Table of contents

By using developed software for solving overdetermined systems of partial differential equations, the authors establish the complete Lie algebra of infinitesimal symmetries of nonlinear diffusion equations.

Three new cases when the Lorenz system has (time dependent) integrals of motion are given.

In finite dimensions every strictly positive boson density is V-representable.

The relation between a quantum dynamical semigroup and the associated quantum stochastic processes can be formulated by an analogy in operator algebras to the martingale problem of Stroock and Varadhan (1969) for classical diffusions. It is shown that under simple circumstances the formulation yields the explicit solution. The authors give three such examples.

An exact solution, compact in form, for the evolution operator of a harmonic oscillator driven by an external chaotic electric field is obtained. It is shown that using the harmonic description of the Ornstein-Uhlenbeck stochastic process, the evolution operator can be calculated explicitly from a normal ordering problem of certain Bose creation and annihilation operators.

The author describes a model of the spread of an epidemic on a lattice, in which sites may be infected by their neighbours, after which they recover and are subsequently immune. It corresponds to a modification of the directed percolation problem, and also to a growth model of clusters. The model is analysed using a continuum field theory, both within a self-consistent approximation and using an epsilon expansion below six transverse dimensions.

The author emphasises the fact that, to obtain the exact solution of Hamiltonian models on Bethe lattices one can apply explicitly the formal method of rigorous statistical mechanics, i.e. the thermodynamic limit of probability measures. This approach solves the well known dichotomy between clashing alternative solutions, and provides a very simple analytic solution for a large class of Hamiltonian models.

Using the Migdal-Kadanoff renormalisation method, the authors study the semi-infinite two-dimensional Ising model with inhomogeneous nearest-neighbour coupling constants, that deviate from the bulk coupling by Am^-y for large m, m being the distance from the edge. The approximation correctly predicts irrelevance of the inhomogeneity for y>v^-1, a surface magnetic phase at the bulk critical temperature for y<v^-1, A>0, and a non-universal A-dependent surface magnetic index in the marginal case y=v^-1.

A Monte Carlo renormalisation group technique is used to calculate time correlation functions for the Baxter-Wu model. Initial lattices of size 24*24 and 48*48 are used. The value obtained for the dynamical exponent z is in good agreement with that obtained by means of standard Monte Carlo methods.

The relation between the density relaxation function Phi and the pair connectedness is shown. Static and dynamical scaling for Phi and quantities related to it are derived from percolation scaling theory. Due to finite clusters Phi contains a non-ergodic singularity even in the conducting phase, whence a Green-Kubo identity does not hold. The form factor of this singularity is discussed. For d>or=3 the static polarisability can be related to a diverging characteristic length also above the threshold. Contributions come from confinement in finite clusters and from the structure of the infinite cluster.

The mean-field equations of the simple cubic ANNNI model are studied on finite lattices. The results are consistent with the sequence of distinct commensurate phases (2^k-13), k=1,2,3,..., springing from the multiphase point, found using low-temperature series expansions. Moreover, evidence for new structure combination branching processes is presented, which generate phases of type ((2^l3)^m(2^l+13)ⁿ) or ((23^l)^m(23^l+1)ⁿ), where l, m and n are integers.

By smearing with a Gaussian distribution a family of single-spin weight functions is constructed which interpolates between an arbitrary single-spin distribution and a pure Gaussian. The observation of Baker and Bishop (1982) concerning the factorisability of the partition function of the double Gaussian model remains valid for all Gaussian-smeared models. The effects of smearing Gaussian and spherical weight functions are studied in further detail.

The authors propose a generalisation of the q-state Potts model in which two neighbouring spins in states s and s' have a coupling-J_s delta (s,s'), and develop an exact graphical expansion of the partition function where each cluster in the graph can be coloured in one of C<or=q colours. The spin model reduces to the standard Potts model (one colour) when J_s=J; it also encompasses many new applications including correlated polychromatic bond percolation, the dilute branched polymer problem, and weighted clusters in the standard Potts model.

The authors simulate the formation of a gel from a mixture of bifunctional and tetrafunctional monomers which may form bonds with the help of initiators on the triangular lattice. Their Monte Carlo results for the average-weight degree of polymerisation exponent, gamma , substantiate the conclusion that this model has distinct critical properties from other models of branched polymers. They observe a crossover to normal bond percolation as the concentration of initiators is increased. However, the cluster number ratio appears to be a universal quantity. They offer an explanation for the differences in phase diagrams of the two- and three-dimensional systems in terms of restricted random walkers.

The authors consider a special class of random anisotropic spin models, characterised by macroscopically preferred directions for spin alignment. Spin wave arguments applied to these models predict that ferromagnetism is unstable below four dimensions. They show that unlike the case of random axes distributed uniformly over the entire unit sphere, this spin-wave result is spurious and ferromagnetism can indeed exist below four dimensions. The ordering is very non-uniform, thus explaining why spin wave theory which is based on the assumption of uniform order yields instabilities.

The total probability outside Wigner's (1962) semicircle is calculated using Mehler's formula for the Hermite polynomials. In the limit N to infinity , N being the dimension of the random Gaussian Hermitian matrix it is shown that this probability is 0.05N^-32/. Some implications of this small probability are discussed.

The low-lying energy levels of electrons moving in a potential of the form V(x)=(1- gamma ²)/(1-2 gamma cos x+ gamma ²) have been calculated. In the limits gamma to 0 and gamma =1, well known results are recovered. The bandgaps are studied as a function of gamma .

A new proof of the Atiyah-Singer index theorem for the Dirac equation in the presence of external gauge and gravitational field is presented.

The authors show that once the motion of a non-holonomic system is known it is possible to reduce the system to the holonomic form. A (singular) Lagrangian function and a Hamiltonian which correctly describe the dynamics of the system can be constructed. The procedure they have developed is applied to a well known system.

The author introduces a new potential and use it with the velocity potential to construct a new variational principle for long surface waves in shallow water. Applying Dirac's theory of constraints, the author casts it into canonical form and obtain an explicit expression for the exact Hamiltonian.

The authors prove, for a particle moving in a plane under the influence of a conservative force, that when the motion is constrained by a 'second' invariant quadratic in the velocities, then the potential allows separability of the Hamilton-Jacobi equation in rectangular, polar, elliptical cylinder or parabolic cylinder coordinates. This link shows the intimate connection between quadratic invariants and the two-dimensional Hamilton-Jacobi equation. They give examples of the utility of parabolic cylinder coordinates in cases of recent study.

Path integration of a general two-time quadratic action characterising memory effects is performed within the framework of Feynman's polygonal path approach. Explicit evaluation of the propagator in exact analytical form is further carried out for the specific kernel used by Feynman in the polaron problem.

The integral of the modulus squared of a radial matrix element, taken over the continuous energy spectrum is considered as a function of the interaction parameter lambda . It is assumed that the matrix element is that of an N-order differential operator with exponentially decreasing coefficients, where 0<or=N<or=2. The analytic properties of this integral in the neighbourhood of a bound-state appearance threshold lambda ₀ are investigated. It is shown that the integral has a branch cut on the complex lambda plane crossing the physical region (the real axis) at the threshold lambda ₀. An analytic continuation of the integral into the second Riemann sheet is found, and it is shown that it contains a bound-state term. Thus the corresponding total quantity (the integral over the continuum and the sum over the bound states) is a smooth function of lambda . The threshold behaviour of the integral and the bound-state term is considered.

General and compact formulae for matrix elements of x^k and X^k X^ke^αX are derived using the second quantisation technique.

For pt.II see Proc. R. Soc. London, Ser. A, vol.384, no.1786, p.89-105 (1982). A procedure for solving the few-particle Schrodinger equation exactly is applied to a model system consisting of two identical particles and a massive third particle. The type of interaction potential is not specified except that it should not diverge more rapidly than r^-2 at the particle positions. Allowable interactions include the Coulomb and the harmonic oscillator potentials. The principles are illustrated by reference to the spatially symmetric states of the system. The solution has the form of a multipole expansion in spherical polar coordinates. The radial dependence for each multipole component is defined by a power series including logarithmic terms. Explicit expressions are given for the coefficients in the expansion. Exact coefficients are obtained for low-order terms in the wavefunction of a two-electron atom.

For pt.III see ibid., vol.16, nol.18, p.4237-53 (1983). Formal series solutions for the Schrodinger equations of few-particle systems contain infinitely many associated with the normalisability, i.e. square-integrability of the wavefunction. The energy is one of these parameters. A method for determining the parameters by examining the asymptotic behaviour of the series has been developed. No integration, matrix inversions or trial and error procedures are involved. The method is directly applicable to the ¹S states of two-electron atoms. If the wavefunction is expressed as a multipole expansion in spherical polar coordinates the radial functions have asymptotic properties which give rise to relations between the parameters. Values assigned to the parameters not specified by these relations (one per multipole) ensure that the wavefunction is normalisable.

The connection between a three-dimensional nonrelativistic hydrogen atom with positive energy and a four-dimensional isotropic harmonic oscillator with repulsive potential is established by applying Jordan-Schwinger boson calculus to the algebra of the Laplace-Runge-Lenz-Pauli vector. The spectrum generating group SO(4,2), both for the bound and free states of the three-dimensional hydrogen atom arises as a quotient of the group Sp(8,R) associated to four-dimensional isotropic harmonic oscillator with constraint.

An earlier study of the role of the poles of the function h(k) in determining details of its inverse Fourier transform h(r), the total correlation function of the hard sphere fluid, is extended to a consideration of their role in determining the details of the structure factor.

The author considers a system with a single locally-conserved field ( identical to density) in a slab geometry with different densities maintained at the two surfaces of the slab. On the basis of fluctuating hydrodynamics, he shows that the static density-density correlations are long ranged and decay as -1/ mod x-y mod ^d-2 for dimension d>or=3 over distances small compared to the size of the slab. This effect vanishes to first order in the density difference. As a particle model he investigates a stochastic lattice gas with Kawasaki dynamics. He establishes the connection to fluctuating hydrodynamics. In the case of hard core interaction only he proves the validity of fluctuating hydrodynamics and obtain, presumably model dependent, corrections.

The authors prove that the fluctuations of the total momentum of a system of quantum mechanical particles at equilibrium obey the classical equipartition law whenever the correlation functions have an integrable clustering. The result holds for a large class of translation invariant two-body potentials and for arbitrary statistics. They also discuss higher moments of the total momentum and position. Finally the behaviour in models with broken symmetry is analysed.

The exactness of the Bogoliubov approximation is discussed from the point of view of the condensation properties of the Bose gas. For the imperfect Bose gas, the authors find that the approximation yields the correct condensate density when the thermodynamic limit is taken by isotropic dilation; this is not the case in general for other ways of going to the infinite volume limit. They also prove the existence of Bose-Einstein condensation for a class of Bose gases with weak interactions and an energy gap.

The authors have proposed new boundary conditions for use in Monte Carlo simulations and have applied them in this instance to the two-dimensional square Ising model. The variables defined on the first layer outside the boundary of the system are determined by the requirement that the nearest-neighbour pair correlation function with the boundary spin as one of the pair, is equal to that calculated from the finite system. The single-site values of the variables outside the boundary are chosen with a probability consistent with the fluctuations in the pair correlation function for the finite system.

The author presents a real space renormalisation group calculation for the adsorption of a single self-avoiding walk in d dimensions at a (d-1)-dimensional impenetrable wall. The case d=3 is used to check the method. For d=2 the result for the crossover exponent phi gives phi =0.55+or-0.15. This, in connection with enumerations of Ishinabe (1982), clearly rules out de Gennes' (1972, 1976, 1977) conjecture phi =1-v also for the case d=2. For d=2 cells up to 6*6 are used, for d=3 cells up to 3*3*3.

The internal spatial distributions of self-avoiding hard sphere sequences are determined for terminally attached and free-floating systems between two rigid boundaries of variable attraction and separation. In all cases the terminally attached distributions are characterised by a pronounced density discontinuity and internal structure, whilst the free-floating sequences form either mono- or biglobular coils depending upon the details of attraction, temperature and boundary separation. The role of entropic and van der Walls contributions to the structure and excess free energy are clearly resolved, and their implications for colloidal stability against flocculation discussed. The mean square molecular span and centre of gravity are also determined as a function of chain length interaction strength and boundary separation.

The scaling behaviour of the hard square lattice gas with diagonal interactions is examined on the basis of Baxter's (1980, 1981, 1982) recent exact solution. It is demonstrated that all the corrections to scaling found may be accounted for in terms of a single irrelevant scaling field that scales simply as a length. The lattice cut-off is proposed as the source of these corrections, although other sources cannot be ruled out. The melting of the 3*1 commensurate ordered phase in this model is examined in a more general context in which the line of Potts critical points found by Baxter appears to be a line of multicritical points on which the effective uniaxially chiral symmetry breaking field vanishes.

It is shown that the chain of coupled particles in the double-well potential introduced by Schmidt (1979) is completely integrable in the static limit. The chaotic behaviour and the associated infinite series of bifurcations found in the related discrete phi ⁴ theory are absent in the model. The solutions are generally unpinned soliton lattices. The model exhibits a bifurcation where a hyperbolic fixed point becomes elliptic and splits into two hyperbolic fixed points. The bifurcation does not lead to chaos.

Exact expressions for the electromagnetic fields of a charged particle through a dispersive and dissipative medium have been used to calculate exact expressions for the energy loss due to distant collisions. These are evaluated using Fermi's one-oscillator approximation to the dielectric constant. For particle velocities with beta < epsilon _o^-1/2 the results are identical to those of Fermi for an electric charge and to those of Ahlen (1980) for a magnetic monopole. For beta > epsilon ₀^-1/2 the present results differ from those of these authors. It is shown that this is due to their incomplete treatment of the conditions imposed by the model for the dielectric constant. The relevance of these results to the detection of magnetic monopoles and of dyons is also discussed.

In the minimal-coupling Lagrangian for the interaction of the electromagnetic field and nonrelativistic charged particles, the charge and current densities are coupled to scalar and vector potentials. In the multipolar Lagrangian, on the other hand, the aggregate or particles is particularly (through sometimes completely) described by polarisation and magnetisation fields and these are coupled to the electric and magnetic induction fields. It is shown that if isotropically averaged polarisation and magnetisation fields are introduced, the minimal-coupling and multipolar Lagrangians are identical, provided also the potentials used are those of the Coulomb gauge. The associated canonical transformation of the Hamiltonian is the identity transformation. Thus the minimal-coupling Hamiltonian can be written directly in multipolar form, without any change in the canonical dynamical variables.

A non-trivial, perturbative solution of the generalised Raman-Nath equation is obtained and its meaning clarified. The connections with earlier work and with possible physical applications are discussed.

Due to an improper interpretation of the relation of first-class constraints to gauge degrees of freedom, many authors have asserted that all primary and secondary first-class constraints are associated with gauge freedom and enter in the generator of the evolution of the system. By presenting several examples, the authors show that secondary first-class constraints are not associated with gauge degrees of freedom, but only intrinsic first-class constraints yield gauge freedom.

Monte Carlo estimates of the heat capacity of the two-dimensional O(n) spin systems with n=3, 4 and 5 are compared to the results given by the spherical model (n to infinity ). The values of the maxima of the heat capacity are approximately equal to ¹/₂n, when the temperature is 2n^-1. The C/n against nT curves of the estimates come close to the curve of the spherical model for nT>or approximately=4.