Dynamics of generalized sine-Gordon soliton in inhomogeneous media

In this paper we introduce a few novel generalized sine-Gordon equations and study the dynamics of its solitons in inhomogeneous media. We consider length, mass, gravitational acceleration and spring stiffness of a coupled pendulums chain as a function of position x. Then in the continuum limit we derive semi-analytical and numerical soliton solutions of the modified sine-Gordon equation in the inhomogeneous media. The obtained results confirm that the behavior of solitons in these media is similar to that of a classical point particle moved in an external potential.     

Construction of convergent simplicial approximations of quantum fields on Riemannian manifolds

We construct simplicial approximations of random fields on Riemannian manifolds of dimensiond. We prove convergence of the fields to the continuum limit, for arbitraryd in the Gaussian case and ford=2 in the non-Gaussian case. In particular we obtain convergence of the simplicial approximation to the continuum limit for quantum fields on Riemannian manifolds with exponential interaction.Dedicated to Res Jost and Arthur WightmanBiBoS Research Centre     

Thermal diffusion of envelope solitons on anharmonic atomic chains

We study the motion of envelope solitons on anharmonic atomic chains in the presence of dissipation and thermal fluctuations. We consider the continuum limit of the discrete system and apply an adiabatic perturbation theory which yields a system of stochastic integro-differential equations for the collective variables of the ansatz for the perturbed envelope soliton. We derive the Fokker-Planck equation of this system and search for a statistically equivalent system of Langevin equations, which shares the same Fokker-Planck equation. We undertake an analytical analysis of the Langevin system and derive an expression for the variance of the soliton position Var[x _s ] which predicts a stronger than linear time dependence of Var[x _s ] (superdiffusion). We compare these results with simulations for the discrete system and find they agree well. We refer to recent studies where the diffusion of pulse solitons were found to exhibit a superdiffusive behaviour on longer time scales.Received: 28 June 2004, Published online: 26 November 2004PACS: 05.10.Gg Stochastic analysis methods - 05.45.Yv Solitons - 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion - 05.50. + q Lattice theory and statistics     

Is the 2D nonlinear model asymptotically free?

We report the results of a Monte Carlo study of the continuum limit of the two dimensional non-linear model. The notable finding is that it agrees very well with both the prediction inspired by Zamolodchikovs' S-matrix ansatz and with the continuum limit of the dodecahedron spin model. The latter finding renders the existence of asymptotic freedom in the model rather unlikely.     

Ideal magnetofluid turbulence in two dimensions

A continuum model of coherent structures in two-dimensional magnetohydro-dynamic turbulence is developed. These structures are macroscopic states which persist among the turbulent microscopic fluctuations, typically as magnetic islands with flow. They are modeled as statistical equilibrium states for the non-dissipative dynamics, which conserves energy and families of cross-helicity and flux integrals. The model predicts that from a given initial state an ideal magnetofluid will evolve into a final state having steady mean magnetic and velocity fields, and Gaussian local fluctuations in these fields. Excellent qualitative and quantitative agreement is found with the known results of direct numerical simulations. A rigorous justification of the theory is also provided, in the sense that the continuum model is derived from a lattice model in a fixed-volume, small-spacing limit. This construction uses the discrete Fourier transform to link the discretization ofx-space with the truncation ofk-space. Under the ergodic hypothesis and a separation-of-scales hypothesis, the lattice model is defined by a mean-field approximation to the Gibbs measure on the discretized phase space. A concentration property shows that this measure is equivalent to the microcanonical measure in the continuum limit.     

Effect of ordering of mobile oxygen on spectra of two-magnon and electronic Raman scattering of light in YBa2Cu3O6+x crystals with different doping levels

The effect of ordering of mobile chain-site oxygen in YBa₂Cu₃O_6+x crystals at different doping levels x on the kinetics of the intensity change of the two-magnon line and the extended structureless electronic continuum in optical Raman spectra and on the superconducting transition temperature T _c , has been studied in detail. An increase in the chain-site oxygen content x leads to a higher contribution of free carriers to the electronic continuum in Raman spectra. The kinetics of the electronic continuum becomes slower with x, whereas the relaxation rate of the two-magnon scattering is a nonmonotonic function of the stoichiometric index. Computer simulations of the relaxation of nonequilibrium states using the Monte Carlo technique qualitatively describe the kinetics observed in experiments. Our results lead us to a conclusion about local inhomogeneities in the electronic and spin systems in CuO₂ planes with scales of several lattice constants. Zh. éksp. Teor. Fiz. 116, 684–703 (August 1999)     

Casimir Effect on a Finite Lattice

Lattice quantum field theory is a well established branch of modern quantum field theory (QFT). However, it has only peripherally been used for the investigation of Casimir systems — i.e. for systems in which quantum fields are distorted by their interaction with classical background objects. This article presents a Hamiltonian lattice formulation of static Casimir systems at a level of generality appropriate for an introductory investigation. Background structure — represented by a lattice potential V(x) — is introduced along one spatial direction with translation invariance in all other spatial directions. It is simple to extend this formulation to include arbitrary background structure in more than one spatial direction. Following some general analysis two specific finite 1D lattice QFT systems are analyzed in detail. The first has three Dirichlet boundaries at the lattice sites x = 0, l and L (L > l > 0) with vanishing lattice potential V(x) everywhere in between. The vacuum energy and vacuum stress tensor T^μν for this system are calculated in 0 < x < L. Very careful attention must be and is given to renormalization in the (continuum) limit of vanishing lattice constant. Globally and locally this lattice system is seen to closely mimic the corresponding 1D continuum system — as one would hope. Then we introduce a lattice potential V(x) = c/(x — x₀)² centered at x = x₀ < 0 to the left of the boundary at x = 0 and extending through this boundary and the middle Dirichlet boundary at x = l out to the right‐hand boundary x = L > l and beyond. The vacuum energy and T^μν are calculated for this far more complicated system in the region 0 〈 x < L, again with very good results. The internal consistency of the lattice version of this system is carefully examined. Our conclusion is that finite‐lattice formulation provides a powerful and effective tool, capable of solving completely many Casimir systems which could not possibly be handled using continuum methods. This is precisely our reason for introducing it. Future investigations (in one and more dimensions and in dynamical as well as static contexts) will display more fully the power of this method.     

On homogenization and scaling limit of some gradient perturbations of a massless free field

We study the continuum scaling limit of some statistical mechanical models defined by convex Hamiltonians which are gradient perturbations of a massless free field. By proving a central limit theorem for these models, we show that their long distance behavior is identical to a new (homogenized) continuum massless free field. We shall also obtain some new bounds on the 2-point correlation functions of these models. This article was processed by the author using the LAT_EX style filepljour1 from Springer-Verlag.     

Ballistic aggregation for one-sided Brownian initial velocity

We study the one-dimensional ballistic aggregation process in the continuum limit for one-sided Brownian initial velocity (i.e. particles merge when they collide and move freely between collisions, and in the continuum limit the initial velocity on the right side is a Brownian motion that starts from the origin x=0). We consider the cases where the left side is either at rest or empty at t=0. We derive explicit expressions for the velocity distribution and the mean density and current profiles built by this out-of-equilibrium system. We find that on the right side the mean density remains constant whereas the mean current is uniform and grows linearly with time. All quantities show an exponential decay on the far left. We also obtain the properties of the leftmost cluster that travels towards the left. We find that in both cases relevant lengths and masses scale as t² and the evolution is self-similar.     

The second Painlevé equation in the electrostatic probe theory: Numerical solutions for the partial absorption of charged particles by the surface

We continue the study of the second Painlevé equation within the framework of the electrostatic probe theory. The integrability conditions for the equation are found for the partial absorption of charged particles by the probe surface. A sets of solutions with the asymptotics y ∼ ν/x for x → +∞ is constructed numerically in a wide range of the free parameter ν. Also, solutions (related to those mentioned above) for half-integer and integer ν, including solutions representable in asymptotic form at x → +∞ through the Airy function y ∼ cAi(x) in the limit ν → 0, are found. The results are discussed from the standpoint of the isomonodromic deformation method.     

A statistical limit in the solution of the nonlinear Schrödinger equation under deterministic initial conditions

We investigate the semiclassical limit for the nonlinear Schrödinger equation in the case of a defocusing medium under oscillating nonperiodic initial conditions specified on the entire x axis. We formulate a system of integral conservation laws in terms of an infinite number of spatially averaged densities explicitly calculated from the initial conditions. We study the direct scattering problem and show that the scattering phase is a uniformly distributed random variable. The evolution of this system leads to the development of nonlinear oscillations, which become statistical in nature on long time scales. A modified inverse scattering method based on constructing a maximizer of the N-soliton solution in the continuum limit for N → is used to obtain an asymptotic solution. Using the maximizer, we found an infinite set of conserved averaged densities in the statistical state. This allowed us to couple the initial state with the limiting statistical steady (for t → ∞) state and, thus, to unambiguously determine the level spectrum in the statistical limit.     

A gas of Brownian particles in statistical dynamics

We consider a large number of particles on a one-dimensional latticel Z in interaction with a heat particle; the latter is located on the bond linking the position of the particle to the point to which it jumps. The energy of a single particle is given by a potentialV(x), x∈Z. In the continuum limit, the classical version leads to Brownian motion with drift. A quantum version leads to a local drift velocity which is independent of the applied force. Both these models obey Einstein's relation between drift, diffusion, and applied force. The system obeys the first and second laws of thermodynamics, with the time evolution given by a pair of coupled non linear heat equations, one for the density of the Brownian particles and one for the heat occupation number; the equation for a tagged Brownian particle can be written as a stochastic differential equation.     

Regge behaviour of distribution functions and t and x-evolutions of gluon distribution function at low-x

In this paper, t and x-evolutions of gluon distribution function from Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equation in leading order (LO) at low-x are presented assuming the Regge behaviour of quarks and gluons at this limit. We compare our results of gluon distribution function with MRST 2001, MRST 2004 and GRV 1998 parametrizations and show the compatibility of Regge behaviour of quark and gluon distribution functions with perturbative quantum chromodynamics (PQCD) at low-x. We also discuss the limitations of Taylor series expansion method used earlier to solve DGLAP evolution equations in the Regge behaviour of distribution functions.      

Quantum transport in dynamically disordered continuum tight binding systems

We consider the continuum limit of three distinct models describing tightly bound electron systems in one dimension. The first model is the usual tight binding hamiltonian for monatomic lattices with nearest-neighbour hopping between sites. The second model describes a two-subband tight binding system involving two different atoms per unit cell. Finally, the third model represents a monatomic system with two energy levels per atomic site and different nearest-neighbour hopping parameters for hopping between equivalent and non-equivalent levels. The continuum limits of these models result in field-theoretic hamiltonians showing similarities with the Dirac hamiltonian. Assuming the different types of site energies to be dynamically disordered with gaussian whitenoise spectra, we calculate exactly the quantum mechanical mean square displacement <x ²(t)>. Due to the use of Novikov's theorem for the evaluation of configuration averages our analysis for the two-band models is restricted to the degenerate case, where the average positions of the two types of atomic levels coincide. Fort we find coherent motion, <x ²(t)>t ², for the one-band model and disorder induced diffusive contributions for the two-band models. However, for the two-level atomic model the diffusive term is dominated by at ²-term describing coherent hopping between equivalent levels. These findings are discussed in relation to previous results for both discrete and continuum models.     

Super-Instantons, Perfect Actions, Finite-Size Scaling, and the Continuum Limit

We discuss some aspects of the continuum limit of some lattice models, in particular the 2DO(N) models. The continuum limit is taken either in an infinitevolume or in a box whose size is a fixed fraction of the infinite-volume correlation length. We point out that in this limit the fluctuations of the lattice variables must be O(1) and thus restore the symmetry which may have been broken by the boundary conditions (b.c.). This is true in particular for the socalled super-instanton b.c. introduced earlier by us. This observation leads to a criterion to assess how close a certain lattice simulation is to the continuum limit and can be applied to uncover the true lattice artefacts, present even in the so-called “perfect actions”. It also shows that David’s recent claim that superinstanton b.c. require a different renormalization must either be incorrect or an artefact of perturbation theory.     

Continuum tight binding models and quantum transport in the presence of dynamical disorder

Continuum limits of various tight binding linear chain lattices used in the author's recent study of quantum transport in the presence of dynamical disorder, are analyzed from the point of view of their energy level spectra when disorder is absent. These spectra show a linear dispersion similar to that in the Luttinger model, and describe the energy levels of the corresponding discrete systems in the range of midband wavevectors. Next, the more conventional longwavelength continuum limit, which describes the energy levels of the actual discrete systems near the bottom of the bands is discussed. On the basis of these properties it is argued that the applicability of the continuum models to the study of dynamical properties is restricted to low frequencies in the range of low lying excitations, near the midband Fermi level in half-filled band situations in the case of the midband models, and near the bottom of a nearly empty band for the longwavelength models. Finally, it is shown that in the presence of dynamic disorder the longwavelength continuum limit of a single-band tight-binding model leads to nondiffusive motion, with a mean squared displacement <x ²(t)>t ³, fort.     

Regge behaviour of distribution functions and evolution of gluon distribution function in next-to-leading order at low-x

Evolution of gluon distribution function from Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equation in next-to-leading order (NLO) at low-x is presented assuming the Regge behaviour of quark and gluon at this limit. We compare our results of gluon distribution function with MRST2004, GRV98LO and GRV98NLO parametrizations and show the compatibility of Regge behaviour of quark and gluon distribution functions with perturbative quantum chromodynamics (PQCD) at low-x.      

On Hilfer's objection to the fractional time diffusion equation

Hilfer [Physica A 329 (2003) 35] claims to give an example of a continuous time random walk (CTRW) model with long-tailed waiting time probability density that approaches a Gaussian behavior in the continuum limit. Rigorous limit theorems, derived previously, show however that in the limit of long-time such a CTRW converges to a non-Gaussian behavior. We discuss two types of continuum limits for the CTRW model: the fractional continuum limit and the one introduced by Hilfer. We show that the fractional limit yields the correct long-time behavior of the CTRW, while Hilfer's continuum limit does not. We discuss a general approach to find a continuum limit of the CTRW process.     

Low-and high-dimension limits of a phase separation model

We study a simple zero-temperature model for phase separation of a binary alloy, in which nearest-neighbor interchange can occur if the fraction of AB pairs is not thereby increased. We present analytic results for the one-dimensional case and numerical results for the infinite dimensionality limit on a Cayley tree. In neither limit does the final fraction of AB pairs agree with the dimension-independent result found previously ind=3, 4, 5.