Possible generalization of Boltzmann-Gibbs statistics

With the use of a quantity normally scaled in multifractals, a generalized form is postulated for entropy, namelyS _q k [1 – _i=1 ^W p _i ^q ]/(q-1), whereq characterizes the generalization andp _i are the probabilities associated withW (microscopic) configurations (W). The main properties associated with this entropy are established, particularly those corresponding to the microcanonical and canonical ensembles. The Boltzmann-Gibbs statistics is recovered as theq1 limit.     

Sensitivity of approximate formulas for determining optical constants of thin films

The sensitivity of approximate formulas for determining the optical constants of thin films using measurement of reflectancesR and transmittancesT at normal incidence have been investigated theoretically. The ranges of refractive indexn, absorption indexk,2nk (=₂) andn ²–k ²(=₁) within relative errors of 5%, 10%, and 20% may be obtained. Selected signs of (₁)⁺ or (₁)^– have been determined. Validity of the condition n₀ A=n _s A has been also evaluated (A=1–R –T andA=1–R–T).     

On the asymptotic behaviour of infinite gradient systems

We consider gradient systems of infinitely many particles in one-dimensional space interacting via a positive invariant pair potential with a hard core. The main assumption is that is strictly convex within the rangeR of (whereR is a fixed number ). Under some technical conditions we prove the following theorems: Let the initial distribution be given by a translation invariant point process onR ¹. Then there exists only one extreme equilibrium state with a given intensityI() satisfyingI()R ^–1, and all ergodic initial distributions with an intensityI()R ^–1 converge weakly ast to the extreme equilibrium state with the same intensity.     

Self-consistent calculation of statics and dynamics of the roughening transition

Statics and dynamics of the modified kinetic discrete Gaussian model are treated selfconsistently using a Gaussian probability assumption. A non-trivial roughening temperatureT _R is found in exactly two dimensions only. The free energyF, the correlation length and the interface roughness h ² are found to behave—lnFlnh ²(T _R –T)^–1 for temperaturesT approachingT _R from below. The linear relaxation rate of the order parameter is found to be proportional to ^–2. As a model for crystal growth, the growth rate depends linearly upon the chemical potential difference aboveT _R , shows a metastable regime belowT _R with a spinodal limit of metastability _c , beyond which oscillatory growth starts. The critical behavior of _c is found to be ln _c –(T _R –T)^–1+O(ln (T _R –T)).     

Small representations of quantum affine algebras

We characterize the finite-dimensional representations of the quantum affine algebra U _q ( _n+1) (whereq ^× is not a root of unity) which are irreducible as representations of U _q (sl _n+1). We call such representations small. In 1986, Jimbo defined a family of homomorphismsev _a from U _q (sl _n+1) to (an enlargement of) U _q (sl_,n+1), depending on a parametera ^·. A second family,ev ^a can be obtained by a small modification of Jimbo's formulas. We show that every small representation of U _q ( _n+1) is obtained by pulling back an irreducible representation of U _q (sl _n+1) byev _a orev ^a for somea ^·.     

Infrared lattice bands of some pearls

From analysis of anisotropical lattice bands properties of 50 reflection spectra both of the CO stretching and bending bands measured from some pearl (Ca⁺⁺CO ₃ ^–– or Ca⁺⁺HCO ₃ ^–– layer) we discussed following subjects.i) Quantized properties present both in reflectivity and in energy. ii) classifications of the Optical Activity. iii) Polar distributions of the CO₃ oscillators in Ca⁺⁺CO ₃ ^–– surface mono-layer. iv) Force constants of these oscillators. v) Step variation of the dipolemoment and their influences to the degree of Optical Activity. vi) Two types of hysteresis loops of the values of Y_N (M_2Jbend ()/M_1Jstret. ()) derived from the oscillators which are at innert-state, at weak active-state and at active-state.     

Schrödinger Operators with Empty Singularly Continuous Spectra

Let H be a semibounded perturbation of the Laplacian H ₀ in L ²( ^d ). For an admissible function sufficient conditions are given for the completeness of the scattering system (H), (H ₀). If is the exponential function and if e^{– H} is an integral operator we denote the kernel of the difference D = e^{– H} – e^{– H} ₀ by D (x, y), > 0. The singularly continuous spectrum of H is empty if^d dx ^d dy |D(x,y)| (1 + |y|²)< for some > 1. This result is applied to potential perturbations and to perturbations by imposing Dirichlet boundary conditions.     

A solvable model exhibiting a glass-like transition

A nonlinear equation of motion of an overdamped oscillator exhibiting a glass-like transition at a critical coupling constant _c is presented and solved exactly. Below _c , in the fluid phase, the oscillator coordinatex(t) decays to zero, while above _c , in the amorphous phase, it decays to a nonzero infinite time limit. Near _c the motion is slowed down by a nonlinear feedback mechanism andx(t) decays exponentially to its long time limit with a relaxation time diverging as (1 – / _c )^–3/2 and (/ _c –1)^–1 for < _c and > _c respectively. At _c x(t) exhibits a power law decay proportional tot ^– with exponent -1/2.     

Excited-state absorption and third-order optical nonlinearities in symmetric π-electron organic molecules

Excited-State Absorption (ESA), Two-Photon Absorption (TPA) and the third-order polarizability (;,, – ) have been investigated for a model dichloride derivative of a symmetrically substituted benzylidene analine (SBAC), using a multielectron configuration-interaction procedure. The calculations indicate that SBAC exhibits ESA across the visible region of the spectrum, but that it is not as extensive as for molecules such as the phthalocyanines. The magnitude of the third-order polarizability is dominated by resonance enhancement from a very strongA _g B _u one-photon absorption. The calculated off-resonance value for (;,, – ) suggests that SBAC is a potential candidate for ultrafast switching applications.     

Note on the interfacial tension of phase-separated polymer solutions

Earlier theoretical calculations of the interfacial tension of phase-separated polymer solutions as a function of the degree of polymerizationN and the temperatureT, based partly on the mean-field approximation, had led toN ^–1/4(1–T/T _c )^3/2 for fixedN1 andT approaching the critical solution temperatureT _c It is here remarked that the scaling procedure of de Gennes then modifies this toN ^–0.37(1–T/T _c )^1.26, which is in close accord with the experimentalN ^–0.44(1–T/T _c )^1.26. The simplest mean-field picture yieldsN ^–1/2(1–T/T _c )^3/2.     

Spectral superbroadening of self-focused picosecond laser pulses in D2O

Intense picosecond light pulses of a mode-locked Nd: glass laser at _L =1054 nm (fundamental wavelength) and _SH =527 nm (second harmonic wavelength) are passed through a sample of D₂O under self-focusing conditions. Spectrally structured superbroadened, spatially bell-shaped emission in the forward direction is obtained. Primary generation processes are pump-pulse-degenerate stimulated parametric four-photon interaction (₁ + _{1 3} + ₄) and stimulated Raman scattering (_{1 R} + ), which occur concurrently (₁= _L or _SH angular pump frequency, #x03C9; _R first Stokes frequency, #x03C9; signal frequency, #x03C9;₃ signal frequency, #x03C9;₄ idler frequency). The parametric four-photon interaction occurs under collinear non-phase-matched conditions and under longitudinally phase-matched, transversally non-phase-matched (erenkov-like) conditions. Subsequent interaction processes are pump-pulse-nondegenerate four-photon interaction of the type ₁ + _{R 3} + ₄, coherent antiStokes Raman scattering (CARS, ₁ + ₁ – _{4 3}), inverse Raman scattering ( _A + ₁ + ), and cascading light up-conversion of the type ₁ + _(i) – _{R (i+1)}.     

On the indistinguishability of classical particles

If no property of a system of many particles discriminates among the particles, they are said to be indistinguishable. This indistinguishability is equivalent to the requirement that the many-particle distribution function and all of the dynamic functions for the system be symmetric. The indistinguishability defined in terms of the discrete symmetry of many-particle functions cannot change in the continuous classical statistical limit in which the number density n and the reciprocal temperature become small. Thus, microscopic particles like electrons must remain indistinguishable in the classical statistical limit although their behavior can be calculated as if they move following the classical laws of motion. In the classical mechanical limit in which quantum cells of volume (2)³ are reduced to points in the phase space, the partition functionTr{exp(–) for N identical bosons (fermions) approaches (2)^–3N(N!) ... d³r₁ d³p₁ ... d³r_N d³p_N exp(–H). The two factors, (2)^–3N and (N!)^–1, which are often added in anad hoc manner in many books on statistical mechanics, are thus derived from the first principles. The criterion of the classical statistical approximation is that the thermal de Broglie wavelength be much shorter than the interparticle distance irrespective of any translation-invariant interparticle interaction. A new derivation of the Maxwell velocity distribution from Boltzmann's principle is given with the assumption of indistinguishable classical particles.     

Local decay of scattering solutions to Schrödinger's equation

The main theorem asserts that ifH=+gV is a Schrödinger Hamiltonian with short rangeV, L _compact ² (IR³), andR>0, then exp(iHt) _{S L} ² _(|x|<R)=O(t ^–1/2), ast where _S is projection onto the orthogonal complement of the real eigenvectors ofH. For all but a discrete set ofg,O(t ^–1/2) may be replaced byO(t ^–3/2).Research supported by the National Science Foundation under grants NSF GP 34260 and MCS 72-05055 A04     

R andRτ ratios at the five-loop level of perturbative QCD

We study perturbative QCD at the five-loop level. In particular we considerR = _tot(e ⁺ e ^– hadrons)/(e ⁺ e ^{– + –}) andR = ( v+hadrons)/( ev). We use our method to estimate the five-loop coefficients. As a result, we obtain _s (M _z ) = 0.1186(11) and _s (34 GeV) = 0.1396(16), which are accurate at the 1% level. We also findR = 3.8350(18), which is consistent withR and is accurate to 0.05%.     

Negative muon precession in nuclei with spin

Polarized negative muons were stopped in various materials containing nuclei with nonzero spin. The TF-SR precession signal of theF ⁺ hyperfine state (frequencyv ₊ > 0) was pronounced for ^–Li and ^–Be, faint for ^–Cl (in NiCl₂), and undetectable for ^–F (in CaF₂ or NiF₂) and ^–P. TheF ^– signal (frequencyv _– < 0) was observed clearly for ^–Be, ^–Al and ^–Na, marginally for ^–K, ^–V and ^–Nb, and not at all for ^–Ga (at 4 kOe) or ^–Co (ferrromagnetic, zero field). In the heavier elements theF ^– signal is fed by transitions from theF ⁺ to theF ^– state at a rateR, as long asR(v ₊ –v _–). [See separate paper on ^–Al in these Proceedings, p. 879.]I am grateful to Alex Schenck and Bruce Patterson for the loan of equipment and samples, to Fred Gygax for helping set up the apparatus, to Jun Imazato for help with the experiment, and to Toshi Yamazaki and Tak Suzuki for valuable comments. I am also indebted to SIN for several weeks of free ^– beam time and to SIN, BOOM, and the University of Tokyo for free time on their VAX computers, which were kept busy for several months in the analysis.     

Ising Models on Hyperbolic Graphs II

We consider Ising models with ferromagnetic interactions and zero external magnetic field on the hyperbolic graph (v, f), where v is the number of neighbors of each vertex and f is the number of sides of each face. Let T _c be the critical temperature and T _c =supTT _c: ^f=( ⁺+ ^–)/2, where ^f is the free boundary condition (b.c.) Gibbs state, ⁺ is the plus b.c. Gibbs state and ^– is the minus b.c. Gibbs state. We prove that if the hyperbolic graph is self-dual (i.e., v=f) or if v is sufficiently large (how large depends on f, e.g., v35 suffices for any f3 and v17 suffices for any f17) then 0<T _c <T _c, in contrast with that T _c =T _c for Ising models on the hypercubic lattice Z ^d with d2, a result due to Lebowitz.⁽²²⁾ While whenever T<T _c , ^f=( ⁺+ ^–)/2. The last result is an improvement in comparison with the analogous statement in refs. 28 and 33, in which it was only proved that ^f=( ⁺+ ^–)/2 when TT _c and it remains to show in both papers that ^f =( ⁺+ ^–)/2 whenever T<T _c . Therefore T _c and T _c divide [0, ] into three intervals: [0, T _c ), (T _c , T _c), and (T _c, ] in which ^{+ –} but ^f =( ⁺+ ^–)/2, ^{+ –} and ^f ( ⁺+ ^–)/2, and ⁺= ^–, respectively.     

Spectral asymptotics for the Schrödinger operator with potential which steadies at infinity

We consider the discrete spectrum of the selfadjoint Schrödinger operatorA _h =–h ² +V defined inL ²(^m) with potentialV which steadies at infinity, i.e.V(x)=g+|x|^– f(1+o(1)) as |x| for>0 and some homogeneous functionsg andf of order zero. Let _h (),0, be the total multiplicity of the eigenvalues ofA _h smaller thanM–, M being the minimum value ofg over the unit sphereS ^m–1 (hence,M coincides with the lower bound of the essential spectrum ofA _h ). We study the asymptotic behaviour of ₁() as0, or of _h () ash0, the number0 being fixed. We find that these asymptotics depend essentially on the structure of the submanifold ofS ^m–1, where the functiong takes the valueM, and generically are nonclassical, i.e. even as a first approximation (2) ^m _h () differs from the volume of the set {(x, )^2m:h ²||²+V(x)<M–}.Partially supported by Contract No. 52 with the Ministry of Culture, Science and Education     

Fractional Moment Estimates for Random Unitary Operators

We consider unitary analogs of d-dimensional Anderson models on l²( $$\mathbb(z)$$^d) defined by the product U=D S where S is a deterministic unitary and D is a diagonal matrix of i.i.d. random phases. The operator S is an absolutely continuous band matrix which depends on parameters controlling the size of its off-diagonal elements. We adapt the method of Aizenman–Molchanov to get exponential estimates on fractional moments of the matrix elements of U(U –z)^–1, provided the distribution of phases is absolutely continuous and the parameters correspond to small off-diagonal elements of S. Such estimates imply almost sure localization for U.     

One-dimensional harmonic lattice caricature of hydrodynamics

We derive the hydrodynamic (Euler) approximation for the harmonic time evolution of infinite classical oscillator system on one-dimensional lattice ¹ It is known that equilibrium (i.e., time-invariant attractive) states for this model are translationally invariant Gaussian ones, with the mean 0, which satisfy some linear relations involving the interaction quadratic form. The natural parameter characterizing equilibrium states is the spectral density matrix function (SDMF)F(), [– , ). Time evolution of a space profile of local equilibrium parameters is described by a space-time SDMFF(t;x, ) t, xR ¹. The hydrodynamic equation forF(t; x, ) which we derive in this paper means that the normal mode profiles indexed by are moving according to linear laws and are mutually independent. The procedure of deriving the hydrodynamic equation is the following: We fix an initial SDMF profileF(x, ) and a familyP ,>0 of mean 0 states which satisfy the two conditions imposed on the covariance of spins at various lattice points: (a) the covariance at points close to the value ^–1 x in the stateP is approximately described by the SDMFF(x, ); (b) The covariance (on large distances) decreases with distance quickly enough and uniformly in. Given nonzerotR ¹, we consider the states P _–1 ,>0, describing the system at the time moments ^–1 t during its harmonic time evolution. We check that the covariance at lattice points close to ^–1 x in the state P _–1 is approximately described by a SDMFF(t;x, ) and establish the connection betweenF(t; x, ) andF(x,).     

Generalized Transvectants-Rankin–Cohen Brackets

We introduce o(p+1q+1)-invariant bilinear differential operators on the space of tensor densities on Rⁿ generalizing the well-known bilinear sl₂-invariant differential operators in the one-dimensional case, called Transvectants or Rankin–Cohen brackets. We also consider already known linear o(p+1q+1)-invariant differential operators given by powers of the Laplacian.