Table of contents

Following recent interest in an earlier work on the Kepler problem (Prince and Eliezer, 1981), a number of remarks are made concerning the many-faceted symmetry-conservation law duality in classical mechanics. In particular, the advantages of associating a number of symmetries with the Runge-Lenz vector are discussed.

An intermittent transition to chaos in the presence of symmetry is investigated. Statistical properties of intermittency are found, including the case of external noise.

A new directed self-avoiding walk (directed SAW) is proposed and studied on a square lattice. The critical behaviour of such directed SAWS (in two dimensions) is established and shown to be of different universality class from that of an ordinary SAW.

The authors simulate a mixture of bifunctional and tetrafunctional molecules which may form bonds, on a triangular lattice, both with and without the help of free radicals. The model of Herrmann et al. (1982, 1983) and normal bond percolation are two special limits of the model. The authors determine the gel point, the fraction of matter in the infinite network or gel, and the weight-average degree of polymerisation, as a function of time. For radical-initiated gelation, the observed cluster distribution confirms the conclusion of Herrmann et al., that this model is different from random restricted-valence percolation. However, the gel point is found to decrease with increasing concentration of bifunctional molecules, quite unlike the behaviour observed in three dimensions.

The Einstein-Kahler equations on a five-dimensional pseudo-Riemannian manifold with Kaluza-Klein symmetry are discussed.

Parisi's replica symmetry breaking solution for spin glasses is extended to finite replica number n. The free energy F_p(n) obtained this way, as well as its first two derivatives with respect to n, are shown to join the corresponding values in the Sherrington-Kirkpatrick (SK) solution at a characteristic value n_s(T), where stability breaks down in the latter. The continuation composed of the SK branch F_SK(n) for n>or=n_s(T) and the Parisi branch F_P(n) for 0<or=n<or=n_S(T) fulfils the requirements of convexity, monotonicity and stability for all n.

Equilibrium configurations of a model for a crystal in one spatial dimension with anharmonic and up to third-neighbour harmonic interactions are related to orbits in a two- or four-dimensional space under a nonlinear symplectic mapping. In this way a connection is given with dynamical systems with two or three degrees of freedom. Depending on the parameters the orbits are smooth or stochastic. Their characterisation in terms of their fixed points and their Ljapunov exponents is given. The mapping shows a large number of bifurcations, of which one can distinguish various types, and these are discussed.

Bifurcations of cubic nonlinear symplectic mappings in two and four dimensions are discussed. Here series of bifurcations are studied by direct numerical calculation and by a renormalisation procedure. It is shown that for period-doubling bifurcations one finds the universal exponent of the quadratic area-preserving map. Other exponents exist for higher multiplicities. The renormalisation transformation has a fixed line in parameter space with an end point. The latter implies that series of period-doubling bifurcations may break off.

Another demonstration of a theorem on the calculus of variations derived recently by Hojman and Harleston (1981), using Helmholtz conditions, is presented.

A hydrogen atom is analysed in spheroidal coordinates. Limiting transformations R to 0 and R to infinity , R being a dimensional parameter characterising the spheroidal coordinates, are considered in trinomial recurrence relations and spheroidal wavefunctions. Expansions are found for normalised spheroidal wavefunctions of the hydrogen atom over spherical and parabolic bases, and relevant limiting transformations are studied.

The authors study the systems of three identical spinless particles moving on R³ and possessing an SU(n) gauge symmetry. Three such systems are possible, corresponding to the three non-isomorphic SU(n) bundles over C₃(R³), the configuration space. The authors retract C₃(R³) to a subcomplex which shows clearly how its homology arises. The three bundles can be realised as pull-backs of the universal bundle S⁷ to S⁴ using three non-homotopic maps C₃(R³) to S⁴. The two non-trivial bundles admit no flat connection; they do not correspond to Bose, Fermi or parastatistics.

A plane rotor, which is an electron constrained to move in a circular orbit, has time-dependent magnetic flux on its axis. By Faraday's law, an induced electric field acts on the electron. The induced electric field exerts a torque on the electron and thus changes its kinetic angular momentum. The induced electric field also does work on the electron and changes its energy. These changes are in agreement with generalised Ehrenfest theorems. The kinetic angular momentum and the energy of the electron depend on the instantaneous flux through the orbit. The Shrodinger equation for the plane rotor is solved exactly in a manifestly gauge-invariant way. The probability that the rotor is in an eigenstate of its energy operator is contrasted with the probability that the system is in an eigenstate of the unperturbed Hamiltonian.

The combined gravitational-neutrino field equations in general relativity are solved under the following two assumptions: (i) space-time is stationary axially symmetric and the line element of the metric can be put into a canonical form, (ii) the energy flow vector of the neutrino field is time-like or null for all observers. The resulting metric is uniquely determined and asymptotically non-flat; the Weyl tensor is of Petrov type D, and the Ricci tensor belongs to the class (2T-2S)₂ in the Plebanski classification.

The author shows that if a_ij=A+B(i+j)+Cij, the solutions of Smoluchowski's coagulation equation (monodisperse case) can be expressed as equilibrium distributions conditioned on a deterministically changing (and known) parameter. A partial converse is also given. Because the equilibrium distributions are already known, this effectively solves the equation for the given a_ij. If C not=0, then there is a finite critical time t=t_c when the moments cease to converge. The author finds t_c for all the given a_ij.

Starting from a causal thermodynamical approach, the authors study the equilibrium fluctuations of relativistic thermoviscous fluids. The results are compared with those of microscopic theory, and the transport coefficients of radiative fluids are obtained. Likewise, non-equilibrium corrections for the fluctuations of the bulk viscous pressure are evaluated for different media.

New high-density series data for the mean number, percolation probability and 'susceptibility' of finite clusters are presented for bond and site percolation on four standard three-dimensional lattices. A Pade approximant analysis of both high- and low-density series makes particular use of the rather precise unbiased estimates of the percolation threshold, p_c, obtained recently by Heermann and Stauffer (1981) for the simple cubic lattice. For this lattice the authors obtain the biased estimate gamma =1.73+or-0.03 for the site problem and a similar estimate but with larger uncertainties for the bond problem. Such a value is significantly larger than earlier series estimates. Assuming gamma to be universal they obtain precise, although biased, estimates of p_c for both bond and site percolation on all four lattices. Using the bond estimates of p_c they find an overall biased estimate of beta =0.454+or-0.008 for bond percolation on all three-dimensional lattices. (The corresponding site problem requires further study.) Scaling estimates of other critical exponents are alpha =-0.64+or-0.05, delta =4.81+or-0.14, Delta =2.18+or-0.04, nu =0.88+or-0.02 and nu =0.03+or-0.03.

The antipercolation problem is solved on the Bethe lattice. The critical exponents are identical to the percolation exponents when the coordination number z is greater than a critical value z_c=3, for which the problem has new exponents satisfying extended universality and below which there is no transition. For alternate lattices the problem may be transformed into a percolation problem with different occupation probabilities on the two sublattices. This allows a connection with an s-state Potts model with different z-spin interactions on the two sublattices. In two dimensions there is no transition on the alternate square and honeycomb lattices whereas a transition exists on the triangular lattice. Using the phenomenological renormalisation group method, the critical concentration is found to be p_c^(a)=0.21 whereas the correlation length exponent nu _a seems to converge towards the accepted percolation value nu _p=1.333.

The q-state ferromagnetic Potts model (FPM) and antiferromagnetic Potts model (APM) are solved on Bethe lattices for all values of the external magnetic field and temperature. The exact expressions of all thermodynamic functions of interest in the FPM and APM are calculated. The authors find the complete phase diagrams for both systems. In the FPM there are first-order phase transitions at the critical point for every q>2. In the APM they find second-order phase transitions along a critical line for every q>or=2.

Two algorithms are formulated for systematically truncating the set of basis states required in lattice Hamiltonian calculations, thus allowing larger lattice abort sizes to be treated. They are tested for the case of the (1+1)-dimensional Ising model. It is shown that smoothly convergent finite lattice sequences are obtained, which can successfully the extrapolated to the bulk limit. The authors point out the relevance of the technique for lattice calculations in higher dimensions.

An approximate method for estimating entropy with computer simulation, suggested recently, is applied here, together with a new method, to the Metropolis Monte Carlo simulation of the simple cubic Ising lattice. Beyond the critical temperature T_c, the accuracy of the best results for the entropy and the free energy is estimated to be better than 0.07% and 0.05% and at T_c itself, it is approximately 1% and 0.2% respectively. These results are in very good agreement with results obtained with the 'stochastic models' method, which is a computer simulation technique, and with series expansion estimates. The results for the entropy are significantly more accurate than results obtained with the commonly used reversible thermodynamic integration.

The classical Yang-Mills theory in the strong-coupling limit is investigated. To do this a formal 1/g expansion for potentials is applied. A recurrence formula for the nth-order term in the expansion is given. Mathematical correctness is also discussed and supported by simple examples. The recurrence equations are shown to be gauge covariant order-by-order, and a local screening effect in the strong-coupling limit is established.

The authors extend Tulczyjew's geometrical formulation of skewsymmetric tensor gauge fields as connections on generalised principal fibre bundles to the category of supermanifolds. Given a smooth d-dimensional manifold M and a k-form field on it, they construct a suitable supersmooth generalised principal fibre bundle P over a (d,2)-dimensional supermanifold M such that the BRS symmetries of the theory have a natural geometrical interpretation.

The octahedral Jahn-Teller system T(X)( epsilon _g(+) tau _2g) is considered, in which the epsilon _g and tau _2g modes correspond to the same frequency and are equally coupled to the electronic state T. A correspondence is drawn between the energy matrix for the J=1 states of this system and the energy matrix for the m=0 states of a three-dimensional harmonic oscillator displaced along the z axis. It is shown that the two sets of matrix elements become asymptotically proportional to each other as one proceeds towards the far interior of each infinite matrix. This permits a perturbation procedure to be set up for the Jahn-Teller system. Orthonormal basis states are established and a number of relations involving Laguerre polynomials are exploited to calculate to first order the energies of the J=1 levels, the Ham reduction factors and the intensities for the vibronic structure of the transition s to p. Analytic expressions are obtained for these quantities as a function of the coupling strength, and good agreement is obtained with the numerical calculations that are available at present.

The author shows that a Backlund transformation between two equations one of which has an auto-Backlund transformation implies an auto-Backlund transformation for the second which can thus be found by simple algebraic operations.