Showing papers in "Physica A-statistical Mechanics and Its Applications in 1991"

Summation of Coulomb fields in computer-simulated disordered systems

Abstract: Formulae are derived for the sums over Coulomb forces exerted on a charged particle by other charged particles, the central cell system being repeated to infinity by periodic boundary conditions. Such sums are needed in molecular dynamics simulations involving either ions or neutral molecules represented as bound conglomerates of charges, and in astrophysical simulations of gravitating masses. The derived sums are rapidly convergent, being expressed in terms of Bessel functions Kr(z), which decrease exponentially with z. The force expressions are integrated analytically to give the potential function, which may be used in Monte Carlo simulations. The geometries considered are: (i) systems confined between two parallel walls, and (ii) unconfined three-dimensional systems.

Lévy walks and enhanced diffusion in Milan stock exchange

Abstract: We perform statistical analyses of the general and sectorial historical M.I.B. indices of the Milan stock exchange. Our analyses show that the price indices have statistical properties which are compatible with a Levy random walk. The time evolution of the daily variations of indices is intermittent on a time scale of years and the variance of almost all indices displays a superdiffusive behavior. By using the theory of enhanced diffusion in Levy walks as theoretical framework we ascribe the superdiffusive behavior to a nonlocal memory coupling price and time.

Quantum theory of non-integrable systems

Abstract: In 1889 H. Poincare introduced a basic distinction between integrable and non-integrable dynamical systems. This distinction refers to the role of resonances which may lead to divergences. A specially important class of non-integrable systems are «large» Poincare systems (LPS) which have a continuous spectrum and present continuous sets of resonances. As has been shown earlier, LPS play an essential role both in classical and quantum mechanics. Essentially all nontrivial problems of field theory as well as kinetic theory belong this class. We consider here the well known Friedrichs model in which an unstable discrete level is coupled to a continuum. Poincare's theorem prevents the existence of solutions of the eigenvalue problem associated to the Hamiltonian which would be analytic in the coupling constant

Ordered phases of DNA in vivo and in vitro

Abstract: In vitro, pure DNA forms multiple liquid crystalline phases when the polymer concentration is increased: precholesteric organization, cholesteric phase and columnar hexagonal phase. Similar organizations of chromatin can be found in vivo: hexagonal packing in bacteriophages and certain sperm heads, cholesteric organization in dinoflagellate chromosomes, bacterial nucleoids and mitochondrial DNA, helical-shaped chromosomes in many species. The different forms of condensed chromatin seem to be related to different local concentrations of DNA. In the highly condensed forms, chromatin is inactive and the double stranded DNA molecule is linear with small amounts of associated proteins. A more detailed analysis is presented in the case of cholesteric structures (helical pitch, defects) in polarizing microscopy and in electron microscopy. Differences observed in vitro and in vivo are probably related to the length of the DNA molecule and to the presence of proteins associated to DNA in the chromosomes.

The Fisher-Hartwig conjecture and generalizations

Abstract: We discuss the status of the Fisher-Hartwig conjecture concerning the asymptotic expansion of a class of Toeplitz determinants with singular generating functions. A counterexample is given for a nonrational generating function; and we formulate a generalized Fisher-Hartwig conjecture.

Integrable mappings derived from soliton equations

Abstract: We derive a hierarchy of ibtegrable mappings (integrable ordinary difference equations) corresponding to solutions of the initial-value problem of an integrable partial difference equation with periodic initial data. For each n ϵ N this hierarchy contains at least one integrable mapping R n→ R n. The integrals of these mappings are constructed using the Lax pair of the underlying partial difference equation. Our approach is illustrated for the integrable partial difference analogues of the sine-Gordon and the (modified) Korteweg-de Vries equations.

Renormalization group for interacting fermions in d>1

Abstract: A renormalization group transformation (RGT) that permits us to analyze the stability of fermionic systems to various perturbations in any number of dimensions is developed. An RGT that leaves invariant the free fermion system (on or off a lattice) is defined and interactions are classified as relevant, irrelevant or marginal. It is shown how the RGT automatically considers competing instabilities simultaneously, in contrast to mean field theory, which focuses on just one. It is shown that at weak coupling only the BCS coupling is relevant for a spherical Fermi surface. Both Landau theory and the Kohn-Luttinger result are viewed in the light of the RG.

Generalized equation of motion for a ferromagnet

Abstract: We derive microscopically an equation of motion for a ferromagnetic substance at nonzero temperatures allowing for both transverse and longitudinal relaxation and generalizing the Landau-Lifshitz equation. The consideration starts from the density matrix equation for a quantum spin interacting with the environment, which is within about 7% accuracy reduced to the closed equation for the first moment of the distribution function — the magnetization. The latter interpolates between the Landau-Lifshitz equation (S ⪢ 1 and low temperatures) and the Bloch equation ( S = 1 2 or high temperatures). For condensed magnetic media (i.e. a ferromagnet) one can replace in the spirit of the mean field theory the magnetic field by the molecular one containing the exchange field acting on a given magnetic ion from its neighbours, which results in a Landau-Lifshitz type equation of motion with a longitudinal relaxation term providing the Curie-Weiss static solution. Further we consider the mobility of a domain wall (DW) in a uniaxial ferromagnet at nonzero temperatures where the magnetization in the middle of the domain boundary is smaller than in the domains (the elliptic DW transforms to the linear one near Tc). It is shown that longitudinal relaxation plays a crucial role in DW dynamics in a wide range of high temperatures.

Multifractal spectra of multi-affine functions

Abstract: Self-affine functions F(x) with multiscaling height correlations cq(x) ∼xqHq are described in terms of the standard multifractal formalism with a modified assumption for the partition. The corresponding quantities and expressions are shown to exhibit some characteristic differences from the standard ones. According to our calculations the f(α) type spectra are not uniquely determined by the Hq spectrum, but depend on the particular choice which is made for the dependence of N on x, where N is the number of points over which the average is taken. Our results are expected to be relevant in the analysis of signal type data obtained in experiments on systems with an underlying multiplicative process.

Numerical solution of the Kardar-Parisi-Zhang equation in one, two and three dimensions

Abstract: On a course-grained level a family of microscopic growth processes may be described by a stochastic differential equation, which is solved numerically for surface dimensions d = 1, 2 and 3. Dimensional analysis shows that the spatial discretization parameter has the meaning of an effective coupling constant. The numerical stability of the Euler integration scheme is discussed. For the strong coupling exponents β defined by surface width ∼ timeβ the following effective values were obtained: β(d = 1) = 0.330 ± 0.004 and β(d = 2) = 0.24 ± 0.005. Considering the width and its ensemble fluctuations at constant dimensionless time the transition between strong and weak coupling phases is located in d = 3. For the largest coupling for which reliable data are available we obtain an effective exponent β close to the best estimates on discrete models, β(d = 3) ∼ 0.17.

Alternative formulation of classical and quantum dynamics for non-integrable systems

Abstract: We continue our work on large Poincare systems (LPS) characterized by a continuous spectrum and a continuous set of resonances. Poincare's theorem prevents the existence of solutions of the eigenvalue problem associated with the Hamiltonian (as well as the Liouville operator) which would be analytic in the coupling constant. In simple cases, such as the Friedrichs model studied in a recent paper, Poincare's «divergences» can be avoided using a natural time ordering of the dynamical states in the Hilbert space. However, in general, the time ordering has to be performed in the Liouville space of density matrices

Critical exponents of the sand pile models in two dimensions

Abstract: We study three sand pile automaton models namely, the critical height, critical slope and the critical Laplacian models in two dimensions, in which the stability criterion of the sand columns depend on the zeroth, first and the second derivatives of the sand height function. We carried out simulations on system sizes up to 2048 × 2048 and up to 108 avalanches were generated. The exponents of the critical height model were calculated by taking into account the strong corrections to scaling. In order to determine the exponents of the critical Laplacian model accurately we introduced a height restriction in the toppling criterion that maintains universality but accelerates convergence to the steady state by orders of magnitude. We see clear scaling in the critical height and the critical Laplacian models and find that they belong to different universality classes. However, we do not find any scaling in the critical slope model.

Statistical foundation of macroscopic balances for liquid crystals in alignment tensor formulation

Abstract: Starting out with the global balance equations of mass, momentum, angular momentum, and energy formulated on the so-called ten-dimensional doubled phase of position, velocity, orientation, and orientation change velocity, the appropriate local balances are derived, which are defined on the five-dimensional half of the doubled phase space including time, position, and the microscopic director. These so-called orientation balance nematic liquid crystals whose alignment need not be uniform as it is presupposed in theories using macroscopic director fields. In R3 we get the usual phenomenological balance equations of micropolar media having the advantage that the balanced quantities are defined statistically. By expanding the orientation distribution function into a series of multipoles we get alignment tensor fields and an additional alignment tensor balance equation on R3.

Covering of a finite lattice by a random walk

Abstract: We study the covering process by a simple random walk of a d-dimensional periodic hypercubic lattice of N sites. In d = 1, the probability LN(X) for site x to be the last site visited in this covering process does not depend on x, as long as x is not the starting point of the walk. We argue that in dimensions d > 2, the probability LN(X) approaches a constant value according to a Coulomb law: L N (x)⋍ 1 2 1− const |x| d−2 valid for ∥x∥ small on the scale N1d, whereas it behaves logarithmically in d = 2. Also, there is a dimension-dependent characteristic time scale on which the last site is visited. The structure of the set of sites not yet visited on this characteristic time scale is fractal-like in d = 2. In d ⩾ 3, on the other hand, this set is essentially distributed randomly through the lattice.

Ordered phases in concentrated DNA solutions

Abstract: Aqueous solutions of concentrated DNA, a strong polyelectrolyte, in 1:1 electrolyte form liquid-crystalline phases analogous to those observed for other semi-rigid neutral polymers and weak polyelectrolytes. Phase transitions were examined in detail for DNA fragments with a contour length (500 A) approximating the persistence length. These fragments form at least three lyotropic phases. The lowest density or “precholesteric” phase appears to be a nematic with a slight and easily variable twist. The intermediate density phase is a true cholesteric with a pitch of ≈2.1 μm. With increasing concentration the cholesteric phase unwinds prior to formation of the third, high density, columnar phase. Phase transition boundaries were determined as functions of DNA concentration from 10 to 300 mg/ml solvent, supporting electrolyte concentration from 0.01 to 1.0 M, and temperature from 20°C to 60°C. Critical concentrations for formation of anisotropic phase (Ci) and disappearance of isotropic phase (Ca) were only moderately dependent on temperature. Ci, Ca, and the pitch of the cholesteric phase were surprisingly insensitive to the supporting electrolyte concentration. The insensitivity can be most simply related to the high concentrations required for anisotropic phase formation by these rather short, highly charged rod-like DNA fragments. At high concentrations the DNA counterions contribute significantly to the effective ionic strength, hence overall charge screening, and the counterion atmosphere as monitored by 23Na NMR, appears to be perturbed by inter-rod interactions.

Block copolymers at interfaces

Abstract: Equilibrium aspects of interfacial activity of block copolymers in incompatible polymer blends are reviewed We discuss in particular a mechanism of reduction of interfacial tension and limitations of emulsifying activity of copolymer chains associated with the formation of intermolecular aggregates (micelles) in bulk phases

Multimode and multiphoton processes in a non-linear Jaynes-Cummings model

Abstract: Exact operator solutions for the generalized Jaynes-Cummings model (JCM) of a two-level atom interacting with a non-linear bosonic field with multimodes has been given. Two different methods have been applied, the Heisenberg equations and the evolution operator. The photon numbers have been given through the characteristic function. Different statistical quantities have been computed. The phenomenon of collapses and revivals in the atomic inversion and the photon number as well as squeezing have been investigated.

Role of interfacial properties on the motion and deformation of capsules in shear flow

Abstract: The influence of interfacial properties on the dynamics of a capsule freely suspended in viscous liquid subjected to flow is assessed. First, the case of a liquid-liquid interface considered: experimental evidence and theoretical models show that the drop deformat depends on the type of flow, the internal to external viscosity ratio and the capillary number which measures the ratio between surface tension and viscous forces. Burst of the drop predicted to occur when the normal stress equilibrium cannot be satisfied everywhere on interface. Next, we consider the case of capsules, which are liquid drops enclosed by a so depormable interface. From experiment and theoretical models, they are known to deform and to burst in some cases. Their motion depends on essentially the same parameters as liquid drops, but different interfacial constitutive laws lead to different types of deformation and motion.

Phase diagrams and tricritical effects in the beg model

Abstract: The critical properties of the spin-1 Ising model on the Bethe lattice with dipolar and quadrupolar exchange interactions are studied exactly. This method is based on recursion relations, which are exact in the infinite-dimensional case, and as a result the phase diagrams are constructed. The λ-line ending in the tricritical point is defined exactly. It is shown that when the quadrupolar exchange interaction is large enough, the tricritical point disappears. The tricritical exponents are calculated exactly.

Phase transitions in the spin-3/2 beg model

Abstract: The spin- 3 2 Ising model with a biquadratic exchange interaction (K) and a single ion anisotropy (D) is studied using two approaches. By means of a variational method we obtain the mean field phase diagram of the model. Besides a second order transition line occurring at high temperatures, the phase diagram exhibits two low temperature lines, which have not been reported previously. In the (T, D)-plane diagram a first order transition line separating the phases ( 3 2 , 3 2 ) and ( 1 2 , 1 2 ) occurs between the points (T = 0, D = −1.0) and (T ≈ 0.76, D ≈ −1.0). In the (T, K)-plane a line running in the non-zero temperature region from (T = 0, K = − 1 3 ) to (T = 0, K = −1.0) separates the region of equal sublattice magnetizations from that of unequal sublattice magnetizations. On the second approach, Monte Carlo simulations have been performed confirming the main qualitative predictions of the mean field calculations.

Nonasymptotic critical thermodynamical behavior of fluids

Abstract: The paper is concerned with some theoretical and empirical attempts to obtain a unified equation for the thermodynamic properties of fluids that incorporates the nonclassical scaling laws near the critical point and crosses over to a classical analytic equation far away from the critical point. Specifically we investigate to what extent the various proposed crossover models agree with the known theoretical predictions for the universal ratios among the asymptotic and correction-to-scaling amplitudes in the power-law expansions of the thermodynamic properties around the critical point.

Quantum Zeno effect

Abstract: In 1977, Misra and Sudarshan showed, based on the quantum measurement theory, that an unstable particle will never be found to decay when it is continuously observed. They called it the quantum Zeno effect (or paradox). More generally the quantum Zeno effect is associated to the inhibition of transitions by frequent measurements. This possibility has attracted much interest over the last years. Recently, Itano, Heinzen, Bollinger and Wineland have reported that they succeeded in observing the quantum Zeno effect. This would indeed be an important step towards the understanding of the role of the observer in quantum mechanics. However, in the present paper, we will show that their results can be recovered through conventional quantum mechanics and do not involve a repeated reduction (or collapse) of the wave function.

The thermal conductivity and heat capacity of fluid nitrogen

Abstract: This paper presents new absolute measurements of the thermal conductivity and the thermal diffusivity of nitrogen made with a transient hot wire instrument. The instrument measures the thermal conductivity with an uncertainty less than ±1% and the thermal diffusivity with an uncertainty of ±5% except at the fluid critical point. The data cover the region from 80 to 300KK pressures to 70 MPa. The data consist of 8 supercritical isotherms, 3 vapor isotherms, and 4 liquid isotherms. A surface fit is developed for our nitrogen thermal conductivity data from 80 to 300 K at pressures to 70 MPa. The data are compared with a recent theory for the first density coefficient of thermal conductivity and a new mode-coupling theory for the thermal conductivity critical enhancement. These data illustrate that it is necessary to study a fluid over a wide range of temperatures and densities in order to characterize the thermal conductivity surface. Isobaric heat capacity results were determined from the simultaneously measured values of thermal conductivity and thermal diffusivity, using the density calculated from an equation of state. The heat capacities obtained by this technique are compared to the heat capacities predicted by a recent equation of state developed specifically for nitrogen.

Collective diffusion of charged spheres in the presence of hydrodynamic interaction

Abstract: The short-time behavior of the scattering intensity, which can be measured by dynamic light scattering, is investigated theoretically. We consider a monodisperse system of charged hard spheres and study the combined effects of electrostatic repulsion and hydrodynamic interactions. The electrostatic repulsion determines the static properties, which are calculated from the thermodynamically consistent Rogers-Young scheme. For comparison, some results obtained within the rescaled mean spherical approximation (RMSA) are shown as well. Many-body hydrodynamic interaction is treated within the renormalization approach due to Beenakker. We compare results depending on the scattering vector to experimental data. Further, we show various trends for the collective diffusion coefficient, which is obtained as the long wavelength limit of the collective diffusion function, when volume fraction of spheres, salinity, and charge of the spheres are changed. The results are aimed at the interpretation of dynamic light scattering properties of systems of weakly charged spheres, which could be ionic micelies or macromolecules at moderate concentrations.

Simple statistical models for river networks

Abstract: The geometric scaling properties associated with simple river network models have been studied using computer simulations. In these models the river networks are comprised of random trajectories which start at randomly selected positions on a square lattice and continue until they either reach an edge of the lattice or join a previous trajectory to form a branched network. For those cases where the trajectories are either self-avoiding random walks (SAWs) or indefinitely growing self-avoiding walks (IGSAWs) the river basins are compact but have fractal basin boundaries with a dimensionally near to 5/4. For the IGSAW model the longest river is a self-similar fractal with a fractal dimensionality close to or equal to that of the IGSAW itself. If the trajectory is a directed walk, then the model is very similar to the Scheidegger river network model. In this case the basin boundary and the longest channel are self-affine Brownian processes. The results obtained from the IGSAW and directed walk models can be described in terms of a simple scaling model. For the SAW models this scaling picture does not apply and it appears that practical simulations are far from the asymptotic (large system size) limit.

The zeroes of the partition function of the random energy model

Abstract: The random energy model is one of the simplest disordered models containing some of the physics of spin glasses. The zeroes of the partition function in the complex temperature plane are determined for this model. They either lie on simple lines or are dense in some regions of the complex plane.

Perturbation theory of isotropic-liquid-crystal phase equilibria in polyelectrolyte solutions

Abstract: The excess Helmholtz free energy ΔF of rodlike and wormlike polyelectrolyte solutions over that of solvent is formulated from a perturbation theory, where the hard core and electrostatic potentials between two polymers are taken as the pair potentials of the reference system and the perturbation, respectively. The free energy of the reference system is obtained from a scaled particle theory for hard spherocylinders, while the perturbation terms are evaluated by the second virial approximation. The polyelectrolyte end effect on the electrostatic binary cluster integral is taken into account along with the contribution of polyions and their counterions to the electrostatic screening. The phase boundary concentrations for the isotropic-liquid-crystal phase equilibria in polyelectrolyte solutions are calculated from ΔF in order to compare them with experimental data for aqueous xanthan (a rigid double helical polysaccharide) and tobacco mosaic virus. The agreement between theory and experiment is almost quantitative for the former system, but not so satisfactory for the latter.

Some facts of life

Abstract: We study the statistics of the time evolution of the Game of Life. We recognize three different time regimes of which the most interesting one is the long time glider regime, which has properties typical of a critical state. We introduce mean field approximations able to give some insights on the time evolution of the density of the density of living cells. Extended simulations are reported which deal with the evolution of the density, damage spreading and the measurements of a finite size exponent. A simple dynamical model explains some aspects of the asymptotic glider regime. We study also the dependence of the asymptotic density on the initial density both analytically and numerically.

Scheidegger's rivers, Takayasu's aggregates and continued fractions

Abstract: A discrete model of (1+1)-dimensional aggregation, equivalent to Scheidegger's 1967 river model, is tackled with continued fractions. An exact expression for the power-law distribution of aggregate masses is found.

Multiscaling analysis of large-scale off-lattice DLA

Abstract: An improved algorithm has been used to simulate off-lattice diffusion-limited aggregation (DLA) in two dimensions. This enables us to grow clusters of up to 6 × 106 particles, which are the largest such clusters made up to now. The density profile in two dimensions is analyzed for the scaling form (multiscaling) g(r, R) = r−d + D(x) C(x) with x = r R , where R is the radius of gyration and r is the distance from the cluster origin. In contrast with previous studies, the dimension D(x) shows no symmetric tendency to drop at values of x but only fluctuates around the global fractal dimension D = 1.712 ± 0.003.