Equilibrium Fluctuations for a Model of Coagulating-Fragmenting Planar Brownian Particles

We study a model of mass-bearing coagulating-fragmenting planar Brownian particles. Coagulation occurs when two particles are within a distance of order ε. Our model is macroscopically described by an inhomogeneous Smoluchowski’s equation in the low ε limit provided that the initial number of particles N is of order |log ε|. When a detailed balance condition is satisfied, we establish a central limit theorem by showing that in the low ε limit, the particle density fluctuation fields obey an Ornstein-Uhlenbeck stochastic differential equation.     

Systematics of perturbative semiclassical quantum defect expansions probed by RKR-QDTand a Fisher-information-based criterion

The systematics of perturbative semiclassical quantum defect expansions corresponding to a hydrogenic potential plus a perturbing term of the form -A/2r^κ, k\geqslant 2\kappa \geqslant 2, are studied as a function of expansion order N. Towards this task the expansions μ _Nare first used as input for constructing associated N-dependent atomic RKR-QDT potential curves. Subsequently the coordinate Fisher information for the energy levels supported by those curves as well as its rate ε with respect to N is semiclassically computed. Then, the plot of relative quantum defect error between successive orders, δμ _N+1,N, with respect to ε serves as convergence indicator for both approximate potentials and quantum defects. For a given κ and when the quantum defect expansion proves to be of limited accuracy the plot reveals an A- and N-dependent scatter of points and “saturation” (the relative error remains almost constant with respect to ε). More importantly, when ε is equal to or lower than the value of ε (N=1) for which πμ ₁\leqslant 1/2_{1}\leqslant 1/2 the relative error exhibits a κ-, A- and N-independent power-law dependence, δμ _N+1,N∝ ε ^m, clearly distinguishing the N=1 order (m=1/2) from all other N>1 orders (m=1). These power-laws may be employed for setting-up confidence level bounds on perturbative expansions.     

A Mathematical Justification for the Herman-Kluk Propagator

A class of Fourier Integral Operators which converge to the unitary group of the Schr?dinger equation in the semiclassical limit ε → 0 in the uniform operator norm is constructed. The convergence allows for an error bound of order O(ε), which can be improved to arbitrary order in ε upon the introduction of corrections in the symbol. On the Ehrenfest-timescale, the result holds with a slightly weaker error bound. In the chemical literature the approximation is known as the Herman-Kluk propagator.     

Driven Tracer Particle in One Dimensional Symmetric Simple Exclusion

Consider an infinite system of particles evolving in a one dimensional lattice according to symmetric random walks with hard core interaction. We investigate the behavior of a tagged particle under the action of an external constant driving force. We prove that the diffusively rescaled position of the test particle εX(ε^-2 t), t > 0, converges in probability, as ε→ 0, to a deterministic function v(t). The function v(⋅) depends on the initial distribution of the random environment through a non-linear parabolic equation. This law of large numbers for the position of the tracer particle is deduced from the hydrodynamical limit of an inhomogeneous one dimensional symmetric zero range process with an asymmetry at the origin. An Einstein relation is satisfied asymptotically when the external force is small. Received: 5 December 1996 / Accepted: 30 June 1997     

Controlling the noise enhanced stability effect via noise recycling in a metastable system

We analyze the role of the delay time τ _d and the fraction ε of recycled noise on the enhancement of the mean first-passage time (MFPT) in a metastable system with recycled noise, generated by the superposition of a primary Gaussian noise source with a second component of constant delay. The results indicate that MFPT as a function of the noise intensity D shows either a non-monotonic behavior with a maximum or a divergent behavior, which is the identifying characteristic of the noise enhanced stability (NES) phenomenon. The increasing of τ _d or ε strengthens the NES effect for ε > 0. However, for ε < 0, there is a critical value of τ _d , below which we observe an increase of MFPT whose maximum goes to infinity, and above which the divergent behavior tends to disappear and MFPT versus D shows a transition from one peak to two peaks and eventually one peak as τ _d or |ε| increases. Moreover, we also discuss the effect of different initial conditions. These observations illustrate that the noise recycling may be used as an effective scheme for controlling the NES effect.     

A solution of the DGLAP equation for gluon at low x

We obtain a solution of the DGLAP equation for the gluon at low x first by expanding the gluon in a Taylor series and then using the method of characteristics. We test its validity by comparing it with that of Glück, Reya and Vogt. The convergence criteria of the approximation used are also discussed. We also calculate εF ₂(x,Q)²/ε In Q ² using its approximate relations with the gluon distribution at low x. The predictions are then compared with the HERA data.     

Generation of neutrons at the relaxation of fast deuterons in deuterium matter

We analyze the fusion process involving two deuterium nuclei in the case of deceleration of a fast deuteron with an energy of approximately 100 keV located in a deuterium target. We calculate the probability ω _fus(ε) of generating a neutron by a fast deuteron with an initial kinetic energy ε during its deceleration. The mean free path of fast deuterons with respect to their relaxation is found for various deuterium targets. The data are analyzed for neutron generation in deuterium cluster beams under laser irradiation. The method of neutron generation in the collision of two deuterium cluster beams is suggested. The text was submitted by the author in English.     

Instanton Calculus for the Self-Avoiding Manifold Model

We compute the normalisation factor for the large order asymptotics of perturbation theory for the self-avoiding manifold (SAM) model describing flexible tethered (D-dimensional) membranes in d-dimensional space, and the ε-expansion for this problem. For that purpose, we develop the methods inspired from instanton calculus, that we introduced in a previous publication (Nucl. Phys. B 534 (1998) 555), and we compute the functional determinant of the fluctuations around the instanton configuration. This determinant has UV divergences and we show that the renormalized action used to make perturbation theory finite also renders the contribution of the instanton UV-finite. To compute this determinant, we develop a systematic large-d expansion. For the renormalized theory, we point out problems in the interplay between the limits ε→ 0 and d→∞, as well as IR divergences when ε=0. We show that many cancellations between IR divergences occur, and argue that the remaining IR-singular term is associated to amenable non-analytic contributions in the large-d limit when ε=0. The consistency with the standard instanton-calculus results for the self-avoiding walk is checked for D=1.     

Time-Scales to Equipartition in the Fermi–Pasta–Ulam Problem: Finite-Size Effects and Thermodynamic Limit

We investigate numerically the common α+β and the pure β FPU models, as well as some higher order generalizations. We consider initial conditions in which only low-frequency normal modes are excited, and perform a very accurate systematic study of the equilibrium time as a function of the number N of particles, the specific energy ε, and the parameters α and β. While at any fixed N the equilibrium time is found to be a stretched exponential in 1/ε, in the thermodynamic limit, i.e. for N→∞ at fixed ε, we observe a crossover to a power law. Concerning the (usually disregarded) dependence of T _eq on α and β, we find it is nontrivial, and propose and test a general law. A central role is played by the comparison of the FPU models with the Toda model.     

Tensor product composition of algebras in Bose and Fermi canonical formalism

We consider a graded algebra with two products (σ, α) over anε-factor commutation. One of the products (σ) isε-commutative, but, in general non-associative; and the other (α) is a graded Lieε-product and a gradedε-derivative with respect to the first (σ). Using the obvious mathematical condition, namely—the tensor product of two graded algebras with the sameε-factors is another with the sameε-factor, we determine the complete structure of a two-product (σ, α) graded algebra. When theε-factors are taken to be unity and the gradation structure is ignored, we recover the algebras of the physical variables of classical and quantum systems, considered by Grgin and Petersen. With the retention of the gradation structure and the possible choice of two ε-factors we recover the algebras of the canonical formalism of boson and fermion systems for the above classical and quantum theories. We also recover in this case the algebra of anticommutative classical systems considered by Martin along with its quantum analogue.     

The widths of single-particle states of anisotropic, strongly correlated electron systems in solids

The damping γ(ε) of electron states in crystals is investigated beyond the phase transition point related to a rearrangement of the Fermi surface. Attention is focused on the alteration of the standard Landau theory due to the emergence of a flat portion in the spectrum ξ(p) of single-particle excitations as a result of the rearrangement. In the limit ε →0, the width γ(ε) of the states in the region of the Brillouin zone where the dispersion of ξ(p) is of an ordinary order of magnitude is found to depend on ε almost linearly, in contrast to the Fermi-liquid-theory result γ(ε)∼ε². Pis’ma Zh. éksp. Teor. Fiz. 70, No. 11, 759–765 (10 December 1999) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Frequency dependence of the complex dielectric constant of poly(ethylene oxide) under hydrostatic pressure

Summary Electrical-impedance measurements have been made in the frequency range 5 Hz to10 MHz in pure poly(ethylene oxide) having a molecular weight of 600 000 from 254 K nearly up to the melting point of the crystalline phase (about 330 K). As the temperature approaches the melting point there are large increases in the realε′ and imaginaryε″ parts of the dielectric constant. The frequency dependence ofε′ is characterized by a primary-relaxation process, whose frequency increases with increasing temperature as a consequence of decrease of the average structural relaxation time. There is strong evidence that this low-frequency dispersion rises mainly from the diffusive transport of localised charge carriers rather than a purely orientation relaxation process. In addition the effects of hydrostatic pressure (0–25 Gpa) on the frequency dependences of the realε′ and imaginaryε″ parts of the dielectric constant have been measured in the same temperature range. Paper presented at the I International Conference on Scaling Concepts and Complex Fluids, Copanello, Italy, July 4–8, 1994.     

Spherically symmetric static solutions of the Einstein-Maxwell equations

We report a new formalism to obtain solutions of Einstein-Maxwell’s equations for static spheres assuming the matter content to be a charged perfect fluid of null-conductivity. Defining three new variablesu=4πεr ²,ν=4πpr ^{2 2} andw=(4π/3)(ρ+ε)r ² whereε, ρ andε denote respectively energy densities of the electric, matter and free gravitational fields whereasp is the fluid pressure, Einstein’s field equations are rewritten in an elegant form. The solutions given by Bonnor [1], Nduka [2], Cooperstock and De la Cruz [3], Mehra [4], Tikekar [5,6], Xingxiang [7], Patino and Rago [8] are all shown to possess simple relations betweenu, v, andw whereas Pant and Sah’s [9] solution for which all the three functions,u, v, andw are constants is a trivial case of the present formalism, We have presented six new solutions with ε = 2ρ. For the first three solutionsw andu are constants withv as a variable whereas the remaining three solutions satisfy the equation of state for isothermal gas;v =kw =-ku where (i)k is an arbitrary constant but not equal to 1 or 1/3 (ii)k = 1 and (iii)k = 1/3. We also obtained a generalization of Cooperstock and De la Cruz’s [3] solution which is regular for 2ρ > ε but singular for 2ρ ≤ ε.     

The Transition to a Giant Vortex Phase in a Fast Rotating Bose-Einstein Condensate

We study the Gross-Pitaevskii (GP) energy functional for a fast rotating Bose-Einstein condensate on the unit disc in two dimensions. Writing the coupling parameter as 1/ε ² we consider the asymptotic regime ε → 0 with the angular velocity Ω proportional to (ε ²|log ε|)⁻¹. We prove that if Ω = Ω₀(ε ²|log ε|)⁻¹ and Ω₀ > 2(3π)⁻¹ then a minimizer of the GP energy functional has no zeros in an annulus at the boundary of the disc that contains the bulk of the mass. The vorticity resides in a complementary ‘hole’ around the center where the density is vanishingly small. Moreover, we prove a lower bound to the ground state energy that matches, up to small errors, the upper bound obtained from an optimal giant vortex trial function, and also that the winding number of a GP minimizer around the disc is in accord with the phase of this trial function.     

Spinodal Decomposition¶for the Cahn–Hilliard–Cook Equation

This paper gives theoretical results on spinodal decomposition for the stochastic Cahn–Hilliard–Cook equation, which is a Cahn–Hilliard equation perturbed by additive stochastic noise. We prove that most realizations of the solution which start at a homogeneous state in the spinodal interval exhibit phase separation, leading to the formation of complex patterns of a characteristic size. In more detail, our results can be summarized as follows. The Cahn–Hilliard–Cook equation depends on a small positive parameter ε which models atomic scale interaction length. We quantify the behavior of solutions as ε→ 0. Specifically, we show that for the solution starting at a homogeneous state the probability of staying near a finite-dimensional subspace ?_ε is high as long as the solution stays within distance r _ε=O(ε ^R ) of the homogeneous state. The subspace ?_ε is an affine space corresponding to the highly unstable directions for the linearized deterministic equation. The exponent R depends on both the strength and the regularity of the noise. Received: 2 May 2000 / Accepted: 8 July 2001     

Kinetic Limit for a System of Coagulating Planar Brownian Particles

We study a model of mass-bearing coagulating planar Brownian particles. The coagulation occurs when two particles are within a distance of order ε. We assume that the initial number of particles N is of order |logε|. Under suitable assumptions of the initial distribution of particles and the microscopic coagulation propensities, we show that the macroscopic particle densities satisfy a Smoluchowski-type equation.     

On Random Field Induced Ordering in the Classical <Emphasis Type="Italic">XY</Emphasis> Model

Consider the classical XY model in a weak random external field pointing along the Y axis with strength ε. We study the behavior of this model as the range of the interaction is varied. We prove that in any dimension d≥2 and for all ε sufficiently small, there is a range L=L(ε) so that whenever the inverse temperature β is larger than some β(ε), there is strong residual ordering along the X direction.     

On the spectrum of narrow periodic waveguides

This is a continuation of [1] and [2]. We consider the spectrum of the Dirichlet Laplacian on the domain {(x, y) : 0 < y < εh(x)}, where h(x) is a positive periodic function. The main assumption is that h(x) has one point of global maximum on the period interval. We study the location of bands and prove that the band lengths decay exponentially as ε → 0. To the memory of Leonid Romanovich Volevich     

Husimi Operator for Describing Probability Distribution of Electron States in Uniform Magnetic Field Studied by Virtue of Entangled State Representation

For the first time we introduce an operator Δ _h (γ,ε;κ) for studying Husimi distribution function in phase space (γ,ε) for electron’s states in uniform magnetic field, where κ is the Gaussian spatial width parameter. The marginal distributions of the Husimi function are Gaussian-broadened version of the Wigner marginal distributions. Using the Wigner operator in the entangled state 〈λ | representation we find that Δ _h (γ,ε;κ) is just a pure squeezed coherent state density operator | γ,ε〉 _{κ κ} 〈γ,ε |, which brings much convenience for studying Husimi distribution, so we name Δ _h (γ,ε;κ) the Husimi operator. We then derive Husimi operator’s normally ordered form that provides us with an operator version to examine various properties of the Husimi distribution. Work supported by the National Natural Science Foundation under the grant: 10775097.