Showing papers in "Journal of Physics A in 1985"

Scaling of the active zone in the Eden process on percolation networks and the ballistic deposition model

Abstract: 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.

Angle variable holonomy in adiabatic excursion of an integrable Hamiltonian

Abstract: 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.

Supersymmetric quantum mechanics of one-dimensional systems

Abstract: It is shown that every one-dimensional quantum mechanical Hamiltonian H1 can have a partner H2 such that H1 and H2 taken together may be viewed as the components of a supersymmetric Hamiltonian. The term 'supersymmetric Hamiltonian' is taken to mean a Hamiltonian defined in terms of charges that obey the same algebra as that of the generators of supersymmetry in field theory. The consequences of this symmetry for the spectra of H1 and H2 are explored. It is shown how the supersymmetric pairing may be utilised to eliminate the ground state of H1, or add a state below the ground state of H1 or maintain the spectrum of H1. It is also explicitly demonstrated that the supersymmetric pairing may be used to generate a class of anharmonic potentials with exactly specified spectra. The complete spectrum of an anharmonic potential so generated consists of all the eigenstates of the simple harmonic oscillator and, in addition, a ground state at a specified energy E which lies arbitrarily below the E=1/2 ground state of the harmonic oscillator.

Classical adiabatic angles and quantal adiabatic phase

Abstract: A semiclassical connection is established between quantal and classical properties of a system whose Hamiltonian is slowly cycled by varying its parameters round a circuit. The quantal property is a geometrical phase shift gamma n associated with an eigenstate with quantum numbers n=(nl); the classical property is a shift Delta theta l(I) in the lth angle variable for motion round a phase-space torus with actions I=(Il); the connection is Delta theta l=- delta gamma / delta nl. Two applications are worked out in detail: the generalised harmonic oscillator, with quadratic Hamiltonian whose parameters are the coefficients of q2, qp and p2; and the rotated rotator, consisting of a particle sliding freely round a non-circular hoop slowly turned round once in its own plane.

A new method for diagonalising large matrices

Abstract: The structure and implementation of a new general iterative method for diagonalising large matrices (the 'residual minimisation/direct inversion in the iterative subspace' method of Bendt and Zunger) are described and contrasted with other more commonly used iterative techniques. The method requires the direct diagonalisation of only a small submatrix, does not require the storage of the large matrix and provides eigensolutions to within a prescribed precision in a rapidly convergent iterative procedure. Numerical results for two rather different matrices (a real 50*50 non-diagonally dominant matrix and a complex Hermitian 181*181 matrix corresponding to the pseudopotential band structure of a semiconductor in a plane wave basis set) are used to compare the new method with the competing methods. the new method converges quickly and should be the most efficient for very large matrices in terms both of computation time and central storage requirements; it is quite insensitive to the properties of the matrices used. This technique makes possible efficient solution of a variety of quantum mechanical matrix problems where large basis set expansions are required.

Intermediate Hamiltonians as a new class of effective Hamiltonians

Abstract: The theory of effective Hamiltonians is well established. However, limitations appear in its applicability for many problems in molecular physics and quantum chemistry. The standard effective Hamiltonians may become strongly non-Hermitian when there is a large coupling between the model space, in which they are defined, and the outer space Moreover, in the presence of intruder states, discontinuities appear in the matrix elements of these effective Hamiltonians as a function of the internuclear distances. To solve these difficulties, a new class of effective Hamiltonians (called intermediate Hamiltonians) is presented; only one part of their roots are exact eigen-energies of the full Hamiltonian. The theory of these intermediate Hamiltonians is presented by means of a new wave-operator R which is the analogue of the wave-operator Omega in the theory of effective Hamiltonians. Solutions are obtained by a generalised degenerate perturbation theory (GDPT) and by iterative procedures. Two model systems are numerically solved which demonstrate the good convergence properties of GDPT with respect to standard degenerate perturbation theory (DPT). Continuity of the solutions is also checked in the presence of an intruder state.

The kinetics of cluster fragmentation and depolymerisation

Abstract: The equation that describes fragmentation kinetics, such as that which occurs in cluster breakup and polymer chain degradation (depolymerisation), is solved for models where the rate of breakup depends upon the size of the object breaking up. The resulting dynamic scaling behaviour is investigated. Both discrete and continuous models are considered.

The electrostatic interaction between interfacial colloidal particles

Abstract: The electrostatic interaction between charged, colloidal particles trapped at an air-water interface is considered using linearised Poisson-Boltzmann results for point particles. In addition to the expected screened-Coulomb contribution, which decays exponentially, an algebraic dipole-dipole interaction occurs that may account for long-range interactions in interfacial colloidal systems.

Epidemic models and percolation

Abstract: The authors argue that local epidemic models with immunisation are in the same universality class as percolation cluster growth models, and show that the static exponents are equal to all orders in the epsilon expansion. They calculate the dynamic exponent nu t=1+1/28 epsilon +O( epsilon 2) and relate this to other exponents involving the chemical distance.

Supersymmetry, factorisation of the Schrodinger equation and a Hamiltonian hierarchy

Abstract: 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.

Computer simulation of chemically limited aggregation

Abstract: Cluster-cluster aggregation is studied via a computer simulation in the chemically limited regime. True polydisperse flocculation is compared with idealised monodisperse aggregation in two and three space dimensions in terms of fractal dimensions and the exponents associated with the scaling of the reaction rates with cluster size.

Supersymmetric quantum mechanics and the inverse scattering method

Abstract: The procedures for finding a new potential (1) by eliminating the ground state of a given potential, (2) by adding a bound state below the ground state of a given potential and (3) by generating the phase equivalent family of a given potential using the supersymmetric pairing of the spectra of the operators A+A- and A-A+ are compared with the application of the Gelfand-Levitan procedure (1955) for the corresponding cases. It is shown how the equivalence of the two procedures may be established. A distinction is made between the modifications of the Jost functions associated with four different types of transformations generated by the concept of a supersymmetric partner to a given Schrodinger equation. It is shown that the Bargmann class of potentials may be generated using suitable combinations of the four types of transformations.

Scaling properties of the surface of the Eden model in d=2, 3, 4

Abstract: The surface of the Eden model is investigated numerically by finite-size scaling, using a strip geometry. Three different versions are studied and it shows that the one mostly used previously exhibits strong finite-size corrections. In two dimensions it is found that the surface thickness simply grows as the square root of the width of the strip for infinitely long strips. This result, as well as the results in d=3 and d=4, suggest that the surface of the Eden shares some properties with the equilibrium models used to describe the roughening transition. Moreover, it is found that the surface thickness scales differently with the height of the cluster for infinitely large strips.

Computational complexity of the ground-state determination of atomic clusters

Abstract: The authors prove that determining the ground state of a cluster of identical atoms, interacting under two-body central forces, belongs to the class of NP-hard problems. This means that as yet no polynomial time algorithm solving this problem is known and, moreover, that it is very unlikely that such an algorithm exists. It also suggests the need for good heuristics.

New lattice model for interacting, avoiding polymers with controlled length distribution

Abstract: A new lattice spin model for many self-avoiding polymers is introduced in which the chain length distribution is fully controllable with a single generating ('magnetic') field. The model utilises spins with additional internal symmetry degrees of freedom to impose a causal connectivity of the polymer bonds on the lattice. Use of the method of random fields then produces an equivalent n to 0 limit field theory. The Flory-Huggins theory for a polymer solution emerges simply from this field theory in the mean field approximation. Polymer-polymer interactions between polymer segments on nearest-neighbour lattice are introduced into the field theory, and the low polymer volume fraction limit of the theory reduces to the Edwards type field theory for dilute through semidilute polymer solutions. A sketch is provided towards the treatment of branched polymers with fully controllable chain and branch length distributions and branching probabilities as well as a kinetic polymerisation system governed by specified propagation and termination probabilities.

Characterisation of intermittency in chaotic systems

Abstract: The authors discuss the characterisation of intermittency in chaotic dynamical systems by means of the time fluctuations of the response to a slight perturbation on the trajectory. A set of exponents is introduced which generalise the maximum Lyapunov characteristic exponent. The link with the statistical mechanics formalism is emphasised and they show that the exponents are connected to a free energy formally defined for chaotic systems. They perform some analytical computations in simple cases and give a few numerical examples.

Ultradiffusion: the relaxation of hierarchical systems

Abstract: The authors show that the dynamics of relaxation in hierarchical structures leads to an anomalous decay process which they term ultradiffusion. Using renormalisation group techniques, they explicitly calculate the behaviour of the autocorrelation function in a one-dimensional hierarchical system and show its universal power law decay as a function of temperature. The authors also demonstrate analytically the emergence of a hierarchy of time scales. This phenomenon is relevant to systems ranging from macromolecules to computing structures.

Critique of the replica trick

Abstract: It is shown that the replica trick fails to give the correct non-perturbative result for the two-point function S2 of the Gaussian unitary ensemble of N*N random matrices. The failure arises from an incorrect description of the symmetries of the random-matrix system in the limit N to infinity . The correct description, which involves integration over both non-compact and compact degrees of freedom, is obtained by using the method of superfields. Some implications for the localisation transition in disordered electronic systems and the theory of the quantised Hall effect are suggested.

On the distribution of a random variable occurring in 1D disordered systems

Abstract: The authors consider the random variable: z=1+x1+x1x2+x1x2x3+ . . . , where the xi are independent, identically distributed variables. They derive some asymptotic properties of the distribution of z, which are related e.g. to the low-temperature behaviour of the random field Ising chain. For a special class of distributions of the xi, exact solutions are presented. They also study the cases where the distribution function of z exhibits a power-law fall-off modulated by a 'periodic critical amplitude'.

Classical billiards in magnetic fields

Abstract: A particle moves in circular arcs with Larmor radius R between specular reflections at the smooth convex boundary of a planar region. The dynamics depends on the value of R in relation to the extreme curvature radii rho min and rho max and the radius R* of the largest circle that can be inscribed in the boundary. For R

The Hahn and Meixner polynomials of an imaginary argument and some of their applications

Abstract: The Hahn and Meixner polynomials belonging to the classical orthogonal polynomials of a discrete variable are analytically continued in the complex plane both in variable and parameter. This leads to the origination of two systems of real polynomials orthogonal with respect to a continuous measure. The Meixner polynomials of an imaginary argument obtained in this manner turned out to be known in the literature as the Pollaczek polynomials. The orthogonality relation for the Hahn polynomials with respect to a continuous measure is apparently new. A close connection between the Hahn polynomials of an imaginary argument and representations of the Lorentz group SO(3,1) is considered.

Rod to coil transitions in nematic polymers

Abstract: The order parameter and anisotropy (elongation) of the configurations of a nematic polymer in the nematic phase are calculated At low temperatures exponentially rapid growth of chain dimensions as a function of inverse temperature is found In the nematic direction the chain eventually adopts a rod-like state The thermal activation of hairpins (abrupt reversals in chain directions) causes this behaviour However, at even lower temperatures the deviation from rod-like alignment is governed by gentle meandering away from the mean field direction A Maier-Saupe mean field theory of the nematic-isotropic transition is constructed calculating for long chains the transition temperature as a function of chain properties and predicting a universal jump in the order parameter at the transition, Delta Sni=1/3

A simple derivation of quasi-crystalline spectra

Abstract: A quasi-crystalline solid is an aperiodic array of atoms which gives rise to Bragg-like diffraction peaks. Using the projection method for generating such structures, the authors provide a simple derivation of this paradoxical behaviour.

Application of a reciprocal transformation to a two-phase Stefan problem

Abstract: A reciprocal transformation is employed to reduce a two-phase Stefan problem in nonlinear heat conduction to a form which admits a class of exact solutions analogous to the classical Newmann solution.

Monte Carlo approach to dendritic growth

Abstract: The connection between diffusion-limited aggregation and the equations of dendritic growth is critically examined. A different type of Monte Carlo simulation is proposed and used to construct two-dimensional dendrite-like patterns. The wavelength occurring for short times is in good agreement with the linear stability analysis. The time dependence of the characteristic wavelengths is also determined.

Supersymmetry and the Dirac equation for a central Coulomb field

Abstract: It is shown that the methods of supersymmetric quantum mechanics can be used to obtain the complete energy spectrum and eigenfunctions of the Dirac equation for an attractive Coulomb potential.

Self-avoiding walks in wedges

Abstract: The authors consider the number of self-avoiding walks confined to a subset Zd(f) of the d-dimensional hypercubic lattice Zd, such that the coordinates (x1,x2, . . .,xd) of each vertex in the walk satisfy x1>or=0 and 0

Continuous Hahn polynomials

Abstract: A slightly more general orthogonality relation for the Hahn polynomials of a continuous variable than the recent one given by Atakishiyev and Suslov, (1985), is given here.

Polymer conformations through 'wiggling'

Abstract: A new Monte Carlo method is proposed which allows for the efficient generation of equilibrium conformations of polymer chains in two and three dimensions. The method treats each site (monomer) as a potential pivot around which a new conformation may be generated by rotating a portion of the chain. The method does not suffer from the severe attrition associated with the simple sampling of self-avoiding walks and may be extended to treat the interacting polymer chain. The authors find in two dimensions that nu =0.748+or-0.005 (exact=0.750) and in three dimensions nu =0.595+or-0.005 (series expansion and renormalisation group predict nu approximately 0.588). The end-end distances calculated for shorter chains are in good agreement with the exact values from enumeration techniques.

An improved calculation for screened Coulomb potentials in Rayleigh-Schrodinger perturbation theory

Abstract: Earlier works on screened Coulomb potentials using Rayleigh-Schrodinger perturbation theory have been re-examined. Instead of working with the usual Hulthen potential as the unperturbed Hamiltonian, the authors propose that a scaled Hulthen potential with modified strength and screening coefficient represents the lowest-order approximation for the static-screened Coulomb and exponential cosine-screened Coulomb potentials. The scale parameter appearing in the new Hulthen potential is then determined from the notion of the virial theorem and intuitive physical arguments. It is found that the accuracy of the predicted energy eigenvalues for the bound s states improves significantly even when the screening parameter is large and quite close to its critical values for which the quantum state becomes just bound. In spite of the simplicity of the approach, the numerical results compare fairly well with those obtained from rigorous analytic approximation methods.