Showing papers in "Journal of Physics A in 1979"

The quantum open system as a model of the heat engine

Abstract: The quantum open system weakly coupled to thermal reservoirs at different temperatures and under the influence of slowly varying external conditions is studied. The famous Carnot inequality for the efficiency of any heat engine is obtained.

Long-range correlations in smoke-particle aggregates

Abstract: Ultrafine smoke particles stick together to form chain-like aggregates. We find that the particle density has long-range correlations of the same form in iron, zinc or silicon dioxide aggregates. The correlation data suggest a power-law spatial dependence giving a Hausdorff dimension between 1.7 and 1.9. We discuss the consistency of these results with a model based on percolation. We also compare our results with a random-walk model, which has a nominal Hausdorff dimension of 2.

Dynamical symmetries in a spherical geometry. I

Abstract: The two potentials for which a particle moving non-relativistically in a spherical space under the action of conservative central force executes closed orbits are found. When the curvature is zero they reduce to the familiar Coulomb and isotropic oscillator potentials of Euclidean geometry. The corresponding vector (for the former) and symmetric tensor (for the latter) constants of motion are constructed. For each system in N dimensions the Poisson bracket algebra in classical mechanics, and the commutator algebra in quantum mechanics, of these constants of motion and the angular momentum components are constructed. It is proved that by an appropriate choice of independent constants of motion these Poisson bracket algebras may be transformed into Lie algebraic structures, those of the symmetry groups SO(N+1) and SU(N) respectively: the Hamiltonian of each system is expressed as a function of the Casimir operators of its symmetry group. The corresponding transformations of the quantum mechanical commutator algebras are performed only for N=2: the corresponding expressions for the Hamiltonian as functions of the Casimir operators yield the energy levels of the two systems.

A relation between the temperature exponents of the eight-vertex and q-state Potts model

Abstract: The mapping of the critical points in the q-state model for q

A spectral entropy method for distinguishing regular and irregular motion of Hamiltonian systems

Abstract: Regular and irregular motions of bounded conservative Hamiltonian systems of N degrees of freedom can be distinguished by the structure of the frequency spectrum of a single trajectory. The spectral entropy S is introduced which provides a measure of the distribution of the frequency components. Numerical calculations on the model Henon and Heiles system and a realistic molecular model are performed. Power spectra are obtained from numerical solutions to Hamilton's equations using fast Fourier transforms and the Hanning method. For regular trajectories S is found to stabilise after a finite time of integration, while for irregular cases S increases erratically. Estimates of the relative volume of regular regions of phase space as a function of energy are given for the two systems.

Dynamical symmetries and conserved quantities

Abstract: The invariance properties of second order dynamical systems under velocity dependent transformations of the coordinates and time are studied. For Lagrangian systems the connection between Noether conserved quantities and dynamical symmetries is not too direct; however, the author shows that for general systems dynamical symmetries always possess associated conserved quantities, which are invariants of the symmetry group itself. In the special case of point symmetries this yields the result that the associated conserved quantity is an invariant of the first extended group.

Dynamical symmetries in a spherical geometry. II

Abstract: For pt.I see ibid., vol.12 (1979). The quantum mechanical Coulomb and isotropic oscillator problems in an N-dimensional spherical geometry, which were shown in the previous paper to possess the dynamical symmetry groups SO(N+1) and SU(N) respectively as classical systems, are analysed by the method used by Pauli to find the energy eigenvalues of the hydrogen atom. This analysis is carried through completely for N=3 to obtain energy eigenvalues and recurrence relations among energy eigenfunctions. It is shown that Pauli's method is equivalent to Schrodinger's method of solving the radial Schrodinger equation by factorisation of the second order differential operator. The latter method is used to find the energy eigenvalues in N dimensions, and the corresponding eigenfunctions are obtained in closed form.

Super Weyl transformations in two dimensions

Abstract: A superspace with two commuting and two anti-commuting co-ordinates is discussed with particular emphasis on its superconformal properties. A complete expansion of the supervierbein is given and the local supersymmetry transformations of the component fields derived. Super Weyl transformations are defined and it is shown that (2+2)-dimensional superspace is superconformally flat. The spinning string is re-examined and the problems of previous approaches resolved.

Evolution of semiclassical quantum states in phase space

Abstract: We derive a semiclassical formula for the Wigner function W(q, p, f) describing the evolution in the two-dimensional phase space qp of a nonstationary quantum state $(q, i) for a system with one degree of freedom The initial state $(q, 0) corresponds to a family of classical orbits represented by a curve V0 in qp Under the classical motion Vo evolves into a curve V,; we show that the region where W is large hugs V, in an adiabatic fashion, and that W has semiclassical oscillations depending only on the geometry of (e, and neighbouring curves As t + CO, V, can get very complicated, and we classify its convolutions as 'whorls' and 'tendrils', associated respectively with stable and unstable classical motion In these circumstances the quantum function W cannot resolve the details of V,, and at time f, there is a transition to new regimes, for which we make predictions about the morphology of $ from the way V, fills regions of phase space as t-r CO The regimes associated with whorls and tendrils are different We expect f, = O(h-2'3) for whorls and I, = O(ln h-') for tendrils

Backlund transformations of axially symmetric stationary gravitational fields

Abstract: A generation theorem for solutions of Einstein's equations is presented. It consists mainly of algebraic steps. With its aid, one obtained from an 'old' solution (e.g. from the Minkowski space) 'new' solutions with an arbitrary number of constants. The method of repeated application of potential and coordinate transformations considered by Geroch (1972) and Kinnerley (1977) is included.

A polychromatic correlated-site percolation problem with possible relevance to the unusual behaviour of supercooled H2O and D2O

Abstract: Introduces a new polychromatic correlated site percolation problem, which has the novel feature that the partitioning of the sites into different species arises from a purely random process-that of random bond occupancy. A particular case of this percolation problem is shown to be of possible relevance in providing a physical mechanism which may contribute to the unusual properties displayed by liquid H2O and D2O under conditions of supercooling below the melting temperature.

Universal metric properties of bifurcations of endomorphisms

Abstract: Endomorphisms of the real axis with one extremum have some universal metric properties which depend only on their analytic dependence near the extremum (bifurcation velocity, reduction parameter). It is shown how this problem is similar to the renormalisation problem, and how the bifurcation velocity may be derived from a fixed-point theory.

Thermodynamics of the conversion of diluted radiation

Abstract: The photon density in diluted black-body radiation is epsilon (0< epsilon <1) times that for the black-body radiation at temperature T from which it originated. If sigma is Stefan's constant and B is a geometrical factor, it is shown that the energy and entropy flux due to such radiation is Phi =B epsilon sigma T4/ pi Psi =4/3B epsilon X( epsilon ) sigma T3/ pi (X(1)=1) where X( epsilon ) is a function calculated here for the first time. A special type of steady-state non-equilibrium situation is defined, and called effective equilibrium, for which the effective temperatures T/X( epsilon ) identical to T* of the various components of a system are equal. In this state the system cannot yield work. The maximum efficiency eta 0 of such systems is investigated. The application to solar radiation (diffuse and direct) proves possible and involves the function lambda (x)=1-4/3x+1/3x4. In order to allow for diffuse and direct radiation the calculation is somewhat more complicated than previous ones. It shows that, for a black absorber, eta 0 approximately 0.7 (diffuse) rises to 0.93 as the radiation becomes more direct. However, for a grey absorber the efficiency might range typically from 60% to 83% for absorptivity alpha =0.9. For one pump p and a black absorber at ambient temperature T, eta 0= lambda (T/Tp*).

The replica method and solvable spin glass model

Abstract: The replica method for random systems is critically examined, with particular emphasis on its application to the Sherrington-Kirkpatrick solution of a 'solvable' spin glass model. The procedure is improved and extended in several ways, including the avoidance of steepest descents and a reformulation which isolates the thermodynamic limit N to infinity . Ideas of analyticity and convexity are employed to investigate the two most dubious steps in the replica method: the extension from an integer number (n) of replicas to real n in the limit n to 0, and the reversal of the limits in n and N. The latter step is proved valid for the Sherrington-Kirkpatrick problem, while the non-uniqueness of the former is held responsible for the unphysical behaviour of the result.

Exact multi-soliton solution of the Benjamin-Ono equation

Abstract: An exact multi-soliton of the Benjamin-Ono equation is presented. The asymptotic form of the solution for large time is also given.

A variational principle for invariant tori of fixed frequency

Abstract: A fixed frequency Lagrangian variational principle is formulated for the invariant tori of conservative dynamical systems. It avoids the singularities due to small frequency divisors, and for pure rotation provides a strict bound which can be used as a basis for an effective variational method.

de Sitter gauge invariance and the geometry of the Einstein-Cartan theory

Abstract: A formulation of general relativity as a gauge theory of the de Sitter group SO(3,2) is used to analyse the geometrical structure of the Einstein-Cartan theory. The SO(3,2) symmetry must be spontaneously broken to the Lorentz group in order to reproduce the usual four-dimensional geometry of gravity. Special emphasis is placed upon the role of the Goldstone field of the symmetry breaking mechanism and also that of the original SO(3,2) gauge fields. The latter are not directly identified with the gravitational vierbein and spin connection, but instead generate a kind of parallel transport known as development which is the necessary construction to interpret the effects of space-time torsion and curvature.

Monte Carlo experiments on cluster size distribution in percolation

Abstract: Cluster statistics in two- and three-dimensional site percolation problems are derived here by Monte Carlo methods. The average number n, of percolation clusters with s occupied sites each is calculated by up to 19 runs on a 4000 X 4000 triangular lattice near pc. Our data support the two-exponent scaling assumption n, as-'f(z'), where z'= ( p/pc - 1)s". At the percolation threshold p = pc we find for s up to lo6 a rough agreement with the expected power law n, as-' over 12 decades in n, ; we can approximate the leading correction term near 5-10' by n,cXs-'(l-1.2 s-~'~). If the ratio U, = n,(p)/n,(p,) is plotted against z', then all data follow the same curve U, = f(z') for different p. This scaling function f(z') has a finite slope at z' = 0, has a maximum f(zk, = -0.8) = 5 for p below pc, and decays rapidly for z'+*m. For 5-+m at fixed p this rapid decay corresponds to In n, Cc -s"' above pc and In n, a --s below pc. Apart from finite-size corrections we find the second moment x =Zs2n, diverges as 1 p -pC(-', with y = 2.4, on both sides of the phase transition; the amplitude ratio x(p p,) is about 200. The fraction of occupied sites belonging to the infinite cluster vanishes as (p -p,)'. with p -0.13. In three dimen- sions using system sizes up to 400 x 400 x 400 the two-exponent scaling function is also supported, with the same universal function f(z') valid for both the simple cubic and BCC lattices. f(z') has a maximum f(z;, = -0.8) = 1.6. The amplitude ratio is approximately 11. Our conclusions are in general consistent with but more complete than other recent Monte Carlo work by Stoll and Domb, Leath and Reich, and Nakanishi and Stanley.

Consistency of spin-1 theories in external electromagnetic fields

Abstract: It is known that while spin-1 (Proca) theory with minimal electromagnetic interaction is marred by the Corben-Schwinger anomaly, the inclusion of specific types of 'anomalous' interactions lead to other difficulties such as non-causality of propagation, and the energy spectrum (in homogeneous magnetic fields of sufficient strength) becoming partially non-real The authors investigate whether, by making the anomalous terms (introduced into the Duffin-Kemmer-Petiau equation) sufficiently general, it is possible to escape these inconsistencies The answer, unfortunately, turns out to be in the negative They comment briefly on other formulations for spin-1

Dimensions of irreducible representations of the classical Lie groups

Abstract: Formulae are derived expressing the dimensions of each irreducible representation of each of the classical Lie groups of transformations in an N-dimensional space as a factored polynomial in N divided by a product of hook length factors.

On renormalisation of the quantum stress tensor in curved space-time by dimensional regularisation

Abstract: Using dimensional regularisation, a prescription is given for obtaining a finite renormalised stress tensor in curved space-time. Renormalisation is carried out by renormalising coupling constants in the n-dimensional Einstein equation generalised to include tensors which are fourth order in derivatives of the metric. Except for the special case of a massless conformal field in a conformally flat space-time, this procedure is not unique. There exists an infinite one-parameter family of renormalisation ansatze differing from each other in the finite renormalisation that takes place. Nevertheless, the renormalised stress tensor for a conformally invariant field theory acquires a nonzero trace which is independent of the renormalisation ansatz used and which has a value in agreement with that obtained by other methods. A comparison is made with some earlier work using dimensional regularisation which is shown to be in error.

On a Lagrangian for non-minimally coupled gravitational and electromagnetic fields

Abstract: An arbitrarily chosen Lagrangian L for non-minimally coupled gravitational and electromagnetic fields will usually lead to higher-order field equations, in the sense that the functional derivatives of L with respect to the gravitational potential gij and the electromagnetic potential phi i will involve at least the third, instead of merely the second, derivatives of these quantities. By temporarily contemplating a five-dimensional formalism this paper uncovers an exceptional case in which one is led to second-order equations. The result obtained is in agreement with the conclusions reached by Horndeski (1976) by quite different means.

Large-order behaviour of the 1/N expansion in zero and one dimensions

Abstract: The large-order behaviour of the 1/N expansion in the zero- and one-dimensional g phi 4 model is investigated. The one-dimensional N-vector model is also considered. The asymptotic behaviour shows oscillations due to complex instantons. The phase and the amplitude of these leading behaviours are determined theoretically and are compared with the explicit numerical calculations of higher orders. In the one-dimensional g phi 4 model the relevance of the large-N instanton is discussed.

Susceptibility and fourth-field derivative of the spin-1/2 Ising model for T>Tc and d=4

Abstract: THe authors investigate the spin-1/2 Ising model with nearest-neighbour interactions on the four-dimensional simple hypercubic lattice. High-temperature series expansions are studied for the zero-field susceptibility chi 0 and the fourth-field derivative of the free energy Xi 0(2) up to order nu 17. The series are analysed for singularities of the form t-1 mod 1nt mod p where t is the reduced temperature. For chi 0 it is found that p=0.33+or-0.07 when q=1, in good agreement with the prediction p=1/3, q=1 of renormalisation group theory. The critical temperature is estimated to be nu c-1=6.7315+or-0.0015. Results for chi 0(2) are more slowly convergent but are not inconsistent with the renormalisation group prediction p=1/3, q=4.

The k dependence of the long-time diffusion in systems of interacting Brownian particles

Abstract: The short- and long-time diffusion coefficients in suspensions of charged polystyrene spheres were determined by means of photon correlation spectroscopy. Five different concentrations were measured. The results are discussed in a memory function formalism. It is shown that all concentrations and previously published data from different particles fit the same universal function.

Nonlinear excitations in classical ferromagnetic chains

Abstract: General single solitary-wave excitations are determined for the classical continuum Heisenberg chain in the presence of external magnetic and anisotropy fields. These include both domain walls and pure solitons as examples. Conditions for propagation are carefully analysed. The complete integrability of the zero-anisotropy limit is suggested as a basis for (i) controlled singular perturbation theory, and (ii) formulation of classical statistical mechanics in a natural configurational (nonlinear normal mode) representation.

Automorphic field theory-some mathematical issues

Abstract: The general structure of field theories on multiply connected spaces is presented using the universal covering space concept. Restriction to rigid gauge theories allows the use of representation theory to answer several relevant mathematical questions.

A classification scheme for non-rotating inhomogeneous cosmologies

Abstract: Describes a classification scheme for irrotational cosmological models which is not based on the existence of a group of local isometries and hence is suitable for studying inhomogeneous cosmologies. The scheme is based on the algebraic structure of three trace-free symmetric two-index tensors which are defined in such models, namely the shear tensor of the fluid congruence, assumed irrotational, and the trace-free Ricci and Cotton-York tensors associated with the hypersurfaces orthogonal to the fluid. The restrictions that are imposed on these tensors by the existence of various groups of local isometries are derived, thereby relating the present approach to the usual classifications involving Killing vectors. These results lead to the conjecture that the algebraic structure of the Cotton-York tensor (whose vanishing is a necessary and sufficient condition for the hypersurfaces to be conformally flat) is related to the nature of the gravitational waves that might be present.

Small g and large λ solution of the Schrodinger equation for the interaction λx2/(1+gx2)

Abstract: Using perturbation theory, asymptotic expansions are derived for the eigenenergies and eigenfunctions of the wave equation for the interaction lambda x2/(1+gx2) in the range of small values of g and large values of lambda . The first few energy eigenvalues are calculated and found to be comparable with the non-perturbative results obtained by Mitra (1978).

Static fluid cylinders and plane layers in general relativity

Abstract: Static perfect fluid distributions in general relativity which possess cylindrical, toroidal or pseudoplanar symmetry (these symmetries are locally equivalent) are considered. Solutions in quadratures are obtained for fluids with an unspecified equation of state and for rho c2=np, where n is a constant, with or without an electromagnetic field Fmu nu compatible with the symmetry assumed. Moreover, for rho c2=np, Fmu nu identical to 0, solutions are given in an explicit closed form. Some physical properties of the solutions are discussed.