Diffusion in a bistable potential: A systematic WKB treatment

We study the distributionP of a single stochastic variable, the evolution of which is described by a Fokker-Planck equation with a first moment deriving from a bistable potential, in the limit of constant and small diffusion coefficient. A systematic WKB analysis of the lowest eigenmodes of the equivalent Schrödinger-like equation yields the following results: the final approach to equilibrium is governed by the Kramers high-viscosity rate, which is shown to be exact in this limit; for intermediate times, we show that Suzuki's scaling statement does give the correct behavior for the transition between the one-peak and the two-peak structure forP. However, the intermediate time domain also contains a second half, whereP enters the diffusive equilibrium regions, characterized by a time scale of the same order as Suzuki's time.     

Entropy of self-gravitating radiation

We examine the entropy of self-gravitating radiation confined to a spherical box of radiusR in the context of general relativity. We expect that configurations (i.e., initial data) which extremize total entropy will be spherically symmetric, time symmetric distributions of radiation in local thermodynamic equilibrium. Assuming this is the case, we prove that extrema ofS coincide precisely with static equilibrium configurations of the radiation fluid. Furthermore, dynamically stable equilibrium configurations are shown to coincide with local maxima ofS. The equilibrium configurations and their entropies are calculated and their properties are discussed. However, it is shown that entropies higher than these local extrema can be achieved and, indeed, arbitrarily high entropies can be attained by configurations inside of or outside but arbitrarily near their own Schwarzschild radius. However, if we limit consideration to configurations which are outside their own Schwarzschild radius by at least one radiation wavelength, then the entropy is bounded and we find S_max MR, whereM is the total mass. This supports the validity for self-gravitating systems of the Bekenstein upper limit on the entropy to energy ratio of material bodies.     

The Navier-Stokes equations on a bounded domain

SupposeU is an open bounded subset of 3-space such that the boundary ofU has Lebesgue measure zero. Then for any initial condition with finite kinetic energy we can find a global (i.e. for all time) weak solutionu to the time dependent Navier-Stokes equations of incompressible fluid flow inU such that the curl ofu is continuous outside a locally closed set whose 5/3 dimensional Hausdorff measure is finite.This research was supported in part by the National Science Foundation Grant MCS-7903361     

An eikonal line-integral model and factorization in QCD

We study an approximation to perturbative QCD in which all quarks (active and spectator) move along eikonal lines. These eikonal lines are described by matrix operatorsU defined by time-ordered and path-ordered integrals. All known examples of Bloch-Nordsieck cancellations follow simply from the unitary properties ofU. In particular, when the eikonal lines are space-like, the time-ordering can be omitted and the unitarity ofU is evident. In this model, it is possible to examine rather explicitly the soft and very-soft gluons whose effects must cancel if factorization is hold (to leading twist). We can show such cancellation if there is a smooth continuation from space-like eikonal lines to time-like ones. We give some examples relevant to this conjecture, but we are unable to prove it in general.     

Peierls Instability for the Holstein Model with Rational Density

We consider the static Holstein model, describing a chain of fermions interacting with a classical phonon field, when the interaction is weak and the density is a rational number p = P/Q, with P, Q relative prime integers. We show that the energy of the system, as a function of the phonon field, has one (if Q is even) or two (if Q is odd) stationary points, defined up to a lattice translation, which are local minima in the space of fields periodic with period equal to the inverse of the density.     

Automatic procedure for stable tetragonal or hexagonal structures: application to tetragonal Y and Cd

A simple effective procedure (MNP) for finding equilibrium tetragonal and hexagonal states under pressure is described and applied. The MNP procedure finds a path to minima of the Gibbs free energy G at T=0 K (G=E+pV, E=energy per atom, p=pressure, V=volume per atom) for tetragonal and hexagonal structures by using the approximate expansion of G in linear and quadratic strains at an arbitrary initial structure to find a change in the strains which moves toward a minimum of G. Iteration automatically proceeds to a minimum within preset convergence criteria on the calculation of the minimum. Comparison is made with experimental results for the ground states of seven metallic elements in hexagonal close-packed (hcp), face- and body-centered cubic structures, and with a previous procedure for finding minima based on tracing G along the epitaxial Bain path (EBP) to a minimum; the MNP is more easily generalized than the EBP procedure to lower symmetry and more atoms in the unit cell. Comparison is also made with a molecular-dynamics program for crystal equilibrium structures under pressure and with CRYSTAL, a program for crystal equilibrium structures at zero pressure. Application of MNP to the elements Y and Cd, which have hcp ground states at zero pressure, finds minima of E at face-centered cubic (fcc) structure for both Y and Cd. Evaluation of all the elastic constants shows that fcc Y is stable, hence a metastable phase, but fcc Cd is unstable.     

Inconsistency of QED in the presence of Dirac monopoles

A precise formulation ofU (1) local gauge invariance in QED is presented, which clearly shows that the gauge coupling associated with the unphysical longitudinal photon field is non-observable and actually has an arbitrary value. We then re-examine the Dirac quantization condition and find that its derivation involves solely the unphysical longitudinal coupling. Hence an inconsistency inevitably arises in the presence of Dirac monopoles and this can be considered as a theoretical evidence against their existence. An alternative, independent proof of this conclusion is also presented.     

Diffusion phenomenon in the hyperbolic and parabolic regimes

We discuss the diffusion phenomenon in the parabolic and hyperbolic regimes. New effects related to the finite velocity of the diffusion process are predicted, that can partially explain the strange behavior associated to adsorption phenomenon. For sake of simplicity, the analysis is performed by considering a sample in the shape of a slab limited by two perfectly blocking surfaces, in such a manner that the problem is one-dimensional in the space. Two cases are investigated. In the former, the initial distribution of the diffusing particles is assumed of gaussian type, centered around the symmetry surface in the middle of the sample. In the latter, the initial distribution is localized close to the limiting surfaces. In both cases, we show that the evolution toward to the equilibrium distribution is not monotonic. In particular, close to the limiting surfaces the bulk density of diffusing particles present maxima and minima related to the finite velocity of the diffusion process connected to the second order time derivative in the partial differential equation describing the evolution of the bulk density in the sample.     

Gold diffusion in silicon by Rapid Optical Annealing

Gold diffusion in silicon is investigated using Rapid Optical Annealing at temperatures in the range of 800°C to 1200°C and annealing times from 300 s down to 1 s. The resulting content of substitutional gold is determined by spreading resistance measurements and analyzed by comparison with extensive numerical simulations.The profiles obtained show a broader spectrum as compared to the U-shapes after long time diffusion. The cooling process affects the profiles significantly, since they depend on the wafer thickness. An unexpected penetration depth was found after 1200°C diffusion in thick wafers, which are subject to small cooling rates. This phenomenon is due to a special combination of reverse kick-out, deep diffusion of highly supersaturated interstitial gold, and again an incorporation in lattice sites, termed the RDI effect.Numerical calculations allow us to reproduce the experimentally observed profiles only if a sensitive balance between the different temperature dependencies is obeyed. These investigations, therefore, yield new information about the equilibrium concentration and diffusion of silicon interstitials. A best set of parameters is presented. The time constant of the kick-out process is quantified for the first time.     

Prescribing topological defects for the coupled Einstein and Abelian Higgs equations

We construct multi-string solutions of the coupled Einstein and Abelian Higgs equations so that the spacetime is uniform along the time axis and a vertical direction and nontrivial geometry is coded on a Riemann surfaceM. We concentrate on the critical Bogomol'nyi phase. WhenM is compact, the Abelian Higgs model is defined by a complex line bundleL overM. We prove that, due to the coupling of the Einstein equations, the Euler characteristic ofM and the first Chern number of the line bundleL identified as the total string number impose an exact obstruction to the existence of a string solution. Such an obstruction leads to some interesting implications. We then study the existence of multi-string solutions which can realize a prescribed string distribution. We show that there are such solutions when the local string winding numbers do not exceed half of the total string number. WhenM is noncompact and globally conformal to a plane, we show that the energy scale of symmetry breaking plays a crucial role and there are finite-energy radially symmetric string solutions realizing a given string number if and only if the symmetry breaking scale is sufficiently small but nonvanishing. Finally, we obtain finite-energy multistring solutions with an arbitrary string distribution and associated local winding numbers. These solutions are not radially symmetric and are regular everywhere and topologically nontrivial so that both the energy of the matter-gauge sector and the energy of the gravitational sector viewed as the total Gauss curvature ofM are quantized.     

On retroactive occurrence and twin events in quantum mechanics

Bohr's well-known claim that only a registered phenomenon is a true phenomenon is further elaborated into occurrence in the past: If ideal occurrence of an eventP ((1–P)) is a state at a timet _i makes another eventQ ((1–Q))certain at a later timet _f, and, finallyU is the evolution operator fromt _i tot _f, then, it is proved that the final collapsed stateQ(U U ⁺)Q/TrQU U ⁺, which comes about in ideal occurrence ofQ att _f,equals the initial collapsed stateU(P P/TrP)U ⁺, which evolves from the state resulting from the ideal occurrence ofP in att _i. Utilizing the latter state is called theretroactive apparent ideal occurrence (RAIO) ofP in. A number of consequences, including the general notion of twin events (the case whent _f=t _i, andU=1) is derived. It is pointed out that RAIO is relevant in second-kind quantum measurement, in Wheeler's delayed-choice experiments in second-kind (or conditional) quantum preparators.     

Computing Freidlin’s Cycles for the Overdamped Langevin Dynamics. Application to the Lennard-Jones-38 Cluster

The large time behavior of a stochastic system with infinitesimally small noise can be described in terms of Freidlin’s cycles. We show that if the system is gradient and the potential satisfies certain non-restrictive conditions, the hierarchy of cycles has a structure of a full binary tree, and each cycle is exited via the lowest saddle adjacent to it. Exploiting this property, we propose an algorithm for computing the asymptotic zero-temperature path and building a hierarchy of Freidlin’s cycles associated with the transition process between two given local equilibria. This algorithm is suitable for systems with a complex potential energy landscape with numerous minima. We apply it to find the asymptotic zero-temperature path and Freidlin’s cycles involved into the transition process between the two lowest minima of the Lennard-Jones cluster of 38 atoms. D. Wales’s stochastic network of minima and transition states of this cluster is used as an input.     

On the change of polarization of positive muons in alkali halides

Positive muons implanted in nonconducting solids form with high probability hydrogenlike muonium atoms (µ ⁺ e ^–) with properties similar to those ofU ₂-centers. The influence of superhyperfine interactions with neighbor nuclei on the evolution of the polarization of the muon is investigated theoretically. The resulting muon polarization in longitudinal magnetic fields is calculated for muonicU ₂-centers in some alkali halides.     

Stochastic operators,information, and entropy

For a stochastic operatorU on andL ₁-space, i.eU is linear, positive, and norm preserving on the positive cone ofL ₁, it is shown thatU decreases relative information between two nonnegativeL ₁-functions. Furthermore it is shown that the following properties ofU are closely related:U is energy decreasing (energy preserving),U isH-decreasing, whereH is Boltzmann'sH-functional, and the Maxwell distributions are fixed points ofU.     

Rank of quantized universal enveloping algebras and modular functions

We compute an intrinsic rank invariant for quasitriangular Hopf algebras in the case of general quantum groupsU _q (g). As a function ofq the rank has remarkable number theoretic properties connected with modular covariance and Galois theory. A number of examples are treated in detail, including rank (U _q (su(3))) and rank (U _q (e ₈)). We briefly indicate a physical interpretation as relating Chern-Simons theory with the theory of a quantum particle confined to an alcove ofg.     

Solvable models of temporally correlated random walk on a lattice

We seek the conditional probability functionP(m,t) for the position of a particle executing a random walk on a lattice, governed by the distributionW(n, t) specifying the probability ofn jumps or steps occurring in timet. Uncorrelated diffusion occurs whenW is a Poisson distribution. The solutions corresponding to two different families of distributionsW are found and discussed. The Poissonian is a limiting case in each of these families. This permits a quantitative investigation of the effects, on the diffusion process, of varying degrees of temporal correlation in the step sequences. In the first part, the step sequences are regarded as realizations of an ongoing renewal process with a probability densityψ(t) for the time interval between successive jumps.W is constructed in terms ofψ using the continuous-time random walk approach. The theory is then specialized to the case whenψ belongs to the class of special Erlangian density functions. In the second part,W is taken to belong to the family of negative binomial distributions, ranging from the geometric (most correlated) to the Poissonian (uncorrelated). Various aspects such as the continuum limit, the master equation forP, the asymptotic behaviour ofP, etc., are discussed.     

Optimal Stability of Advection-Diffusion Lattice Boltzmann Models with Two Relaxation Times for Positive/Negative Equilibrium

Despite the growing popularity of Lattice Boltzmann schemes for describing multi-dimensional flow and transport governed by non-linear (anisotropic) advection-diffusion equations, there are very few analytical results on their stability, even for the isotropic linear equation. In this paper, the optimal two-relaxation-time (OTRT) model is defined, along with necessary and sufficient (easy to use) von Neumann stability conditions for a very general anisotropic advection-diffusion equilibrium, in one to three dimensions, with or without numerical diffusion. Quite remarkably, the OTRT stability bounds are the same for any Peclet number and they are defined by the adjustable equilibrium parameters. Such optimal stability is reached owing to the free (“kinetic”) relaxation parameter. Furthermore, the sufficient stability bounds tolerate negative equilibrium functions (the distribution divided by the local mass), often labeled as “unphysical”. We prove that the non-negativity condition is (i) a sufficient stability condition of the TRT model with any eigenvalues for the pure diffusion equation, (ii) a sufficient stability condition of its OTRT and BGK/SRT sub-classes, for any linear anisotropic advection-diffusion equation, and (iii) unnecessarily more restrictive for any Peclet number than the optimal sufficient conditions. Adequate choices of the two relaxation rates and the free-tunable equilibrium parameters make the OTRT sub-class more efficient than the BGK one, at least in the advection-dominant regime, and allow larger time steps than known criteria of the forward time central finite-difference schemes (FTCS/MFTCS) for both, advection and diffusion dominant regimes.     

Path integral formulation of general diffusion processes

A method is given for the derivation of a path integral representation of the Green's function solutionP of equationsP/t=L P,L being some Liouville operator. The method is applied to general diffusion processes.Feynman's path integral representation of the Schrödinger equation and Stratonovich's path integral representation of multivariate Markovian processes are obtained as special cases if the metric of the general diffusion process is flat. For curved phase spaces our result is a nontrivial generalization and new. New applications e.g. to quantized motion in general relativity, to transport processes in inhomogeneous systems, or to nonlinear non-equilibrium thermodynamics are made possible. We expect applications to be fruitfull in all cases where (continuous) macroscopic transport processes in Riemann geometries have to be considered.     

Low-frequency instabilities of electron-hole plasmas in crossed fields

Using local point-contact probes, we observed two types of low-frequency instabilities inn-InSb at 85 K if the samples were exposed to crossed fields. One is a local density instability with threshold frequencies off = 1 20 Mc, the other a more turbulent current instability. The threshold values ofU ₀ andB for the onset of these instabilities and the dependence of their amplitudes on the fields have been measured.If a rectangular semiconductor slab is placed in crossed fields, regions of high electric field strength at opposite edges of the contacts are caused by the distortion of the Hall field, giving rise to the generation of electron-hole plasmas by impact ionization. These plasmas are the sources of the observed instabilities. This is especially evident in the case of the local density instability, which originates at the anode high field corner. Several possible reasons for the development of the instabilities are discussed.     

Entanglement of electronic subbands and coherent superposition of spin states in a Rashba nanoloop

The present work is concerned with an analysis of the entanglement between the electronic coherent superpositions of spin states and subbands in a quasi-one-dimensional Rashba nanoloop acted upon by a strong perpendicular magnetic field. We explicitly include the confining potential and the Rashba spin-orbit coupling into the Hamiltonian and then proceed to calculate the von Neumann entropy, a measure of entanglement, as a function of time. An analysis of the von Neumann entropy demonstrates that, as expected, the dynamics of entanglement strongly depends upon the initial state and electronic subband excitations. When the initial state is a pure one formed by a subband excitation and the z-component of spin states, the entanglement exhibits periodic oscillations with local minima (dips). On the other hand, when the initial state is formed by the subband states and a coherent superposition of spin states, the entanglement still periodically oscillates, exhibiting stronger correlations, along with elimination of the dips. Moreover, in the long run, the entanglement for the latter case undergoes the phenomenon of collapse-revivals. This behaviour is absent for the first case of the initial states. We also show that the degree of entanglement strongly depends upon the electronic subband excitations in both cases.