Geometric expansion of the boundary free energy of a dilute gas

We consider a dilute classical gas in a volume ^–1 which tends to ^d by dilation as 0. We prove that the pressurep(^–1) isC ^q in at =0 (thermodynamic limit), for anyq, provided the boundary isC ^q and provided the Ursell functionsu _n (x ₁, ...,x _n) admit moments of degreeq and have nice derivatives.     

Finite size scaling at first order phase transitions?

It is demonstrated that finite size scaling at first order phase transitions is something basically very simple: As the number of particlesN in the system goes to infinity,s _N (), the entropy per particle, rapidly approaches its limiting behaviours (). Onces () has been determined, the thermal behaviour of the infinite system is completely known and in case of a first order phase transition the specific heat exhibitis a -function singularity. If, however, the specific heatc _N (T) per particle is calculated from the canonical partition functionZ _N ()=d exp {N[s _N ()-]}, then even ifs _N () is replaced by its limiting forms (),c _N (T) only exhibits a peak with a finite maximum value proportional toN which is due to the explicit factorN in front of the angular bracket in the exponent. This is theN-dependence which has recently been called finite size scaling at first order phase transitions. The entropys _N () can very efficiently be determined in the dynamical ensemble.     

Photonic pseudogaps for periodic dielectric structures

We consider the problems of existence and structure of gaps (pseudogaps) in the spectra associated with Maxwell equations and equations that govern the propagation of acoustic waves in periodic two-component media. The dielectric constant is assumed to be real and positive, and the value of = _b on the background is supposed to be essentially larger than the value of = _a on the embedded component. We prove the existence of pseudogaps in the spectra of the relevant operators. In particular, we give an accurate treatment of the term pseudogap. We also show that if the contrast _b / _a approaches infinity, then the bands of the spectrum shrink to a discrete set which can be identified with the set of eigenvalues of a Neumann-type boundary value problem and thus can be effectively calculated.     

Ising-Type and Other Transitions in One-Dimensional Coupled Map Lattices with Sign Symmetry

We consider a one-dimensional lattice of expanding antisymmetric maps [–1, 1][–1, 1] with nearest neighbor diffusive coupling. For such systems it is known that if the coupling parameter is small there is unique stationary (in time) state, which is chaotic in space-time. A disputed question is whether such systems can exhibit Ising-type phase transitions as grows beyond some critical value _c. We present results from computer experiments which give definite indication that such a transition takes place: the mean square magnetization appears to diverge as approaches some critical value, with a critical exponent around 0.9. We also study other properties of the coupled map system.     

Unmixing of binary alloys by a vacancy mechanism of diffusion: a computer simulation

The initial stages of phase separation are studied for a model binary alloy (AB) with pairwise interactions _AA , _AB , _BB between nearest neighbors, assuming that there is no direct interchange of neighboring atoms possible, but only an indirect one mediated by vacancies (V) occurring in the system at a concentrationc _v and which are strictly conserved, as are the concentrationsc _A andc _B of the two species.A-atoms may jump to vacant sites with jump rate _A , B-atoms with jump rate _B (in the absence of interactions). Particular attention is paid to the question to what extent nonuniform distribution of vacancies affects the unmixing kinetics. Our study focuses on the special case _A = _B on a square lattice, considering three different choices of interactions with the same = _AB – ( _AA + _BB )/2: (i) _AB =, _AA = _BB = 0; (ii) _AA = 0, _AA = _BB ; = ; (iii) _AB = _BB = 0, _AA = –2. We obtain both the time evolution of the structure factorS(k,t) following a quench from infinite temperature to the considered temperature, and the timedependence of the mean cluster size and the various neighborhood probabilities of a vacancy. While in case (i) forc _V 0.16 the distribution of vacancies in the system stays nearly random, in case (ii) the vacancies cluster in theA-B interfacial region, and in case (iii) they get nearly completely expelled from theA-rich regions. While phase separation proceeds in case (i) only slightly faster than in case (ii), a significant slowing down of the relaxation is observed for case (iii), which shows up in a strong reduction of the effective exponents describing the growth.     

On the Hamiltonian interpolation of near-to-the identity symplectic mappings with application to symplectic integration algorithms

We reconsider the problem of the Hamiltonian interpolation of symplectic mappings. Following Moser's scheme, we prove that for any mapping , analytic and -close to the identity, there exists an analytic autonomous Hamiltonian system, H such that its time-one mapping _H differs from by a quantity exponentially small in 1/. This result is applied, in particular, to the problem of numerical integration of Hamiltonian systems by symplectic algorithms; it turns out that, when using an analytic symplectic algorithm of orders to integrate a Hamiltonian systemK, one actually follows exactly, namely within the computer roundoff error, the trajectories of the interpolating Hamiltonian H, or equivalently of the rescaled Hamiltonian K=^-1H, which differs fromK, but turns out to be ⁵ close to it. Special attention is devoted to numerical integration for scattering problems.     

Band-gap structure of the spectrum of periodic Maxwell operators

We investigate the band-gap structure of some second-order differential operators associated with the propagation of waves in periodic two-component media. Particularly, the operator associated with the Maxwell equations with position-dependent dielectric constant (x),xR ³, is considered. The medium is assumed to consist of two components: the background, where (x) = _b , and the embedded component composed of periodically positioned disjoint cubes, where (x) = _a . We show that the spectrum of the relevant operator has gaps provided some reasonable conditions are imposed on the parameters of the medium. Particularly, we show that one can open up at least one gap in the spectrum at any preassigned point provided that the size of cubesL, the distancel=L betwen them, and the contrast = _b / _a are chosen in such a way thatL ^–2, and quantities ^-1-3/2 and ² are small enough. If these conditions are satisfied, the spectrum is located in a vicinity of widthw(^3/2)^-1 of the set {² L ^-2 k ²:kZ³}. This means, in particular, that any finite number of gaps between the elements of this discrete set can be opened simultaneously, and the corresponding bands of the spectrum can be made arbitrarily narrow. The method developed shows that if the embedded component consists of periodically positioned balls or other domains which cannot pack the space without overlapping, one should expect pseudogaps rather than real gaps in the spectrum.     

s-Polarized nonlinear surface polaritons

An exact solution of Maxwell's equations is found, corresponding to ans-polarized nonlinear surface polarition, at the planar interface between two dielectric media, one of which is optically unaxial and is characterized by a diagonal dielectric tensor whose elements depend on the amplitude of the electric field according to ₁₁=₂₂=₀()+a() (|E ₁|²+|E ₂|²), ₃₃=(). Such modes have no counterpart in the corresponding linear system.     

Random shearing direction models for isotropic turbulent diffusion

Recently, a rigorous renormalization theory for various scalar statistics has been developed for special modes of random advection diffusion involving random shear layer velocity fields with long-range spatiotemporal correlations. New random shearing direction models for isotropic turbulent diffusion are introduced here. In these models the velocity field has the spatial second-order statistics of an arbitrary prescribed stationary incompressible isotropic random field including long-range spatial correlations with infrared divergence, but the temporal correlations have finite range. The explicit theory of renormalization for the mean and second-order statistics is developed here. With the spectral parameter, for –<<4 and measuring the strength of the infrared divergence of the spatial spectrum, the scalar mean statistics rigorously exhibit a phase transition from mean-field behavior for <2 to anomalous behavior for with 2<<4 as conjectured earlier by Avellaneda and the author. The universal inertial range renormalization for the second-order scalar statistics exhibits a phase transition from a covariance with a Gaussian functional form for with <2 to an explicit family with a non-Gaussian covariance for with 2<<4. These non-Gaussian distributions have tails that are broader than Gaussian as varies with 2<<4 and behave for large values like exp(–C _c |x|^4–), withC _c an explicit constant. Also, here the attractive general principle is formulated and proved that every steady, stationary, zero-mean, isotropic, incompressible Gaussian random velocity field is well approximated by a suitable superposition of random shear layers.     

A stochastic return map for stochastic differential equations

A method is presented for constructing a stochastic return map from a stochastic differential equation containing a locally stable limit cycle and small-amplitude [O()] additive Gaussian colored noise. The construction is valid provided the correlation time isO() orO(1). The effective noise in the return map has nonzeroO( ²) mean and is state dependent. The method is applied to a model dynamical system, illustrating how the effective noise in the return map depends on both the original noise process and the local deterministic dynamics.     

Power law scaling of the top Lyapunov exponent of a Product of Random Matrices

A sequence of i.i.d. matrix-valued random variables with probabilityp and with probability 1–p is considered. Leta() = a ₀ + O(), c() = c ₀ + O() lim ₀ b() = Oa ₀,c ₀, >0, andb()>0 for all >0. It is shown show that the top Lyapunov exponent of the matrix productX _n X _n-1...X ₁, = lim_n (1/n) n X _n X _n-1...X ₁ satisfies a power law with an exponent 1/2. That is, lim ₀(ln /ln ) = 1/2.     

Reflection and Absorption Spectra of Fluorite in the UV Spectral Region

We have calculated the R(E) and ₁(E) spectra from the theoretical ₂(E) spectra of five models in the region 8–27 eV and the ₂(E) and ₁(E) spectra from the experimental R(E) spectrum in the region 6–35 eV. The results are compared with the known theoretical ₂(E) spectra of five models. The basic features of all of the R(E), ₂(E), and ₁(E) spectra have been revealed. It is established that the experimental R(E) spectrum and the ₂(E) and ₁(E) spectra calculated with the use of experimental data are in good agreement with the results of theoretical calculations for the models of ₂(E). On the basis of the known theoretical calculations of the fluorite zones, a scheme of the nature of the principal maxima of R(E) and ₂(E) is suggested.     

Fine Structures of the Permittivity and Characteristic Losses of Electrons in Fluorite

Experimental-computational spectra of the permittivity and characteristic losses –Im^–1 for energies in the range 5–21 eV at a temperature of 4.2 K and theoretical spectra of and –Im^–1 of a fluorite crystal are resolved into elementary transition bands. The parameters of transition bands (energies of their maxima E _i, band halfwidths H _i and areas S _i, and oscillator forces f _i) are determined. A correlation of the spectral bands of and –Im^–1is established, and their specific features are elucidated.     

A renormalization group analysis of turbulent transport

The field-theoretic renormalization group is used to derive scaling relations for the transport of passive scalars by an incompressible velocity field with a specified energy spectrum. Results are obtained with the analog of the expansion of critical phenomena and compared to exact results which are available for shear flows in two dimensions.A 1/N expansion is proposed for the regions in which the expansion fails.     

Frequency Dependence of the Components of Dielectric Permeability and Characteristic Losses of Fluorite

The ₁(E), –Im^–1, and Re^–1 spectra of the fluorite crystal are calculated on the basis of the experimental (10–35 eV) and theoretical spectra ₂(E) (10–27 and 8–20 eV). They were employed to decompose the ₂(E) and –Im^–1 spectra into elementary components. The most intense transverse and longitudinal components of transitions and their parameters have been determined. The correlation between two types of components of transitions and their distinguishing features have been established.     

Some remarkable degenerations of quantum groups

We show that an irreducible representation of a quantized enveloping algebraU at a ^th root of 1 has maximal dimension (= ^N ) if the corresponding symplectic leaf has maximal dimension (=2N). The method of the proof consists of a construction of a sequence of degenerations ofU , the last one being aq-commutative algebraU ^(2N) . This allows us to reduce many problems concerningU to that concerningU ^(2N) .To Armand Borel on his 70^th birthdaySupported in part by the NSF grant DMS-9103792     

Invariant curves for area-preserving twist maps far from integrable

One-parameter families of area-preserving twist maps of the formF (x, y)=(x +y +f(x),y +f(x)) are considered. Various invariant curves, for the maps corresponding tof(x)=sin andf(x)=sinx+(1/50) sin(5x), are rigorously constructed forlarge values of the nonlinearity parameter . For larger values of , close to critical, some numerical experiments are briefly discussed.     

P-Polarized nonlinear surface polaritons in layered structures

We found an exact solution of Maxwell's equations, which describes the propagation ofp-polarized nonlinear surface polaritons and ofp-polarized nonlinear guided wave polaritons in two cases:i) in a film of a surface active material placed on a substrate described by a diagonal dielectric tensor whose elements depend on the amplitude of the electric field according to _{1 1}=_{2 2}= + (|E ₁|² + |E ₂|²), _{3 3}=, andii) in a film described by the same dielectric tensor (optically uniaxial nonlinear crystal) placed on a substrate with dielectric constant ₃ (optically linear medium). The power carried in the surface waves has also been exactly calculated.     

Gravitational collapse with charge and small asymmetries I. Scalar perturbations

A recent analysis by Richard Price of spherical collapse with small nonspherical perturbations is here generalized to the case of an electrically charged collapsing star (0¦Q¦-M). The perturbations are confined to a scalar field generated by a nonspherical distribution of scalar charge in the star. As in the electrically neutral case, the scalar perturbations are probably a prototype for all others — electromagnetic, gravitational, and higherspin. The collapse is shown to produce a Reissner-Nordström black hole, and the scalar-field perturbations are shown to radiate completely away; but they die out more slowly the larger is the star's electric charge. For charge ¦Q¦M, the -pole part of the perturbation at fixedr and late times is dominated by a tail that dies out ast ^{–(2+ 2)}. But for ¦Q¦=M, the primary outgoing waves emitted from the star's surface are everywhere larger than the tail. At fixedr and late times they die as t^–(+2). Also, a calculation of the redshift shows that a collapsing star becomes black more slowly the larger is the star's electric charge.Supported in part by the National Science Foundation (GP-27304, GP-28027, GP-19887).     

Dielectric constant characterization of large-area substrates in millimeter wave band

A possibility of the dielectric constant measurement for substrates with permittivity=+i without an essential restriction on their area has been shown experimentally. The method uses frequency measurement of quasioptical dielectric resonator (QDR) with two slots oriented along the QDR radius with a dielectric substrate in one of them. Taking QDR of teflon in 8mm waveband as an example it is found that measurable values of can ran up 15 _q , where _q is the QDR material permittivity. Absolute error of the measurements is determined by an accuracy with which the permittivity of calibrated (standard) samples is known. The relative measuring error is determined by the accuracy of the QDR frequency measurement and can be quite a small. As an example the method is demonstrated forLaAlO ₃ single crystals.