Showing papers in "Journal of Physics A in 1996"

Models of traps and glass phenomenology

Abstract: We study various models of independent particles hopping between energy `traps' with a density of energy barriers , on a d-dimensional lattice or on a fully connected lattice. If decays exponentially, a true dynamical phase transition between a high-temperature `liquid' phase and a low-temperature `aging' phase occurs. More generally, however, one expects that for a large class of , `interrupted' aging effects appear at low enough temperatures, with an ergodic time growing faster than exponentially. The relaxation functions exhibit a characteristic shoulder, which can be fitted as stretched exponentials. A simple way of introducing interactions between the particles leads to a modified model with an effective diffusion constant in energy space, which we discuss in detail.

Langevin equation for the density of a system of interacting Langevin processes

Abstract: We present a simple derivation of the stochastic equation obeyed by the density function for a system of Langevin processes interacting via a pairwise potential. The resulting equation is considerably different from the phenomenological equations usually used to describe the dynamics of non-conserved (model A) and conserved (model B) particle systems. The major feature is that the spatial white noise for this system appears not additively but multiplicatively. This simply expresses the fact that the density cannot fluctuate in regions devoid of particles. The steady state for the density function may, however, still be recovered formally as a functional integral over the coursed grained free energy of the system as in models A and B.

Quantum KZ equation with |q| = 1 and correlation functions of the XXZ model in the gapless regime

Abstract: An integral solution to the quantum Knizhnik - Zamolodchikov (qKZ) equation with |q|=1 is presented. Upon specialization, it leads to a conjectural formula for correlation functions of the XXZ model in the gapless regime. The validity of this conjecture is verified in special cases, including the nearest-neighbour correlator with an arbitrary coupling constant and general correlators in the XXX and XY limits.

Cellular automata models of traffic flow along a highway containing a junction

Abstract: We examine various realistic generalizations of the basic cellular automaton model describing traffic flow along a highway. In particular, we introduce a slow-to-start rule which simulates a possible delay before a car pulls away from being stationary. Having discussed the case of a bare highway, we then consider the presence of a junction. We study the effects of acceleration, disorder, and slow-to-start behaviour on the queue length at the entrance to the highway. Interestingly, the junction's efficiency is improved by introducing disorder along the highway, and by imposing a speed limit.

Thermal fluctuations in quantized chaotic systems

Abstract: We consider a quantum system with N degrees of freedom which is classically chaotic. When N is large, and both and the quantum energy uncertainty are small, quantum chaos theory can be used to demonstrate the following results: (i) given a generic observable A, the infinite time average of the quantum expectation value is independent of all aspects of the initial state other than the total energy, and equal to an appropriate thermal average of A; (ii) the time variations of are too small to represent thermal fluctuations; (iii) however, the time variations of can be consistently interpreted as thermal fluctuations, even though these same time variations would be called quantum fluctuations when N is small.

Representations of the quadratic algebra and partially asymmetric diffusion with open boundaries

Abstract: We consider the one-dimensional partially asymmetric exclusion model with open boundaries. The model describes a system of hard-core particles that hop stochastically in both directions with different rates. At both boundaries particles are injected and extracted. By means of the method of Derrida et al the stationary probability measure can be expressed as a matrix-product state involving two matrices forming a Fock-like representation of a general quadratic algebra. We obtain the representations of this algebra, which were unknown in the mathematical literature and use the two-dimensional one to derive exact expressions for the density profile and correlation functions. Using the correspondence between the stochastic model and a quantum spin chain, we obtain exact correlation functions for a spin- Heisenberg XXZ chain with non-diagonal boundary terms. Generalizations to other reaction - diffusion models are discussed.

Special solutions from the variable separation approach: the Davey - Stewartson equation

Abstract: A variable separation procedure for the Davey - Stewartson (DS) equation is proposed by using a prior ansatz to its bilinear form The reduced equations for two variable separated fields have the same trilinear form although they possess different independent variables The trilinear equation can be changed to a spacetime symmetric form and can be solved by means of a Boussinesq-type equation system Whenever a pair of solutions of the reduced fields are obtained, a corresponding solution of the DS equation can be obtained algebraically The single dromion solution and some kinds of positon solutions are obtained explicitly

Quantum fractals in boxes

Abstract: A quantum wave with probability density , confined by Dirichlet boundary conditions in a D-dimensional box of arbitrary shape and finite surface area, evolves from the uniform state . For almost all positions , the graph of the evolution of P is a fractal curve with dimension . For almost all times t, the graph of the spatial probability density P is a fractal hypersurface with dimension . When D = 1, there are, in addition to these generic time and space fractals, infinitely many special `quantum revival' times when P is piecewise constant, and infinitely many special spacetime slices for which the dimension of P is 5/4. If the surface of the box is a fractal with dimension , simple arguments suggest that the dimension of the time fractal is , and that of the space fractal is .

Shocks in the asymmetry exclusion model with an impurity

Abstract: We consider the one-dimensional asymmetric exclusion process with an impurity. This model describes particles hopping in one direction with stochastic dynamics and a hard core exclusion condition. The impurity hops with a rate different from that of the normal particles and can be overtaken by these particles. We solve this model exactly and give its phase diagram. In one of the phases the system presents a shock, i.e. a sharp discontinuity between a region of high density of particles and a region of low density. Density profiles and relevant exponents are explicitly calculated. These exact results for systems of finite size are consistent with anomalous diffusion laws observed in infinite systems.

Phase transitions in driven diffusive systems with random rates

Abstract: We study a one-dimensional driven lattice gas model in which quenched random jump rates are associated with the particles. Under suitable conditions on the distribution of jump rates the model displays a phase transition from a high-density `laminar' phase with product measure to a low-density `jammed' phase in which the interparticle spacings have no stationary distribution. Using a waiting time representation the phase transition is shown to be equivalent to a pinning transition of directed polymers with columnar defects. The phenomenon is argued to have a natural realization in traffic flow.

Phase space geometry and slow dynamics

Abstract: We describe a non-Arrhenius mechanism for the slowing down of dynamics that is inherent to the high dimensionality of the phase space. We show that such a mechanism is at work both in a family of mean-field spin-glass models without any domain structure and in the case of ferromagnetic domain growth. The marginality of spin-glass dynamics, as well as the existence of a `quasi-equilibrium regime' can be understood within this scenario. We discuss the question of ergodicity in an out-of equilibrium situation.

A crossing probability for critical percolation in two dimensions

Abstract: Langlands et al considered two crossing probabilities, and , in their extensive numerical investigations of critical percolation in two dimensions. Cardy was able to find the exact form of by treating it as a correlation function of boundary operators in the limit of the Q-state Potts model. We extend his results to find an analogous formula for which compares very well with the numerical results.

Contact interactions on graph superlattices

Abstract: We consider a quantum mechanical particle living on a graph and discuss the behaviour of its wavefunction at graph vertices. In addition to the standard (or -type) boundary conditions with continuous wavefunctions, we investigate two types of a singular coupling which are analogous to the interaction and its symmetrized version for a particle on a line. We show that these couplings can be used to model graph superlattices in which point junctions are replaced by complicated geometric scatterers. We also discuss the band spectra for rectangular lattices with the mentioned couplings. We show that they roughly correspond to their Kronig - Penney analogues: the lattices have bands whose widths are asymptotically bounded and do not approach zero, while the lattice gap widths are bounded. However, if the lattice-spacing ratio is an irrational number badly approximable by rationals, and the coupling constant is small enough, the lattice has no gaps above the threshold of the spectrum. On the other hand, infinitely many gaps emerge above a critical value of the coupling constant; for almost all ratios this value is zero.

The magnetization of the 3D Ising model

Abstract: We present highly accurate Monte Carlo results for simple cubic Ising lattices containing up to spins. These results were obtained by means of the Cluster Processor, a newly built special-purpose computer for the Wolff cluster simulation of the 3D Ising model. We find that the spontaneous magnetization M(t) is accurately described by , where , in a wide temperature range 0.0005 < t < 0.26. Any corrections to scaling with higher powers of t could not be resolved from our data, which implies that they are very small. The magnetization exponent is determined as . An analysis of the magnetization distribution near criticality yields a new determination of the critical point: , with a standard deviation of .

Coherent states for the hydrogen atom

Abstract: The long-standing problem of finding coherent states for the (bound state portion of the) hydrogen atom is positively resolved. The states in question (i) are normalized and parametrized continuously, (ii) admit a resolution of unity with a positive measure, and (iii) enjoy the property that the temporal evolution of any coherent state by the hydrogen atom Hamiltonian remains a coherent state for all time.

Chaotic analytic zero points: exact statistics for those of a random spin state

Abstract: A natural statistical ensemble of 2J points on the unit sphere can be associated, via the Majorana representation, with a random quantum state of spin J, and an exact expression is obtained here for the general k point correlation function in this ensemble. The pair correlation in the large-J limit takes the relatively simple form where and is the angular separation of the pair of points on the sphere. It appears (from the numerical work of others) that, in this limit, these statistics are typical of the zero points of analytic functions associated with chaotic quantum dynamical systems.

Coherent states for a quantum particle on a circle

Abstract: The coherent states for a quantum particle on a circle are introduced. The Bargmann representation within the actual treatment provides the representation of the algebra , where U is unitary, which is a direct consequence of the Heisenberg algebra , but it is more adequate for the study of circular motion.

Analytical inversion of symmetric tridiagonal matrices

Abstract: In this paper we present an analytical formula for the inversion of symmetrical tridiagonal matrices. The result is of relevance to the solution of a variety of problems in mathematics and physics. As an example, the formula is used to derive an exact analytical solution for the one-dimensional discrete Poisson equation with Dirichlet boundary conditions.

Non-equilibrium thermodynamics and anomalous diffusion

Abstract: The convenience of a new thermodynamic frame for the description of anomalous diffusion is explored. Our research, which makes use of a recent new definition for entropy arising from multifractal analysis, shows that both dynamical and thermodynamical effects may contribute to non-classical diffusion.

Cellular automata model of car traffic in a two-dimensional street network

Abstract: We propose a road traffic cellular automata model suitable for an urban environment. North, east, south and west car displacements are possible and road crossings are naturally implemented as rotary junctions. We consider the traffic in a Manhattan-like city and study the flow diagram and the car density profile along road segments. We observe that the length of the car queues obeys a complex dynamics and is not uniform across the network. The street length between two junctions and the turning strategies at rotaries are relevant parameters of the model. Our results are also confirmed by fully continuous traffic simulations.

Exhaustive search for low-autocorrelation binary sequences

Abstract: Binary sequences with low autocorrelations are important in communication engineering and in statistical mechanics as ground states of the Bernasconi model. Computer searches are the main tool in the construction of such sequences. Owing to the exponential size of the configuration space, exhaustive searches are limited to short sequences. We discuss an exhaustive search algorithm with run-time characteristic and apply it to compile a table of exact ground states of the Bernasconi model up to N = 48. The data suggest F > 9 for the optimal merit factor in the limit .

Constant mean curvature surfaces via an integrable dynamical system

Abstract: It is shown that the equation which describes constant mean curvature surfaces via the generalized Weierstrass - Enneper induction has Hamiltonian form. Its simplest finite-dimensional reduction is the integrable Hamiltonian system with two degrees of freedom. This finite-dimensional system admits -action and classes of -equivalence of its trajectories are in one-to-one correspondence with different helicoidal constant mean curvature surfaces. Thus the interpretation of well known Delaunay and do Carmo - Dajczer surfaces via an integrable finite-dimensional Hamiltonian system is established.

Darboux transformations for the nonlinear Schrödinger equations

Abstract: Darboux transformations for the AKNS/ZS system are constructed in terms of Grammian-type determinants of vector solutions of the associated Lax pairs with an operator spectral parameter. A study of the reduction of the Darboux transformation for the nonlinear Schr¨ odinger equations with standard and anomalous dispersion is presented. Two different families of new solutions for a given seed solution of the nonlinear Schr ¨ odinger equation are given, being one family related to a new vector Lax pair for it. In the first family and associated to diagonal matrices we present topological solutions, with different asymptotic argument for the amplitude and nonzero background. For the anomalous dispersion case they represent continuous deformations of the bright n-soliton solution, which is recovered for zero background. In particular these solutions contain the combination of multiple homoclinic orbits of the focusing nonlinear Schrequation. Associated with Jordan blocks we find rational deformations of the just described solutions as well as pure rational solutions. The second family contains not only the solutions mentioned above but also broader classes of solutions. For example, in the standard dispersion case, we are able to obtain the dark soliton solutions.

Discrete gradient methods for solving ODEs numerically while preserving a first integral

Abstract: We show that all direct methods for preserving a first integral during the numerical integration of an ordinary differential equation fit into the unified framework of discrete gradient methods. Using this framework we construct several new integral-preserving schemes.

Symmetries and integrability of difference equations

Abstract: The concept of integrability was introduced in classical mechanics in the 19th century for finite dimensional continuous Hamiltonian systems. It was extended to certain classes of nonlinear differential equations in the second half of the 20th century with the discovery of the inverse scattering transform and the birth of soliton theory. Also at the end of the 19th century Lie group theory was invented as a powerful tool for obtaining exact analytical solutions of large classes of differential equations. Together, Lie group theory and integrability theory in its most general sense provide the main tools for solving nonlinear differential equations. Like differential equations, difference equations play an important role in physics and other sciences. They occur very naturally in the description of phenomena that are genuinely discrete. Indeed, they may actually be more fundamental than differential equations if space-time is actually discrete at very short distances. On the other hand, even when treating continuous phenomena described by differential equations it is very often necessary to resort to numerical methods. This involves a discretization of the differential equation, i.e. a replacement of the differential equation by a difference one. Given the well developed and understood techniques of symmetry and integrability for differential equations a natural question to ask is whether it is possible to develop similar techniques for difference equations. The aim is, on one hand, to obtain powerful methods for solving `integrable' difference equations and to establish practical integrability criteria, telling us when the methods are applicable. On the other hand, Lie group methods can be adapted to solve difference equations analytically. Finally, integrability and symmetry methods can be combined with numerical methods to obtain improved numerical solutions of differential equations. The origin of the SIDE meetings goes back to the early 1990s and the first meeting with the name `Symmetries and Integrability of Discrete Equations (SIDE)' was held in Esterel, Quebec, Canada. This was organized by D Levi, P Winternitz and L Vinet. After the success of the first meeting the scientific community decided to hold bi-annual SIDE meetings. They were held in 1996 at the University of Kent (UK), 1998 in Sabaudia (Italy), 2000 at the University of Tokyo (Japan), 2002 in Giens (France), 2004 in Helsinki (Finland) and in 2006 at the University of Melbourne (Australia). In 2008 the SIDE 8 meeting was again organized near Montreal, in Ste-Adele, Quebec, Canada. The SIDE 8 International Advisory Committee (also the SIDE steering committee) consisted of Frank Nijhoff, Alexander Bobenko, Basil Grammaticos, Jarmo Hietarinta, Nalini Joshi, Decio Levi, Vassilis Papageorgiou, Junkichi Satsuma, Yuri Suris, Claude Vialet and Pavel Winternitz. The local organizing committee consisted of Pavel Winternitz, John Harnad, Veronique Hussin, Decio Levi, Peter Olver and Luc Vinet. Financial support came from the Centre de Recherches Mathematiques in Montreal and the National Science Foundation (through the University of Minnesota). Proceedings of the first three SIDE meetings were published in the LMS Lecture Note series. Since 2000 the emphasis has been on publishing selected refereed articles in response to a general call for papers issued after the conference. This allows for a wider author base, since the call for papers is not restricted to conference participants. The SIDE topics thus are represented in special issues of Journal of Physics A: Mathematical and General 34 (48) and Journal of Physics A: Mathematical and Theoretical, 40 (42) (SIDE 4 and SIDE 7, respectively), Journal of Nonlinear Mathematical Physics 10 (Suppl. 2) and 12 (Suppl. 2) (SIDE 5 and SIDE 6 respectively). The SIDE 8 meeting was organized around several topics and the contributions to this special issue reflect the diversity presented during the meeting. The papers presented at the SIDE 8 meeting were organized into the following special sessions: geometry of discrete and continuous Painleve equations; continuous symmetries of discrete equations—theory and computational applications; algebraic aspects of discrete equations; singularity confinement, algebraic entropy and Nevanlinna theory; discrete differential geometry; discrete integrable systems and isomonodromy transformations; special functions as solutions of difference and q-difference equations. This special issue of the journal is organized along similar lines. The first three articles are topical review articles appearing in alphabetical order (by first author). The article by Doliwa and Nieszporski describes the Darboux transformations in a discrete setting, namely for the discrete second order linear problem. The article by Grammaticos, Halburd, Ramani and Viallet concentrates on the integrability of the discrete systems, in particular they describe integrability tests for difference equations such as singularity confinement, algebraic entropy (growth and complexity), and analytic and arithmetic approaches. The topical review by Konopelchenko explores the relationship between the discrete integrable systems and deformations of associative algebras. All other articles are presented in alphabetical order (by first author). The contributions were solicited from all participants as well as from the general scientific community. The contributions published in this special issue can be loosely grouped into several overlapping topics, namely: •Geometry of discrete and continuous Painleve equations (articles by Spicer and Nijhoff and by Lobb and Nijhoff). •Continuous symmetries of discrete equations—theory and applications (articles by Dorodnitsyn and Kozlov; Levi, Petrera and Scimiterna; Scimiterna; Ste-Marie and Tremblay; Levi and Yamilov; Rebelo and Winternitz). •Yang--Baxter maps (article by Xenitidis and Papageorgiou). •Algebraic aspects of discrete equations (articles by Doliwa and Nieszporski; Konopelchenko; Tsarev and Wolf). •Singularity confinement, algebraic entropy and Nevanlinna theory (articles by Grammaticos, Halburd, Ramani and Viallet; Grammaticos, Ramani and Tamizhmani). •Discrete integrable systems and isomonodromy transformations (article by Dzhamay). •Special functions as solutions of difference and q-difference equations (articles by Atakishiyeva, Atakishiyev and Koornwinder; Bertola, Gekhtman and Szmigielski; Vinet and Zhedanov). •Other topics (articles by Atkinson; Grunbaum; Nagai, Kametaka and Watanabe; Nagiyev, Guliyeva and Jafarov; Sahadevan and Uma Maheswari; Svinin; Tian and Hu; Yao, Liu and Zeng). This issue is the result of the collaboration of many individuals. We would like to thank the authors who contributed and everyone else involved in the preparation of this special issue.

The schouten-nijenhuis bracket, cohomology and generalized poisson structures

Abstract: Newly introduced generalized Poisson structures based on suitable skew-symmetric contravariant tensors of even order are discussed in terms of the Schouten - Nijenhuis bracket. The associated `Jacobi identities' are expressed as conditions on these tensors, the cohomological contents of which is given. In particular, we determine the linear generalized Poisson structures which can be constructed on the dual spaces of simple Lie algebras.

Supersymmetry for systems with unitary disorder: circular ensembles

Abstract: A generalized Hubbard - Stratonovitch transformation relating an integral over random unitary matrices to an integral over Efetov's unitary -model manifold, is introduced. This transformation adapts the supersymmetry method to disordered and chaotic systems that are modelled not by a Hamiltonian but by their scattering matrix or time-evolution operator. In contrast to the standard method, no saddle-point approximation is made, and no massive modes have to be eliminated. This first paper on the subject applies the generalized Hubbard - Stratonovitch transformation to Dyson's circular unitary ensemble. It is shown how a supersymmetric variant of the Harish-Chandra - Itzykson - Zuber formula can be used to compute, in the large-N limit, the n-level correlation function for any n. Non-trivial applications to random network models, quantum chaotic maps, and lattice gauge theory, are expected.

Entropic uncertainty relations for a quantum oscillator

Abstract: We calculated the Shannon entropy of position and momentum for the stationary quantum states of the harmonic oscillator as a function of its energy and determined the corresponding entropic uncertainty relations for them. We found an approximate phenomenological function for the dependence of position and momentum entropies on the large quantum numbers and the corresponding asymptotic entropy - energy relation for the stationary harmonic oscillator. We also studied the time evolution of the position and momentum entropies of the non-stationary harmonic oscillator for the coherent states, squeezed vacuum and Schrodinger cat states.

Two-level system in a one-mode quantum field: numerical solution on the basis of the operator method

Abstract: Accurate eigenvalues and eigenfunctions of a two-level system interacting with a one-mode quantum field are calculated numerically. A special iteration procedure based on the operator method permits one to consider the solution within a wide range of the Hamiltonian parameters and to find the uniformly approximating analytical formula for the eigenvalues. Characteristic features of the model are considered, such as the level intersections, the population of the field states and the chaotization in the system through the doubling of the frequencies.

On matrix product ground states for reaction-diffusion models

Abstract: We discuss a new mechanism leading to a matrix product form for the stationary state of one-dimensional stochastic models The corresponding algebra is quadratic and involves four different matrices For the example of a coagulation - decoagulation model explicit four-dimensional representations are given and exact expressions for various physical quantities are recovered We also find the general structure of n-point correlation functions at the phase transition