Table of contents

The finite-size scaling spectra of the XXZ Heisenberg chain are presented for even and odd numbers of sites, respectively. The operator content is given for free as well as for toroidal boundary conditions.

A generic formula for the Hausdorff and box counting dimensions of self-affine fractals is presented. Extensions to the non-linear situation and to repellers in discrete dynamical systems are discussed.

The authors investigate numerically how typical trajectories fill the phase space in low-dimensional symplectic (Hamiltonian) maps with finite phase space. They do not find any sign of a 'chaos threshold' as reported by other authors when the non-linearity parameters are increased. Instead, as expected from Arnold diffusion, they find that single trajectories fill most (if not all) of the coarse-grained phase space even for very small non-linearities. Due to the 'stickiness' of tori also observed in two-dimensional maps, this filling is much slower than what one might expect naively and is possibly described by power laws. The 'chaos threshold' observed in a previous paper is explained as a trivial effect.

The energy spectrum of a potential consisting of an array of barriers is obtained by solving the Schrodinger equation. The barriers have the same height and constant separation, but their widths are incommensurately modulated. The hierarchical band splittings obey the same rules which were derived earlier for the tight-binding Aubry model. The authors have also demonstrated their equivalence to Hofstadter's rules which were derived by Stinchcombe and Bell (1987).

Using the Walsh-Fourier transform of the interface function the author introduces a new method for the analysis of the morphology of the complex interfacial patterns evolving with time on discrete lattices in a wide class of physical systems including crystal growth models. For the crystal growth models he also generalises an old technique for characterising the interfacial patterns on small length scales to patterns evolving from initial states far from equilibrium.

The mean square vertical displacement W² is considered for (1+1)-dimensional interfaces of the SOS type which make an angle theta with respect to the horizontal axis. It is proved that W² per unit horizontal length is proportional to the inverse of beta gamma ( theta )+ beta gamma "( theta ), where beta is the inverse temperature, gamma ( theta ) is the surface tension of the interface at angle theta and gamma "( theta ) is the second derivative of gamma with respect to its argument. This result establishes, on a microscopic basis, the validity of the corresponding formula proposed by Akutsu and Akutsu (1986) using thermodynamical arguments.

The behaviour of a chain in a d-dimensional space interacting with a generalised surface (a plane, a rod or a point) of dimensionality d/sub /// (2, 1 or 0) is described. Fixed points of the system are calculated to first order in in =(4-d)/2 and in _{perpendicular to} =2-d+d/sub /// using a multiple in expansion procedure, and the relevance of the surface and the excluded volume interactions is discussed.

The results of numerical simulations of diffusion-limited aggregation on the torus are presented. The usual random walk was generalised by allowing the particle to perform jumps of length equal to s lattice spacings, s>or=1. Patterns with periodic structure were obtained.

The authors have calculated the number of self-avoiding polygons on the square lattice to 56 steps, and the caliper size to 54 steps. Analysis of the generating function permits estimates of the connective constant, mu =2.638 1585+or-10^-6 and the critical exponents alpha =0.500 06+or-0.000 06 and v=0.753+or-0.007. The singularity structure of the polygon generating function is found to be consistent with a correction to the scaling exponent Delta =1.5, as predicted by Nienhuis (1982, 1984). The confluent part, however, maps into the additive analytic term due to the value of the exponent alpha .

Analytic expressions are derived for the leading finite-size corrections to a class of energy eigenvalues of the Potts and Ashkin-Teller quantum chains, making use of their equivalences with a modified XXZ Heisenberg chain. Assuming conformal invariance, exact results are thence obtained for some bulk scaling dimensions and surface exponents in these models.

The results of Monte Carlo simulations of the four-state Potts model on a square lattice at the bond percolation threshold are presented. Estimates are given for the new dynamical exponents A and B. The result for A is in clear contradiction with a recent conjecture of Nunes da Silva and Lage (1987).

The eigenvalue spectra of generalised corner transfer matrices are calculated for an Ising model on a square lattice. The predictions of conformal invariance at the critical point are thereby verified.

According to the Jerne hypothesis, the control of the immune response is due to interactions among anti-idiotypic antibodies which recognise each other. The authors present a model based on a network of interacting binary automata and compare its dynamics with the evolution of a typical auto-immune disease, experimental auto-immune encephalomyelitis.

The author shows that a scheme based on the critical Andrews-Baxter-Forrester model may be used to construct all known representations of operators obeying the Temperley-Lieb relations. Using associated groups he shows, for a small number of operators, that this scheme gives the complete set of irreducible representations. He obtains the degeneracies of the transfer matrix spectrum for the q-state Potts model.

Let X(a,b,c) be a ternary vector cross product for eight-dimensional Euclidean space E. An identity is derived which expresses (X(a,b,c), X(u,v,w)) in terms of the Spin(7)-invariant scalar quadruple product Phi (a,b,c,d)=(a,X(b,c,d)). The proof of the identity is coordinate free, and starts out from an explicit expression for X, with E viewed as complex four-dimensional Hilbert space.

Some properties of unitary representations of the de Sitter group SO₀(1,4) are considered. In the case of spin zero and positivity of the mass operator, these representations involve a decomposition which ensures the fulfilment of the cluster separability condition and allows for the conventional interpretation of the de Sitter analogues of momentum and boost operators. The decomposition of the tensor product of representations belonging to the principal series contains the representations of the principal series with all masses. Under natural assumptions concerning the interaction operators, there are no bound states in the theory, which is due to the universal de Sitter antigravity.

The author generalises a method derived from the theorem on the separation of coupled equations which has been stated previously to solve the case of two equations. It will be shown that this generalisation is always possible with a sequence of special transformations in terms of scaling parameters. The case of a system of three coupled differential equations is considered in some detail including a discussion on the search and use of these parameters.

For a continuum version of Frohlich's (1968) pumped phonon system the author shows that (i) there exists a critical value of the pumping above which the stationary state displays condensation; (ii) the stationary state is globally stable with respect to perturbations; and (iii) the relaxation times for the condensate diverge at the onset of condensation.

The Wood-Kirkwood equations model ZND detonation waves in cylindrical geometry where the flow is restricted to the central stream tube. The entire class of self-similar solutions is obtained and in a special limit the governing equations are reduced to a single first-order non-linear equation. Solutions are obtained in the case in which the radial divergence of the flow is constant.

Dynamical and kinematical symmetries and supersymmetries of the n-dimensional harmonic oscillator are discussed in connection with two different supersymmetrisation procedures: the so-called standard and spin-orbit coupling procedures. The largest invariance structures appear within the standard procedure dealing with the same numbers of bosonic and fermionic degrees of freedom. The authors also obtain meaningful substructures within the spin-orbit coupling procedure dealing with fermionic degrees of freedom which are half of the bosonic ones. Finally the n=2 case is connected with the study of the motion in a constant magnetic field. Each procedure leads to a nice correspondence between the two systems under consideration.

A simple dynamical system having a Berry phase component completely analogous to the Aharonov-Bohm circuit phase is presented. The underlying 'magnetic-vector-potential-like' object is identified. This parameter-space vector plays a vector-potential-like role at both quantal and classical levels.

An analytic continuation procedure using Taylor series is utilised to produce very accurate wavefunctions and eigenvalues for the Schrodinger equation.

The Bohr-Sommerfeld quantisation condition has a meaningful extension to classically chaotic systems whose periodic (unstable) orbits are isolated. It provides a semiclassical Euler factorisation for the functional determinant of the quantal Hamiltonian, in contrast to the Hadamard infinite product over the eigenvalues by which the exact determinant is defined.

Experiments have been proposed to observe non-linear QED effects using intense plane-wave fields. An external photon propagating in the field develops an index of refraction and a reversed polarisation component. The author calculates simple analytic expressions for these effects in the low-energy low-intensity approximation and compares with the constant crossed-field limit and with numerical results.

A new theoretical treatment of scattering in the presence of a low-frequency radiation field is presented. A modification of the Kroll-Watson formula, caused by the stochastic properties of the radiation field, is obtained. It is shown that the radiationless cross section remains on the mass shell, but an appropriate shift of the initial and final momenta strongly depends on the stochastic model used. An analytical calculation is carried out for the chaotic field model.

The conservation of the momentum density on the basis of the Pokrovsky-Khalatnikov Hamiltonian technique, suitable for superfluids, is examined. It is demonstrated that the momentum density j(t,x) is conserved when the local behaviour of the curl of the superfluid velocity v^(s) is described for ³He-A by the Mermin-Ho formula and for superfluid ⁴He, in a simply connected region, by the Landau formula. It is conserved also in the case of special relations between j and curl v^(s). In the case of the existence of the vortex lines the translational invariance of the system is violated and equations of motion for the momentum density contain extra terms with 'sources' of the momentum.

The authors investigate the calculational scheme for an eventual first-principles description of the ground-state energy of the many-boson Schrodinger equation. They employ a model ⁴He intermolecular pair potential consisting of a hard core surrounded by an attractive rectangular well. The study is based on the well known low-density expansion for such a system, where three coefficients are known. The first corrections to the fourth coefficient are determined on the basis of the rearranged series being a thermodynamic perturbation expansion, namely one representing perturbation not about the ideal gas but about the hard-sphere fluid. Pade approximants, and variations thereof, are employed to extrapolate to physical regimes in both resulting series, in density as well as in the attractive coupling constant.

With reference to the anharmonic oscillator problem, a temperature-dependent Hartree approach that is able to deal with quite general asymmetric potentials is presented. The mean-field approximation is seen to exhibit interesting phenomena associated with temperature changes.

The author presents a new algorithm to simulate 2D ballistic aggregates with one-sided rain on a seed. This algorithm makes use of the non-local shadowing effect by which a particle in the aggregate gets shadowed by a chain of particles which have originated elsewhere in the cluster. This algorithm has far less memory requirement as compared to the conventional algorithm. Physical implications of the triangle rule proposed as a part of the algorithm are discussed. Possible generalisations of the algorithm are indicated.

For pt.I see ibid., vol.20, p.257 (1987). The authors suggest a picture of the intrinsic surface width in Eden models and show that it is a major source of corrections to scaling of the surface roughness. Using the multiple-hit noise reduction method, they can control the intrinsic width and thereby improve the scaling behaviour systematically as is demonstrated in detailed calculations on the square lattice. They calculate the number of excess perimeter sites as a function of time and find that its asymptotic value decays with a power law as a function of increasing hitting number. Substrate effects and anisotropy become more apparent if noise reduction is applied.

It is shown that Glauber dynamics in 1D Ising spin systems is not universal. This is illustrated on a periodic model with a basic unit cell (J₁,. . ., J_n) containing an arbitrary set of n ferromagnetic coupling constants. The dynamic critical exponent z is calculated exactly as z=1+max(J_i)/min(J_i), the known value z=2 is recovered only for n=1. The extension of this result to other types of dynamics is briefly discussed.

The dynamical critical exponent z is obtained using the finite-size scaling method for the two-dimensional three-state Potts model with conserved dynamics. It is observed that there is a change in the dynamical behaviour of the system at the size n=15. For finite lattices of sizes n<or=15, the value of z is predicted as z=2.0+or-0.1. For n>or=15, the asymptotic behaviour is observed and z is calculated as z=2.78+or-0.20.

As the fugacity of intersections in trails (intersecting but non-overlapping lattice walks) is increased, the configurations change from swollen ones, with the scaling exponents of self-avoiding walks, to compact ones. Separating the two regimes is a potentially new tricritical point with no perturbative renormalisation fixed point associated with it. Supporting evidence for the existence of a tricritical point, its likely location and exponents are computed for the first time from exact enumeration of all trails up to length of fifteen lattice constants on the triangular lattice. The divergence of the specific heat indicates the location of the tricritical point. Generalised ratio and Pade methods are used to extract the scaling exponents for the number of configurations and their end-to-end distance.

The criticality of the anisotropic Ising model on a three-node hierarchical lattice is investigated by an exact renormalisation group transformation. The phase diagram exhibits three physically different phases, namely a paramagnetic one and surface and bulk ferromagnetic ones. When J₁ not=J/sub /2 not=J₃, the system orders in the direction with the largest J before it orders in the bulk. The bulk para-ferromagnetic transition is separated into three different universality classes, i.e. one isotropic and two anisotropic. This phenomenon, which is quite different from that on anisotropic Bravais lattices and seems analogous to that of a semi-infinite Ising model, is analysed.

A class of random-site mean-field Potts models is introduced and solved exactly. The bifurcation properties of the resulting mean-field equations are analysed in detail. Particular emphasis is put on the relation between the solutions and the underlying symmetries of the model. It turns out that, in contrast to the Ising case, the introduction of randomness in the Mattis-Potts model can change the order of the transition. For q<or=6 the transition becomes second order.

The Edwards-Anderson spin-glass model with m-component spins is studied in the low-temperature phase within the context of a q³ theory and the limit m to infinity is taken. Perturbation theory for the order parameter reveals at one-loop order the following unusual properties: (i) the perturbation expansion fails due to the appearance of infrared singularities for dimensionalities d<or=8 where d_c=8 is also the upper critical dimension of the theory, and (ii) for d=8 the expected terms involving ln(T_c-T) are absent thus implying a breakdown of scaling.

The author treats the probability density of homogeneous functions of independent variables. Assuming these variables are narrowly distributed, he finds an explicit approximate form for the Laplace transform of the function's density. This expression is used to show that, if the variables are all Gaussian, then so is the global function. The latter result is also claimed to hold for the total resistance of any random resistor network between two arbitrary terminals. He further utilises the result to explicitly find the Laplace transform of the conductance density of (i) a hierarchical structure and (ii) a random network in 1+ in dimensions using the Migdal-Kadanoff procedure. In the last case he also finds the scaling of all the cumulants and finds that they exhibit a constant gap.

Two efficient ways to accelerate on-lattice DLA simulations without introducing any bias are presented. The first requires one-dimensional tables but little arithmetic; the second needs no tables but necessitates the evaluation of a few elementary functions.

The author determines the surface fractal dimension D_s for two-dimensional percolation from Monte Carlo data in finite systems. The results agree with the prediction from conformal invariance, D_s=2/3.

The author gives a short proof of the global existence and uniqueness of solutions of Smoluchowski's coagulation equation with monodisperse initial conditions.

For original paper see ibid. vol.20, p.915, (1987). The authors show that energy fluctuations, and thus the higher-order moments of energy, contain essential information and cannot be neglected even in the thermodynamical limit. In contrast with some of Jaworski's statements, they point out how all the thermodynamical properties depend in a crucial way on the fluctuations on the basis of realistic physical assumptions. Indeed, phenomenological thermodynamics allows one to conclude that the Gibbs canonical distribution is the only possible probability distribution which does not violate the second principle. It follows that the knowledge of the free energy as a function of temperature beta ^-1 is equivalent to that of the probability law governing fluctuations. This probability law is therefore a characteristic of a body which can be investigated by measuring either energy moments at fixed beta or the mean values of entropy and energy at varying beta .

The author makes a conjecture for the temperatures at which the slowing-down processes occur in random systems. The estimated values in simple models are in good agreement with the existing simulations.

Random walks' on Sierpinski carpets are studied with bond moving renormalisation. Using the results, the author constructs fractal lattices whose fractal dimension of random walk is two. Such fractal lattices are called hyper-Euclidean lattices. It turns out that hyper-Euclidean lattices have a connectivity which is one smaller than the fractal dimension.

Different diffusion processes can be defined on random networks like the infinite incipient clusters at percolation threshold. The long-time behaviour of two such processes is shown to be the same. In particular the mean-square displacements and the autocorrelation function scale with the same exponents in the two cases.