Showing papers in "Journal of Physics A in 1981"

The mechanism of stochastic resonance

Abstract: It is shown that a dynamical system subject to both periodic forcing and random perturbation may show a resonance (peak in the power spectrum) which is absent when either the forcing or the perturbation is absent.

Geometrical description of quantal state determination

Abstract: Under the assumption that every quantal measurement may give data about the post-measurement state of the inspected ensemble, the problem of the state determination is reconsidered. It is shown that orthogonal decomposition of the set of complex, n*n, Hermitian matrices into the commutative subsets allows operators to be found such that post-measurement information on these observables allows a partial (in some cases total) determination of the pre-measurement state to be effected.

Symmetries and differential equations

Abstract: The knowledge of the maximal Lie group or abstract monoid of symmetries of an ordinary non-singular differential equation (or system of equations) allows us to obtain solutions of them. Traditional similarity analysis of point transformations is extended to non-point transformations (inclusion of derivatives), giving analytic expressions for solutions, where previously only numerical methods were used. Examples are given, and the didactic aspect is emphasised.

Singularities in the kinetics of coagulation processes

Abstract: The authors consider a system of substances Ai reacting according to the following scheme: Ak+Al-RKl to k+l, (Rkl=Rlk>or=0). (The reaction is taken as irreversible.) They discuss the existence of global solutions of the kinetic equations derived for the concentrations. It is shown that one cannot expect the total number of monomers to remain constant. Rather, it can decrease as the result of the formation of infinite clusters (gelation). With this restriction, they obtain that a physically reasonable global solution exists if Rkl

Continuum percolation in two dimensions: Monte Carlo tests of scaling and universality for non-interacting discs

Abstract: Detailed results are reported for the connectivity properties of a system of discs of unit radius free to be situated anywhere within a square of area 2L2. Ordinary lattke percolation would correspond to the discs being situated on the vertices of a J2L x J2L lattice. Computer simulations are carried out for a sequence of increasing system sizes ranging from L = 20 to L = 1000; for each value of L a large number of realisations are generated for 25 values of the disc concentration x. We calculate a variety of estimates for the threshold parameter x,, as well as the critical exponents p, y, T and v. Our exponent estimates are in close agreement with accepted values for ordinary lattice percolation; therefore, this continuum system appears to be in the same 'universality class' as lattice percolation.

Nonlinear differential difference equations as Backlund transformations

Abstract: Shows that the best known nonlinear differential difference equations associated with the discrete Schrodinger spectral problem and also with the discrete Zakharov-Shabat spectral problem can be interpreted as Backlund transformations for some continuous nonlinear evolution equations.

Order propagation near the percolation threshold

Abstract: A new Monte Carlo method for studying bond percolation clusters is developed and used to identify new critical quantities associated with the percolation threshold. The bonds in each cluster are partitioned into three distinct connectivity classes, 'red' (singly connected backbone bonds), 'blue' (multiply connected backbone bonds) and 'yellow' (non-backbone bonds, often called dangling ends). Among the new cluster properties studied are the mean number of red bonds, a critical quantity diverging at p_c with exponent y_R approximately ≃ 1, and the length of the shortest connected path through the cluster which is critical with exponent y_(min)=1.35 ± 0.02. For all cluster properties studies, the authors also compute averages over only the largest clusters; the corresponding critical exponents are found to be significantly different from those obtained by averaging over clusters of all sizes.

On the differential geometry of the Euler-Lagrange equations, and the inverse problem of Lagrangian dynamics

Abstract: The conditions for a system of second-order differential equations to be derivable from a Lagrangian-the conditions of self-adjointness, in the terminology of Santilli (1978) and others-are related, in the time-independent case, to the differential geometry of the tangent bundle of configuration space. These conditions are simply expressed in terms of the horizontal distribution which is associated with any vector field representing a system of second-order differential equations. Necessary and sufficient conditions for such a vector field to be derivable from a Lagrangian may be stated as the existence of a two-form with certain properties: it is interesting that it is a deduction, not an assumption, that this two-form is closed and thus defines a symplectic structure. Some other differential geometric properties of Euler-Lagrange second-order differential equations are described.

A simple nonlinear dissipative quantum evolution equation

Abstract: The author has considered a nonlinear dissipative evolution equation that generalises the Schrodinger equation. In the corresponding evolution all the stationary states of the usual Schrodinger equation have a behaviour of semistable limit cycles, except the ground state which is stable. The model is applied to the spin-1/2 and to the damped harmonic oscillator. For the latter it is shown that the coherent states remain coherent and evolve as in the corresponding classical problem.

Renormalised perturbation series

Abstract: Renormalised perturbation series are obtained directly (by the hypervirial method) for the perturbed oscillator and hydrogen atom, and indirectly (by series transformation) for the helium atom.

Critical exponents for the percolation problem and the Yang-Lee edge singularity

Abstract: Gives details of a calculation of critical exponents for a class of field theory models that have an interaction cubic in the fields. The results have already been reported, give the exponents to third order in epsilon where epsilon =6-d and d is the dimensionality of space. The class of models includes the percolation problem and the Yang-Lee edge singularity, so the authors give explicit results for the exponents to order epsilon 3 in these cases. By resummation methods, based on the symptotic behaviour of the epsilon expansion, they obtain numerical estimates for these exponents for a number of interesting values of d.

A gravitational analogue of the Aharonov-Bohm effect

Abstract: It is shown that particles constrained to move in a region where the Riemann tensor vanishes may nonetheless exhibit physical effects arising from nonzero curvature in a region from which they are excluded. This is a gravitational analogue of the Aharonov-Bohm effect.

A quantum Hamiltonian approach to the two-dimensional axial next-nearest-neighbour Ising model

Abstract: A quantum Hamiltonian analogue of the two-dimensional axial next-nearest-neighbour Ising (ANNNI) model is presented. The phase diagram is investigated by the analysis of perturbation series and by finite-lattice methods.

Observation of a long-time tail in Brownian motion

Abstract: By photon correlation dynamic laser light scattering the authors have measured the time dependence of the mean-square displacement ( Delta x2(t)) of spherical particles (radius approximately 1.7 mu m) in Brownian motion. Clear evidence was found for the existence of a t1/2 term in ( Delta x2(t)) which corresponds to the expected t-3/2 'long-time tail' in the particle velocity autocorrelation function. The experimentally determined amplitude of the t1/2 term was about 74+or-3% of the value predicted theoretically. Despite detailed consideration of possible systematic errors the authors were unable to explain the magnitude of this disagreement.

Non-linear dynamics of the fermion-boson model: interference between revivals and the transition to irregularity

Abstract: Extends studies of the quantised fermion-boson model. Parallel numeric and approximate analytic advances allow greater understanding of the 'collapse' regions between successive 'revivals' in the fermion energy evolution. The authors give an estimate of the time for the onset of continuous irregularity as a function of the average boson number.

Finite-lattice extrapolations for Z3 and Zs models

Abstract: A method is presented to accelerate the convergence of finite-lattice sequences to their bulk limit. The calculation of highly accurate estimates of the critical parameters of the bulk system is then possible. Applied to the Hamiltonian version of the Z3 model (three-state Potts model) in (1+1) dimensions, these techniques yield estimates for the exponents gamma =1.444+or-0.0001, nu =0.8333+or-0.0003 and alpha =0.33+or-0.01. For the Z5 model, the presence of a Kosterlitz-Thouless transition is confirmed.

Directed percolation: a finite-size renormalisation group approach

Abstract: The finite-size renormalisation group technique introduced by Nightingale (1976) is applied to the directed percolation problem. The decay of correlations is anisotropic in this model and finite-size scaling is extended to treat such anisotropy. Precise estimates for critical exponents and percolation probabilities are obtained for site, bond and site-bond percolation on the square lattice with bonds directed along the positive axes. Both free boundary conditions for which the results converge linearly with 1/n as n to infinity , and helical boundary conditions, for which, unexpectedly, the results converge linearly with 1/n3, are considered.

One-dimensional models with 1/r2 interactions

Abstract: The critical behaviour of a general discrete one-dimensional model with inverse square interactions is discussed, using renormalisation group methods.

Phenomenological renormalisation of the self avoiding walk in two dimensions

Abstract: Phenomenological renormalisation is used to calculate the exponent v and the connective constant of the self-avoiding walk problem on a square lattice. A transfer matrix technique is developed for the polymer problem. The results indicate that Flory's value v = 0.75 is true in two dimensions to extremely high accuracy.

Localisation-delocalisation transition in a solid-on-solid model with a pinning potential

Abstract: The influence of pinning forces on domain-wall fluctuations is studied in a continuous planar solid-on-solid model with a one-dimensional interface. The system is simple enough so that exact results can be obtained for a variety of pinning forces. The pinning of the interface is formally equivalent to the binding of a quantum mechanical particle in a temperature-dependent effective potential. In the case of a short-range pinning force applied a finite distance from the edge of the system, there is a localisation-delocalisation transition at a finite temperature. The transition is qualitatively similar to that studied in the d=2 Ising model by O.B. Abraham (see Phys. Rev. Lett., vol.44, p.1165-8, 1980).

Comparison of classical and quantum spectra for a totally bound potential

Abstract: The quantal energy spectrum is compared with the classical motion for the totally bound potential 1/2(x2+y2)+ax2y2. The classical phase space is filled with regular trajectories at lower energies, but as the energy is increased both regular and irregular trajectories are observed to coexist. At very high energies the classical phase space is almost totally filled with irregular trajectories. The work reported here is similar to that performed by the authors on the Henon-Heiles potential with the purpose of testing the prediction by Percival (1973) that there is good agreement between the amount of classical irregular motion and the portion of energy eigenvalues sensitive to small changes in the perturbation parameter. However, the potential investigated has several computational advantages over the Henon-Heiles potential as well as avoiding complications due to quantum mechanical tunnelling. The results show good agreement with Percival's predictions.

Diagram and superfield techniques in the classical superalgebras

Abstract: Introduces the concept of 'graded permutation group' in the analysis of tensor operators in the classical superalgebras. For U(m/n) and SU(m/n), irreducible tensor representations correspond to classes of Young tableaux with definite graded symmetry type. Diagram techniques are given for Kronecker products, dimensions, and branching rules. The tensor techniques are complemented by the introduction of a superfluid formalism, in which U(m/n) and SU(m/n) act on (polynomial) functions over the appropriate superspace. Such superfluids may admit constraints. A general superfield interpolates between the classes of Young tableaux which correspond to particular types of constraint. The tensor and superfield techniques are illustrated with case studies of SU(2/1) and SU(m/1).

A method of integration over matrix variables: II

Abstract: The integral over twon ×n hermitan matricesZ(g, c)=∫dAdBexp{−tr[A2+B2−2cAB+g/n(A4+B4)]} is evaluated in the limit of largen For this purpose use is made of the theory of diffusion equation and that of orthogonal polynomials with a non-local weight The above integral arises in the study of the planar approximation to quantum field theory

On the Lie symmetries of the classical Kepler problem

Abstract: The standard version of Noether's theorem, when applied to the classical Kepler problem, leads to the constants of energy and angular momentum, but does not give the 'hidden symmetry' known as the Runge-Lenz vector. Lie's theory of differential equations is used to obtain all three constants of motion. The transformations of solutions under the point transformations to which these constants correspond are studied. The results are generalised to n dimensions.

Exact spatially inhomogeneous cosmologies

Abstract: Gives an overview of the derivation and properties of exact solutions of the Einstein field equations which are spatially inhomogeneous with the source assumed to be an irrotational perfect fluid. It is shown that the known such spatially inhomogeneous solutions either admit a group of isometries with 2D orbits or are algebraically special. The solutions are related to a classification scheme which is based on the intrinsic and extrinsic geometry of the hypersurfaces orthogonal to the fluid flow.

Position-space renormalisation group for isolated polymer chains?

Abstract: We develop a cell position-space renormalisation group (PSRG) with which we study the scaling properties of isolated polymer chains. We model a chain by a self-avoiding walk constrained to a lattice. For rescaling factors b S 6, we calculate recursion relations analytically on the square lattice with several different choices for the PSRG weight function. We also calculate implicit cell-to-cell transformations in which a cell of size b is rescaled to a cell of size b'. The results of these PSRGS improve both as b increases and as b/b'+ 1. In addition, we construct a true infinitesimal PSRG transformation, which appears to become exact as the dimensionality d approaches 1; we obtain the closed-form expression for the correlation length exponent, U = (d - l)/(d In d). The Flory formula deviates from this already at first order in (d - 1). We also develop a constant-fugacity Monte Carlo method which enables us to simu- late-in an unbiased way within the grand canonical ensemble-chains of up to lo3 bonds. With this method, we extend the PSRG to larger cells (b s 150) on the square lattice. Our numerical method provides high statistical accuracy for all cell sizes. However, in the range of b we study, the asymptotic behaviour of our results appears to depend on the choice of weight function. One weight function provides smooth behaviour as a function of b, and with it we extrapolate to find U = 0.756* 0.004. Further work is required to resolve the apparent anomalies in the results based on the other weight functions.

The square-lattice Ising model with first and second neighbour interactions

Abstract: The critical behaviour of the square-lattice Ising model, with nearest and next-nearest neighbour interactions of either sign, has been investigated by means of high-temperature series. The location of the critical lines in the coupling constant plane has been accurately determined. Along the critical line which corresponds to transitions to the layered or superantiferromagnetic state a breakdown of universality is observed and explicit numerical estimates obtained for the exponent of the ordering susceptibility.

Hydrogen atom in a strong magnetic field: on the existence of the third integral of motion

Abstract: The problem of the existence of the third integral of motion for the classical Hamiltonian describing a hydrogen atom in a magnetic field is studied by numerical methods. It is found that the third integral is isolating for all initial conditions for which the energy is lower than a critical energy, beyond which the phase orbits are unstable and the Hamilton system can behave stochastically. This critical energy depends upon the strength of the magnetic field and the value of the z component of the angular momentum. The critical energy approaches the (classical) ionisation energy in the weak-field and strong-field limits, while it is lowest in the transition region. The consequences for the quantum mechanical energy spectrum of the hydrogen atom are discussed: the existence of this approximate dynamical symmetry would allow for close anti-crossings of levels, and might facilitate the analytic calculations of the energy levels below the critical energy. In discussion of the correspondence diagram a criticism of an earlier paper is given.

Harmonic oscillator with exponentially decaying mass

Abstract: The problem of a harmonic oscillator with varying mass parameter is reduced by canonical transformation to the corresponding constant mass problem and is solved in the case of an exponentially decaying mass. The constructed canonical Hamiltonian has time-independent eigenvalues and eigenvectors. The cases of undercritical and overcritical damping are considered in detail. The Green function is calculated and the behaviour of coherent states is discussed. The theory is related to the case of a cavity oscillator with a decaying field as in threshold laser operation. In particular, the energy of the field is considered.

Monte Carlo renormalisation-group approach to percolation on a continuum: test of universality

Abstract: It is shown that a Monte Carlo renormalisation-group technique can be employed for direct renormalisation of a system on a continuum, i.e. without restriction to a periodic lattice. For the problem of overlapping discs the authors find the critical area fraction sc=0.688+or-0.005 and the correlation-length critical exponent nu =1.33+or-0.07. The latter result indicates the regular-irregular lattice universality.