Showing papers in "Journal of Physics A in 1988"
TL;DR: In this paper, a new class of boundary conditions for quantum systems integrable by means of the quantum inverse scattering (R-matrix) method is described, which allows the author to treat open quantum chains with appropriate boundary terms in the Hamiltonian.
Abstract: A new class of boundary conditions is described for quantum systems integrable by means of the quantum inverse scattering (R-matrix) method. The method proposed allows the author to treat open quantum chains with appropriate boundary terms in the Hamiltonian. The general considerations are applied to the XXZ and XYZ models, the nonlinear Schrodinger equation and Toda chain.
1,774 citations
TL;DR: In this paper, the typical fraction of the space of interactions between each pair of N Ising spins which solve the problem of storing a given set of p random patterns as N-bit spin configurations is considered, and a local iterative learning algorithm for updating the interactions is given which will converge to a solution of given kappa provided such solutions exist.
Abstract: The typical fraction of the space of interactions between each pair of N Ising spins which solve the problem of storing a given set of p random patterns as N-bit spin configurations is considered. The volume is calculated explicitly as a function of the storage ratio, alpha =p/N, of the value kappa (>0) of the product of the spin and the magnetic field at each site and of the magnetisation, m. Here m may vary between 0 (no correlation) and 1 (completely correlated). The capacity increases with the correlation between patterns from alpha =2 for correlated patterns with kappa =0 and tends to infinity as m tends to 1. The calculations use a saddle-point method and the order parameters at the saddle point are assumed to be replica symmetric. This solution is shown to be locally stable. A local iterative learning algorithm for updating the interactions is given which will converge to a solution of given kappa provided such solutions exist.
900 citations
TL;DR: The authors calculate the number, p= alpha N of random N-bit patterns that an optimal neural network can store allowing a given fraction f of bit errors and with the condition that each right bit is stabilised by a local field at least equal to a parameter K.
Abstract: The authors calculate the number, p= alpha N of random N-bit patterns that an optimal neural network can store allowing a given fraction f of bit errors and with the condition that each right bit is stabilised by a local field at least equal to a parameter K. For each value of alpha and K, there is a minimum fraction fmin of wrong bits. They find a critical line, alpha c(K) with alpha c(0)=2. The minimum fraction of wrong bits vanishes for alpha alpha c(K). The calculations are done using a saddle-point method and the order parameters at the saddle point are assumed to be replica symmetric. This solution is locally stable in a finite region of the K, alpha plane including the line, alpha c(K) but there is a line above which the solution becomes unstable and replica symmetry must be broken.
462 citations
TL;DR: In this paper, it was shown that almost all sufficiently long self-avoiding walks on the 3-dimensional simple cubic lattice contain a knot, using Kesten's (1963) pattern theorem.
Abstract: Discusses the existence of knots in random self-avoiding walks on a lattice. Using Kesten's (1963) pattern theorem, it is shown that almost all sufficiently long self-avoiding walks on the three-dimensional simple cubic lattice contain a knot.
311 citations
TL;DR: In this article, the density of the final configuration in the sequence of cubes (L/2, L/2) d typically undergoes an abrupt transition, from being close to 0 to the value 1.
Abstract: Bootstrap percolation models, or equivalently certain types of cellular automata, exhibit interesting finite-volume effects. These are studied at a rigorous level. The authors find that for an initial configuration obtained by placing particles independently with probability p or=2), the density of the 'bootstrapped' (final) configurations in the sequence of cubes (-L/2, L/2)d typically undergoes an abrupt transition, as L is increased, from being close to 0 to the value 1. With L fixed at a large value, the mean final density as a function of p changes from 0 to 1 around a value which varies only slowly with L-the pertinent parameter being lambda =p1(d-1)/ln L. The driving mechanism is the capture of a 'critical droplet'. This behaviour is analogous to the decay of a metastable state near a first-order phase transition, for which the analysis offers some suggestive ideas.
286 citations
TL;DR: In this article, the wavefunctions of all known shape-invariant potentials were given explicit expressions for all the known wave functions by using the operator method, and the expressions for the wave functions of all possible wave functions were obtained for all possible shapes.
Abstract: The authors obtain explicit expressions for the wavefunctions of all the known shape-invariant potentials by using the operator method.
244 citations
TL;DR: Two algorithms for sequence extrapolation, van den Broeck and Schwartz (1979) and Bulirsch and Stoer (1964), are reviewed and critically compared in this paper, and it is shown that Bulirsch's algorithm is superior to Schwartz's algorithm if only very few finite-lattice data are available.
Abstract: Two algorithms for sequence extrapolation, due to van den Broeck and Schwartz (1979), and Bulirsch and Stoer (1964), are reviewed and critically compared. Applications to three-state and six-state quantum chains and to the (2+1)D Ising model show that the algorithm of Bulirsch and Stoer is superior, in particular if only very few finite-lattice data are available.
173 citations
IBM1
TL;DR: In this article, a detailed study of the emergence and the long-time behaviour of coherent vortices in two-dimensional decaying turbulence is presented, where the authors find that the coherent structures are self-similar, i.e. their energy, enstrophy and size satisfy scaling laws.
Abstract: The authors present a detailed study of the emergence and the long-time behaviour of coherent vortices in two-dimensional decaying turbulence. By high-resolution numerical experiments they find that the coherent structures are self-similar, i.e. their energy, enstrophy and size satisfy scaling laws. Moreover, the knowledge of the statistical distribution of the size of these vortices is sufficient to compute the energy spectrum of the 2D turbulent flow and to explain the significant deviations from the Kraichnan-Batchelor theory. At long times the motion of the fluid is dominated by the vortex dynamics, which is strikingly similar to the Hamiltonian motion of few point vortices, as is confirmed by a comparison between the numerical simulations of the two systems.
167 citations
TL;DR: In this article, an elementary introduction of quantum-state-valued Markovian stochastic processes (QSP) for N-state quantum systems is given, and it is pointed out that a so-called master constraint must be fulfilled.
Abstract: An elementary introduction of quantum-state-valued Markovian stochastic processes (QSP) for N-state quantum systems is given. It is pointed out that a so-called master constraint must be fulfilled. For a given master equation a continuous and, as a new alternative possibility, a discontinuous QSP are derived. Both are discussed as possible models for state reduction during measurement.
161 citations
TL;DR: The Bohr-Sommerfeld quantisation condition has a meaningful extension to classically chaotic systems whose periodic (unstable) orbits are isolated as discussed by the authors, providing a semiclassical Euler factorisation for the functional determinant of the quantal Hamiltonian, in contrast to the Hadamard infinite product over the eigenvalues by which the exact determinant is defined.
Abstract: The Bohr-Sommerfeld quantisation condition has a meaningful extension to classically chaotic systems whose periodic (unstable) orbits are isolated. It provides a semiclassical Euler factorisation for the functional determinant of the quantal Hamiltonian, in contrast to the Hadamard infinite product over the eigenvalues by which the exact determinant is defined.
145 citations
TL;DR: In this paper, the scattering amplitudes for all currently known supersymmetric shape-invariant potentials are calculated by analytically continuing the explicit wavefunctions obtained via super-symmetric operator techniques.
Abstract: The scattering amplitudes for all currently known supersymmetric shape-invariant potentials are calculated by analytically continuing the explicit wavefunctions obtained via supersymmetric operator techniques. The procedure is illustrated in detail for the superpotential W(x)=A tanh alpha x+B sech alpha x, for which the S matrix has not been previously calculated.
TL;DR: In this article, the authors considered the effect of slowly varying the parameters Xi of a finite-sized quantum mechanical system, and showed that when the spectral statistics are of those of the Gaussian orthogonal ensemble, the rate of dissipation is proportional to Xi2.
Abstract: Considers the effect of slowly varying the parameters Xi of a finite-sized quantum mechanical system. The system is excited to higher energies by Landau-Zener transitions at avoided crossing; since this increases the energy of the system, it has the effect of dissipation of the driving motion. The rate of dissipation depends on the level spacing distribution of the system. When the spectral statistics are those of the Gaussian unitary ensemble, the rate of dissipation is proportional to Xi2, i.e. there is viscous or ohmic damping. When the spectral statistics are of those of the Gaussian orthogonal ensemble, the rate of dissipation is proportional to Xi32/.
TL;DR: The symmetry group of the generalised non-linear Schrodinger equation in three dimensions was shown to be the extended Galilei group G(3), for a 1a2 not=0, and the Galilei-similitude group Gd(3, including a dilation) for a1=oor a2=0.
Abstract: The symmetry group of the generalised non-linear Schrodinger equation i psi t+ Delta psi =a0 psi +a1 mod psi mod 2 psi +a2 mod psi mod 4 psi in three space dimensions is shown to be the extended Galilei group G(3), for a1a2 not=0, and the Galilei-similitude group Gd(3) (including a dilation) for a1=oor a2=0. All Lie subgroups of G(3) and Gd(3) are found. They will be used in a subsequent paper to obtain group invariant solutions of the equation.
TL;DR: In this article, the fractal dimension d(min) of the shortest path l between two points on a percolation cluster was calculated, where l is the Pythagorean distance between the points.
Abstract: The authors calculate the fractal dimension d(min) of the shortest path l between two points on a percolation cluster, where l approximately rd(min) and r is the Pythagorean distance between the points. They find d(min)=1.130+or-0.002 for d=2 and 1.34+or-0.01 for d=3.
TL;DR: In this paper, a ring-shaped potential was obtained by replacing the Coulomb part of the Hartmann potential by a harmonic oscillator term, and the Schrodinger equation was solved in spherical, circular cylindrical, prolate and oblate spheroidal coordinates.
Abstract: A new ring-shaped potential, obtained by replacing the Coulomb part of the Hartmann potential by a harmonic oscillator term, is investigated. The Schrodinger equation is solved in spherical, circular cylindrical, prolate and oblate spheroidal coordinates. As in the case of the Hartmann potential, the 'accidental' degeneracies occurring in the spectrum are shown to be due to an su(2) dynamical invariance algebra. This establishes a close connection between both ring-shaped potentials.
TL;DR: In this article, a method of constructing (2+1)-dimensional non-linear integrable equations and their solutions by means of the non-local delta problem is developed, where a "basic set" of equations is obtained by using different normalisations of the NLD problem and the Lagrangian of the set is found.
Abstract: A method of constructing (2+1)-dimensional non-linear integrable equations and their solutions by means of the non-local delta problem is developed. A 'basic set' of equations is obtained by using different normalisations of the non-local delta problem and the Lagrangian of the set is found. Other integrable equations, which are degenerate cases of the basic set, are also Lagrangian.
TL;DR: Behaviour consistent with finite-size scaling, characteristic of a first-order phase transition, is shown to be exhibited by the basins of attraction of the stored patterns both in the case of the Hopfield model and for systems using a local iterative learning algorithm designed to optimise the basin of attraction.
Abstract: The content-addressability of patterns stored in Ising-spin neural network models with symmetric interactions is studied. Numerical results from simulations on the ICL distributed array processor (DAP) involving systems with up to 2048 neurons are presented. Behaviour consistent with finite-size scaling, characteristic of a first-order phase transition, is shown to be exhibited by the basins of attraction of the stored patterns both in the case of the Hopfield model and for systems using a local iterative learning algorithm designed to optimise the basins of attraction. Estimates are obtained for the critical minimum overlaps which an input pattern must have with a stored pattern in order to successfully retrieve it.
TL;DR: In this paper, hydrodynamic dispersion in flow through a disordered porous medium is investigated and a Monte Carlo simulation approach is used to study dispersion processes in random network models of porous media.
Abstract: The authors study hydrodynamic dispersion in flow through a disordered porous medium. The main goals are to investigate the condition(s) under which a convective-diffusion equation (CDE) cannot describe dispersion processes and to investigate the effect of the disordered morphology of the pore space on dispersion processes. They first use simple models of porous media and study dispersion processes analytically and compare the results with the predictions of the CDE. The results show that the morphology of the porous medium can strongly affect dispersion processes. They then use a Monte Carlo simulation approach to study dispersion processes in random network models of porous media which are made of interconnected capillary tubes with distributed effective radii. A percolation network is used as a prototype of porous media with disordered topology. They show that, as the percolation threshold Xc of the network is approached, there exists an anomalous and length-dependent dispersion regime that cannot be described by the CDE. They propose a generalisation of the Gaussian distribution to describe dispersion in the anomalous regime, and confirm it by Monte Carlo simulations.
TL;DR: In this paper, the second half of the Lax pair, the associated Hamiltonian structures, and an infinite hierarchy of Poisson commuting Hamiltonians were derived for delta 2+ Sigma 1N upsilon i lambda i) Psi = alpha Psi.
Abstract: The authors consider the isospectral flows of ( delta 2+ Sigma 1N upsilon i lambda i) Psi = alpha Psi . Using an unusual form of the 'Lax approach' they derive in a particularly simple manner: (a) the 'second half' of the Lax pair; (b) the associated Hamiltonian structures; (c) an infinite hierarchy of Poisson commuting Hamiltonians. In this way they show that these equations possess (N+1) compatible, purely differential Hamiltonian structures. The case N=1 is just the bi-Hamiltonian Harry Dym hierarchy. They thus extend the recent results on multi-Hamiltonian coupled KdV equations.
TL;DR: In this article, a subadditive thermodynamic formalism for fractals is presented, for which a variational principle is derived and used to study the dynamics of nonconformal transformations.
Abstract: The thermodynamical description of fractals that has recently attracted much interest both experimentally and theoretically in the study of dynamical systems is, in some ways, limited, being essentially an additive theory. The author presents a subadditive thermodynamic formalism for which he derives a variational principle and shows how it may be used to study the dynamics of non-conformal transformations. In particular the author discusses an analogue of Bowen's formula for the dimension of a mixing repeller.
TL;DR: In this article, a method for finding the evolution operator for the Schrodinger equation for the Hamiltonian expressible as H(t)=a1(t)J4+a2(t),J0+a3(t)) J- where J+, J0 and J- are the SU(2) group generators is presented.
Abstract: The authors present a method for finding the evolution operator for the Schrodinger equation for the Hamiltonian expressible as H(t)=a1(t)J4+a2(t)J0+a3(t) J- where J+, J0 and J- are the SU(2) group generators. Such a method is applied to the disentangling technique for exponential operators which are not necessarily unitary. As a demonstration of our general approach, they solved the problem of a harmonic oscillator with a varying mass.
TL;DR: In this article, the authors studied mutual synchronisation in a model of interacting limit cycle oscillators with random intrinsic frequencies, and showed rigorously that the model exhibits no long-range order in one dimension, and that in higher-dimensional lattices, large clusters of synchronised oscillators necessarily have a sponge-like structure.
Abstract: The authors study mutual synchronisation in a model of interacting limit cycle oscillators with random intrinsic frequencies. It is shown rigorously that the model exhibits no long-range order in one dimension, and that in higher-dimensional lattices, large clusters of synchronised oscillators necessarily have a sponge-like structure. Surprisingly, the phase-locking behaviour of the mean-field model is completely different from that of any finite-dimensional lattice, indicating that d= infinity is the upper critical dimension for phase locking.
TL;DR: The generating functions for the number of convex polygons on the square and honeycomb lattices were derived rigorously by Guttmann and Enting (1988).
Abstract: The generating functions for the number of convex polygons on the square and honeycomb lattices are derived rigorously. These functions were found by Guttmann and Enting (1988). Their calculation is based on the series expansions up to the 64th order and is nonrigorous. The asymptotic form of the mean-squared radius of gyration of n-step convex polygons on the square lattice is determined and the critical exponent v is 1.
TL;DR: In this article, the authors suggest a picture of the intrinsic surface width in Eden models and show that it is a major source of corrections to scaling of the surface roughness, using the multiple-hit noise reduction method, they can control the intrinsic width and thereby improve the scaling behavior systematically as demonstrated in detailed calculations on the square lattice.
Abstract: For pt.I see ibid., vol.20, p.257 (1987). The authors suggest a picture of the intrinsic surface width in Eden models and show that it is a major source of corrections to scaling of the surface roughness. Using the multiple-hit noise reduction method, they can control the intrinsic width and thereby improve the scaling behaviour systematically as is demonstrated in detailed calculations on the square lattice. They calculate the number of excess perimeter sites as a function of time and find that its asymptotic value decays with a power law as a function of increasing hitting number. Substrate effects and anisotropy become more apparent if noise reduction is applied.
TL;DR: The authors give an exact solution of the model for connectivity K=1, valid everywhere in the frozen phase and at a critical point, valid for finite as well as for infinite networks.
Abstract: Kauffman's model is a randomly assembled network of Boolean automata. Each automaton receives inputs from at most K other automata. Its state at discrete time t+1 is determined by a randomly chosen, but fixed, Boolean function of the K inputs at time t. The resulting quenched, random dynamics of the network demonstrates two phases: a frozen and a chaotic phase. The authors give an exact solution of the model for connectivity K=1, valid everywhere in the frozen phase and at a critical point, valid for finite as well as for infinite networks. They discuss the network's critical behaviour and finite-size effects. The results for the frozen phase presented complement recent exact results for the chaotic phase obtained for K= infinity .
TL;DR: In this article, it was shown that for the general dynamics of a quantum spin, there exists a stochastic process in an extended phase space which at each time allows them to compute correlations between the different components of the spin.
Abstract: The authors prove that, for the general dynamics of a quantum spin, there exists a stochastic process in an extended phase space which at each time allows them to compute correlations between the different components of the spin. The scheme is limited to integer spins, but some possibilities of extension are discussed as well as the connection with Nelson's stochastic dynamics (1985).
TL;DR: The phase structure is found to be independent of the updating scheme used in the dynamical law for the network and an order parameter for a phase transition well known from Kauffman's model is found.
Abstract: An exact polynomial equation is given for the size of the stable core of networks of automata with random connections. When the connectivity K of a network equals 1, 2, 3, 4 or 5 this equation is exactly solvable. It is found that the size of the stable core is an order parameter for a phase transition well known from Kauffman's model. A new derivation of critical parameter values follows. The phase structure is found to be independent of the updating scheme used in the dynamical law for the network.
TL;DR: For an asymmetric version of the McCulloch-Pitts neural network (1943) and for Kauffman's infinite-range Boolean network model (1984), the time evolution of the Hamming distances between two different initial configurations are compared in the thermodynamic limit as mentioned in this paper.
Abstract: For an asymmetric version of the McCulloch-Pitts neural network (1943) and for Kauffman's infinite-range Boolean network model (1984), the time evolution of the Hamming distances between two different initial configurations are compared in the thermodynamic limit. It is shown that in both models phase transitions occur for corresponding values of the transition parameters and that their Hamming distances can have the same time evolution leading to quantitatively the same dynamics, as known from time-dependent Landau theory for phase transitions.
TL;DR: Extended series expansions for the mean size and the first and second moments of the pair connectedness for both bond and site percolation on the directed square and triangular lattices have been obtained.
Abstract: Extended series expansions for the mean size and the first and second moments of the pair connectedness for both bond and site percolation on the directed square and triangular lattices have been obtained. Analysis based on differential approximants allows the critical percolation probabilities and exponents to be estimated, and as a result the critical exponents are conjectured to be gamma =41/18, nu perpendicular to =79/72 and nu /sub ///=26/15. Scaling then gives beta =199/720, alpha -=-299/360 and delta =1839/199.
TL;DR: Several models for the dynamic growth of percolation clusters were introduced and analysed in this paper, where a random walker is allowed to step off such clusters and add new sites to them if certain conditions are met.
Abstract: Several models for the dynamic growth of percolation clusters, or 'diffusion percolation' (DP), are introduced and analysed. In these models a random walker (an 'ant') walks on percolation clusters (which are occupied with initial site concentration, pi). The 'ant' is allowed to step off such clusters and add new sites to them if certain conditions are met. Some of these models are shown to have a one-to-one correspondence with models of bootstrap percolation (BP), in which sites which do not have a required number of neighbours are successively culled. Two new percolation thresholds have been calculated for two diffusion percolation models on the square lattice.