Table of contents

The authors establish a connection between the non-unitary transformation scheme which breaks the time reversibility at the microscopical level and an algebraic deformation scheme which generalises the Lie product. This formal approach is further supported by an explicit construction of the various operators involved. The compatibility conditions are considered for this case and are found to be automatically satisfied by the algebraic deformation scheme in the particular case of the Fokker-Planck equation. In this way, a particular model of the non-unitary transformation scheme is given constructively.

Invariant Backlund transformations of reciprocal type are derived for a class of (n+1)th-order conservation laws. A new auto-Backlund transformation for the Harry-Dym equation is presented as a special case of the analysis.

Lattice models interpolating between free and self-avoiding random walks are investigated. Generating functions are constructed for k-tolerant walks, various trail problems, etc. For trail problems the effective field theory describing the global behaviour is analysed in the vicinity of the upper critical dimension d_c=4. Their asymptotic large-scale behaviours are the same as those of the self-avoiding walk. Arguments are presented to support the same conclusion for much more general classes of walks including k-tolerant walks. One of the models exhibits a new tricritical point of order epsilon ^1/2 if the fugacity for crossing is increased.

The authors apply scaling and crossover arguments to the one-dimensional true self-avoiding walk, which avoids itself with strength g. The problem is formulated in terms of a grand canonical ensemble, a real-space renormalisation-group analysis shows that for large repulsion, x=e^-g is a relevant variable causing crossover from the self-repelling chain (SRC) limit, with crossover exponent phi =1. A physical interpretation of this result is given in terms of competition between the correlation length and the average distance before the walk turns back on itself. The resulting flow suggests the existence of an intermediate attractive fixed point which makes the exponent v different from SRC and random-walk values for all x between 0 and 1, in agreement with recent Monte Carlo results.

In order to find out as precisely as possible the site percolation threshold in the square lattice, a Fortran high speed Monte Carlo program has been developed (2.2-2.4 mu s per site on CDC Cyber 76) for simulating site percolation in L*L square lattice with L=50 to 50000. Using L=750 to 16000 we found p_c=0.59277+or-0.00005. This value is within the confidence limits of values published earlier (p_c=0.5931+or-0.0006, p_c=0.5923+or-0.0007 and p_c=0.5927+or-0.0002). Three test checks with L=50000 confirmed this result within the above error bars.

Diffusion on a percolative lattice is formulated as a theory of interacting Bose-Fermi fields. The effective medium approximation (EMA) of Odagaki and Lax (1981) is obtained directly from the functional integral and is used to provide an EMA approximation for the diffusion constant and the DC conductivity. Simple corrections to the usual EMA produce exponents near the percolation threshold that agree with the best available data for d=2 and 3. This agreement reveals a strong connection between the dynamics of percolative diffusion and percolation itself.

The authors study the low temperature phases of the two-dimensional ANNNI model in an external field. A new type of melting mechanism due to hard walls is found and special Lifshitz-type multicritical points are predicted. For strong attraction between elementary units of different periodicity the dimerisation process leads to a phase diagram typical for an XXZ chain.

The authors consider regular crystal lattices in which the bonds are either diodes (probability p) or insulators. The mean number of backbone bonds L_BB(p) when averaged over lattice points is found to be related to the mean size of clusters S(p) by L_BB(p)=zpS(p) where z is the number of bonds directed away from any lattice site. Thus L_BB diverges at p_c with the mean size exponent gamma . The resistive susceptibility chi _R(p) of Harris and Fisch (1977, 1978) is expanded graphically as a power series in p and it is found that term by term chi _R(p)= Psi _R(p)(S(p))² where Psi _R(p) is obtained from chi _R(p) by ignoring contributions from nodal graphs. The above results are valid for bond and site dilution in any dimension. For bond dilution on the square lattice the authors have determined Psi _R(p) to p¹⁷ and Pade analysis of the resulting series for chi _R(p) shows that it diverges with exponent gamma _R=3.654+or-0.017. Using a scaling relation the exponent t for the conductivity of the infinite cluster is estimated to be t=0.75+or-0.02.

A model is proposed to describe the growth of clusters by a mechanism of irreversible clustering of clusters. The fractal exponent is extracted by means of numerical simulations on small systems, both directly and in the form of a renormalisation-group analysis. The results are in excellent agreement with previous Monte Carlo simulations. The numerical precision is better due to the simpler (hierarchical) formulation of the model. In the present form the model looks like the diffusion limited aggregation model, which for the sake of comparison is treated by the same renormalisation-group method.

Finite size scaling is applied to the one-dimensional Hubbard model with the half filled energy band at zero temperature. It is shown that, even for the scaling between small blocks of four, six and eight atoms, the essential singularity of the weak coupling limit is reproduced with remarkable accuracy, the error being less than 1% for the exponent and a few percent for the multiplicative constant. The results are discussed in comparison with the quantum renormalisation group approach, which fails to give the right exponent, and the advantages of the present method are pointed out.

The mean-field renormalisation group of Indekeu et al. (1982) is applied to the antiferromagnetic nearest-neighbour Ising model on the triangular lattice. The resulting phase diagram in the temperature-field plane is in good agreement with other calculations, while the predicted specific-heat index, alpha , is negative and thus qualitatively wrong. The study indicates that the method may be a useful approach, at least for determining phase diagrams, for frustrated systems and systems with competing interactions where conventional mean-field theory is dubious.

The conflict of basic conservation laws in cyclotron radiation is considered in more general terms, taking into account relativistic effects of the electron (i.e. synchrotron radiation). The authors also investigate effects due to the most important approximation in cyclotron theory, namely the omission of radiation back reaction. The conclusions are (i) the disagreement is of a magnitude considerably larger than any errors introduced by the approximation; (ii) the 'degree of conflict' attains its maximum in relativistic velocities, when the energy loss to radiation can approach the total energy of the electron.

A new approach to finding unique labels for many-electron states in multicentre systems is described and a particular example (eight electrons in s-states on eight centres) is worked out in detail. This approach is shown to provide a labelling with a more direct physical interpretation than has previously been found.

Exact upper and lower bounds are obtained for the class of stochastic processes described in terms of Pauli master equations with a tridiagonal infinitesimal stochastic matrix. The sum of time-scales, each equal to a reciprocal eigenvalue of this matrix, is expressed in terms of the elements of arbitrary matrices. This forms the upper bound on the longest time-scale, while the same quantity, divided by the number of time-scales minus one, is the average time-scale, and forms a lower bound. The longest time-scale determines the rate of approach to equilibrium. The result is valid for any number of states, and in particular provides recurrence criteria for the infinite chain, which are consistent with known results.

The even and odd parity eigenvalues for the bounded potential mu x²+ lambda x⁴, with both positive and negative values of mu and lambda >0, are obtained by the method of series solution.

Practical methods for the numerical implementation of the uniform swallowtail approximation have been developed. This approximation arises in the uniform asymptotic evaluation of oscillating integrals with four coalescing saddle points. A complex contour quadrature technique has been used to evaluate the swallowtail canonical integral S(x,y,z) and its partial derivatives delta S/ delta x, delta S/ delta y, delta S/ delta z. This method has the advantage that it is straightforward to implement on a computer and results of high accuracy are readily obtained. A comparison is made with other methods that have been reported in the literature for the evaluation of S(x,y,z). Isometric plots of mod S(x,y,z) mod , mod delta S/ delta x mod , mod delta S/ delta y mod , mod delta S/ delta z mod are presented and some properties of the zeros of S(0,y,z) that lie on the line y=0 are also discussed. Two methods for the evaluation of the mapping parameters (x,y,z) are described: an iterative method that is valid when (x,y,z) is not close to the swallowtail caustic and an algebraic method valid for (x,y,z) on the caustic and for y=0. Symbolic algebraic computer programs have been used to carry out the necessary algebraic manipulations. In practice both methods for determining (x,y,z) are complementary. An application of the uniform swallowtail approximation to the butterfly canonical integral has been made. The uniform asymptotic swallowtail approximation can now be regarded as a practical tool for the evaluation of oscillating integrals with four coalescing saddle points.

The large scale chaotic motion of a charged particle in a homogeneous static magnetic field and a longitudinal electrostatic wave is discussed. A formula for estimating the stochasticity threshold alpha _thr of the wave amplitude from the wave frequency nu and the propagation angle phi is proposed. For ϕ = π/2 and phi approximately = π/4 this formula reduces to known results for these special cases. It is shown that the essential characteristics of the ϕ = π/2 case persist for angles π/2 > ϕ > π/3.

Through the introduction of two complex stereographic variables it is shown that the Schrodinger equation for the bound states of hydrogen reduced to the wave equation for two coupled two-dimensional harmonic oscillators. The wavefunctions obtained by this means are the same as those which arise using parabolic coordinates.

For pt.I see ibid., vol.15, p.3481-90 (1982). Asymptotic formulae of the energy eigenvalues for the states which 'originate' from the middle well of the one-dimensional Schrodinger equation with the potential U(x)=x²(R²-x²)²/8R⁴ are analysed. In particular, the relation between eigenvalues for modal and non-modal solutions is determined analytically to the next higher approximation, and compared with available numerical data.

A quantum mechanical model of an oscillator interacting linearly with an environment is treated by the method of perturbation series expansion. For a special class of environments and interactions, this series is summed up to all orders. An integral equation for the time dependence of the coordinate operator of the oscillator is obtained, which is solved analytically by the method of Laplace transformations. General conditions are stated for a dissipative behaviour of the special class of environments considered. An example, which is widely applicable, is discussed.

This is an investigation of the energy levels and wavefunctions of an anharmonic oscillator characterised by the potential 1/2 (omega ²q² + lambda q⁴). As a lowest-order approximation an extremely simple formula for energy levels, E_i⁽⁰⁾ = (i + 1/2)1/4(3/alpha_i + alpha_i), is derived (i being the quantum number of the energy level), which covers any (lambda, i). alpha_i is the real positive root of a cubic equation: y_i alpha_i³ + alpha_i² - 1 = 0, with y_i = 6lambda(2i² + 2i + 1)/(2i+1). This formula reproduces the exact energy levels within an error of about 1% for any ( lambda ,i) (the worst case is 2% for i = 0, lambda to infinity). Systematically higher orders of this perturbation theory are developed, which contains the 'usual' perturbation theory for the limiting case of small lambda , but this perturbation theory is valid for any (i, lambda). The second-order perturbation theory reduces the errors of our lowest-order results by a factor of about 1/5 in general. Various ranges (large, intermediate, small) of (i, lambda ) are investigated and compared with the exact values obtained by the Montroll group (1975, 1978). For i = 0, 1, even the fourth-order perturbation calculation can be elaborated explicitly, which reduces the error to about 0.01% for any lambda . For small lambda it gives correct numerical coefficients up to lambda⁴ terms, as it should.

The renormalised series technique is applied to two quantum mechanical problems, the octically perturbed oscillator and the hydrogen Stark effect. Modification of the method is required for the latter problem; the results yield the Stark shifted energies and an estimate of the widths of the states studied.

Feynman's path integral for the one-dimensional Dirac particle uses paths for which Delta x approximately Delta t. For non-relativistic path integrals, typical paths satisfy ( Delta x)² approximately Delta t (as in Brownian motion). The authors demonstrate the consistency of these two stochastic schemes and show how the relativistic formalism contains within itself the scale of the transition regime, namely Delta x approximately h(cross)/mc, the Compton wavelength.

The author investigates both analytically and numerically the one-point Friedrichs model characterised by a lambda V interaction with a non-square integrable coupling function nu ( omega )=1. If the continuous spectrum of the Hamiltonian is bounded from below, a form factor is needed to avoid some infinities and the solution of the spontaneous emission problem then depends on an unphysical cut-off parameter alpha . This solution is denoted by p(t; alpha ) and its alpha -dependence may be removed by taking the appropriate limit. Indeed, the author finds that: Lim_{alpha to infinity} p(t; alpha )=1. If, on the contrary, the continuous spectrum is assumed to be infinite, then no form factor is needed and p(t)=exp(-2 pi lambda ²t).

A droplet theory for Ising-like systems valid in low spatial dimension can be formulated in terms of a surface tension Hamiltonian. The author presents an explicit two-loop calculation of the partition function for a single non-spherical droplet of one phase in a background of the other phase. The resulting beta -function for the spherical interface is found to agree with the beta -function of the planar interface. Corrections to the one-loop results are discussed.

The two-dimensional version of a generalised Ising gauge theory in which the gauge part of the action involves a product of Ising link variables around every rectangular 2*1 face of the lattice is studied. The pure gauge theory is related to the Ising model and is self-dual. The gauge plus matter theory is also self-dual. For negative gauge coupling an Ising critical line in the ( beta _g, beta _m) plane separates a phase with staggered frustration from another which is homogeneously frustrated.

The authors present a generating function formalism for growth processes in general, and introduce a new cell renormalisation scheme. As examples, they treat the 'true' self-avoiding walk and the Eden process (or growing animals) in two and three dimensions using small cells. The results in both cases indicate a substantial increase in the fractal dimension D compared with their equilibrium counterparts. The case of the Eden process may, however, suggest a peculiar convergence behaviour as the cell size b tends to infinity.

For pt.I see ibid., vol.16, p.1267 (1983). The authors construct and investigate a family of fractals which are generalisations of the Sierpinski gaskets (SGs) to all Euclidean dimensionalities. These fractal lattices have a finite order of ramification, and can be considered 'marginal' between one-dimensional and higher-dimensional geometries. Physical models defined on them are exactly solvable. The authors argue that short-range spin models on the SG show no finite-temperature phase transitions. As examples, they solve a few spin models and study the resistor network and percolation problems on these lattices.

Close-packed self-avoiding walks and circuits, as models for condensed polymer phases, are studied on the square-planar and honeycomb lattices. Exact solutions for strips from these lattices are obtained via transfer matrix methods. Extrapolations are made for the leading asymptotic terms in the count of compact conformations on the square-planar lattice. The leading asymptotic term for each lattice is bounded from below, and it is noted that boundary effects can be important.

Following Nienhuis' (1982) exact evaluation of the connective constant of the honeycomb lattice self-avoiding walk model, and the exact exponent values he has conjectured, the author has re-examined the available data on all three regular two-dimensional lattices. In the case of the triangular lattice the author has additionally corrected and extended the extant series. The author finds support for Nienhuis' value gamma =1¹¹/₃₂ for all three lattices, and further finds that alpha =¹/₂ for all three lattices, as given by Nienhuis' result nu =³/₄ and the hyperscaling relation d nu =2- alpha . The author is unable to find any consistent evidence of a 'correction-to-scaling' exponent Delta ₁<1 from the walk generating function, though other workers have found such an exponent for the mean square end-to-end distance series. But an exponent Delta ₁>1 cannot be ruled out.

The effect of detuning on the squeezing obtained in a prototype model for squeezing, namely the degenerate parametric amplifier, is investigated. A squeezed minimum uncertainty state is only obtained in a set of variables related to the original variables by a time dependent transformation. The required transformation is obtained using a symplectic decomposition of the dynamic evolution matrix.