Showing papers in "Journal of Physics A in 1976"
TL;DR: In this article, the possible domain structures which can arise in the universe in a spontaneously broken gauge theory are studied, and it is shown that the formation of domain wall, strings or monopoles depends on the homotopy groups of the manifold of degenerate vacua.
Abstract: The possible domain structures which can arise in the universe in a spontaneously broken gauge theory are studied. It is shown that the formation of domain wall, strings or monopoles depends on the homotopy groups of the manifold of degenerate vacua. The subsequent evolution of these structures is investigated. It is argued that while theories generating domain walls can probably be eliminated (because of their unacceptable gravitational effects), a cosmic network of strings may well have been formed and may have had important cosmological effects.
2,994Â citations
TL;DR: The second quantization method is applied to classical many-particle systems and is especially useful for the system in which the number of the composite molecules changes with time, e.g. the system including chemical reaction.
Abstract: The second quantization method is applied to classical many-particle systems. Statistical quantities such as free energy and time correlation functions are expressed in terms of creation and annihilation operators. The method is especially useful for the system in which the number of the composite molecules changes with time, e.g. the system including chemical reaction.
586Â citations
TL;DR: A stochastic theory of a diffusion-controlled reaction is developed with the emphasis on the many-body aspects which rigorous Stochastic theories inevitably encounter as mentioned in this paper, and the classical Smoluchowski theory is shown to be strictly valid in the long-time scale.
Abstract: A stochastic theory of a diffusion-controlled reaction is developed with the emphasis on the many-body aspects which rigorous stochastic theories inevitably encounter. The field operator method developed in the previous paper (see ibid., vol.9, no.9, p.1465 (1976)) is extensively used in the analysis. The classical Smoluchowski theory is shown to be strictly valid in the long-time scale, and its relation to the Boltzmann equation is discussed.
461Â citations
TL;DR: Tem Temperley and Lieb (Proc. R. Soc., vol.A322, p.251 of 1971) have used operator methods to show that, for a square lattice, this problem is in turn equivalent to a staggered ice-type model as discussed by the authors.
Abstract: The partition function of the Potts model (1952) on any lattice can readily be written as a Whitney polynomial (1932). Temperley and Lieb (Proc. R. Soc., vol.A322, p.251 of 1971) have used operator methods to show that, for a square lattice, this problem is in turn equivalent to a staggered ice-type model. Here the authors rederive this equivalence by a graphical method, which they believe to be simpler, and which applies to any planar lattice. For instance, they also show that the Potts model on the triangular or honeycomb lattice is equivalent to an ice-type model on a Kagome lattice.
230Â citations
TL;DR: In this article, it is shown that supersymmetry may be applied to spin systems and a simple algebra is proposed and various examples of supersymmetric spin systems can be found.
Abstract: It is shown that supersymmetry may be applied to spin systems. A simple algebra is proposed and various examples are discussed. It is argued that certain correlation functions must vanish on account of supersymmetry.
199Â citations
TL;DR: In this article, a Gaussian probability density function with the same mean and variance is used to calculate the eigenvalue spectrum of a large symmetric square matrix, each of whose upper triangular elements is described by the Gaussian distribution.
Abstract: A new and straightforward method is presented for calculating the eigenvalue spectrum of a large symmetric square matrix each of whose upper triangular elements is described by a Gaussian probability density function with the same mean and variance. Using the n to 0 method, the authors derive the semicircular eigenvalue spectrum when the mean of each element is zero and show that there is a critical finite mean value above which a single eigenvalue splits off from the semicircular continuum of eigenvalues.
187Â citations
TL;DR: In this paper, the derivation of low-density series expansions for the mean cluster size in random site and bond mixtures on a two-dimensional lattice is described briefly, and new data are given for the triangular, simple quadratic and honeycomb lattices.
Abstract: The derivation of low-density series expansions for the mean cluster size in random site and bond mixtures on a two-dimensional lattice is described briefly. New data are given for the triangular, simple quadratic and honeycomb lattices.
156Â citations
TL;DR: In this paper, the authors studied the Potts model with a general number of states and showed that the critical exponents can be computed to O((6-d)2) when there is a fixed point.
Abstract: The author studies the Potts model with a general number of states. First, he discusses the situation in which the Landau theory leads to a first order transition, but which does show fixed points of the renormalization group. Here, there are many questions which need further clarification. Then he describes, rather pedagogically, the logic behind the application of dimensional regularization to critical phenomena. He argues that this is a particularly natural approach. This technique is then applied to the Potts model, for which the critical exponents are computed to O((6-d)2), when there is a fixed point. The one state results, which correspond to the percolation problem, are compared with other calculations and with numerical simulation.
122Â citations
TL;DR: In this paper, series data for the mean cluster size for site mixtures on a d-dimensional simple hypercubical lattice are presented and it is concluded that dc=6.
Abstract: Series data for the mean cluster size for site mixtures on a d-dimensional simple hypercubical lattice are presented. Numerical evidence for the existence of a critical dimension for the cluster growth function and for the mean cluster size is examined and it is concluded that dc=6. Exact expansions for the mean number of clusters K(p) and the mean cluster size S(p) in powers of 1/ sigma where sigma =2d-1 and p
106Â citations
TL;DR: In this article, the authors derived low-density series expansions for the mean cluster size in random site and bond mixtures on a three-dimensional lattice, and concluded that the data are reasonably consistent with the hypothesis that the mean clique size S(p) approximately =C(pc-p)- gamma as p to pc-with gamma a dimensional invariant, gamma =1.07 in three dimensions.
Abstract: The derivation of low-density series expansions for the mean cluster size in random site and bond mixtures on a three-dimensional lattice is described briefly. New data are given for the face-centred cubic, body-centred cubic, simple cubic and diamond lattices. The critical concentrations for the site problem is estimated as pc=0.198+or-0.003 (FCC), pc=0.245+or-0.004 (BCC), pc=0.310+or-0.004 (SC), pc=0.428+or-0.004 (D); for the bond problem as pc=0.119+or-0.001 (FCC), pc=0.1785+or-0.002 (BCC), pc=0.247+or-0.003 (SC), pc=0.388+or-0.005 (D). It is concluded that the data are reasonably consistent with the hypothesis that the mean cluster size S(p) approximately=C(pc-p)- gamma as p to pc-with gamma a dimensional invariant, gamma =1.66+or-0.07 in three dimensions. Estimates of the critical amplitude C are also given.
98Â citations
TL;DR: In this paper, the magnetostatic field of a current loop surrounding a black hole is given in integral form using some results of Copson (1928) in the Schwarzschild metric.
Abstract: The electrostatic field of a point charge at rest in the Schwarzschild metric is given in algebraic form using some results of Copson (1928). It is possible also to determine the magnetostatics. As an example, the magnetostatic field of a current loop surrounding a black hole is given in integral form.
TL;DR: In this paper, the mean cluster size for site and bond mixtures in two dimensions was examined and the critical concentration for the site problem on the simple quadratic lattice was estimated as pc=0.593+or-0.003.
Abstract: For pt.I see ibid., vol.9, p.87 (1976). New series data are examined for the mean cluster size for site and bond mixtures in two dimensions. The critical concentration for the site problem on the simple quadratic lattice is estimated as pc=0.593+or-0.002 and on the honeycomb lattice as pc=0.698+or-0.003. It is concluded that the data are reasonably consistent with the hypothesis that the mean cluster size S(p) approximately=C(pc-p)- gamma as p to pc- with gamma a dimensional invariant, gamma =2.43+or-0.03 in two dimensions. Estimates of the critical amplitude C are also given.
TL;DR: In this article, the Casimir effect is discussed and calculated using covariant methods using Ford's results (see Phys Rev, volD11, p3370 (1975)) on vacuum energy in curved space are considered and criticized from the point of view of noncovariance.
Abstract: The Casimir effect is discussed and calculated using covariant methods Ford's results (see Phys Rev, volD11, p3370 (1975)) on vacuum energy in curved space are considered and criticized from the point of view of non-covariance The relevance of the 'conformal anomaly' of Fulling and Davies (1976) is noted
TL;DR: In this article, the coherent state for charged bosons is constructed, its properties are investigated and the corresponding classical model is discussed, and its properties and corresponding classical models are discussed.
Abstract: The coherent state for charged bosons is constructed, its properties are investigated and the corresponding classical model is discussed.
TL;DR: In this article, the authors applied the Lie theory of differential equations to the equation of motion of the classical one-dimensional harmonic oscillator and found that the equation is invariant under a global Lie group of point transformations.
Abstract: Lie's theory of differential equations is applied to the equation of motion of the classical one-dimensional harmonic oscillator. The equation is found to be invariant under a global Lie group of point transformations that is shown to be SL(3, R). The physical significance of the analysis and the results is considered. It is shown that the periodicity of the motion is a local topological property of the equation, while the length of the periods depends upon global properties.
TL;DR: Tangent space null rotations are used to give a straightforward classification of the Ricci tensor in general relativity theory as discussed by the authors, and they are used for the classification of Ricci Tensor.
Abstract: Tangent space null rotations are used to give a straightforward classification of the Ricci tensor in general relativity theory.
TL;DR: In this paper, a canonical transformation which removes the coherent oscillatory motion of a particle in a stochastic potential (the renormalised oscillation-centre transformation) is constructed by a new classical perturbation method using Lie operators and Green function techniques.
Abstract: A canonical transformation which removes the coherent oscillatory motion of a particle in a stochastic potential (the renormalised oscillation-centre transformation) is constructed by a new classical perturbation method using Lie operators and Green function techniques. A frequency and wavevector dependent particle-wave collision operator is calculated explicitly for stationary, homogeneous electrostatic turbulence in the short wavelength limit. The width of the resonance is proportional to the one-third power of the quasilinear diffusion coefficient, in agreement with Dupree's result (1966). However the k dependence is quite different from that expected from a simple Wiener process model. At large k spatial diffusion dominates over velocity diffusion in sharp contrast with previous theories.
TL;DR: In this paper, the authors derived series expansions for the mean size of finite clusters in the Ising model and concluded that for a two-dimensional lattice in zero magnetic field, Tc, as (Tc-T)- theta, with theta = 1.91+or 0.01.
Abstract: The derivation of series expansions for the mean size of finite clusters in the Ising model is described briefly. From the analysis of low temperature series it is concluded that for a two-dimensional lattice in zero magnetic field the mean size probably diverges at the Ising critical temperature, Tc, as (Tc-T)- theta , with theta =1.91+or-0.01. It appears therefore that theta < gamma '=1.75 the corresponding Ising susceptibility exponent. For a three-dimensional lattice it is tentatively concluded that the mean size diverges at some temperature T*
TL;DR: In this paper, the authors modified the droplet model of condensation to take account of ramified clusters whose surface/volume ratio tends to a finite limit as the number n of constituent molecules becomes large.
Abstract: The droplet model of condensation as developed by Fisher (see Physics, vol.3, p.255 (1967)) is modified to take account of: (a) ramified clusters whose surface/volume ratio tends to a finite limit as the number n of constituent molecules becomes large; (b) the excluded volume interactions between clusters. It is found that the interactions change the position of the first singularity in the activity series so that it no longer coincides with the phase boundary but is located beyond it. A thermodynamic metastable state can then be defined as in the classical Gibbs picture.
TL;DR: In this article, a geometrical description of how the strata with r = 1, 2, 3 fit together in the 8-dimensional convex set of a quantum system of spin 1/2 is presented.
Abstract: The states of a quantum mechanical system form a convex set with the pure states as extremals. For a system of spin-1/2, the set is a 3-dimensional ball. For spin j, the convex set is stratified by rank r, 1
TL;DR: In this article, the normal electromagnetic modes at small cubes and rectangular parallelepipeds are discussed and an expansion in terms of external multipoles is used; this allows an analytical evaluation of all elements of the interaction matrix.
Abstract: The normal electromagnetic modes at small cubes and rectangular parallelepipeds are discussed An expansion in terms of external multipoles is used; this allows an analytical evaluation of all elements of the interaction matrix A cluster point of eigenvalues arises at the eigenvalue epsilon int( omega )+ epsilon ext( omega )=0 of a half-space Only a few terms are needed to obtain convergence of the isolated eigenvalues which give rise to optical absorption peaks The maximum dipole absorption peak moves from the bulk eigenvalue epsilon int( omega )=0 to the monopole eigenvalue epsilon int( omega )=- infinity with increasing extension of the parallelepiped in the direction of the dipole
TL;DR: In this paper, the spatial and temporal correlation functions of the scattered field and intensity in the Fraunhofer and Fresnel regions were derived for a rigid deep random-phase screen in uniform linear motion.
Abstract: The scattering of radiation by a rigid deep random-phase screen in uniform linear motion is studied and formulae are derived for the spatial and temporal correlation functions of the scattered field and intensity in the Fraunhofer and Fresnel regions. A joint Gaussian distribution is used to represent the phase screen and the illuminating beam is assumed to have a curved wavefront and Gaussian intensity profile. It is shown that the coherence properties of the scattered radiation depend on an apparent area of illumination which is a function both of the actual width of the illuminating beam and of the slope distribution of the scattered wavefront. The dependence of the second intensity moment on distance from the screen is discussed and compared with that predicted by previous authors.
TL;DR: In this paper, the percolation probability for site and bond mixtures in two dimensions was examined and it was concluded that the data are reasonably consistent with the hypothesis that P(p) approximately=B(qc-q)beta as q to qc-with beta a dimensional invariant, beta = 0.138+or 0.175.
Abstract: For pt.III see ibid., vol.9, p.175. New series data are examined for the percolation probability P(p) for site and bond mixtures in two dimensions. It is concluded that the data are reasonably consistent with the hypothesis that P(p) approximately=B(qc-q)beta as q to qc-with beta a dimensional invariant, beta =0.138+or-0.007 in two dimensions. Estimates of the critical amplitude B are also given. Series data for the mean cluster size S(p) in the high density region are examined and it is tentatively concluded that S(p) approximately=C"(qc-q)- gamma 'as q to qc- and that the data are not inconsistent with the hypothesis gamma '= gamma .
TL;DR: The sensitivity of the Gibbs-DiMarzio theory for the glass transition of polymers to its basic assumptions is analyzed in this article, where it is shown that the value of the flexing energy parameter epsilon calculated from a measurement of the temperature Tg, is dominated by the result appropriate to the limiting case when the concentration of holes is zero, the number of chains unity, and the chain length goes to infinity.
Abstract: The sensitivity of the Gibbs-DiMarzio theory for the glass transition of polymers to its basic assumptions is analysed. The underlying model, and all the problems it raises, are graph-theoretical in nature. It is shown that the value of the flexing energy parameter epsilon calculated from a measurement of the glass transition temperature Tg, is dominated by the result appropriate to the limiting case when the concentration of holes is zero, the number of chains unity, and the chain length goes to infinity. Accordingly, the problem is dominated by the classical Hamiltonian-walk problem on a lattice graph. The nature of the lattice graph, its coordination number, and the boundary conditions, are examined. The dimensionality of the embedding space (e.g. the distinction between 'two-dimensional' and 'three-dimensional' lattices) is discarded in favour of the parameter actually relevant, called the r-degree of the lattice graph. Asymptotic results on the enumeration of Hamiltonian walks are presented for the unoriented honeycomb and for the oriented square and other lattices, including the covering lattices of certain orientations of the diamond and cubic lattices.
TL;DR: In this paper, the scaling function describing the crossover from Gaussian to Heisenberg behavior in the susceptibility of an isotropic n-component spin system was determined to second order in epsilon = 4-d.
Abstract: Renormalisation group methods are used to determine, to second order in epsilon =4-d, the scaling function describing the crossover from Gaussian to Heisenberg behaviour in the susceptibility of an isotropic n-component spin system. The results are used in conjunction with an earlier Feynman graph calculation to obtain an O( epsilon 2) representation of the n-to-m-component susceptibility crossover function, and the corresponding effective exponents, for an anisotropic n-component system.
TL;DR: In this paper, an elementary proof is given that the statistical mechanical transfer matrix, when symmetric, has a maximum eigenvalue which is non-degenerate and larger than the absolute value of any other eigen value.
Abstract: An elementary proof is given that the statistical mechanical transfer matrix, when symmetric, has a maximum eigenvalue which is non-degenerate and larger than the absolute value of any other eigenvalue. Moreover, the corresponding eigenvector can be chosen so that all its entire entries are strictly positive.
TL;DR: In this paper, a lower bound renormalization transformation of the type introduced by Kadanoff (see J. Statist. Phys., vol.14, p.171, 1976) is applied to the three-state Potts model.
Abstract: A lower-bound renormalisation transformation of the type introduced by Kadanoff (see J. Statist. Phys., vol.14, p.171, 1976) is applied to the three-state Potts model. For both d=2 and d=3 the Kadanoff transformation predicts a continuous transition. Values for the critical exponents and the critical temperature are reported. The consistency of the d=2 results with series estimates gives the authors some confidence in the predictions for d=3.
TL;DR: In this article, a brief analysis of the statistics of lattice animals (connected clusters) of n cells is undertaken, and the results are applied to the site percolation problem.
Abstract: A brief analysis is undertaken of the statistics of lattice animals (connected clusters) of n cells, and the results are applied to the site percolation problem. Recent proposals by Stauffer (1975) and Leath (1976) are examined, and an alternative interpretation is offered of the relation between percolation critical exponents and cluster statistics.
TL;DR: In this article, the authors investigated the relation between the self-avoiding walk problem and the trail problem on the square lattice and showed that certain critical exponents obey the same values for both problems.
Abstract: The trail problem on the square lattice is studied by the method of exact enumeration and its relation to the self-avoiding walk problem is pointed out. The number of N-stepped trails and their mean-square sizes are enumerated on a computer up to N=17. An asymptotic analysis of the numerical data suggests that certain critical exponents obey the same values for both the trail and the self-avoiding walk problem on the square lattice.
TL;DR: In this paper, a standard transformation was proposed for expressing Cartesian tensors and tensor expressions in spherical form, and vice versa, and some properties of the transformation coefficients were derived.
Abstract: In a previous paper (see Molec. Phys., vol.29, p.1461 (1975)), a standard transformation was proposed for expressing Cartesian tensors and tensor expressions in spherical form, and vice versa. Some properties of the transformation coefficients are derived. A recursion formula is given, which provides a simple means of generating the coefficients from those of the next rank below. The coefficients ( alpha 1,... alpha n mod j1...jn; m) are tabulated for n