Marek Wolf

Chaos — The Interplay Between Stochastic and Deterministic Behaviour

The study of chaotic behaviour of dynamical systems has triggered new efforts to reconcile deterministic and stochastic processes as well as classical and quantum physics. New efforts are made to understand complex and unpredictable behaviour. The papers collected in this volume give a broad overview of these activities. Readers will get a glimpse of the growing importance of Lvy processes for physics. They will find new views on fundamental concepts of quantum physics and will see many applications of chaotic and essentially random phenomena to a number of physical problems.

1/ƒ noise in the distribution of prime numbers

The Fourier transform of the “signal” given by the number of primes contained in the successive intervals of equal length l = 216 = 65 536 up to N = 238 ≈ 2.749 × 1011 was performed. It turns out that the power spectrum displays the 1/ƒβ behaviour with the exponent β ≈ 1.64. This slope β does not depend on the length of the sampled intervals, which suggests some kind of self-similarity in the distribution of primes.

Applications of statistical mechanics in number theory

The links between statistical physics and number theory are discussed. First the attempts to prove the Riemann Hypothesis by means of the suitable spin model and the Lee–Yang theorem about zeros of the partition function are shortly reviewed. Next, the analogies between random walks and prime numbers are mentioned. In the last section the partition function of the system whose energies are defined by the distances between consecutive primes is calculated. The arguments are given that such a “prime numbers gas” behaves like a set of noninteracting harmonic oscillators.

Random walk on the prime numbers

The one-dimensional random walk (RW), where steps up and down are performed according to the occurrence of special primes, is defined. Some quantities characterizing RW are investigated. The mean fluctuation function F(l) displays perfect power-law dependence F(l)∼l1/2 indicating that the defined RW is not correlated. The number of returns of this special RW to the origin is investigated. It turns out that this single, very special, realization of RW is a typical one in the sense that the usual characteristics used to measure RW, take values close to the ones averaged over all random walks. This fact suggests that random numbers of good quality could be obtained by means of RW on prime numbers. The fractal structure on the subset of primes is also found.

MULTIFRACTALITY OF PRIME NUMBERS

Abstract The multifractal formalism is applied to prime numbers. The spectrum of critical indices is found to be contained in the interval ( α min , 1), where α min tends to 1 for increasing sets of numbers. Besides the scaling of moments with respect to the length of intervals the scaling with respect to the sizes of subsets of natural numbers is also considered. We have found the cusps in the plots of the functions ƒ(α) and we claim that they are not caused by numerical roundings but they are a real effect. Besides the computer method, some analytical calculations are presented.

Central Charges in the Massive Supersymmetric Quantum Theory of Scalar Spinor and Scalar - Spinor - Vector Fields

The problem of central charges in supersymmetric models consisting of massive scalar and spinor fields (scalar models) as well as of massive scalar, spinor and vector fields (vector model) is investigated. To probe this problem the models of incoming or outgoing free fields, asymptotic for the interacting fields, are examined as the variety of symmetries of the free fields is much greater than that of the interacting fields. For the free scalar model it was found that the number of spinorial charges N ⩽ 2; when N = 2 there has to be one central charge (z = 1) and the number of spinor fields must be even. The commutators of the central charge with the fields can not vanish. The models always have a solution. For the free vector model one has N ⩽ 4. If z = 0, then N ⩽ 2; for one central charge (z = 1) or two central charges (z = 2) linked to a common spinorial charge one has N ⩽ 3. IF N = 3, then z = 1,2 or 3. For N = 4 one has either z = 2 and the central charges can not be linked to the same spinorial charge, or z = 4 and each spinorial charge is linked to exactly two central charges, or z = 6. If a central charge acts trivially on spinorial or scalar fields, it has to vanish. This is also partially true for vector fields. The necessary condition for a model to exist is that the number of vector fields ν does not exceed the number of scalar fields s(ν ⩽ s). The models with ν = s (N ⩽ 2) as well as with ν < s and N = 1 always have a solution. A discussion of the case ν = 1 and s = 4N − 3 is also presented. The characteristic of the interacting scalar model is the same as of the free one. For the interacting vector model with two S-matrix spinorial charges (Nint = 2) one has at most one S-matrix central charge (zint ⩽ 1); if Nint = 3 then zint = 1,2 or 3; for Nint = 4 the characteristic of the interacting model is similar to that of a free one.

Diffusion limited aggregation : a paradigm of disorderly cluster growth

The purpose of this talk is to present a brief overview of our groups recent research into dynamic mechanisms of disorderly growth, an exciting new branch of condensed matter physics in which the methods and concepts of modern statistical mechanics are proving to be useful. Our strategy has been to focus on attempting to understand a single model system — diffusion limited aggregation (DLA). This philosophy was the guiding principle for years of research in phase transitions and critical phenomena. For example, by focusing on the Ising model, steady progress was made over a period of six decades and eventually led to understanding a wide range of critical point phenomena, since even systems for which the Ising model was not appropriate turned out to be described by variants of the Ising model (such as the XY and Heisenberg models). So also, we are optimistic that whatever we may learn in trying to “understand” DLA will lead to generic information helpful in understanding general aspects of dynamic mechanisms underlying disorderly growth.

Nearest-neighbor-spacing distribution of prime numbers and quantum chaos.

We give heuristic arguments and computer results to support the hypothesis that, after appropriate rescaling, the statistics of spacings between adjacent prime numbers follows the Poisson distribution. The scaling transformation removes the oscillations in the nearest-neighbor-spacing distribution of primes. These oscillations have the very profound period of length six. We also calculate the spectral rigidity Δ(3) for prime numbers by two methods. After suitable averaging one of these methods gives the Poisson dependence Δ(3)(L) = L/15.

MULTIFRACTALITY OF SNOWFLAKES

The results of the extensive numerical simulations of the Dielectric Breakdown Model (DBM) with noise reduction on the hexagonal lattice are presented. Seventy-five clusters grown under different boundary conditions consisting of 16 000 particles on the lattice 1001×1001 were generated. The simulations were done for the noise reduction parameter s equal to 200 and two values of the parameter η, namely for η=0.5 and 1. For the latter case, two boundary conditions were considered: the DBM and DLA b.c. Such a growth model leads to the formation of the fractal objects resembling real snowflakes. The growth probabilities were stored at the five stages of the clusters growth: at the masses of 1000, 2000, 4000, 8000 and 16 000 particles. Multifractal analysis was performed and obtained results are presented. The comparison of two methods: the histogram and moments methods, is provided. We discover that for the η=0.5 parameter, there is a phase transition, while for η=1, there appears to be no phase transition. Besides the usual growth probability measure, the measure given by the noise reduction counters is considered and multifractality of it is presented.

Central charges and Lorentz and internal symmetries in massive supersymmetric quantum field theory

Abstract The properties of central charges in the framework of the massive supersymmetric quantum field theory related to internal symmetries, Lorentz covariance and locality of the fields are investigated. It is shown that in the presence of z central charges the largest semisimple part of the internal symmetry algebra is a direct sum of z compact symplectic group algebras and possibly an additional term representing the unimodular unitary group algebra. Next it is shown that 4 j ⩾ N + K , where j is the highest spin value of the underlying fields, N is the number of spinorial charges and K the number of these spinorial charges which are not linked to other spinorial charges by a central charge. It is further demonstrated that, in general, the central charge can not be redefined in such a way that it is at the same time real and preserves the locality principle. The discussion of the obtained results concludes the paper.