Table of contents

Some exact solutions of the three-dimensional non-linear Schrodinger equation are found. The formulae for generating solutions of the Schrodinger-invariant equations are adduced.

The problem of a formulation of Nelson's (1967) stochastic mechanics of scalar fields in the context of general relativity is considered. The simple example of the scalar field in the Wightman vacuum state on the Rindler wedge is examined, the stochastic counterpart of the Fulling ambiguity of canonical quantisation is formulated, and the role of the stochastic mechanics of thermal mixtures, as formulated by Guerra and Loffredo (1981), is analysed, in the spirit of Davies (1975) and Unruh (1976), in the solution of the above ambiguity. An overall picture emerges which, both in the explicit example considered here and in its straightforward generalisations to static submanifolds of more general spacetimes, confirms Smolin's point of view that stochastic quantisation is a very natural conceptual frame in which to study the general non-covariance of the distinction between quantum and thermal fluctuations.

The devil's staircase spectrum of a Fibonacci chain is obtained using a simple non-linear map. This map exhibits intermittency for all Fibonacci number iterates of the function whenever the frequency lies in a gap of the spectrum. The self-similar nature of the eigenfunctions is shown to be a consequence of the intermittency behaviour. Certain simple patterns that emerge in the gap-labelling theorem are discussed.

The authors numerically determine the f( alpha ) spectrum of scaling indices for critical KAM tori in the area-preserving standard map. Similarities and differences between the corresponding orbits of circle maps are discussed.

A strict solution of Boltzmann's equation without collision term has been found under the initial condition of a uniform spherical group of particles, between which there is only gravitation interaction and the initial velocity of which obeys the Maxwell velocity distribution law. No approximation, except for the collision term, is made in deriving the solution.

We have developed an efficient quantisation scheme for highly excited states of the anisotropic Kepler problem. This applies to the level spectrum of a donor impurity in semiconductors. Results for silicon and germanium are presented.

The authors study the geometric and thermal properties of a model of self-interacting linear polymers (self-avoiding walks) on a fractal lattice, the three-dimensional modified Sierpinski gasket of base b=3. As a consequence of the topological structure of this lattice, the polymer is 'frustrated' and cannot fill the available space with a finite density. When strong attractive interactions between monomers are present the polymer cannot reach the compact globule state usually observed. Rather, it shrinks into a novel phase, the 'semicompact' state, below a finite critical temperature. In this phase, the monomer density per lattice site vanishes asymptotically for large polymers. The authors give the exact values of the critical exponents at this transition and discuss its possible relevance for polymers placed in a random matrix.

The hierarchical cluster-cluster aggregation model is extended in the presence of dipolar interactions between the magnetic dipoles attached to each individual particle. It is found that the fractal dimension of the resulting clusters decreases when the intensity of the momenta increases, as observed in recent experiments.

A noise-driven model is developed to describe the role of fluctuations in sidebranch phenomena in growth patterns for the fluid displacement problem and for dendritic crystal growth. Simulation results are compared with recent experiments on NH₄Br dendrites. It is found that the RMS sidebranch amplitude is an exponential function of distance from the tip, with no apparent onset threshold. Moreover, the sidebranches are non-periodic (at all distances from the tip) with apparently random variations in amplitude.

A very large class of kinetic growth processes manifest oscillatory behaviour in a variety of physical quantities such as the propagation of the growing interface and the density of the resulting cluster. The authors demonstrate that these oscillations should generically display quasiperiodic behaviour. Using a formalism based on the projection method for quasicrystalline spectra, they elucidate general features such as the dominant frequency and amplitude of the oscillations. They also briefly discuss the effects of interfacial roughening on the spectra of the oscillations.

The dynamics of the q-state Potts model on a fractal lattice is studied using Monte Carlo simulations. The Glauber dynamics is used leading to an effective temperature-dependent critical exponent of the form z=AK+B implying the breakdown of conventional dynamic scaling. The value of A is shown to be independent of q, within the error bars.

The question of the existence of multiple pure states in spin glasses and other disordered systems is considered. The authors show that the distribution of overlaps, P(q), is an unreliable indicator of how many distinct pure states a system has. They discuss the definition of, and possible relations between, pure states. They also discuss the spin glass on a Bethe lattice and the infinite-range model, pointing out why their ordered phases are likely to be unlike that of realistic spin glass models. Various possibilities for the behaviour of finite-range spin glasses are discussed.

Based on a simple scaling ansatz, the authors argue that no pure thermodynamic states other than the paramagnetic state and a pair of states in zero magnetic field which are related by a global spin flip can exist in short-range Ising spin glass models in any dimension. An analogous result should hold for XY and Heisenberg spin glasses, as well as for square-integrable long-range interactions.

The author studies the threshold concentrations, P_q, for critical behaviour at zero temperature in the correlation functions χ^(2q) for an Ising spin glass in which nearest-neighbour interactions randomly assume the values +J, 0, and -J with respective probabilities p/2, 1-p and p/2. Here χ^(q)≡σ _j(s_is_j)^q)_av, where ( ) denotes an average over the ground states for fixed configuration of J and ( )_av an average over all such configurations. Due to frustration effects p_c<p₂<p₄<p₆. . .<p_infinity , where p_c is the percolation threshold. Thus spin-glass theory with only χ⁽²⁾ critical (at p=p₂) applies and the critical exponents along the T=0 axis are the same as thermal critical exponents for p=1. When the values +J and -J are replaced by distributions of narrow width, the frustration is removed and percolation exponents are expected at p=p_c.

The author investigates the instability of the high-temperature phase of the Sherrington-Kirkpatrick model at finite magnetic fields. He finds that the relevant function for this analysis is q⁽¹²⁾(h⁽¹⁾,h⁽²⁾) identical to (m_i(h⁽¹⁾)m_i(h ⁽²⁾))_av with two external fields h⁽¹⁾ and h⁽²⁾ and reveals its non-analytic structure at the critical points. The author indicates that the field breaking the 'hidden' symmetry of the high-temperature solution and causing a breakdown of linear response theory is v identical to h⁽¹⁾-h⁽²⁾.

More than 1000 h on a high-speed special computer were used to analyse the global limit cycle period of the Q2R automata in 2D at various energies. An exponential increase of the periods with system size N was found even at energies below the Curie temperature. Even for T to infinity , i.e. in the fully chaotic regime, the paramagnetic phase of Q2R was found to be non-ergodic.

With the use of a large-cell Monte Carlo renormalisation group method, the quantum percolation threshold is determined to be 0.42 for the site problem and 0.30 for the bond problem in a simple cubic lattice.

The distribution of Lyapunov exponents of the N-channel Anderson problem is investigated. The Kolmogorov entropy and the maximum Lyapunov exponent are calculated in the thermodynamical limit at weak disorder. The ratio I of these two quantities, which characterises the convexity of the distribution, is shown to depend on the disorder, the energy and the energy spectrum of the noiseless system. Moreover, I is found to be in general greater than two, in agreement with the convex spectrum obtained previously by some authors through numerical experiments. It is also shown that, in contrast to the one-dimensional case, the 'generalised Lyapunov exponent' of order two and the largest Lyapunov exponent are equal in the limit N to infinity .

The algebraic structure of the Schrodinger equation is discussed. The author takes advantage of Lie algebra to derive the kernels of the Schrodinger equation.

The generating function for the number of independent Hermitian SO(3) scalar operators in the enveloping algebra of U(6) restricted to totally symmetric U(6) representations is constructed. This predicts the number of interaction terms that may appear in the most general Hamiltonian for the interacting boson approximation (IBA) model. A complete analysis up to cubic interaction terms is then given.

Group properties are investigated for unsteady axisymmetric incompressible viscous flow by means of the kinematic approach of Pillow and Paull (1985). The full symmetry group and Lie algebra for the original system of three partial differential equations is derived and is shown to be infinite dimensional. Further group reductions are possible and some solutions are constructed.

The authors discuss interrelations between generating invariants (GI) for the reduction Pi ^m_i=1 D(P_i) to D(O_n-1) of the m-fold tensor product of SU(n) irreducible representations (irreps) D(P_i) and polynominal bases of the SU(m) irreps D(O_n-1J, O_m-n-1), O_k=0, 0,. . .0 (k times). A realisation of the SU(m) irrep D(OJ,O_m-3) bases is given in terms of GI for SU(2) coupling (Wigner) coefficients. As a byproduct an expression is obtained for SU(2) 6-j symbols in terms of only two Wigner coefficients. The authors also discuss some possibilities of the analysis involved in solving the Wigner-Biedenharn problem (construction of orthonormal sets of the Wigner coefficients) for SU(n) groups (n>or=3).

Transport properties for simple ring polymers are derived using renormalisation group techniques. The translational diffusion constant and the relaxational spectrum in the presence of hydrodynamic interactions are calculated to O( in ) ( in identical to 4-d, d being the spatial dimensionality). Explicitly time-dependent correlations are presented in universal functional form to O( in ).

The total number of three-dimensional integrable potentials which possess one linear and one quadratic integral of motion in involution in the momenta is found. These potentials, which are given by a Darboux equation, belong to four distinct classes and involve two arbitrary functions in their expression.

The authors show the importance of the quadratic term in the lattice action for Monte Carlo simulations. The effects of this action modification are analysed through the calculations of some simple non-relativistic quantum systems.

Numerical methods for obtaining lattice Green functions are presented. The whole array to arbitrary distance on an infinite lattice can be generated in approximately 50 mu s CPU (VAX) per site. The accuracy is close to machine error. For small mass, the coefficients of the power series expansion can be generated similarly. Results are given in two and three dimensions.

The authors study the concept of projectability by the Legendre transformation FL:TQ to T*Q induced by a singular Lagrangian, and the restrictions on this concept when submanifolds of TQ are considered. The FL projectability of the characteristic vector fields and constraint functions relative to such a submanifold is analysed, thus giving a way of characterising the submanifolds of TQ according to their behaviour under FL, by means of the study of the tangency of the fields belonging to ker FL*. The application of some of the results to the theory of constrained systems is discussed.

Numerical studies of two-dimensional turbulence show the importance of localised structures, both in the small scales where intermittent transfers occur and in the large scales dominated by coherent vortices. This work is an attempt to establish a more appropriate theoretical framework than the usual Fourier representation. The authors present a dual representation, mixing continuous field and point vortices. The difficulties arising from the redundancy of the representation are solved through a hypothesis of mutual advection. The resulting equations conserve the natural invariants of the system. As a first application, the authors show that introducing a weakly charged point vortex in a continuous field with quasiperiodic behaviour induces a purely Lagrangian chaos. As the point vortex grows in intensity, it becomes quasistationary and the induced phase mixing dominates the behaviour of the continuous field.

A novel perturbative method based on the non-trivial decomposition of the Hamiltonian into the unperturbed part and the perturbation is proposed. The method admits using practically any approximate solution of the Schrodinger equation as a starting point by constructing a perturbative expansion. The resulting series is convergent. The procedure of calculation of its terms is purely algebraic. The method is available for calculations in both the weak- and strong-coupling regimes, and for estimating and increasing the accuracy of variational analysis results.

The authors provide a detailed analysis of properties of the Schrodinger operator in L²(R) which formally can be written as -d²/dx²+ Sigma _{y in gamma} nu _y delta '(.-y) where delta ' is the derivative of Dirac's delta -function and Y contained in/implied by R is discrete. This model allows for an explicit calculation of spectral properties. Special emphasis is given to the periodic case Y=Z, nu = nu _j, j in Z where the spectrum and the density of states are explicitly computed. Also the spectrum for a half-crystal is given. The authors study in detail spectral consequences when various defects and impurities are added to the periodic case.

Non-relativistic wavefunctions of one-dimensional rings threaded by a magnetic flux are characterised by a quantised vortex structure, which is visualised in the plane into which the ring is embedded. With increasing flux, vortex cores traverse the perimeter of the ring periodically at flux values where anticrossing of the energy levels occurs. The topology of the vortex pattern is related to measurable physical quantities.

The Casimir surface force density on a compact spherical ball composed of matter satisfying the condition εμ=1 (ε being the permittivity and μ the permeability) is calculated. The matter is taken to be non-dispersive. Schwinger's source theory is employed. The present calculation improves a previous calculation of the same effect by Brevik and Kolbenstvedt (1985) by four orders of magnitude in the Debye expansion. As before, the surface force is found to be repulsive.

Formulae for the phase of each off-diagonal element in terms of the amplitudes of the diagonal elements of the 3*3 unitary symmetric S matrix, together with the inverse relations, are given. The formulae have been used to check numerical calculations by applying them to three coupled channel potential wells.

In the autoimmune disease systemic lupus erythematosus the DNA molecules of an individual are attacked by its own antibodies. As these antibodies are bivalent they can crosslink different DNA molecules which can lead to the formation of DNA-antibody complexes and gels. Statistical properties of these complexes are derived and evaluated analytically in the limit of very long DNA molecules, as well as the concentrations at which a gel is being formed. The authors also present various numerical results for DNA molecules of intermediate lengths. This work can also be considered as a theory of the crosslinking and gelation of linear polymers.

The author studies conformal invariance in the RSOS models of Andrews et al (1985) at both the ferromagnetic and antiferromagnetic transitions, using finite-size scaling of the transfer matrix. The author confirms the conjecture of Huse (1984) concerning the conformal classification of the ferromagnetic transitions and gives evidence that the antiferromagnetic transitions lie in the universality class of Z_n models. In addition, he shows that conformal results can be applied to anisotropic systems if logarithmic corrections are taken into account.

The stability of a chain reaction with respect to global and local fluctuations of the reaction parameters is discussed within a linear model for the reaction kinetics. The spatial correlations in the system are studied for the case when the local fluctuations are smaller than the global one. The time dependence of the correlation length is presented.

Evaluating the q colourings of a lattice is equivalent to solving the q-state zero-temperature antiferromagnetic Potts model. This has recently been done exactly for an infinite triangular lattice with q real. Here the results are extended to the full complex q plane, giving the limiting distribution of the zeros of the chromatic polynomial. The results are compared with finite lattice calculations and the occurrence of isolated real zeros converging on the Beraha numbers is noted.

A set of generalised percolation probabilities P_n are defined for the dichromatic polynomial formulation of the Potts model. A generating function for these P_n is calculated at the self-dual temperature. P₁ and P₂ are explicitly given and the behaviour of P_n is investigated.

The statistical properties of the multivalley structure of disordered systems and of randomly broken objects have many features in common. For all these problems, if W_s denotes the weight of the sth piece, the authors show that the probability distributions P₁(W₁) of the largest piece W₁, P₂(W₂) of the second largest piece, and Pi (Y) of Y= Sigma _s W²_s always have singularities at W₁=1/n, W₂=1/n and Y=1/n, n=1,2,3,. . .

The authors study the problem of a travelling salesman who must visit a randomly chosen subset of sites of a d-dimensional lattice. The average length of the shortest path per chosen site is alpha (q) where (1-q) is the density of chosen sites. For a triangular lattice, they show that alpha (q) differs from 1 only by terms of order q⁵. For the square lattice, they show that, to first order in q, optimal paths can be found from the dynamics of a model of a one-dimensional gas of kinks and antikinks. The authors find alpha (q)<or=1+terms of order q^3/2. They also obtain a constructive upper bound valid for all q, which gives alpha (q)>or=L/3(1-q)^-1/2 as q tends to 1.

The exact calculation of an infinite number of correlation functions for general cubic and HCP anisotropic Ising models on their 'disorder variety' is described. The intra-plane correlation functions are found to be exactly the same as for a two-dimensional free-fermion Ising model. This allows the authors to characterise the intersection between the disorder and critical variety which displays in general anisotropic scaling with non-trivial exponents.

A water-like lattice gas on the square lattice is introduced. The asymmetry of the hydrogen bond is ignored. Molecules, pictured as having four square-planar oriented bonding arms, may occupy lattice sites in either of two possible orientations, + or *. The bonding arms of a molecule in the + orientation point toward nearest-neighbour (NN) lattice sites and the bonding arms of a molecule in the * orientation point toward next-nearest-neighbour (NNN) lattice sites. A hydrogen bond, energy in ₁<0, occurs between two + oriented molecules on NN lattice sites and a hydrogen bond, energy in ₂<0, occurs between two * oriented molecules on NNN lattice sites. Two molecules on NN sites repel with an energy gamma >0, regardless of their relative orientations. Reflection positivity combined with the Peierls argument is used to prove that phase transitions to both open- and close-packed ice phases occur in the model.

An Ising model with three-spin interaction is investigated by finite-size scaling applying free boundary conditions. The surface critical properties of the model were determined by conformal invariance. Taking into account logarithmic corrections the surface critical exponents obtained are in accordance with the four-state Potts values.

Using conformal invariance the author calculates finite-size corrections to the scaled spectra of one-dimensional quantum chains at the critical point. An explicit application to the Ising and three-state Potts model gives good agreement with analytical and numerical results, respectively.

For pt.I see ibid, vol.20, p.1043, 1987. For a hopping model on a square lattice with a fraction c of sites randomly replaced by impurities, the static conductivity and diffusion coefficients can be expanded in powers of the concentration c. The authors calculate the exact coefficients of O(c) and O(c²) using analytical and numerical methods, and the results are compared with computer simulations.

Zeta-function regularisation (of functional determinants) is used to derive the exact high-temperature expansions of the thermodynamic potentials for the ideal massive Bose and Fermi gases for non-zero chemical potential. The purpose is to show agreement with the results obtained by the Mellin transform method, and by the method of zeta-function regularisation of infinite series. The generalised zeta functions calculated are the simplest examples of Feynman diagram zeta functions. The extension of these calculations to Euclidean manifolds M*Eⁿ more complicated than the finite-temperature cylinder S¹*Eⁿ is discussed.

The finite-size corrections approach to the calculation of the asymptotics of correlation functions containing the oscillating terms in the asymptotics in (1+1)-dimensional theories is presented. Explicit formulae for the critical exponents in integrable models are obtained.

Starting from a Carnot cycle with entropy production in the heat reservoirs, the authors suggest a physical interpretation of a non-equilibrium absolute temperature.

An expression for the square-law fluctuation of internal energy is obtained in the framework of the local equilibrium approach with the aid of the non-equilibrium distribution function presented by Zubarev (1974). It is shown that this fluctuation satisfies the necessary conditions for its use as Liapounov's function. It is demonstrated that Liapounov's stability requirement leads to a fluctuating functional which has a minimum for the hydrodynamic stage. The infinitesimality estimates of the right parts are found for two balance equations: energy and entropy. It is shown that the local potential of Prigogine and Glansdorff (1971) can be derived from perturbed fluctuating functional obtained here. With the aid of the well known formula of the Onsager-Machlup postulate for the density probability transition of a thermodynamic system from some non-equilibrium state, the path integrals are obtained for the evolution of the system in the hydrodynamic stage.

The authors give an extension to non-conservative Lagrangian systems of a previous work (Sarlet et al, ibid vol.17, p.1999-2099, 1984).

The idea of generating a family of isospectral Hamiltonians from a given Hamiltonian using supersymmetric quantum mechanics is exploited in constructing the 'partner' stability equation for the phi ⁴ soliton stability equation. From the 'partner' stability equation, the soliton profile and the potential that admits the soliton solution are derived.

By transforming the radial equation of the isotropic oscillator to the Coulomb form, the authors show that the energy degeneracy of the three-dimensional oscillator may be understood on account of supersymmetry.

Averages over percolation clusters Gamma of the form Sigma _Gamma P( Gamma )X( Gamma )n( Gamma )^k identical to chi _k(p) are considered, where n( Gamma ) is the number of sites in the cluster, X( Gamma ) is some property of the cluster and P( Gamma ) is the probability per site of forming such a cluster at bond concentration p.

The author re-examines the two-parameter position space renormalisation group calculation for Eden growth process on the square Manhattan lattice (SML), first studied by Chernoutsan and Milosevic (ibid vol.19, p.2237, 1986), and find that, contrary to the finding of the above authors, Eden processes on SML and on the ordinary square lattice are in the same universality class.

Simulation results of Q2R cellular automata on the simple cubic lattice are compared with usual Monte Carlo simulations of the Ising model. Curie temperature, spontaneous magnetisation and specific heat agree, but at low temperatures the susceptibility and the distribution of local energies differ.

An approximate mapping between a random Potts model with a different number (q and p) of Potts spin states and the Ising model is used to locate the concentration at which the change of the order of the phase transition occurs. Results are obtained for d=2, q=3, p=5,6,7, and d=3, q=2, p=3.

A power law is found for a steady state of a cascade of fracture. A master equation of the number of fragments is presented. It is linear in the number and has a homogeneous kernel. The kernel stands for a statistical mechanism of brittle fracture on the basis of randomly distributed flaws and their spatially scale-invariant activation. It includes extended Gilvarry's theory if this could be scale invariant.