Ne Frankel

CHAOS AND FRACTALS AROUND BLACK HOLES

Fractal basin boundaries provide an important means of characterizing chaotic systems. We apply these ideas to general relativity, where other properties such as Lyapunov exponents are difficult to define in an observer independent manner. Here we discuss the difficulties in describing chaotic systems in general relativity and investigate the motion of particles in two and three black hole spacetimes. We show that the dynamics is chaotic by exhibiting the basins of attraction of the black holes which have fractal boundaries. Overcoming problems of principle as well as numerical difficulties, we evaluate Lyapunov exponents numerically and find that some trajectories have a positive exponent.

Fractal basins and chaotic trajectories in multi-black-hole spacetimes.

We investigate the phase-space for trajectories in multi-black hole spacetimes. We find that complete, chaotic geodesics are well described by Lyapunov exponents, and that the attractor basin boundary scales as a fractal in a diffeomorphism invariant manner.

Potential theory and analytic properties of a Cantor set

The authors consider the electrostatic potential due to a uniform distribution of charge on a Cantor set. Mellin transforms are used to find expansions for the potential at a distance r from the end of the charge distribution. It is found that the potential is a power law multiplied by a function periodic in ln r, together with a power series in r. A recursion relation for the moments of the distribution is used to reveal a similar structure in these moments. Finally Fourier transform techniques are used to find explicit representations for the distribution and its moments.

Potential Theory and Analytic Properties of Self-Similar Fractal and Multifractal Distributions

By the use of recursion relations and analytic techniques we deduce general analytic results pertaining to the electrostatic potential, moments, and Fourier transform of exactly self-similar fractal and multifractal charge distributions. Three specific examples are given: the binomial distribution on the middle-third Cantor set, which is a multifractal distribution, the uniform distribution on the Menger sponge, which illustrates the added complication of higher dimensionality, and the uniform distribution on the von Koch snowflake, which illustrates the effect of rotations in the defining transformations.

Relativistic spin-1 bosons in a magnetic field

The quantum mechanics of charged, massive, spin-1 bosons in the presence of a homogeneous magnetic field (HMF) is studied using a six-component wavefunction formalism. The energy eigenvalues are compared with those previously obtained via other formalisms, the equations of motion of certain operators are given, and the positive and negative energy eigensolutions are obtained by the use of a ladder operator method. The six-component current for the case of general external electromagnetic fields is also displayed and finally, the employment of the eigensolutions and current in a study of a spin-1 boson-antiboson plasma is discussed.

STRUCTURE FACTOR OF DETERMINISTIC FRACTALS WITH ROTATIONS

We derive a recursion relation for the Fourier transform of any self-similar multifractal mass distribution. This is then used to find sufficient conditions under which S(k)↛0 as |k|→∞. Among two-dimensional distributions for which the similarity transformations contain 2π/n rotations, it is found that for values of n equal to 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18 and 30, distributions may be constructed satisfying the above condition. The possible scaling factors in the similarity transformations are strongly constrained by the value of n. In three dimensions, the equivalent condition is that all rotations/reflections are elements of a finite group, together with similar constraints on the scaling factors.

Probability distributions for the overlaps and self-correlations of the pure states of an n-vector model

The probability distributions for the overlaps between and the self-correlations of the pure states of the Stanleyn-vector model with infinite-range interactions are derived. These probability distributions represent two new order parameters for the model and are intimately related to the parameters which arise naturally within the replica formalism for the treatment of the corresponding quenched random-bond model. In contrast to then = 1 Ising case, the probability distributions are nontrivial whenn > 1 and an additional parameter for self-correlation has to be introduced.

Quenched n-Vector p-Spin Model

A disorderedn-vector model withp spin interactions previously introduced is studied for the quenched case by means of the replica method and a generalized Parisi theory. We present formal solutions for generaln andp and then study the casep → ∞. The high-temperature solution is stable at all temperatures and there is only one phase transition at a temperatureTg. Only longitudinal lowtemperature solutions are possible. There is one spin-glass solution, and it is stable for allT<Tg. The phase transition atTg is of first order and displays a jump discontinuity in the order parametersqj(L) andd. The spin-glass free energy is temperature dependent forn > 1 while it is constant whenn = 1.

Annealedn-vectorp-spin model

A disorderedn-vector model withp spin interactions is introduced and studied in mean field theory for the annealed case. We present complete solutions for the casesn=2 andn=3, and have obtained explicit order parameter equations for all the stable solutions for arbitraryn. For alln andp we find one stable high-temperature phase and one stable low-temperature phase. The phase transition is of first order. Forn=2, it is continuous in the order parameters for p⩽4 and has a jump discontinuity in the order parameters ifp>4. Forn=3, it has a jump discontinuity in the order parameters for allp.

Inclusion of short-range order in a mean field theory for the disordered magnetic lattice gas

The disordered magnetic lattice gas (DMLG) as a unifying description of many simpler random spin systems has been investigated in an attempt to devise a mean field theory which goes beyond the infinitely-long-ranged model by incorporating short-range order (SRO). The authors have shown rigorously that the local thermodynamic properties of the DMLG on a Cayley tree of finite coordination number z are identical to the thermodynamic properties of the DMLG in a pair approximation obtained by using the method of the distribution function. Further, a modified pair approximation for the DMLG is presented which is exactly solvable. It is formulated for general random bond distribution functions, and is then examined for the special case of Gaussian distributions.