On a result of R. R. Hall

AbstractAn upper bound for the mean value of a non-negative submultiplicative function by R. R. Hall [3] is sharpened and generalised. Hall's inequality implies a certain rather accurate upper sieve estimate, and this aspect of Hall's result is exploited in deriving good lower bounds for π(x) via the sieve

Combinatorial sieves of dimension exceeding one

AbstractA general sieve for each dimension κ > 1 is given which improves the sieve estimates of Ankeny and Onishi. The work depends on a combinatorial identity which is invariant under Buchstab iteration and on the solution of a pair of differential-difference equations with side conditions

Topologically disordered systems at the glass transition

The thermodynamic approach to the viscosity and fragility of amorphous oxides was used to determine the topological characteristics of the disordered network-forming systems. Instead of the disordered system of atoms we considered the congruent disordered system of interconnecting bonds. The Gibbs free energy of network-breaking defects (configurons) was found based on available viscosity data. Amorphous silica and germania were used as reference disordered systems for which we found an excellent agreement of calculated and measured glass transition temperatures. We reveal that the Hausdorff dimension of the system of bonds changes from Euclidian three-dimensional below to fractal 2.55 ± 0.05-dimensional geometry above the glass transition temperature

Renormalisation and fixed points in Hilbert Space

The energies of low-lying bound states of a microscopic quantum many-body system of particles can be worked out in a reduced Hilbert space. We present here and test a specific non-perturbative truncation procedure. We also show that real exceptional points which may be present in the spectrum can be identified as fixed points of coupling constants in the truncation procedure.Comment: 4 pages, 1 tabl

Finite size effects and the order of a phase transition in fragmenting nuclear systems

We discuss the implications of finite size effects on the determination of the order of a phase transition which may occur in infinite systems. We introduce a specific model to which we apply different tests. They are aimed to characterise the smoothed transition observed in a finite system. We show that the microcanonical ensemble may be a useful framework for the determination of the nature of such transitions.Comment: LateX, 5 pages, 5 figures; Fig. 1 change

Corresponding States of Structural Glass Formers

The variation with respect to temperature T of transport properties of 58 fragile structural glass forming liquids (68 data sets in total) are analyzed and shown to exhibit a remarkable degree of universality. In particular, super-Arrhenius behaviors of all super-cooled liquids appear to collapse to one parabola for which there is no singular behavior at any finite temperature. This behavior is bounded by an onset temperature To above which liquid transport has a much weaker temperature dependence. A similar collapse is also demonstrated, over the smaller available range, for existing numerical simulation data.Comment: 6 pages, 2 figures. Updated References, Table Values, Submitted for Publicatio

Relationship between dynamical heterogeneities and stretched exponential relaxation

We identify the dynamical heterogeneities as an essential prerequisite for stretched exponential relaxation in dynamically frustrated systems. This heterogeneity takes the form of ordered domains of finite but diverging lifetime for particles in atomic or molecular systems, or spin states in magnetic materials. At the onset of the dynamical heterogeneity, the distribution of time intervals spent in such domains or traps becomes stretched exponential at long time. We rigorously show that once this is the case, the autocorrelation function of the renewal process formed by these time intervals is also stretched exponential at long time.Comment: 8 pages, 4 figures, submitted to PR

Statistical Mechanics of Glass Formation in Molecular Liquids with OTP as an Example

We extend our statistical mechanical theory of the glass transition from examples consisting of point particles to molecular liquids with internal degrees of freedom. As before, the fundamental assertion is that super-cooled liquids are ergodic, although becoming very viscous at lower temperatures, and are therefore describable in principle by statistical mechanics. The theory is based on analyzing the local neighborhoods of each molecule, and a statistical mechanical weight is assigned to every possible local organization. This results in an approximate theory that is in very good agreement with simulations regarding both thermodynamical and dynamical properties

Critical temperature for the nuclear liquid-gas phase transition (from multifragmentation and fission)

Critical temperature Tc for the nuclear liquid-gas phase transition is stimated both from the multifragmentation and fission data. In the first case,the critical temperature is obtained by analysis of the IMF yields in p(8.1 GeV)+Au collisions within the statistical model of multifragmentation (SMM). In the second case, the experimental fission probability for excited 188Os is compared with the calculated one with Tc as a free parameter. It is concluded for both cases that the critical temperature is higher than 16 MeV.Comment: 15 pages, 8 figure

Heterogeneities in systems with quenched disorder

We study the strong role played by structural (quenched) heterogeneities on static and dynamic properties of the Frustrated Ising Lattice Gas in two dimensions, already in the liquid phase. Differently from the dynamical heterogeneities observed in other glass models in this case they may have infinite lifetime and be spatially pinned by the quenched disorder. We consider a measure of local frustration, show how it induces the appearance of spatial heterogeneities and how this reflects in the observed behavior of equilibrium density distributions and dynamic correlation functions.Comment: 8 page