Bifurcation Cascades and Self-Similarity of Periodic Orbits with Analytical Scaling Constants in Hénon–Heiles Type Potentials

We investigate the isochronous bifurcations of the straight-line librating orbit in the Hénon–Heiles and related potentials. With increasing scaled energy e, they form a cascade of pitchfork bifurcations that cumulate at the critical saddle-point energy e=1. The stable and unstable orbits created at these bifurcations appear in two sequences whose self-similar properties possess an analytical scaling behavior. Different from the standard Feigenbaum scenario in area preserving two-dimensional maps, here the scaling constants and corresponding to the two spatial directions are identical and equal to the root of the scaling constant that describes the geometric progression of bifurcation energies e_n in the limit n. The value of is given analytically in terms of the potential parameters.     

Derivations commuting with abelian gauge actions on lattice systems

Let be an action of a compact abelian groupG on aC*-algebraA, and assume that the fixed-point subalgebraA is an AF-algebra. We show that if is a closed *-derivation onA commuting with , and the restriction of toA generates a one-parameter group of *-automorphisms, then itself is a generator. In particular, the result applies if is an infinite product action ofG on a UHF algebra. Furthermore, if in this situation ₁ and ₂ are two derivations both satisfying the hypotheses on , and ₁ and ₂ have the same restriction toA , then there exists a one-parameter subgroup of the action with generator ₀ such thatD(₁)D(₂)D(₀) is a joint core for the three derivations, and ₂=₁+₀ on this core.     

What determinates the chemical changes of the internal conversion coefficients for inner atomic shells?

Calculations of internal conversion coefficients (ICC) of the E1–E4 and M1–M4 transitions for nuclei in ions show that the relative changes _i / _i of the ICC _i for deep inner subshells can differ significantly from the relative changes _i/_i of the electron densities _i at the nucleus. For the K conversion _i/ _i are many times greater than _i/_i. Especially large deviations of _i/ _i are characteristic of transitions of high multipolarity; however, for the M1 transitions they can also be significant. Illustrations of various dependencies of _i/ _iare presented for the conversion in the regionZ-50.     

Anomalous fluctuation and relaxation in unstable systems

The essential ideas of the scaling theory of transient phenomena proposed by the author for a single macrovariable near the instability point are extended to multi-macrovariables in nonequilibrium systems. The time region is divided into three regimes according to the scaling behavior of the fluctuating parts of the macrovariables. In the first regime, the fluctuation is Gaussian and it is described by the linearized stochastic equation (or linear Fokker-Planck equation). In the second regime, the fluctuation is non-Gaussian, but it is probabilistic or stochastic (not dynamical) in the sense that the stochastic nature comes from the probability distribution in the initial regime and that each representative motion is deterministic, namely a random force can be neglected asymptotically in the second regime. In the final regime, the fluctuation is again Gaussian. A fluctuation-enhancement theorem for multi-macrovariables is given, which states that the fluctuation becomes enhanced by the order of the system size in the second regime, which is of order log , if the initial system is located just at the unstable point. An anomalous fluctuation theorem for multi-macrovariables is also proven, which states that the fluctuation is anomalously enhanced in proportion to ^–2 at times of order log if the initial system deviates by from the unstable point.This work is partially financed by the Scientific Research Fund of the Ministry of Education.     

The Micro-Canonical Point Vortex Ensemble: Beyond Equivalence

The fluid limit N is constructed for a sequence of ensembles of N classical point vortices in a finite domain ² whose ensemble densities (w.r.t. Liouville measure) are Gaussian approximations to (E-H). Letting the variance 0 after N has been taken, one recovers the special class of nonlinear stationary Euler flows that is expected from the micro-canonical ensemble. The construction improves over previous ones which either had to regularize the logarithmic singularities of the point vortex Hamiltonian or had to assume equivalence of ensembles. In particular, nonequivalence between micro-canonical and canonical ensemble prevails for certain geometries where conditionally stable configurations with negative 'global vortex pair-specific heat' can and do exist in the micro-canonical but not in the canonical ensemble.     

The Scaling Limit of the Incipient Infinite Cluster in High-Dimensional Percolation. I. Critical Exponents

This is the first of two papers on the critical behavior of bond percolation models in high dimensions. In this paper, we obtain strong joint control of the critical exponents and for the nearest neighbor model in very high dimensions d6 and for sufficiently spread-out models in all dimensions d>6. The exponent describes the low-frequency behavior of the Fourier transform of the critical two-point connectivity function, while describes the behavior of the magnetization at the critical point. Our main result is an asymptotic relation showing that, in a joint sense, =0 and =2. The proof uses a major extension of our earlier expansion method for percolation. This result provides evidence that the scaling limit of the incipient infinite cluster is the random probability measure on ^d known as integrated super-Brownian excursion (ISE), in dimensions above 6. In the sequel to this paper, we extend our methods to prove that the scaling limits of the incipient infinite cluster's two-point and three-point functions are those of ISE for the nearest neighbor model in dimensions d6.     

High-temperature series expansion for the relaxation times of the two dimensional Ising model

We derive the high temperature series expansions for the two relaxation times of the single spin-flip kinetic Ising model on the square lattice. The series for the linear relaxation time _l is obtained with 20 non-trivial terms, and the analysis yields 2.183±0.005 as the value of the critical exponent _l , which is equal to the dynamical critical exponentz in the two-dimensional case. For the non-linear relaxation time we obtain 15 non-trivial terms, and the analysis leads to the results _nl = 2.08 ± 0.07. The scaling relation _l – _nl = ( being the exponent of the order parameter) seems to be fulfilled, though the error bars of _nl are still quite substantial. In addition, we obtain the series expansion of the linear relaxation time on the honeycomb lattice with 22 non-trivial terms. The result for the critical exponent is close to the value obtained on the square lattice, which is expected from universality.     

Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation

For independent translation-invariant irreducible percolation models, it is proved that the infinite cluster, when it exists, must be unique. The proof is based on the convexity (or almost convexity) and differentiability of the mean number of clusters per site, which is the percolation analogue of the free energy. The analysis applies to both site and bond models in arbitrary dimension, including long range bond percolation. In particular, uniqueness is valid at the critical point of one-dimensional 1/x–y² models in spite of the discontinuity of the percolation density there. Corollaries of uniqueness and its proof are continuity of the connectivity functions and (except possibly at the critical point) of the percolation density. Related to differentiability of the free energy are inequalities which bound the specific heat critical exponent in terms of the mean cluster size exponent and the critical cluster size distribution exponent ; e.g., 1+ (/2–1)/(–1).Research supported in part by NSF Grant PHY-8605164Research supported in part by the NSF through a grant to Cornell UniversityResearch supported in part by NSF Grant DMS-8514834     

Nonlinear properties of the laser as an amplifier

Summary The paper presents the results of a theoretical analysis of an optical traveling-wave amplifier, to whose input are applied monochromatic pulses of frequency equal to the transition frequency. At distances zp(c/2) |2N₂–N₁|/ /(N₁–N₂) where p>1, and for <2₀aN₁ all input pulses assume the form of a stationary pulse (see Eq. (33)). The stationary signal intensity, duration and total energy are substantially dependent on the nonresonant loss in the medium, the relaxation time T₂ and the initial inversion population difference N₁. For loss values >2₀aN₁ the input pulses are attenuated.In the amplifier model considered we have assumed the laser medium to have a uniformly broadened line. This enables us to apply the results of the investigation to a ruby laser which satisfies this condition (by excluding low temperatures).The neglect of pumping and the lifetime T₁ is permissible for values of not too close to 2₀aN₁.Izvestiya VUZ. Radiofizika, Vol. 8, No. 5, pp. 899–908, 1965     

Zero value of the Schwarzschild mass for an asymptotically Euclidian system of gravitational waves

A class of the asymptotically Euclidian space-times is shown to exist for which the Schwarzschild mass is equal to zero. The coordinate atlases of these space-times satisfy two additional conditions: _k (-gg ^0k )=0 and _ik ⁰ _0g ^ik - _ik ^k _0g ⁰ⁱ =0. In aT-orthogonal metricgs ²=g ₀₀ dt ² -g dx dx these conditions take a simple form: ₀(detg )=0 and (₀ g )(₀ g )=0.     

Mean-Field Theory for Percolation Models of the Ising Type

The q=2 random cluster model is studied in the context of two mean-field models: the Bethe lattice and the complete graph. For these systems, the critical exponents that are defined in terms of finite clusters have some anomalous values as the critical point is approached from the high-density side, which vindicates the results of earlier studies. In particular, the exponent ~ which characterizes the divergence of the average size of finite clusters is 1/2, and ~, the exponent associated with the length scale of finite clusters, is 1/4. The full collection of exponents indicates an upper critical dimension of 6. The standard mean field exponents of the Ising system are also present in this model (=1/2, =1), which implies, in particular, the presence of two diverging length-scales. Furthermore, the finite cluster exponents are stable to the addition of disorder, which, near the upper critical dimension, may have interesting implications concerning the generality of the disordered system/correlation length bounds.     

Eigenresonance states in the Kronig-Penney model

We investigate a sandwich of three layer systems with Dirac -functions in the Kronig-Penney model. The inner system ofN=5 atomic layers is enclosed by the two outer systems with different potential strength. The numberM of the atomic layers in the outer system is varied betweenM=9 and infinity, whereas the numberN of the inner layers is held fixed. We obtain the transmission coefficient for the finite system in the region of scattering energies (E>0). An alternating set of transmission gaps, transmission bands and bands of eigenresonance states is obtained. The normalizable eigenresonances occur (forM going to infinity), if a transmission band of the inner system overlaps a transmission gap of the outer systems. The reason for obtaining solutions of standing waves in the band of eigenresonances is the rapid change of the wave phase of a traveling wave, which occurs in a transmission band of the inner system.     

On the finite-size scalling equation for the spherical model

The mean spherical model with an arbitrary interaction potential, the Fourier transform of which has a long-wavelength exponent , 0<2, is considered under periodic boundary conditions and fully finite geometry ind dimensions, when <d<2. A new form of the finite-size scaling equation for the spherical field in the critical region is derived, which relates the temperature shift to Madelung-type lattice constants. The method of derivation makes use of the Poisson summation formula and a Laplace transformation of the momentumspace correlation function.On leave of absence from Institute of Mechanics and Biomechanics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria.     

Fractal drums and then-dimensional modified Weyl-Berry conjecture

In this paper, we study the spectrum of the Dirichlet Laplacian in a bounded (or, more generally, of finite volume) open set R ⁿ (n1) with fractal boundary of interior Minkowski dimension (n–1,n]. By means of the technique of tessellation of domains, we give the exact second term of the asymptotic expansion of the counting functionN() (i.e. the number of positive eigenvalues less than ) as +, which is of the form ^/2 times a negative, bounded and left-continuous function of . This explains the reason why the modified Weyl-Berry conjecture does not hold generally forn2. In addition, we also obtain explicit upper and lower bounds on the second term ofN().     

Isomorphism of critical vaporization phenomena in pure substances and binary mixtures

By integration with respect to a noncritical order parameter (the concentration of the mixture), a model Hamiltonian of Landau-Ginzburg type for a binary mixture is reduced to the Hamiltonian of a single-component fluid. The conditions under which this isomorphism is possible are considered. It is shown that taking into account fluctuations of a critical order parameter (the total density of the mixture) by Wilson's approximate method leads, in an approximation linear in ( =4 – d; d is the dimensionality of the space), to the renormalization of the Hamiltonian parameters and to values of the critical indices, , , , which for = 1 are close to those observed experimentally.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 35–40, November, 1976.     

Critical behavior of betaine arsenate

The critical exponents , and of undeuterated and partially deuterated ferroelectric betaine arsenate are estimated. Small but remarkable deviations from the tricritical values are observed. The obtained values are tested via scaling of the isothermals.At some distance aboveT _c a cross-over of the effective critical exponent to a value typically for short-range forces is observed. A suitably defined cross-over temperature range decreases with increasing deuteration and seems to vanish near the antiferroelectric phase boundary.     

Direction-Dependent Free Energy Singularity of the Antiferroelectric Asymmetric Six-Vertex Model

The transition from the ordered commensurate phase to the incommensurate Gaussian phase of the antiferroelectric asymmetric six-vertex model is investigated by keeping the temperature constant below the roughening point and varying the external fields (h, v). In the (h, v) plane, the phase boundary is approached along straight lines v = k h, where (h, v) measures the displacement from the phase boundary. It is found that the free energy singularity displays the exponent 3/2 typical of the Pokrovski–Talapov transition f const(h)^3/2 for any direction other than the tangential one. In the latter case f shows a discontinuity in the third derivative.     

Cooperative plasticity due to the motion of disorientation and phase separation boundaries

Conclusions As has already been noted above, the theory of planar defects organically includes the mechanics of twinning, grain boundaries, Somigliani dislocations, translational dislocations, disclination, and dispiration. The fundamental propositions of the theory and methods of giving the tensor T are listed in Table 4. The mathematical formalism remains the same throughout, and it is applicable to both discrete objects (it is then necessary to conserve the -function apparatus), and to a continuous (then appropriate smoothing is needed, which usually reduces to replacement of the multiplication procedure by the normal n or by the direction , to operations of finding the gradient, divergence, and curl of regular expressions, and discarding the -functional), In particular, the problem of thermoelasticity is formulated successfully by such a method in the terminology of the present theory.In a broad sence of the word, the development of the theory should be perceived as an extension of the concept of imperfection to defects of sufficiently arbitrary origin. A completely developed formalism was worked out earlier for just linear defects; in the symbols used here, for the case b=b₀ + X (r – r₀) for constant b₀,, and r₀, and without taking account of processes on the boundary S if the linear defect contained such a feature. Let us emphasize that to describe three-dimensional defects occurring because of homogeneous distortion = (V)., it is sufficient to use the apparatus of just the theory of planar defects since the fundamental phenomena are associated with precisely the presence of boundaries and in a formal plane, with the spatial derivatives of , they are always expressed in terms of the functional (S), while in the case of finite surface gradients in terms of (L). The time derivatives of the distortion T, i.e., is written down in the developed representations in terms of the form with all the resulting consequences.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 83–102, June, 1981.     

Novel perturbation expansion for the Langevin equation

We discuss the randomly driven systemdx/dt= -W(x) +f(t), wheref(t) is a Gaussian random function or stirring force withf(t)f(t)=2(t–t), andW(x) is of the formgx ¹⁺². The parameter is a measure of the nonlinearity of the equation. We show how to obtain the correlation functionsx(t)f(t)···x(t( n)) _f as a power series in. We obtain three terms in the expansion and show how to use Padé approximants to analytically continue the answer in the variable. By using scaling relations, we show how to get a uniform approximation to the equal-time correlation functions valid for allg and.     

A lower bound for the memory capacity in the Potts-Hopfield model

We consider Potts-Hopfield networks of sizeN. We prove the result: _c >0 such that for all 0<< _c we can find, >0 in such a way that, whenN, we can store N patterns, all of them being sorrounded by -energy barriers at distance.