On the Wang–Uhlenbeck Problem in Discrete Velocity Space

The arguably simplest model for dynamics in phase space is the one where the velocity can jump between only two discrete values, ±v with rate constant k. For this model, which is the continuous-space version of a persistent random walk, analytic expressions are found for the first passage time distributions to the origin. Since the evolution equation of this model can be regarded as the two-state finite-difference approximation in velocity space of the Kramers–Klein equation, this work constitutes a solution of the simplest version of the Wang–Uhlenbeck problem. Formal solution (in Laplace space) of generalizations where the velocity can assume an arbitrary number of discrete states that mimic the Maxwell distribution is also provided.     

A fractal approach to the distribution function of a paramagnetic system

A similarity between the random walk problem and a paramagnetic system has been established. The distribution functions of the stationary states have been obtained by making the Tsallis entropy a maximum, belonging to the statistical ensemble of a paramagnetic system, under suitable constraints using the variational methods. The asymptotic form of the distribution of the magnetic moments has been determined from the behaviour of the Lévy distribution. For the paramagnetic system which has been considered as the fractally structured system, following the way used by Alemany and Zanette [1] Tsallisq index has been related to the fractal dimension and the interval of the values ofq has also been determined.     

Optical surface <Emphasis Type="Italic">TM</Emphasis>-waves at the boundary of a nonlinear Kerr-type medium

Solutions for a nonlinear system of Maxwell’s equations and for a corresponding boundary problem describing the propagation of a surface TM-wave (p-polarization) along an interface with an optically nonlinear Kerr-type medium are presented. An analytical expression for the intensity of an integrated flux J₀ carried by such a wave along the interface between two media has been derived as a function of both the wave propagation constant ξ and the optical characteristics of the adjacent media. It is shown that flux intensities corresponding to the same values of the propagation constant ξ are much smaller for p-polarization than for s-polarization.     

Nonlinear evolutionary processes in a free-electron laser generator of diffracted radiation

Nonlinear evolutionary processes with two control parameters, one of which is related to the electrodynamic structure (positive feedback) and the other is related to the constant electric field applied to the electron flux, are studied in a free-electron laser, which is a diffracted-radiation oscillator. To this end, first, the linear spectral problem for an open-cavity resonator is investigated and the dispersion relation near the Morse critical point is established. Then the nonlinear evolutionary equation with two control parameters is constructed. Analysis of the latter makes it possible to determine the properties of the parametric dependence of the bifurcation and structural stability, which are determined by small variations of the control parameter (tuning of the cavity). This explains the operating efficiency of a diffracted-radiation oscillator in the millimeter and submillimeter wavelength ranges. Zh. éksp. Teor. Fiz. 111, 2243–2262 (June 1997)     

On the Quantum Mechanical Scattering Statistics of Many Particles

The probability of a quantum particle being detected in a given solid angle is determined by the S-matrix. The explanation of this fact in time-dependent scattering theory is often linked to the quantum flux, since the quantum flux integrated against a (detector-) surface and over a time interval can be viewed as the probability that the particle crosses this surface within the given time interval. Regarding many particle scattering, however, this argument is no longer valid, as each particle arrives at the detector at its own random time. While various treatments of this problem can be envisaged, here we present a straightforward Bohmian analysis of many particle potential scattering from which the S-matrix probability emerges in the limit of large distances.     

First-Passage and Extreme-Value Statistics of a Particle Subject to a Constant Force Plus a Random Force

We consider a particle which moves on the x axis and is subject to a constant force, such as gravity, plus a random force in the form of Gaussian white noise. We analyze the statistics of first arrival at point x ₁ of a particle which starts at x ₀ with velocity v ₀. The probability that the particle has not yet arrived at x ₁ after a time t, the mean time of first arrival, and the velocity distribution at first arrival are all considered. We also study the statistics of the first return of the particle to its starting point. Finally, we point out that the extreme-value statistics of the particle and the first-passage statistics are closely related, and we derive the distribution of the maximum displacement m=max  _t [x(t)].     

The Discrete N-Vector Ferromagnet: Connection to a Percolation with Frustration Features

We extend the Kasteleyn–Fortuin formalism to the discrete N-vector ferromagnet. We show that the free energy and the correlation functions of this model are related, when the number of states tends to 1, to the mean number of clusters and to the pair connectedness of a polychromatic bond percolation type problem which combines frustration and connectivity features.     

Dirac Operators on Taub-NUT Space: Relationship and Discrete Transformations

It is shown that the N = 4 superalgebra of the Dirac theory in Taub-NUT space has different unitary representations related among themselves through unitary U(2) transformations. In particular the SU(2) transformations are generated by the spin-like operators constructed with the help of the same covariantly constant Killing-Yano tensors which generate Dirac-type operators. A parity operator is defined and some explicit transformations which connect the Dirac-type operators among themselves are given. These transformations form a discrete group which is a realization of the quaternion discrete group. The fifth Dirac operator constructed using the non-covariant constant Killing-Yano tensor of the Taub-NUT space is quite special. This non-standard Dirac operator is connected with the hidden symmetry and is not equivalent to the Dirac-type operators of the standard N = 4 supersymmetry.     

Nonintersecting Brownian Motions on the Half-Line and Discrete Gaussian Orthogonal Polynomials

We study the distribution of the maximal height of the outermost path in the model of N nonintersecting Brownian motions on the half-line as N→∞, showing that it converges in the proper scaling to the Tracy-Widom distribution for the largest eigenvalue of the Gaussian orthogonal ensemble. This is as expected from the viewpoint that the maximal height of the outermost path converges to the maximum of the Airy₂ process minus a parabola. Our proof is based on Riemann-Hilbert analysis of a system of discrete orthogonal polynomials with a Gaussian weight in the double scaling limit as this system approaches saturation. We consequently compute the asymptotics of the free energy and the reproducing kernel of the corresponding discrete orthogonal polynomial ensemble in the critical scaling in which the density of particles approaches saturation. Both of these results can be viewed as dual to the case in which the mean density of eigenvalues in a random matrix model is vanishing at one point.     

Dynamics of a magnetic flux jump in a composite superconductor

An attempt is made at direct experimental verification of the theory of thermomagnetic instability in composite superconductors under conditions of external magnetic field or transport current variations. The development of thermomagnetic instability in the form of a magnetic flux jump is experimentally studied in a bulk low-temperature composite niobium-tin superconductor. The liquid-helium-cooled sample representing a compressed tape helix (helicoid) is placed in an external magnetic field orthogonal to the turn plane and varying with a constant rate. For the first time, both the magnetic induction inside the sample and its temperature are simultaneously detected in experiments. The sample overheat preceding the magnetic flux jump is measured to be 0.23 + 0.02 K. This value is found to be independent of the rate of the external magnetic field variation and the value of the jump itself and coincides, within the experimental accuracy, with the temperature parameter of electric field buildup involved in the general exponential I-V characteristic of the composite superconductor, which depends on temperature and magnetic induction.     

Light scattering by the apical oxygen sublattice in a thin layer of YBaCuO crystal

Peculiarities of scattering of a TM-polarized light wave by the apical oxygen sublattice, which are associated with dispersion of a phonon n-polariton localized in an ultrathin YBaCuO layer, are studied. The difference between the angle of scattering of the luminous flux from the angle of reflection is estimated for the maximum of the Raman frequency shift. It is shown that the amplitude of g-oscillations of bridge oxygen ions in the vicinity of the resonance frequency increases sharply; consequently, it becomes possible to observe additional (in respect to bulk) scattering of coherent electromagnetic waves at the Stokes and anti-Stokes frequencies.     

Pertubation analysis of the dynamical behavior of two-junction interferometers

The dynamical properties of symmetric quantum interferometers with equal junctions of negligible capacitance have been studied by means of perturbation analysis in the limit of small values of the parameter β. In this limit, two characteristic time constants arise. These quantities may be linked to two different dynamical processes in the system: the first is related to the time evolution of the average superconducting phase difference across the two junctions; the second defines the time scale for flux motion. The response of the system to constant and time-dependent externally applied magnetic fields is considered and a general perturbed solution for the average superconducting phase difference and the fluxon number variable is derived to first order in β.     

Wetting Transition on a One-Dimensional Disorder

We consider wetting of a one-dimensional random walk on a half-line x≥0 in a short-ranged potential located at the origin x=0. We demonstrate explicitly how the presence of a quenched chemical disorder affects the pinning-depinning transition point. For small disorders we develop a perturbative technique which enables us to compute explicitly the averaged temperature (energy) of the pinning transition. For strong disorder we compute the transition point both numerically and using the renormalization group approach. Our consideration is based on the following idea: the random potential can be viewed as a periodic potential with the period n in the limit n→∞. The advantage of our approach stems from the ability to integrate exactly over all spatial degrees of freedoms in the model and to reduce the initial problem to the analysis of eigenvalues and eigenfunctions of some special non-Hermitian random matrix with disorder-dependent diagonal and constant off-diagonal coefficients. We show that even for strong disorder the shift of the averaged pinning point of the random walk in the ensemble of random realizations of substrate disorder is indistinguishable from the pinning point of the system with preaveraged (i.e. annealed) Boltzmann weight.     

Transport in a disordered one-dimensional system: A fractal view

We reexamine the calculation of the transmission coefficient of a random array ofN isotopic defects in an otherwise perfect, harmonic, one-dimensional crystal lattice. The thermal conductivity of this model system has been studied under steady state conditions in which there is a kinetic temperature difference across, and an associated energy flux through, the array of defects. An exact expression for the transmission coefficient is obtained in terms of the magnitude of anNth-order determinant. Rubin reduced the evaluation of the determinant to the evaluation of a sequence ofN–1 nonlinear transformations drawn from a set of transformations parametrized by the nearest-neighbor spacing of the isotopic defects. These transformations are self-inverse and provide an example of what Mandelbrot has termed aself-inverse fractal. The variety of limiting distributions of values obtained under these transformations will be illustrated.     

Flux to a trap

The flux of particles to a single trap is investigated for two systems: (1) particles in 3D space which jump a fixed step lengthl (the Rayleigh flight) and are adsorbed by a spherical surface, and (2) particles on a lattice, jumping to nearest neighbor sites, with a single adsorbing site. Initially, the particles are uniformly distributed outside the traps. When the jump length goes to zero, both processes go over to regular diffusion, and the first case yields the diffusive flux to a sphere as solved by Smoluchowski. For nonzero step length, the flux for large times is given by a modified form of Smoluchowski's result, with the effective radius replaced byR-cl, wherec=0.29795219 andcl is the Milne extrapolation length for this problem. For the second problem, a similar expression for the flux is found, with the effective trap radius a function of the lattice (sc, bcc, fcc) being considered.     

The Planck Constant for the Definition and Realization of the Kilogram

On May 20, 2019 the values of the Planck, h, and Avogadro, N_A, constants will be fixed, revising our measurement system, the International System of Units (SI), and providing a new way to get mass traceability. While the famous energy relations mc² and hf may remind many that the Planck constant is indeed related to mass, it is less recognized how this is also true for the Avogadro constant. These concepts are reviewed in the context of the upcoming revision of the SI. How the fixed values were chosen and how mass traceability will be maintained with the smallest uncertainty are also discussed.     

Magnetic flux penetration into high-temperature superconductors in the vortex liquid state under the blow-up conditions

The problem of magnetic field penetration into a type-II high-temperature superconductor that is in the weakly pinned vortex-liquid phase is considered. A magnetic field on the superconductor boundary rises with time in the blow-up regime. A model hydrodynamic equation describing the magnetic induction distribution in the vortex-liquid phase for thermomagnetic motion of the flux is derived. Analytical expressions for the depth and rate of magnetic field penetration into the superconductor are found. It is demonstrated that these quantities depend on parameters of the problem: index of power n in the boundary regime characterizing the penetration rate of vortices into the superconducting half-space and a parameter describing the effect of random pinning forces and thermal fluctuations on the magnetic flux distribution.     

Depinning of a Discrete Elastic String from a Random Array of Weak Pinning Points with Finite Dimensions

In possible connection with dislocation pinning by foreign atoms in alloys and vortex pinning in type II superconductors, we compute the external force required to drag an elastic string along a discrete two-dimensional random array with finite dimensions. The obstacles, with a maximum pinning force f _m are distributed randomly on a rectangular lattice with square symmetry. The system dimensions are fixed by the total course of the elastic string L _x and the string length L _y . Our study shows that Larkin’s length is larger than L _y when f _m is less than a certain bound depending on the system size as well as on the obstacle density c _s . Below such a bound an analytical theory is developed to compute the depinning threshold. Some numerical simulations allow us to demonstrate the accuracy of the theory for an obstacle density ranging from 1 to 50% and for different geometries.     

Interacting fermions on a random lattice

We extend previous work on the properties of the Dirac lagrangian on two-dimensional random lattices to the case where interaction terms are included. Although for free fermions the chiral symmetry of the doubles is spontaneously broken by their interaction with the lattice and they decouple from long-distance physics, our results in this paper show that all is undone by quantum corrections in an interacting field theory and that the end result is very similar to what is found with Wilson fermions. Two field-theoretical models with interacting fermions are studied by perturbation expansion in the field theory coupling constant. These are a model with one fermion and one boson species interacting via a scalar Yukawa coupling and the massive Thirring model. It is shown that on the random lattice ultraviolet finite diagrams and finite parts of ultraviolet divergent diagrams have the correct continuum limit. Ultraviolet divergent parts can be removed by the same renormalisation procedure as in the continuum, but do not exhibit the same dependence on the lagrangian mass. In the case of the massive Thirring model this causes a fermion mass correction of order the cut-off scale, which breaks the chiral symmetry of the remaining light fermion; there is consequently a fine-tuning problem. In the context of the same model we discuss the effect of the Goldstone boson associated with the spontaneous breakdown of the chiral symmetry of the doubles on two-dimensional models with vector couplings.     

The construction of thed+1-dimensional gaussian droplet

The aim of this note is to study the asymptotic behavior of a gaussian random field, under the condition that the variables are positive and the total volume under the variables converges to some fixed numberv>0. In the context of Statistical Mechanics, this corresponds to the problem of constructing a droplet on a hard wall with a given volume. We show that, properly rescaled, the profile of a gaussian configuration converges to a smooth hypersurface, which solves a quadratic variational problem. Our main tool is a scaling dependent large deviation principle for random hypersurfacesDedicated to the memory of Roland Dobrushin, who passed away on 13 November 1995