Dynamical theory of polymer melt morphology

A general kinetic equation of the monomer density variables for polymer blends and block copolymer melts is obtained which describes slow morphology variations. The general theory is applied to a polymer blend adopting the biased reptation model of a polymer chain under mean field. We obtain an equation of motion of interfaces in a phase-separated polymer blend, which contains an interface reaction term for length scales shorter than l_c ≡ R²_G/ξ, where R_G is the gyration radius of a polymer chain and ξ the interfacial width. We also discuss some problems associated with the incompressibility requirement for phase separation kinetics of binary systems not limited to polymers. For length scales greater than l_c the interface dynamics involves diffusion in bulk pure phases even in the strong segregation limit in a way different from that for the usual time-dependent Ginzburg-Landau equation for the conserved order parameter. Implications of the existence of the new term on the late stage phase separation kinetics of polymer blend are discussed.A phenomenological model to study morphology dynamics not relying on the reptation model is also proposed.     

Phenomenology of local scale invariance: from conformal invariance to dynamical scaling

Statistical systems displaying a strongly anisotropic or dynamical scaling behaviour are characterized by an anisotropy exponent θ or a dynamical exponent z. For a given value of θ (or z), we construct local scale transformations, which can be viewed as scale transformations with a space–time-dependent dilatation factor. Two distinct types of local scale transformations are found. The first type may describe strongly anisotropic scaling of static systems with a given value of θ, whereas the second type may describe dynamical scaling with a dynamical exponent z. Local scale transformations act as a dynamical symmetry group of certain non-local free-field theories. Known special cases of local scale invariance are conformal invariance for θ=1 and Schrödinger invariance for θ=2.The hypothesis of local scale invariance implies that two-point functions of quasiprimary operators satisfy certain linear fractional differential equations, which are constructed from commuting fractional derivatives. The explicit solution of these yields exact expressions for two-point correlators at equilibrium and for two-point response functions out of equilibrium. A particularly simple and general form is found for the two-time autoresponse function. These predictions are explicitly confirmed at the uniaxial Lifshitz points in the ANNNI and ANNNS models and in the aging behaviour of simple ferromagnets such as the kinetic Glauber–Ising model and the kinetic spherical model with a non-conserved order parameter undergoing either phase-ordering kinetics or non-equilibrium critical dynamics.     

Curvature autocorrelations in domain growth dynamics

We show how the interface curvature autocorrelation function (ICAF) and associated structure factor (ICSF), of relevance in non-equilibrium pattern-formation problems where sharp interfaces are present, provide new and interesting information on domain structure, as yet not visible via the order-parameter structure factor (OPSF). This is done by discussing numerical simulations of model A (non-conserved relaxational phase-ordering kinetics) in two-dimensional systems. The ICAF is Gaussian over short distances and exhibits dynamical scaling and t ^1/2 power-law growth. We use it to show what the typical length-scale in the model A dynamics corresponds to physically and how it can be obtained uniquely, rather than simply within a multiplicative constant. Experimental methods to measure the ICAF and/or ICSF are still needed at this point. Received 22 April 1998     

A self-consistent test of the rescaling model of the EMC effect

We examine predictions of the dynamical rescaling model for the dueteron. An alternate expression for F₂^D is derived by explicitly including the six-quark state. This lead to a new self-consistence test for dynamical rescaling which is well established. However, discrepancies remain between the experimental F₂^D and the predictions of the model.     

Effect of aerosil dispersions on the nematic-to-isotropic interface

The effect of silica aerosils on the kinetics of the first-order nematic-isotropic (NI) phase transition is phenomenologically described in the framework of the time-dependent Landau-Ginzburg equation. A steady-state solution to the equation is presented such that the NI interface may propagate with a solitary-like wave profile under constant quenching. The results provide a plausible basis for the interpretation of the dynamical effects of quenched disorder in the liquid-crystal systems, caused by randomly interconnected porous media, such as aerosils. In the low silica aerosil ρ_s ( ≤0.1 g/cm^3) regime, the calculated values of the interface velocity v(T,ρ_s), the interface thickness κ(T,ρ_s), and the critical radius of a spherical nucleus of new nematic phase in a bulk isotropic environment, composed of polar molecules, such as 4-n-octyl- 4^′- cyanobiphenyl and 4-n-heptyl- 4^′- cyanobiphenyl shows that the effect of silica aerosils on the kinetics is reflected in a shifting of the set of temperature-dependent curves to lower temperature values.-1     

Strange objects in the complex plane

Julia sets are examined as examples of strange objects which arise in the study of long time properties of simple dynamical systems. Technically they are the closure of the set of unstable cycles of analytic maps. Physically, they are sets of points which lead to chaotic behavior. The mapf(z)=z²+p is analyzed for smallp where the Julia set is a closed curve, and for largep where the Julia set is completely disconnected. In both cases the Hausdorff dimension is calculated in perturbation theory and numerically. An expression for the rate at which points escape from the neighborhood of the Julia set is derived and tested in a numerical simulation of the escape.     

Theory of tensor invariants of integrable Hamiltonian systems. I. Incompatible Poisson structures

This paper develops a new theory of tensor invariants of a completely integrable non-degenerate Hamiltonian system on a smooth manifoldM ⁿ. The central objects in this theory are supplementary invariant Poisson structuresP _c which are incompatable with the original Poisson structureP ₁ for this Hamiltonian system. A complete classification of invariant Poisson structures is derived in a neighbourhood of an invariant toroidal domain. This classification resolves the well-known Inverse Problem that was brought into prominence by Magri's 1978 paper deveoted to the theory of compatible Poisson structures. Applications connected with the KAM theory, with the Kepler problem, with the basic integrable problem of celestial mechanics, and with the harmonic oscillator are pointed out. A cohomology is defined for dynamical systems on smooth manifolds. The physically motivated concepts of dynamical compatibility and strong dynamical compatibility of pairs of Poisson structures are introduced to study the diversity of pairs of Poisson structures incompatible in Magri's sense. It is proved that if a dynamical systemV preserves two strongly dynamically compatible Poisson structuresP ₁ andP ₂ in a general position then this system is completely integrable. Such a systemV generates a hierarchy of integrable dynamical systems which in general are not Hamiltonian neither with respect toP ₁ nor with respect toP ₂. Necessary conditions for dynamical compatibility and for strong dynamical compatibility are derived which connect these global properties with new local invariants of an arbitrary pair of incompatible Poisson structures.Supported by NSERC grant OGPIN 337.     

Topological entropy for endomorphisms of localC *-algebras

A notion of topological entropy for endomorphisms of localC ^*-algebras is introduced as a generalisation of the topological entropy of classical dynamical systems. The basic properties are derived and a series of calculations are presented.     

Theory of Tensor Invariants of Integrable Hamiltonian Systems. II. Theorem on Symmetries and Its Applications

The theorem on symmetries is proved that states that a Liouville-integrable Hamiltonian system is non-degene\-rate in Kolmogorov's sense and has compact invariant submanifolds if and only if the corresponding Lie algebra of symmetries is abelian. The theorem on symmetries has applications to the characterization problem, to the integrable hierarchies problem, to the necessary conditions for the strong dynamical compatibility problem, and to the problem on master symmetries. The invariant necessary conditions for the non-degenerate C-integrability in Kolmogorov's sense of a given dynamical system V are derived. It is proved that the C-integrable Hamiltonian system is non-degenerate in the iso-energetic sense if and only if the corresponding Lie algebra of the iso-energetic conformal symmetries is abelian. An extended concept of integrability of Hamiltonian systems on the symplectic manifolds M ⁿ , n= 2k, is introduced. The concept of integrability describes the Hamiltonian systems that have quasi-periodic dynamics on tori or on toroidal cylinders of an arbitrary dimension . This concept includes, as a particular case, all Hamiltonian systems that are integrable in Liouville's classical sense, for which . The A-B-C-cohomologies are introduced for dynamical systems on smooth manifolds. Received: 16 January 1996 / Accepted: 3 July 1996     

Dynamical symmetries of the Geodesic equation

A class of dynamical symmetries for the Euler-Lagrange equations corresponding to the LagrangianL=(1/2)g _ab q ^a q ^b is determined. The members of the class are closely related to tensor fields defined on the configuration space. First integrals generated by the dynamical symmetries through deformation of a given first integral are then examined. Noether-type conserved quantities whose expression depends only on the dynamical symmetry are also explicitly exhibited. Applications to general relativity are also pointed out in the course of the discussion.     

K + production inp-nucleus collisions

We present a non-perturbative dynamical study ofK ⁺ meson production in proton-nucleus collisions from 1.2 to 2.5 GeV bombarding energy. The dynamical evolution of the proton-nucleus collision is described by a transport equation of the Boltzmann-Uehling-Uhlenbeck type evolving phase-space distribution functions for nucleons, δ’s,N(1440)’s,N(1535)’s, poins and η’s with their isospin degrees of freedom. We incorporate all known sources forK ⁺ production and study their momentum and angular distributions, and the excitation function. We show that at lower energies (E _b<1.5 GeV)NΔ andNN* channels dominate the kaon yield for light systems, and the πN channel for heavy systems. At higher bombarding energies the directNN channel accounts for almost the whole cross-section.     

Intermediate mechanisms in fission-like fragmentation in the 32S + 59Co and 32S + 63Cu reactions

The fission-like fragmentation of the ³²S + ⁵⁹Co and ³²S + ⁶³Cu composite systems has been studied at incident energies E/A ≈ (4–5) MeV/nucleon. Mass, energy and angular distributions of the fission-like fragments were obtained from time-of-flight measurements. The mean total kinetic energies of the fragments are found to be fully relaxed, whereas a strong angular dependence is observed in the mass distributions. The data are interpreted in terms of a dynamical model based on the transport theory.     

Reduction of the Lane-Robson “calculable” reaction theory to standard R-matrix form

General dynamical equations derived from the Lane-Robson calculable reaction formalism are cast into a form amenable to standard R-matrix treatment, permitting the resonance content of the equations to be made explicit. Formulae are given which enable the collision matrix and the amplitudes of physical eigenfunctions to be calculated directly from the R-matrix with or without the isolation of resonance contributions. The present methodology permits a significant reduction of effort in numerical investigations of the energy dependence inherent in dynamical models of the nucleus. The formalism is illustrated by calculational results obtained from a potential model fitted to ¹⁶O + n scattering data.     

Nucleation field, hysteresis loop, coercivity mechanism and critical thickness in composite multilayers with perpendicular anisotropy

Analytical expression for the nucleation field has been derived for a hard/soft multilayer system with anisotropy perpendicular to the film plane, which depends on the soft thickness L^s, the interface exchange coupling constant Jⁱ and the intrinsic material parameters. Both nucleation field and coercivity decrease as L^s increases. For very small L^s, the coercivity mechanism is pure nucleation and the hysteresis loops are square. As L^s rises, the coercivity mechanism changes from nucleation to pinning gradually, where the hysteresis loops have to be calculated numerically. The critical thickness at which the mechanism varies has been discussed in detail on the basis of easy axis orientation and the interface exchange coupling constant.     

Measure theoretic ergonic theory in quantum mechanics

We apply the measure theoretic ergonic theory to quantum systems. The time evolution is considered as a point transformation in the set of states of a C¹-algebra. Invariant Borel measures on this set are shown to exist, so there is a measure theoretic dynamic system. Interrelations between this system and the corresponding quantum dynamical system are established.     

Instantons: Dynamical mass generation,chiral Ward identities and the topological charge correlation function

When dynamical mass generation resulting from the breakdown of chiral symmetry is taken into account, instanton dynamics treated within the dilute gas approximation may satisfy the constraints on the quark condensates and the topological charge correlation function derived by Crewther from an analysis of the chiral Ward identities assuming the absence of a physical axial U(1) Goldstone boson. From a consideration of the contribution of the η′ to the topological charge correlation function, a relationship is derived in which m_η′²f_η′² is proportional to the vacuum energy density.     

Field of linear frames as a fundamental self-interacting system

We present some philosophical and physical arguments supporting the hypothesis that the most fundamental self-interacting field in an amorphous space-time is the field of linear frames, i.e. the quadruple of vector bosons. We construct a wide class of Lagrangian dynamical models invariant under the total group of diffeomorphisms and under the natural action of the proper linear group GL⁺ (4, R) on the tetrad field. There exist some links between these models and the Hamiltonian dynamical systems on GL⁺ (3, R) (the mechanics of affinely-rigid bodies [23] [27]). We present the general form of field equations, conservation laws and Bianchi identities. There exist some formal similarities between our Lagrangians and those used in non-linear electrodynamics, in particular in the Born–Infeld theory [21]. We also give a few rough remarks concerning models invariant under natural subgroups of GL⁺ (4, R), i.e. under SL(4, R) and SO(1, 3; R) (special linear group and Lorentz group). The latter class includes the conventional Einstein relativity and the more general metrical-parallelism models. It turns out that there are GL⁺ (4, R)-invariant Lagrangians which are structurally alike the conventional Einstein Lagrangian.We have not derived as yet either mathematical or physical consequences of the presented model. Nevertheless, it seems to follow from our discussion that, a priori, the GL⁺ (4, R)-invariant tetrad models could be competitive with the Einstein theory. The next thing to be done would be a careful mathematical analysis of these models and attempts to compare their consequences with those of the Einstein relativity and of other field theories.     

Production of large p⊥ single particles and multiparticle systems (JETS) in K−p interactions at 110 GeV/c

Differential cross sections of single particles and multiparticle systems produced in K^?p interactions at 110 GeV/c are presented. At large values of transverse momentum (p_⊥) we find the multiparticle cross section to be an order of magnitude larger than that for single particles. This and other features of our data are consistent with results obtained at 200 GeV/c, that have been interpreted as evidence of a new dynamical mechanism in hadron-hadron interactions. However, similar features are observed in a K^?p experiment at 16 GeV/c.     

Interface resistances and AC transport in a Luttinger liquid

We consider a Luttinger liquid (LL) connected to two reservoirs when the two sample-reservoir interface resistances R_S and R_D are arbitrary (not necessarily quantized at half-the-quantum of resistance). We compute exactly the dynamical impedance of a Luttinger liquid and generalize earlier expressions for its dynamical conductance in the following situations. (i) We first consider a gated Luttinger liquid. It is shown that the Luttinger liquid parameters u and K and the interface resistances R_S and R_D can be experimentally determined by measuring both the dynamical conductance and impedance of a gated wire at second order in frequency. The parallel law addition for the charge relaxation resistance R_q is explicitly recovered for these non-trivial interface resistances as R_q^-1 = R_S^-1 + R_D^-1. (ii) We discuss the AC response when only one electrode is connected to the LL. (iii) Thirdly we consider application of an arbitrary AC electric field along the sample and compute the dynamical response of the LL with arbitrary interface resistances. The discussion is then specialized to the case of a uniform electric field.Received: 19 May 2003, Published online: 30 January 2004PACS: 73.23.-b Electronic transport in mesoscopic systems - 71.10.Pm Fermions in reduced dimensions(anyons, composite fermions, Luttinger liquid, etc.) - 72.30. + q High-frequency effects; plasma effects     

On non-local representations of the ageing algebra

The ageing algebra is a local dynamical symmetry of many ageing systems, far from equilibrium, and with a dynamical exponent z=2$z=2$. Here, new representations for an integer dynamical exponent z=n$z=n$ are constructed, which act non-locally on the physical scaling operators. The new mathematical mechanism which makes the infinitesimal generators of the ageing algebra dynamical symmetries, is explicitly discussed for an n-dependent family of linear equations of motion for the order-parameter. Finite transformations are derived through the exponentiation of the infinitesimal generators and it is proposed to interpret them in terms of the transformation of distributions of spatio-temporal coordinates. The two-point functions which transform co-variantly under the new representations are computed, which quite distinct forms for n even and n odd. Depending on the sign of the dimensionful mass parameter, the two-point scaling functions either decay monotonously or in an oscillatory way towards zero.