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Karmaşık sistemlerde faz değişimi

Başlık çevirisi mevcut değil.

  1. Tez No: 39152
  2. Yazar: METİN HÜNER
  3. Danışmanlar: DOÇ.DR. AYŞE ERZAN
  4. Tez Türü: Yüksek Lisans
  5. Konular: Fizik ve Fizik Mühendisliği, Physics and Physics Engineering
  6. Anahtar Kelimeler: Belirtilmemiş.
  7. Yıl: 1993
  8. Dil: Türkçe
  9. Üniversite: İstanbul Teknik Üniversitesi
  10. Enstitü: Fen Bilimleri Enstitüsü
  11. Ana Bilim Dalı: Belirtilmemiş.
  12. Bilim Dalı: Belirtilmemiş.
  13. Sayfa Sayısı: 35

Özet

ÖZET Bu tezde, bir örgü üzerinde birbirleriyle 'çadır1 fonksiyonlarıyla etkileşen hücrelerden oluşan, lineer olmayan dinamik bir sistemin kritik noktadaki fraktal boyutu analitik olarak hesaplanmıştır. Bunun için Sabit ölçek Dönüşümü metodu uygulanmıştır. Bu metodun kullanılabilmesi için belli konflgurasyonlara sahip üç komşu hücrenin bir zaman adımı sonrasında, hangi olasılıkla, hangi değeri alacağının bilinmesi gerekmektedir. Bu olasılığın hesaplanabilmesi için, iki bilinmeyen parametreye sahip bir koşullu olasılık dağılım fonksiyonu önerilmiştir. İkili hücreye komşu olan hücrenin olasılık dağılım fonksiyonu da benzer bir fonksiyonla verilmiştir, iki normalizasyon şartı ve komşu hücrenin doluluğunun verildiği bağıntı ile SÖD metodunun aynı anda kullanılmasıyla bilinmeyen parametreler, dolayısıyla fraktal boyut hesaplanmıştır. Turbulans probleminin SÖD metoduyla incelenmesi için umut verici bir adım atılmakla kalınmamış,aynı zamanda olasılık dağılım fonksiyonunun saptanmasıyla sistem hakkında bilgi edinilmiştir. iv

Özet (Çeviri)

SUMMARY PHASE TRANSITIONS IN COMPLEX SYSTEMS Spatiotemporal chaos is a complex dynamical phenomenon with many degrees of freedom, emerging in spatially extended systems. It appears in a broad area of natural phenomena, including fluid turbulance, chemical reactions, solid state physics such as Josephson junction array, charge density wave (CDW), spin density wave, liquid crystal convection, biological information processing and so on, Qualitative, quantitative and theoretical understanding of spatiotemporal chaos remains one of the most important problems in nonlinear dynamics. Coupling together identical simple dynamical systems appears to be a conceptually simple way to increase the number of degrees of freedom When a spatial meaning is given to the coupling, they can be used to investigate spatiotemporal disorder. Chains of maps with diffusive coupling and nearest neighbour interaction will be considered here. A general expression for such systems reads *r1=o-*)/(«r)+|[/«+i)+/«i)] where the subscripts denote the site indeces, the superscripts the discrete time, f(x) is the local evolution law (the elementary map) and s is the coupling strength. The iterates of such high dimensional functions are viewed as crude approximations to a Poincare mapping describing the evolution of a spatially extended system. They provide an efficient tool for the study of space-time chaos. A number of studies have indicated that phase transitions reminiscent of critical phenomena are present in these models. The change of behaviour that occurs as a parameter is varied is a non-equilibrium phenomena, involving the formation of complex spatiotemporal patterns. The systematic approach begins with the definition of a particular map which fulfills what are believed to be the minimal requirements necessary to observe spatiotemporal intermittency. These are mainly, for an isolated map, the existence of a laminar or regular asymptotic state together with the possibility of a transient chaotic behaviour. One further requirement is that the laminar state be stable with respect to infinitesimal perturbations of finite amplitude introduced by the coupling. Keeping the approach as simple as possible, a piecewise linear local evolution law f(x) is chosen:/(*)= rx x e [0, 1/2] r(l-x) *e[l/2,l] x x e[l,r/2] with r>2. The uncoupled dynamics of such a mapping is chaotic as long as f(x) remains in the unit interval since Lyapinov exponent is positive everywhere in this domain. However, as soon as f(x)>l the iteration is locked. The local phase space can thus be seen as a continuum of stable laminar fixed points (x s[l, r/2]) connected to a chaotic repellor (x e[0,l]). At the beginning all sites are initialized randomly between 0 and r/2. Then at every discrete time step they are updated. The order parameter is NT m- - - N where Nj is the number of turbulant sites and N is the number of the total sites on the chain. At the critical point fractal dimension of the system is D'~d-&y where d is the euclidean dimension where the system, is embedded. The aim of this work is to find the fractal dimension analytically. The fixed-scale transformation (FST) which was first developed [12] to compute the fractal dimension of Laplacian growth models such as diffusion-limited aggregation and dielectric breakdown, has since been successfully applied to a series of equilibrium phase transitions. In this work we apply FST to computing yhe fractal dimension of the system described above. The intersection of a given fractal structure of dimension Dp with a line perpendicular to the growth direction rise to a set of points of dimension DF = DF - 1. By analyzing this set of points with a procedure of box covering we assign a black dot to a box if this contains some point of the set and a white dot otherwise. The process by which a black box is subdivided into two leads to two possible configurations indicated as type 1 (one black and one white subbox) and type 2 (both subboxes black) The corresponding probabilities in this process of fine graining are indicated by C\ and C2 respectively. The average number of black subboxes that appear at the next level of fine graining from one black box is (n) = Cl+2C2 In this process of box covering the total number N^Q of boxes of size 1 needed to cover a fractal structure of dimension Dp is given viFor each iteration of the process 1 -> 1 / 2 N^ is increased exactly by a factor . The fractal dimension of the structure is directly related to the values (Ci,C2) by DF = l + ^> F In 2 Given a certain level of fine graining the distribution (Ci,C2) corresponds to the density of pairs of type 1 and 2 respectively. The natural subject for the iterative fixed point problem is the distribution of elementary configurations {Q} that appear from one scale to the next in the process of fine graining. It is natural therefore to look for an iterative equation of type -m+l Mn M21 M, M L/"12 22. C where n is the index of iteration. The fixed point of this iteration corresponds to the fractal properties of the system. The problem is therefore to define the matrix elements My = probability (i -> j) which define the conditional probability to have a cell of type i followed by one of ypej in the growth direction. X X is the adjacent site to the initial cell. X=T if the boundary is 'closed' or X=L if the boundary is open (the adjacent cell is empty). We have to consider all the graphs originating from an occupied site in the initial cell to an occupied site in the final cell in order to conserve the connectivity of the fractal set. Thus M\ ı is the probability that at t=n a cell of type 1 is followed by one of type 1 at t=n+l. For the closed and open boundary conditions matrix elements (M» and M?) must be calculated. The matrix element My is found only if M-j1 and Mjj are multiplied with the weights which correspond to the probability to have closed or open boundary condition, Mij=M°P°+M°lPcl P° = 3Q 3-C, pd _ 2C2 3-C, It can easily be seen that we have to calculate the probability of the transition of an initial cell consisting of three sites to next configuration. Thus S)'*!'5^ denotes the transition of the site 1 to 1' when the indices of the next nearest neigbours of site 1 are 0 and 2. vnWhen calculating these transition probabilities we assumed in the zeroth approximation that there is no correlation between sites and x is distributed uniformly. We calculated the fractal dimension to be 1,43 but simulations indicated that it has to be approximately 1,80. As it was pointed out earlier we want to calculate C\ and C2. C2 is the conditional probability of a turbulant site to have a turbulant neighbour site. Thus we are forced to consider conditional probabilities. With the aid of simulations we found that at criticallity the conditional probability distribution function has a peak around one. If the system goes into the laminar state all x values would be distributed between 1 and r/2. If the systems order parameter is finite most of the sites would be turbulant. Thus it is reasonable to expect a peak around one at criticallity. The calculations have to take this fact into account. We tried a few conditional probability distribution functions all of which had a peak around one. All of these but one failed. This is a very important result because after the calculations we gained more information then we had expected, namely we would be able to find a conditional distribution function which caracterizes the dynamical system. The conditional distribution function we used is f{x0 ?7^) = A xf -(*!-&)12 We also have to take boundary fluctuations into account. For this reason we propose a similar distribution function for X2 which also have a peak around one. f(x2) = C x62-{x2-d)n With these distribution functions transition probabilities are calculated. Because of computational difficulties we replace these functions with their mean values. We denote the area under the distribution function I(x). We find a parameter a such that I(b*)- 1(1) = 1(1) -1(a) where b denotes the x value where the distribution function becomes zero. The amplitude is determined via «2 =t;j! Âxo eTfcJ&i o -a vıııThe amplitude for the other interval (l/2,a) can be computed in a similar way. A similar simplification for the distribution function for boundary conditions is also made. With these modified distribution functions transition probabilities can be computed according to the formula jdx0 /X2 xoeT ATjec, x2ex2 -^ jdx0 \dc^ \dx2 f(x0 efyi) f(x2) xqeT ^«i x2ex2 We have four unknowns A,b,C,d and four equations JJ/4^«oer|x1)tfc1 = l f0i2p(x2)dx2 = l jl0Pix2)dx2 = Pd = 3C2 3-C, C, Mn M2l Mn M22 cx C 2J After solving these equations self consistently, we found fractal dimensions at four critical points. The values of the calculated fractal dimensions deviates from the ones obtained with simulation at most 13 %. We think that to assume a distribution function with unknown parameters and to insert it into the FST equations is a new approach to FST calculations. IX

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