On the layering transition of an SOS surface interacting with a wall. I. Equilibrium results

We consider the model of a 2D surface above a fixed wall and attracted toward it by means of a positive magnetic fieldh in the solid-on-solid (SOS) approximation when the inverse temperature is very large and the external fieldh is exponentially small in . We improve considerably previous results by Dinaburg and Mazel on the competition between the external field and the entropic repulsion with the wall, leading, in this case, to the phenomenon of layering phase transitions. In particular, we show, using the Pirogov-Sinai scheme as given by Zahradník, that there exists a unique critical valueh _k ^* () in the interval (1/4e ^–4k , 4e ^–4k ) such that, for allh(h _k+1 ^* ,h _k ^* ) and large enough, there exists a unique infinite-volume Gibbs state. The typical configurations are small perturbation of the ground state represented by a surface at heightk+1 above the wall. Moreover, for the same choice of the thermodynamic parameters, the influence of the boundary conditions of the Gibbs measure in a finite cube decays exponentially fast with the distance from the boundary. Whenh=h _k ^* () we prove instead the convergence of the cluster expansion for bothk andk+1 boundary conditions. This fact signals the presence of a phase transition. In the second paper of this series we will consider a Glauber dynamics for the above model and we will study the rate of approach to equilibrium in a large finite cube with arbitrary boundary conditions as a function of the external fieldh. Using the results proven in this paper, we will show that there is a dramatic slowing down in the approach to equilibrium when the magnetic field takes one of the critical values and the boundary conditions are free (absent).     

Rayleigh-Taylor instability of a composite medium with variable viscosity in hydromagnetics

The frictional effect of collisions of ionized with neutral atoms on the Rayleigh-Taylor instability of a composite medium with variable viscosity is considered in the presence of a horizontal magnetic field. It is found that the simultaneous presence of viscosity, magnetic field and collisions has a stabilizing effect and completely stabilizes the wave-number bandk _* wherek _*=(k _x ² V²L/g). The collisions have no effect as such on the stratification, i.e., stable configuration remains stable and unstable configuration remains unstable. However the growth rate, under either of conditions (25), decreases with the increase of collisions.     

Quantum Ergodicity of Boundary Values of Eigenfunctions

Suppose that ⁿ is a bounded, piecewise smooth domain. We prove that the boundary values (Cauchy data) of eigenfunctions of the Laplacian on with various boundary conditions are quantum ergodic if the classical billiard map on the ball bundle B^*() is ergodic. Our proof is based on the classical observation that the boundary values of an interior eigenfunction , =² is an eigenfunction of an operator F_h on the boundary of with h=^–1. In the case of the Neumann boundary condition, F_h is the boundary integral operator induced by the double layer potential. We show that F_h is a semiclassical Fourier integral operator quantizing the billiard map plus a small remainder; the quantum dynamics defined by F_h can be exploited on the boundary much as the quantum dynamics generated by the wave group were exploited in the interior of domains with corners and ergodic billiards in the work of Zelditch-Zworski (1996). Novelties include the facts that F_h is not unitary and (consequently) the boundary values are equidistributed by measures which are not invariant under and which depend on the boundary conditions. Ergodicity of boundary values of eigenfunctions on domains with ergodic billiards was conjectured by S. Ozawa (1993), and was almost simultaneously proved by Gerard-Leichtnam (1993) in the case of convex C^1,1 domains (with continuous tangent planes) and with Dirichlet boundary conditions. Our methods seem to be quite different. Motivation to study piecewise smooth domains comes from the fact that almost all known ergodic domains are of this form.The first author was partially supported by an Australian Research Council Fellowship.The second author was partially supported by NSF grant #DMS-0071358 and DMS-0302518.     

Trapped modes in a T-shaped waveguide

We study the discrete spectrum of the Laplace operator with the Dirichlet boundary conditions in a plane waveguide made of a band of unit width and half-band of width h perpendicular to it. The existence of critical width h _* ∈ (1,2] is established: in the case of h ∈ (0, h_*), the discrete spectrum σ _d ^h of the problem contains only one eigenvalur λ₁ ^h , but σ _d ^h = ⊘ at h ≥ h _*. The asymptotics of the eigenvalue λ₁ ^h at h → + 0 is found.     

Interferometric phase measurements of average field cubeE ω 2 E 2ω *

The method of interferometric measurements of the field cube evolutionE ² E ₂ ^* due to propagation in air and other media was investigated experimentally by the use of a pair of nonlinear crystals. A gas cell with variable pressure turned out to be the most convenient among different methods of changing the phase of the field cube. The phase shift by rad of the field cube in passing through the focus was demonstrated experimentally.     

Equivalences of the large deviation principle for Gibbs measures and critical balance in the Ising model

In this paper we obtain the equivalence of the large deviation principle for Gibbs measures with and without an external field. For the Ising model, the equivalence allows us to study the result of competing influences of a positive external fieldh and a negative boundary condition in the cube ((B/h) ash0 for variousB. We find a critical balance at a valueB ₀ ofB.     

Monte Carlo renormalization of hard sphere polymer chains in two to five dimensions

A renormalization group for polymer chains with hard-core interaction is considered, where a chain ofN ₀ links of lengthl ₀ and hard-core diameterh ₀ is mapped onto a chain ofN ₁=N ₀/s links of lengthl ₁ and hard-core diameterh ₁. The lengthl ₁ is defined in terms of suitable interior distances of the original chain, andh ₁ is found from the condition that the end-to-end distance is left invariant. This renormalization group procedure is carried through by various Monte-Carlo methods (simple sampling is found advantageous for short enough chains or high dimensionalities, while dynamic methods involving kinkjumps or reptation are used else). Particular attention is paid to investigate systematic errors of the method by checking the dependence of the results on bothN ₀ ands. It is found that for dimensionalitiesd=2, 3 only the nontrivial fixed-point is stable, where upon iteration the ratio _k =h _k /l _k tends to nonzero fixed-point value ^*, while ford=4,5 the method converges to the gaussian fixed point with ^*=0. Taking both statistical and systematic errors into account, we estimate the exponentv asv=0.74±0.01 (d=2) andv =0.59±0.01 (d=3). The results are consistent with the expected crossover exponents =1/2 (d=3) and =1 (d=2), respectively.     

Dynamic regimes in films with a periodic array of antidots

We have studied the dynamic response of Pb thin films with a square array of antidots by means of ac susceptibility measurements. At low enough ac drive amplitudes h, vortices moving inside the pinning potential give rise to a frequency- and h-independent response together with a scarce dissipation. For higher amplitudes, the average distance travelled by vortices surpasses the pinning range and a critical state develops. We found that the boundary h^*(H,T) between these regimes smoothly decreases as T increases whereas a step-like behavior is observed as a function of field. We demonstrate that these steps in h^*(H) arise from sharp changes in the pinning strength corresponding to different vortex configurations. For a wide set of data at several fields and temperatures in the critical state regime, we show that the scaling laws based on the simple Bean model are satisfied.Received: 13 October 2003, Published online: 19 February 2004PACS: 74.78.Na Mesoscopic and nanoscale systems - 74.25.Sv Critical currents - 74.78.Db Low-T_c films     

The Dynamics of Intramolecular Excited State Relaxation of <Emphasis Type="Italic">N</Emphasis>-Anthryl Substituted Pyridinium Cations

N-(1-Anthryl)-2,4,6-trimethyl-pyridinium (I), N-(2-anthryl)-2,4,6-trimethyl-pyridinium (II) and 10-(1-anthryl)-1,2,3,4,5,6,7,8-octahydro-acridinium cations (III) with anomalously high fluorescence Stokes shift have been investigated. Fluorescence kinetics analysis at various temperatures showed that in the range 293–77 K, the radiative deactivation rate constants (k_f) increase by 5.5 to 30 times. The low-temperature time-resolved emission spectra of I–III were found to be consistent with the model: A A^* B^* where A^* is the local excited twisted form and B^* is the relaxed more planar, bent conformer of the molecule. The rate constants of the excited relaxed state formation (k₁) and back reaction (k_–1) of compounds studied were estimated.     

Image-potential surface states on Ag(100): A reinvestigation

Employing inverse photoemission we have remeasured the energy dispersionE(k) of the lowest-lying image state on Ag(100) with improved energy resolution (electrons and photons, E=0.35 eV) andk-resolution (k<0.1 Å^–1). In a least-square fit with the binding energyE _B atk=0 and the effective massm ^* as parameters we obtainE _B =E _vac –0.67 eV andm ^*=1.5 m in agreement with our earlier findings but differing from the two-photon photoemission values (0.53 eV and 1.15 m).     

Phase diagram and scattering intensity of binary amphiphilic systems

A Ginzburg-Landau model with a scalar and a vector order parameter, which describe the concentration and orientation of the amphiphile, respectively, is used to study the phase diagram and the scattering intensity of binary amphiphilic systems. With increasing amphiphile concentration, the calculated phase diagram shows the typical sequence of ordered phases observed experimentally, that is micellar liquid cubic micellar hexagonal lamellar cubic bicontinuous invers hexagonal. The scattering intensity in the homogeneous phase is calculated in the oneloop approximation. In the vicinity of a phase transition to an ordered phase, the intensity is found to show a 1/q behavior for not too small wave vectorsq, followed by a small peak, and a 1/q ² decay for large wave vectors, in agreement with experimental observations in theL ₃-(or sponge-)phase.Dedicated to Prof. H. Wagner on the occasion of his 60th birthday     

Estimates for Non-Resonant Normal Forms in Hamiltonian Perturbation Theory

We make a remark about an estimate of the rest for the non-resonant action-angle normal forms and exhibit a simple example suggesting the optimality of this estimate when there are no small divisors. Given a polynomial perturbation of degree P and an integer k, calling the size of the small denominators up to order k, we prove that the kth order remainder is bounded by (2/ ₀) ^k+1 with ₀=const ²/(kP ²). Thus, fixing the degree of the perturbation, if is independent of k (i.e., if there are no small divisors), we obtain a rest bounded by (const k) ^k+1. These estimates are also applied to the case in which the small divisors are absent, and they are conjectured to be optimal in this context. To support this idea we present a simplified model problem with no small denominators, formally related to the above calculations, and we show that it indeed has factorial divergence of its Birkhoff series. We also obtain Nekhoroshev's Theorem for harmonic oscillators. We hope that our simple approach makes more accessible to a general audience this important (although quite technical) topic.     

The Spectral Gap for the Kawasaki Dynamics at Low Temperature

In this paper we analyze the convergence to equilibrium of Kawasaki dynamics for the Ising model in the phase coexistence region. First we show, in strict analogy with the nonconservative case, that in any lattice dimension, for any boundary condition and any positive temperature and particle density, the spectral gap in a box of side L does not shrink faster than a negative exponential of the surface L ^d–1. Then we prove that, in two dimensions and for free boundary condition, the spectral gap in a box of side L is smaller than a negative exponential of L provided that the temperature is below the critical one and the particle density satisfies (*_–, *₊), where *_± represents the particle density of the plus and minus phase, respectively.     

Discontinuous behavior of effective transport coefficients in quasiperiodic media

We investigate the effective conductivity ^* of a quasiperiodic medium in ^d and the discontinuous dependence, found in ref. 1, of ^* on the wavelengths of the system. It was shown there, for example, that the effective conductivity ^*(k) for a layered medium with a one-dimensional local conductivity _k (x)=A+cosx+coskx, A>2, is discontinuous ink. An explicit class of higherdimensional examples which exhibit the discontinuity is constructed here. The conductivity ^*(k, L) of a sample of lengthL in one dimension asL is also analyzed and shown to have a plateau structure for any irrationalk well approximated by rationals.     

Incidence of electromagnetic waves on a plasma-vacuum boundary

An electromagnetic wave polarized in a plane normal to the magnetostatic field, parallel to the plasma boundary, falls from a vacuum on a half-space filled with magnetoactive plasma and bounded by a wall. The coefficient of wave reflection is computed for plasmas with ₀ ² / _h ² 1.The authors express their sincere thanks to Mr. J. Václavík for his discussions of the study.     

Integration theory of observables

Representations of abstract observables on a generalised logic are given in terms of bounded vector-valued Borel measures on the real line whose ranges are in the dual spaceX ^* of the Banach space of statesX. Each bounded observable is furthermore represented by an elementu ^* ofX ^* such that for any proper statep X, u ^* (p) is the expectation value ofu when the system is in the statep.     

Rotational energy transfer in I 2 * (B 3 Π) colliding with I2(X 1 Σ), Ar and He: Experiment and fitting law

Using the LIF method, we have measured the rate constants of rotational energy transfer for I ₂ ^* (B ³) in several rovibrational levels and for changes inj up to 32, with I₂(X ¹), Ar and He as collision partners. A new approach of data processing has been employed to deduce the rate constants from the experimental data. These rate constants were then correlated with fitting laws. It was found that the I ₂ ^* -I₂ and I ₂ ^* -Ar pairs follow the power gap law with the restriction of m _j=0, but the I ₂ ^* -He system has to be represented by an intermediate case between the power gap and exponential gap laws.     

Invariant subspaces of clustering operators

A class of clustering operators is defined which is a generalization of a transfer matrix of a Gibbs lattice field with an exponential decay of correlations. It is proved that for small values of the clustering operator has invariant subspaces which are similar tok-particle subspaces of the Fock space. The restriction of the clustering operator onto these subspaces resembles the operator exp(-H _k, whereH _k is thek- particle Schrödinger Hamiltonian in nonrelativistic quantum mechanics. The spectrum of eachH _k,k1, is contained in the interval (C ₁^k,C ₂^k). These intervals do not intersect with each other.     

A Second Eigenvalue Bound for the Dirichlet Schrödinger Operator

Let λ _i (Ω,V) be the i ^th eigenvalue of the Schrödinger operator with Dirichlet boundary conditions on a bounded domain $\Omega \subset \mathbb{R}^nLet λ _i (Ω,V) be the i ^th eigenvalue of the Schr?dinger operator with Dirichlet boundary conditions on a bounded domain and with the positive potential V. Following the spirit of the Payne-Pólya-Weinberger conjecture and under some convexity assumptions on the spherically rearranged potential V _*, we prove that λ₂(Ω,V) ≤ λ₂(S ₁,V _*). Here S ₁ denotes the ball, centered at the origin, that satisfies the condition λ₁(Ω,V)=λ₁(S ₁,V _*).Further we prove under the same convexity assumptions on a spherically symmetric potential V, that λ₂(B _R , V) / λ₁(B _R , V) decreases when the radius R of the ball B _R increases.We conclude with several results about the first two eigenvalues of the Laplace operator with respect to a measure of Gaussian or inverted Gaussian density.R.B. was supported by FONDECYT project # 102-0844.H.L. gratefully acknowledges financial support from DIPUC of the Pontifí cia Universidad Católica de Chile and from CONICYT.     

Spectral duality for planar billiards

For a bounded open domain with connected complement inR ² and piecewise smooth boundary, we consider the Dirichlet Laplacian- on and the S-matrix on the complement ^c . We show that the on-shell S-matricesS _k have eigenvalues converging to 1 askk ₀ exactly when-- has an eigenvalue at energyk ₀ ² . This includes multiplicities, and proves a weak form of transparency atk=k ₀. We also show that stronger forms of transparency, such asS _{k 0} having an eigenvalue 1 are not expected to hold in general.