Table of contents

The author presents a systematic procedure for constructing a hierarchy of non-relativistic Hamiltonians with the property that the adjacent members of the hierarchy are 'supersymmetric partners' i.e. they share the same eigenvalue spectrum except for the 'missing' ground state and the eigenvectors are simply related.

A partially directed self-avoiding walk model with the 'kinetic growth' weighting is solved exactly, on the square lattice and for two restricted, strip geometries. Some finite-size effects are examined.

Manna has recently studied two extensions of spiral self-avoiding walks on the square lattice, with spiral constraints after East and West steps, but not after North and South steps (see ibid., vol.17, p.L899, 1984). Here the author calculates the connective constants for these two models.

The number of independent self-avoiding walk (SAW) loop configurations (C_N) for fixed perimeter length N on the square lattice is fitted here to the scaling form C_N approximately mu ^NN^h, for SAWS with various constraints. For two-choice SAWS, extrapolation of enumeration results gives h=-1.5+or-0.1 (which falls in the same universality class as ordinary SAWS) and exact calculations for spiral SAWS give h=3.

The interface of the Eden clusters on percolation networks and the ballistic deposition model is studied by Monte Carlo simulations, using a simple definition for the surface thickness. The width of the active zone in the ballistic deposition model is found to diverge differently from the mean height, indicating breakdown of the single scaling length assumption in this model. The exponents nu and nu ' describing, respectively, the divergence of the radius and the active zone of the Eden clusters on percolation networks appear to be the same within the statistical errors. The central value of nu ', however, is slightly, but systematically, less than nu . The surface thickness of ballistic deposits is shown to exhibit finite-size scaling.

The authors consider systems at the bulk critical temperature with an ordering field applied at the boundary. For various boundary geometries, universal expressions for the spatial dependence of the order parameter are obtained by conformal mapping of the known results for the half space. Analytical results for strip, wedge and other two-dimensional geometries and for systems with spherical boundaries in arbitrary dimension are given.

The authors have employed a novel type of analogue electronic simulator for an experimental study of a nonlinear stochastic differential equation representative of a large class of bistable systems. The results obtained are in good agreement with theoretical predictions derived from the Stratonovic version of the white noise Fokker-Planck equation. The work is believed to be of relevance to a wide range of topics in physics, chemistry, biology and other branches of science.

For pt.I see J. Phys. C., vol.17, p.5049, 1984. The author extends the previous calculation on a dynamical mean-field theory of realistic spin glasses (Chowdhury and Mookerjee) so as to incorporate the effect of the reaction field. The 'ordering temperature' of a mode is shown to depend not only on the corresponding eigenvalue of the random matrix J but also on the structure of the whole spectrum.

The Clebsch-Gordan (CG) coefficients of the permutation groups S(2)-S(6) are calculated by using the eigenfunction method. Phases have been chosen in a consistent way to exhibit as many symmetries for the coefficients as is possible, including non-multiplicity free cases. The CG coefficients of S(2)-S(5) and those for (321)*(321) to (321)⁵ of S(6) are tabulated in the square root form of rationals. These tables together with the S(6) contains/implies S(5) isoscalar factors published earlier provide a complete tabulation of the S(2)-S(6) CG coefficients given in the Young-Yamanouchi basis.

The authors show, on a set of examples containing the hierarchies of equations associated with the Zakharov and Shabat spectral problem (1972), the modified Korteweg-de Vries-Burgers spectral problem and the chiral field equation, that one can easily prove the strength or weakness of a Backlund transformation for a whole hierarchy of equations by the use of the Darboux matrix approach and thus with just the knowledge of half a Backlund transformation.

If an integrable classical Hamiltonian H describing bound motion depends on parameters which are changed very slowly then the adiabatic theorem states that the action variables I of the motion are conserved. Here the fate of the angle variables is analysed. Because of the unavoidable arbitrariness in their definition, angle variables belonging to distinct initial and final Hamiltonians cannot generally be compared. However, they can be compared if the Hamiltonian is taken on a closed excursion in parameter space so that initial and final Hamiltonians are the same. The result shows that the angle variable change arising from such an excursion is not merely the time integral of the instantaneous frequency omega =dH/dI, but differs from it by a definite extra angle which depends only on the circuit in parameter space, not on the duration of the process. The 2-form which describes this angle variable holonomy is calculated.

The relation between some normal ordering forms for boson operator functions and generalised Stirling numbers is shown. In particular, the ordering of the operator function a^k(a^r+N+s)ⁿ is obtained for positive integers k, r, n and an arbitrary integer s. This is a generalisation of the recent result of Katriel (ibid., vol.16, p.4171-3, 1983). Some other normal ordering formulae are presented. It was shown how to obtain antinormal forms from several normal expansions given here.

A quantum mechanical plane rotor with time-dependent magnetic flux on its axis is considered in a representation in which the wavefunction is multivalued. If the representation is chosen such that the new kinetic angular momentum is equal to the old canonical angular momentum, the Schrodinger equation is simplified. An energy eigenfunction expansion in the new representation is used to solve the Schrodinger equation exactly.

A new application of the Hellmann-Feynman theorem is shown to allow accurate calculation of expectation values along with energies for the Schrodinger equation, for variable boundary position and boundary conditions. Power series and finite difference versions of the method are developed and applied to perturbed oscillator and perturbed hydrogen atom problems.

The basic contributions to the concept of the many-worlds interpretation of quantum mechanics are analysed. It is found necessary to divide them into two classes corresponding to (a) the relative-states interpretation (RSI), which avoids problems of measurement, but is found generally unconvincing, and (b) the many-worlds interpretation (MWI), which is more comprehensible, but has all the problems of the von Neumann scheme. The EPR problem is tackled using each interpretation. The RSI finds no problem, while the MWI meets the same difficulties as conventional interpretations.

The asymptotic behaviour of the S-matrix in the case of repulsive singular potential at the origin for large mod k mod is presented in a more precise form. It is shown that the WKB method in the 'Langer' form yields explicit values of the first- and second-order terms of the S-matrix expansion.

Within a fully relativistic framework, the thermodynamics of a classical dilute arbitrarily hot plasma in equilibrium is studied. The internal energy of the plasma is calculated to all orders in KT/mc². The case of a one-component plasma immersed in a static background is also studied. Up to order kT/mc² the results given previously by the authors (see ibid., vol.28, p.3030, 1983) are recovered. On the other hand the authors also give explicit expressions for the thermodynamic functions of a high-temperature electron-positron plasma. Some important questions concerning the coherence of their calculations and those of other authors are discussed.

It is proved that there exists a general decomposition of elements of the algebra A" into two terms which could be viewed, in a certain physical approximation, to (i) create quasi particles and annihilate holes (ii) annihilate quasi particles and create holes, respectively. The mathematical and physical relevance of the construction, which appears to be an extension of the early results of Araki and Woods (1963) and Araki and Wyss (1964) to the general situation, will be discussed.

The Onsager-Machlup approximation for a discrete Markov chain, representing multistationary state transitions, is formulated in the limit of small thermal fluctuations. The principle of minimal correlational entropy is used to determine the state in which the invariant probability measure tends to concentrate as the intensity of the fluctuations tends to zero. It is this principle, rather than that of maximum entropy, which is valid in open systems without microscopic reversibility. In systems characterised by microscopic reversibility, the two principles give the same predictions. The principle of minimal correlational entropy is shown to be the statistical analogue of the thermodynamic principle of least dissipation of energy. The principle of minimal correlational entropy is applied to the problem of stochastic exit from domains enclosing multistationary states; it reduces to a minimum entropy difference principle for systems satisfying microscopic reversibility.

There has been considerable controversy in recent years over the value of the conductivity exponent t. This exponent can be deduced from series expansions via the scaling relations, t= zeta +(d-2) nu , where zeta is deduced from differences between the exponents of the resistive ( gamma _r), percolative ( gamma _p) and conductive ( gamma _c) susceptibility. The author finds that allowance for non-analytic confluent corrections to scaling and the use of recent p_c estimates leads to estimates for gamma _r, gamma _p and gamma _c that are somewhat different to those of Fisch and Harris (1978); however, the differences between these exponents do not change significantly. Moreover the change in accepted estimates of nu in the last five years cancels some of this remaining discrepancy and she concludes (using the relation zeta = gamma _r- gamma _p), that t=1.31, d=2; t=2.04, d=3; t=2.39, d=4; t=2.72, d=5; with an error of about +or-0.10 in each case. The d=2 estimate is in significantly better agreement with those of other methods than that of Fisch and Harris.

The random field Ising model is solved numerically in the Bethe-Peierls approximation. For a model with a two-peak delta -distribution, the transition is first order at low temperatures and second order at high temperatures, and the tricritical point appears as an inflection point of the transition curve. The behaviour at low temperatures is analysed analytically as a function of the coordination number, and compared with the mean-field prediction.

The Smoluchowski equations of coagulation are solved analytically in two cases involving a finite cut-off of the system: the constant kernel set to zero for any j>N on the one hand, and the general three-particle case on the other. Both are seen to exhibit rather unusual large-time behaviour. The first model can be used to account for large particles precipitating out of a system and its behaviour is therefore of particular interest.

It is shown that the Bethe lattice with coordination number 4 is a dual lattice for Sierpinski's gasket.

This paper defines and studies a general algorithm for constructing new families of fractals in Euclidean space. This algorithm involves a sequence of linear interscale transformations that proceed from large to small scales. The authors find that the fractals obtained in this fashion decompose in intrinsic fashion into linear combinations of a variable number of 'addend' fractals. The addends' relative weights and fractal dimensionalities are obtained explicitly through an interscale matrix, which they call the transfer matrix of the fractal (TMF). The authors first demonstrate by a series of examples, then prove rigorously, that the eigenvalues of our TMFs are real and positive, and that the fractal dimensions of the addend fractals are the logarithms of the eigenvalues of their TMF. They say that these dimensions form the overall fractal's eigendimensional sequence. The eigenvalues of their TMF are integers in the non-random variants of the construction, but are non-integer in the random variants. A geometrical interpretation of the eigenvalues and the eigenvectors is given. Their TMF have other striking and very special properties that deserve additional attention.

Analysis of low-density series for site-bond percolation on the directed square (SQ) and simple cubic (SC) lattice (and related series for bond percolation on the honeycomb (H) and diamond (Di) lattices) is found to be consistent with p_c(SQ, site-bond)=p_c(H, bond)=0.8228+or-0.0002, p_c(SC, site-bond)=p_c(Di, bond)=0.637+or-0.002 and previous estimates of gamma , nu /sub /// and nu _{perpendicular to} . Analysis of the square lattice series supports the validity of the scaling relation gamma ₀= gamma -(d-1) nu _{perpendicular to} for the two-dimensional lattices. Site percolation on the honeycomb and diamond lattices is also considered.

The author has derived eleven-term high temperature series for the wavevector dependent susceptibility chi (q) of the ANNNI model in two and three dimensions. In three dimensions the locations of the phase boundaries of the ferromagnetic and modulated phases, and the location of the Lifshitz point, are found to be in good agreement with previous work. In two dimensions the analysis is less precise, but yields results consistent with the currently accepted picture in which the paramagnetic phase extends to zero temperature at the multiphase point.

The manifestly gauge invariant relativistic electrodynamics of neutral atoms is derived from a path-dependent Lagrangian. A second quantised spinor field is used for the description of electrons, while the infinitely heavy nucleus is represented by a static Coulomb field. Classical Lagrangian formalism is constructed. Generalised Poisson brackets are obtained from canonical formalism. Canonical quantisation is performed. The whole description is without any reference to the electromagnetic potentials. A comparison with other formulations is given.