar with structure at all levels of magni cation Rather than try to tighten down on this de nition it is more useful to generate some examples Example 1 In Fig 3ila we show an interval of length 1 ln going from a to b we remove the middle half of this interval This leaves two intervals each of equal length i This rst step is the generating step The second step from b to c is a repeated application of the generating step We remove the middle half of each of the two subintervalsi This leaves 4 22 intervals all of equal length i We continue in the obvious way At the n step we have 2 intervals each of length This process continues foreveri 32 Fractal Dimension A convenient way to de ne the dimension of a geometric object is to cover it with boxes whose edge length is e 239e small In Fig 32 we show how this process works for some familiar geometric objects two points a smooth curve and a simple area In these three examples the number of boxes Ne required to cover the geometric objects behaves like Geometric Object Ne Points P N KeD Smooth Curves C N Ke1 Simple Areas A N K62 where K is an unimportant constant The number of boxes required to cover the geometric object behaves like e d where d is the dimension of the object We can turn this observation around and use this type of computation to de ne the dimension of peculiar objectsi 2 CHAPTER 3 FRACTALS 12 0 l l a b l r 39 39 0 r 1 I 1 a H H H H I 4 H H H Figure 31 A middle half fractal is constructed by repeated application of the rst generating step The middle half of the interval of length one a is removed lo At each succeeding step the middle half of each interval is removed This continues forever 321 Definition of Dimension Box Counting De nition We de ne the dimension d of a geometric in terms of e and N e as follows d 2 hm log N gtO Example 1 Continued At the nth step of the generation process of the middle half fractal there are 2 boxes each of length The fractal dimension is therefore l 2 l 2 1 322 Dimension of the Middle 1 p Fractal Example 2 We can generalize this to fractals These are fractals in Which the middle of the interval is removed in the generating step Each interval obtained during the generating step has length 6 Then log2 d 1 2 19 0sp1 33 TWO SCALE FRACTALS 3 For p 2 37 4 these dimensions are p Dimension 2 5 3 Ioglt2gt loge 4 Ioglt2gt Ioglt83gt s Ioglt2gt Ioglts2gt We plot the fractal dimension7 d as a function of f 117 in Fig 33 323 Direct Product Spaces Direct Sum Dimensions Fractals in higher dimensional spaces can be built up systematically as direct products of fractals in lower dimensional spaces If a fractal is a direct product of two fractals with dimensions d1 and d2 then its dimension is the direct sum of the dimensions of the two fractals dd1d2 As an example7 a fractal in the plane can be constructed as the direct product of the middle half fractal along each of the axes The dimension of this direct product fractal is then 1 1 d 7 7 1 2 T 2 It is clear from this example that fractals can have integer dimension 33 Two Scale Fractals 331 Construction Another way to build up fractals is shown in Fig 34 In the generating step7 an interval of length 1 is reproduced twice7 once reduced by the scale factor A1 the other time reduced by the scale factor A2 These reduced in tervals are shown on the left and right in Fig 34b The process is re peated in the second generation This produces four subintervals7 of lengths A Ang AgAl Ag proceeding from left to right In the third generation the distribution is A 3A A27 3A1Ag A3 You can see the binomial distribution of lengths emerging from this process7 which of course continues forever7 as before 332 Dimension The dimension of this two scale fractal can be computed as follows Assume that at level Is Nk 6 boxes of length 6 are required to cover the 2k intervals At the next level Is 17 the structure on the left is a scaled down version of the entire structure at level 16 Therefore the number of boxes of length 6 required 4 CHAPTER 3 FRACTALS l AA e 7 b 2 Q c Figure 32 Ott p 70a Two boxes cover two points no matter how small the boxes are b The number of boxes required to cover a smooth curve is proportional to the length of the curve and inversely proportional to the box size that is N E N 16 C The number of boxes required to cover the area behaves like Ne N 162 33 TWO SCALE FRACTALS 5 Dimension of Middle 1p Fractal 09 08 07 06 05 04 Fractal Dimension 03 02 01 0 i i i i i i i i i 0 01 02 03 04 05 06 07 08 09 1 f1lp Figure 33 The dimension of a middle 117 fractal is plotted as a function of f 19 6 CHAPTER 3 FRA CTALS a3 s 1 l c I E 3 3912 c 1 397 m39 m 1 2 Figure 34 Construction of a two scale fractal proceeds as shown Each of the two subintervals in the generating stage a a b is a replica of the original reduced in scale by the scale factors A1 and A2 If A1 is negative 1 lt A1 lt O the orientation of an interval is reversed when scaled down by A1 to cover the left half of the structure at level k 1 is equal to the number of larger boxes of length 6 A1 required to cover the structure at level k Nk1left E Nk EAl A similar argument holds for the half on the right at level k 1 Thus we have Nk1 Nltk1zeft 6 Nk1mght 6 NickA1 NickA2 If we assume as usual that Ne N Ked then Ke d Ke1 dKe2d leads directly to a simple expression defining the fractal dimension d A A lzl Fractals obtained from three or more scaling transformations in the generating step have dirnensions determined by similar expressions 333 Feigenbaurn Fractal The Feigenbaurn attractor is the fractal which exists at the accumulation point of the period doubling cascade Figure 35a shows the locations of orbits of periods 1 2 4 8 The locations suggest that a scaling exists This scaling is reinforced in Fig 3 5b which shows the locations of points in the orbits of 2 These occur alternately in the left and the right halves of the return plot and seem to obey scaling 1042 on the left and 1d on the right We now describe how to view the Feigenbaurn attractor as a two scale fractal Begin by connecting the two points in the period two orbit by a line Next 8 CHAPTER 3 FRACTALS Figure 36 The beginning of the bifurcation diagram is shown At the bifur cation 2 a 2 271 1 segments are drawn which connect adjacent points on the 2 orbit 334 Two Scale Fractal Dimensions Fractal dimension is not generally a constant To illustrate this idea we consider a unit square which is mapped into itself according to the following rules In the generating step two images are created In the first image the m axis 3 O is mapped to itself the upper side at y 1 is mapped to the parabola y 001 2 X m 072 The V ll11 is unchanged and 3 values are linearly scaled between the boundaries In the second image the side 3 1 is mapped to itself and the side 3 O is mapped to the straight line 3 099 O80gtIltQ The boundaries of these scaling regions are shown with light lines in Fig 38 The fractal dimension in the y direction varies as a function of m The dimension is shown by the heavy line in this figure A histogram of the fractal dimension distribution for this fractal is shown in Fig 39 The fractal dimension of the two scale fractal built up by the generating step shown in Fig 3 8 is Dimension 2 1 d 32 The 1 comes from the m direction which is smooth The fractal structure is 34 OTHER DIMENSIONS 9 C program fraCdimif January 307 2001 C This program computes the fraCtal dimension of the C Feigenbaum attraCtor by the 7divide and Conquer7 methodi impliCit none inte er i real8 lam17lam2 real8 alpha7delta7X7y C begin alpha 2150290 78750 95892 8485 1 input data delta 4166920 16091 029 lam1 1 0alpha l establish sCaling lam2 1 0alpha2 dmin 010 l initialize dmax 110 do i125 l begin divide and Conquer d 015dmindmax ylam1d lam2d 1 ifyigt1010dmind ifyilt1010dmaxd end do ll end divide and Conquer Write72x2f12187dy 1 output result7 error stop end Figure 317 This short FORTRAN Code Computes the fraCtal dimension of the Feigenbaum attraCtor to 25 bits d 0152450 830401 1 only in the y direCtioni SinCe the fraCtal dimension varies along the zaxis7 the average dimension ltdgt is taken The average is Computed by interpreting the histogram in Fig 319 as a probability distribution ltdgt fol zp2dzi 34 Other Dimensions A number of other dimensions have been introduCed in an attempt to distinguish geometry from dynamiCsi Almost all of these are based on the invariant measure over a strange attraCtori ReCall that this is de ned as M M10 Ci lim M 33 Taco T Here 10 is an initial Condition for the dynamiCs7 Ci is box 239 in a very re ned partition of the phase spaCe7 T measures the temporal length of a trajeCtory7 and 77 measures the total time the trajeCtory is in Cube ii Remark The quantities M or their limits p17 are Called measures They are invariant measures if Mr for all 1 lnvariant measures are Closely 10 CHAPTER 3 FRACTALS Dimension for Two Scale Fractal ambda ambda2 Fractal Dimension 0 O O O O O O I 0 Jgt II D l 00 O Figure 38 The fractal dimension of a twoscale fractal is plotted as a function of position along the zaxis The two scales are zdependenti They are the distance below the parabola and the distance above the straight line The third curve is the fractal dimension related to discussions of ergodicity the equality of time averages With space av erages for almost all initial conditions The ergodic hypothesis is usually assumed as a foundation for statistical physics The existence of invariant measures is a necessary but not sufficient condition for the proof of the ergodic theoremi Other conditions irreducibility77 in some sense are necessary and not usually met in statistical physics 341 Information Dimension The information dimension is de ned as H 61131 Zioilogoi 3 4 34 OTHER DIMENSIONS 11 Distribution of Fractal Dimensions Two Scale Fractal 1 i i i 09 08 07 06 05 04 Frequency of Occurrence 03 02 01 0 i i i i i i i i i 0 01 02 03 04 05 06 07 08 09 1 Fractal Dimension Figure 39 Histogram of the relative occurrence of fractal dimension of the two scale fractal is plotted as a function of the fractal dimension The van Hove singularity is characteristic of a quadratic turnaround7 and occurs in one dimensional Quantum Mechanical lattice models This is the Shannon Boltzmann entropy function 342 Correlation Dimension The correlation dimension is the fractal dimension Which is most often used in the analysis of data It is de ned as follows Count the number of points Within a distance 6 of each other NltegtZZelteeixrx1igt 35gt ifj 1 In this expression7 xi are points on an attractor in an n dimensional phase space7 y is the Heaviside theta function y 0 if y lt 0 and y 1 if y 2 0 it is the integral of the Dirac delta function y film 6zdz This number 12 CHAPTER 3 FRACTALS 8 10 7 10 4 1 5 5 4 4 IIII u95 9 0 so logzr Figure 310 BMC 93 pg 36 Correlation dimension computations are shown for the X ray binary Her X 1HZ Her A and background data The em bedding dimension ranges from one bottom curve both cases to 20 top The correlation integral is capable of distinguishing deterministic chaos from noise decreases as 6 decreases One hopes that this number decreases exponentially Ne N edC If so then the ratio dC lim 10gN e gtO should might exist This limit defines the correlation dimension The correla tion dimension is generally not the same as either the box counting dimension or the information dimension Some samples of the use of this statistic are shown in Figs 310 and 311 Fig 310a is a correlation dimension calculation for data from the X ray binary Her X 1HZ Her The scalar time series data are embedded as vectors in spaces of dimension 1 2 20 one curve for each dimension from bottom to top The correlation integral is carried out and its slope is plotted as a function of 6 Some of the curves converge to a more or less constant slope over a limited region of the size parameter 6 The converged slope is interpreted as the correlation dimension In Fig 310b the same computation is repeated on background data assumed to be gaussian noise This series of 20 curves behaves remarkably different If the correlation dimension computation does not provide a convincing quantitative value for a dimension at least it provides a mechanism to distinguish between processes with low dimensional deterministic structure and those without Fig 311 shows two additional attempts to determine correlation dimen sions In these cases embeddings of dimensions 1 2 40 were made The curves on the left were computed for numerically generated data from the Lorenz attractor The curves on the right were made from data taken on a far infrared 36 34 OTHER DIMENSIONS 13 3 CW 1 l l I 39339 lll 335 AW 2339 39 13 3 h N 39 g quot 4 N a N m M Q quot1 Cl V 39 y z 0 l 1 J l l 39 quot 0 1 2 3 4 5 0 6 1099 if quot gt Figure 311 BMC 93 pg 135 Correlation dimension computations are shown for numerically generated data Lorenz model and experimental data r laser Correlation dimensions are computed using embeddings ranging from dimen sions one bottom line to 40 top The peak on the right is not a signal since the dimension is de ned in the limit of 6 7 small A reasonable small 7 limit is obtained for the numerical data but not for the experimental data Where the noise daemon kills the computation laser In both cases the behavior at large 6 is not useful The de nition in volves the 6 a 0 limit so the behavior on the right is of little interest anyway Computations based on numerically generated data seem to converge to a value slightly above 2 in the small 6 limit This is not the case for computations based on experimental data The problem here is that noise kills the computation With results like this it is not too surprising that theorists demand enormously long data sets With unbelieveable signal to noise ratios for correlation dimension computations While experimentalists view these computations With a jaundiced eye 343 Dq Dimensions There is an entire one parameter family of dimensions based on the invariant measure m These dimensions are de ned by the limit 1 l 1 Dq Z hm MEN 1 q gtO 37 Here 6 is the diameter of the largest box C in the partition of the phase space Three special cases of this dimension have already been introduced D0 Box Counting Dimension D1 Information Dimension D2 Correlation Dimension 14 CHAPTER 3 FRA CTALS The information dimension can be obtained from the de nition above by taking a delicate limit at q 1 This family is monotonic decreasing or at least monotonic non increasing For example D0 2 D1 2 D2 A plot of Dq vs q is shown in Fig 312 for the Feigenbaum attractor The scaling function f oz is shown for this spectrum in Fig 313 06 05 Figure 312 Schuster Fig 84 pg 130 The one parameter family of dimen sions Dq is plotted vs q for the Feigenbaum attractor 344 Multifractal Scaling and fd A fractal for which the spectrum Dq of dimensions is not constant but depends on q is called a multifractal A formalism has been developed to describe multifractals In this formalism the multifractal scaling function f oz plays a prominent role We introduce this function as follows The invariant measure a scales with the smallness parameter 6 with some power law dependence u N 60 It is possible to build up a histogram of the distribution over the exponents oz in the usual way Call this histogram N oz 34 OTHER DIMENSIONS 15 Then the multifractal scaling function fa is related to the histogram Na as follows fa 7 IogNltagt 38gt A theory can be built up more rigorously as follows De ne a function 739 from Dq as follows 7 7 10gEi Mg 739 i 4 1Dq i 5113 W 39 Then de ne a as the slope of 739 d d7 1 E 4 1Dql dig 310 Finally de ne fa as the Legendre transform of 739 d7 fa 74317701 311 The monotonic decreasing property of Dq translates into a concavity property on fa In Fig 312 we show the monotonic decreasing spectrum of fractal dimensions for the Feigenbaum attractor In Fig 313 we show the multifractal scaling function fa for the same attractor Because of the close relation of the two the transformations are invertible many properties of one are reflected in the properties of the other as suggested in Fig 313 The multifractal scaling function is dif cult to determine from experimental data and in the end provides little leverage for distinguishing one dynamical system from another 345 Thermodynamic Formalism There is a 11 relationship between the multifractal scaling formalism and clas sical thermodynamics which is as breathtaking in its elegance and beauty as it is useless in distinguishing among different mechanisms for generating fractal strange attractors let alone distinguishing among different dynamical systems The identi cations are shown in the Table below Thermodynamics Multi 7 Fractal Formalism 4 flogZ F TqilDq UilogZ a 017 S UlogZ faqa7739 d 5 4 Here we use standard nomenclature for the thermodynamic functions 6 lkT Z is the partition function U is the internal energy F is the Gibbs fre energy and S is the entropy 16 CHAPTER 3 FRACTALS 3 4 6 Lyapunov Dimension The most useful of all the dimensions is the Lyapunov dimensionl It is not based on geometery but rather on dynamics It is de ned in terms of Lyapunov exponents Which describe the stability of the dynamical systeml Since all of the fractals discussed in this chapter are geometric and have no dynamical origins it is not possible to de ne a Lyapunov dimension for any of theml Unfortunately we have not yet reached the point Where we are able to de ne the Lyapunov exponent of a dynamical systeml The de nition must wait 34 OTHER DIMENSIONS 17 06 04 02 0 02 01 06 08 Figure 313 Schuster7 Fig 84 pg 130 The scaling function f 04 is plotted VS 04 for the Feigenbaum attractor
Are you sure you want to buy this material for
You're already Subscribed!
Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'