Showing papers in "Physica A-statistical Mechanics and Its Applications in 1992"

A rigorous derivation of the chemical master equation

Abstract: It is widely believed that the chemical master equation has no rigorous microphysical basis, and hence no a priori claim to validity. This view is challenged here through arguments purporting to show that the chemical master equation is exact for any gas-phase chemical system that is kept well stirred and thermally equilibrated.

Conformal weights of RSOS lattice models and their fusion hierarchies

Abstract: The finite-size corrections, central charges c and conformal weights Δ of L-state restricted solid-on-solid lattice models and their fusion hierarchies are calculated analytically This is achieved by solving special functional equations, in the form of inversion identity hierarchies, satisfied by the commuting row transfer matrices at critically The results are all obtained in terms of Rogers dilogarithms The RSOS models exhibit two distinct critical regimes For the regime III/IV critical line, we find c = [3p/(p + 2)][1 − 2(p + 2)/r(r − p)] where L = r − 1 is the number of heights and p = 1, 2, … is the fusion level The conformal weights are given by a generalized Kac formula Δ = {[rt − (r − p)s]2 − p2}/ 4pr(r − p) + (s0 − 1)(p − s0 + 1)/ 2p(p + 2) where s = 1, 2, …, r − 1; t = 1, 2, …, r − p − 1; 1 ⩽ s0 ⩽ p + 1 and s0 − 1 = ±(t − s) mod 2p For p = 1, 2, these models are described by the unitary minimal conformal series and the discrete superconformal series, respectively For the regime I/II critical line, we obtain c = 2(N − 1)/(N + 2) and Δ = l(l + 2)/4(N + 2) − m2/4N for the conformal weights, independent of the fusion level p, where N = L − 1, l = 0, 1, …, N and m = −l, −l + 2, …, l − 2, l In this critical regime the models are described by ZN parafermion theories

Equivalence between the Abelian sandpile model and the q→0 limit of the Potts model

Abstract: We establish an equivalence between the undirected Abelian sandpile model and the q→0 limit of the q-state Potts model. The equivalence is valid for arbitrary finite graphs. Two-dimensional Abelian sandpile models, thus, correspond to a conformal field theory with central charge c = −2. The equivalence also gives a Monte Carlo algorithm to generate random spanning trees. We study the growth process of the spread of fire under the burning algorithm in the background of a random recurrent configuration of the Abelian sandpile model. The average number of sites burnt upto time t varies at ta. In two dimensions our numerically determined value of a agrees with the theoretical prediction a = 8 5 . We relate this exponent to the conventional exponents characterizing the distributions of avalanche sizes.

Crossover approach to global critical phenomena in fluids

Abstract: Long-range critical fluctuations appear to affect the thermodynamic properties of fluids in a very large range of temperatures and densities around the critical point. To treat these effects the universal scaling laws, which are valid in the immediate vicinity of the critical point, need to be extended to account for a crossover to classical thermodynamic behavior far away from the critical point. The paper considers approaches for dealing with this problem and attempts to elucidate the physical features of the crossover from singular asymptotic critical thermodynamic behavior to classical thermodynamic behavior.

Fractional diffusion equation for transport phenomena in random media

Abstract: A differential equation for diffusion in isotropic and homogeneous fractal structures is derived within the context of fractional calculus. It generalizes the fractional diffusion equation valid in Euclidean systems. The asymptotic behavior of the probability density function is obtained exactly and coincides with the accepted asymptotic form obtained using scaling argument and exact enumeration calculations on large percolation clusters at criticality. The asymptotic frequency dependence of the scattering function is derived exactly from the present approach, which can be studied by X-ray and neutron scattering experiments on fractals.

Differential operator technique for higher spin problems

Abstract: A new effective-field theory for the Blume-Capel model with a high spin value S is developed by making use of exact spin identities and taking advantage of the differential operator technique. The general formulation for evaluating the transition line and the tricritical point is derived. In particular, the phase diagrams are examined for S = 32 and S = 2. Our results show that the tricritical behavior does not exist in the spin-32 Blume-Capel model but does exist in the spin-2 Blume-Capel model. The tricritical point in the S = 2 system is found at D/zJ≅−0.498, where z is the coordination number, D the crystal-field constant and J the exchange interaction.

Equilibrium properties of ferrocolloids

Abstract: We consider a thermodynamic model of a stable macroscopically homogeneous ferrocolloid containing identical spherical particles with permanent moments suspended in a non-dissociated liquid on the basis of the hard sphere perturbation theory. Magnetostatic properties and the phase diagram of the ferrocolloid are shown to agree very well with experimental evidence. Dipole interparticle interactions result in an effective attraction between the particles which increase, as the strength of an externally applied magnetic field grows, favours the phase separation and cause a reduction in the coefficient of mutual Brownian diffusion of the particles.

Fractional diffusion equation and relaxation in complex viscoelastic materials

Abstract: A fractional equation describing relaxation phenomena in complex viscoelastic materials is derived by employing a formal analogy between linear viscoelasticity and difusion in a disordered structure. From this analogy, a power-law relaxation follows which is in agreement with experimental results obtained in many complex viscoelastic materials.

Line tension near the wetting transition: results from an interface displacement model

Abstract: An interface displacement model is employed for calculating the line tension of a contact line where three phases meet. At a first-order wetting transition the line tension reaches a positive and finite limit if the intermolecular potentials decay faster than r−6. In contrast, for non-retarder Van der Waals forces, and forces of still longer range, the line tension diverges at first-order wetting. The boundary tension along the prewetting line is positive and finite. Approaching wetting, it increases (with diverging slope) and converges to the value of the line tension at first-order wetting. Approaching first-order wetting at bulk phase coexistence, the line tension is finite provided the potentials decay faster than r−5, and increases (with diverging slope) towards its limit at wetting. In contrast, at a critical wetting transition the line tension vanishes. Comparison with recent results from alternative microscopic mean-field approximations is favourable.

Adaptive self-organization during growth of bacterial colonies

Abstract: We present a study of interfacial pattern formation during diffusion-limited growth of Bacillus subtilis It is demonstrated that bacterial colonies can develop patterns similar to morphologies observed during diffusion-limited growth in non-living (azoic) systems such as solidification and electro-chemical deposition The various growth morphologies, that is the global structure of the colony, are observed as we vary the growth conditions These include fractal growth, dense-branching growth, compact growth, dendritic growth and chiral growth The results demonstrate the action of a singular interplay between the micro-level (individual bacterium) and macro-level (the colony) in selecting the observed morphologies as is understood for non-living systems Furthermore, the observed morphologies can be organized within a morphology diagram indicating the existence of a morphology selection principle similar to the one proposed for azoic systems We propose a phase-field-like model (the phase being the bacterial concentration and the field being the nutrient concentration) to describe the growth The bacteria-bacteria interaction is manifested as a phase dependent diffusion constant Growth of a bacterial colony presents an inherent additional level of complexity compared to azoic systems, since the building blocks themselves are living systems Thus, our studies also focus on the transition between morphologies We have observed extended morphology transitions due to phenotypic changes of the bacteria, as well as bursts of new morphologies resulting from genotypic changes In addition, we have observed extended and heritable transitions (mainly between dense branching growth and chiral growth) as well as phenotypic transitions that turn genotypic over time We discuss the implications of our results in the context of the evolving picture of genome cybernetics Diffusion limited growth of bacterial colonies combined with new understanding of pattern formation in azoic systems provide new tools for the study of adaptive self-organization and mutation in the presence of selective pressures We include brief reviews of both the recent developments in the study of interfacial pattern formation in non-living systems and the current trends in the view of mutation dynamics

A new approach to the determination of the surface fractal dimension of porous solids

Abstract: A new approach to the determination of surface fractal dimension is proposed. It is based on the approximation of a given surface by a set of inscribed equicurvature surfaces. The surface fractal dimension, d fs , is determined from the relationship between the area, S c , and the mean radius of curvature, a c , of these surfaces, S c ∼ a 2− d fs . It is the common relationship for the area of a fractal surface measured by a yardstick of varying size, whose role here is played by a c . The equicurvature surfaces can be realized in practice as the interfaces between fluids at the conditions of capillary equilibrium in the vicinity of a given surface. The area and the mean radius of curvature of equilibrium interfaces can be calculated on the basis of experimental data by using general thermodynamic relationships. Corresponding thermodynamic methods for calculating the surface fractal dimension are developed for capillary condensation and intrusion of a nonwetting fluid.

Novel phase behaviour of a confined fluid or Ising magnet

Abstract: The phase behaviour of a simple fluid or Ising magnet (at temperatures above its roughening transition) confined between parallel walls that exert opposing surface fields h 2 = - h 1 is found to be markedly different from that which arises for h 2 = h 1 . Whereas critical wetting plays little role for confinement by identical walls, it is of crucial importance for opposing surface fields. Analysis of a Landau functional and other mean-field treatments show that if h 1 is such that critical wetting occurs at a single wall ( L = ∞) at a transition temperature T w , then phase coexistence, for finite wall separation L , is restricted to temperatures T T c , L , where the critical temperature T c , L lies below T w . In the temperature range T c , b > T > T w there is a single soft mode phase that is characterized, for zero bulk field and large L , by a +- interface located at the centre of the slit, a transverse correlation length ξ ∼≈ e L and a solvation force that is repulsive. For large h 1 , T w can lie arbitrarily far below the bulk critical temperature T c , b . Scaling arguments, whose validity we have confirmed in two dimensions by comparison with exact solutions for interfacial Hamiltonians, predict that such behaviour persists beyond mean-field for systems with short-ranged forces. They also predict similar phase behaviour for long-ranged forces, but with ξ ξ ∼ increasing algebraically with L in the soft mode phase. The solvation force tf s changes from repulsive to attractive (at large L ) as the temperature is reduced below T w , i.e. the sign of tf s reflects wetting characteristics.

Hard and soft multifractal processes

Abstract: We show that multifractal notions encompass a wider variety of phenomena than often believed. Ranked by increasing highest order of singularities we have geometric, then microcanonical and finally canonical multifractals. They are respectively localized and “calm”, delocalized and “calm”, and delocalized and “wild”. Canonical multifractals may also involve rare violent (“hard”) singularities which cause high order statistical moments to diverge. We demonstrate this classification in the framework of universal multifractals characterized by three fundamental exponents.

Statistical properties of deterministic threshold elements - the case of market price

Abstract: We analyze statistical properties of a set of deterministic threshold elements which is introduced as a model for the stock market. The macroscopic variable of the stock market price shows seemingly stochastic fluctuation with a f-2 power spectrum consistent with real economic fluctuations. The maximum Lyapunov exponent is estimated to be zero indicating that the system is at the edge of chaos.

About entropy and correlations in a fluid of hard spheres

Abstract: A synthesis of the structural and dynamical phenomenology exhibited by hard spheres at equilibrium is presented, with an emphasis on the relation between entropy and multiparticle correlations. A novel criterion for the freezing of the fluid is also proposed.

Interrelations between stochastic equations for systems with pair interactions

Abstract: Several types of stochastic equations are important in thermodynamics, chemistry, evolutionary biology, population dynamics and quantitative social science. For systems with pair interactions four different types of equations are derived, starting from a master equation for the state space: First, general mean value and (co)variance equations. Second, Boltzmann-like equations. Third, a master equation for the configuration space allowing transition rates which depend on the occupation numbers of the states. Fourth, a Fokker-Planck equation and a “Boltzmann-Fokker-Planck equation”. The interractions of these equations and the conditions for their validity are worked out clearly. A procedure for a self-consistent solution of the nonlinear equations is proposed. Generalizations to interactions between an arbitrary number of systems are discussed.

Crossover behavior of the susceptibility and the specific heat near a second-order phase transition

Abstract: Simple crossover equations for the susceptibility and the specific heat in zero field have been obtained on the basis of the renormalization-group method and e-expansion. The equations contain the Ginzburg number as a parameter. At temperatures near the critical temperature, scaling behavior including the first Wegner corrections is reproduced. At temperatures far away from the critical temperature the classical Landau expansion with square-root corrections is recovered. For small values of the Ginzburg number the crossover equations approach a universal form. The equations are applied to represent experimental specific heat data for CH 4 , C 2 H 6 , Ar, O 2 and CO 2 along the critical isochore in a universal form.

Self-organizing criticality in cloud formation?

Abstract: Starting from the phenomenology of cloud formation, we derive a cellular automaton model which turns out to have a strong resemblance to percolation-based growth models. Projections of thereby produced clusters show a realistic fractal dimension of the perimeter, but only at or near the critical percolation threshold pc. A mechanism based on the continuity equation is proposed and numerically tested, which possibly explains why the atmosphere acts in fact near its critical threshold during cloud formation (self-organizing criticality). Numerically, this is achieved by a feedback of the number of ‘active’ sites on the occupation concentration p.

Derivation of microscopic expressions for the rigidity constants of a simple liquid—vapor interface

Abstract: We derive equations for the spontaneous curvature and rigidity constants of a simple liquid-vapor interface analogous to the virial expression for the surface tension given by Kirkwood and Buff [J. Chem. Phys. 17 (1949) 338]. Using an approximation for the pair density of a curved interface we give expressions for these quantities in terms of an integral over the interaction potential and the pair correlation function in the liquid region. Near the critical point the rigidity constants are explicitely calculated to yield k = 0.631kBTc and k = 0.239kBTc.

Competition in ramped Turing structures

Abstract: Stationary pattern selection and competition in the uniform Brusselator in two (2D) and three (3D) dimensions are reviewed, including reentrant hexagonal and striped zig-zag phases. Influences of linear or chain-like profiles of the pool chemicals on this selection are presented in the form of numerical experiments. The relation with the recent experimental patterns obtained with the CIMA reaction is discussed.

Hydrodynamic interaction of a spherical particle with a planar boundary: II. Hard wall

Abstract: Using a general method developed in a previous paper we construct the grand resistance and mobility matrices for a hard sphere moving in an incompressible viscous fluid bounded by a hard wall. Starting from an explicit form of the Green tensor for the semi-infinite fluid we compute the matrices whose matrix elements give the scalar resistance and mobility functions describing translation and rotation of the sphere in a general direction and orientation relative to the hard wall. We give explicit series expansions for these functions in inverse powers of the normal distance from the sphere centre to the wall. We find that the series expansion for the friction functions are better behave than are those for the mobility functions. By combining our series results with known asymptotic formulae valid for close approach of sphere and wall we obtain explicit expressions accurate at all separations of particle and wall.

Modeling forest fire by a paper-burning experiment, a realization of the interface growth mechanism

Abstract: We present an experiment on the propagation front of a flameless fire on a thin piece of paper. We find that fire fronts on a piece of paper follow quite well self-affine scaling statistics, with the roughening exponent χ around 0.70, well above the value 1 2 in the theoretical interface growth model using Gaussian noise. This discrepancy may be due to an anomalously singular behavior of the noise distribution, consistent with some recent studies.

Generalized spectral decomposition of the β-adic baker's transformation and intrinsic irreversibility

Abstract: We construct a generalized spectral decomposition of the Frobenius-Perron operator of the β-adic baker's transformation using a general iterative operator method applicable in principle for any mixing dynamical system. The eigenvalues in the decomposition are related to the decay rates of the autocorrelation functions and have magnitudes less than one. We explicitly define appropriate generalized function spaces, which provide mathematical meaning to the formally obtained spectral decomposition. The unitary Frobenius-Perron evolution of densities, when extended to the generalized function spaces, splits into two semigroups, one decaying in the future and the other in the past. This split, which reflects the asymptotic evolution of the forward and backward K-partitions, shows the instrinsic irreversibility of the baker's transformation.

Second acoustic virial coefficients of nitrogen between 80 and 373 K

Abstract: The speed of sound in nitrogen has been measured in the temperature range 80 to 373 K and in the density range 10 to 200 mol/m 3 using a spherical resonator. Values of the second acoustic virial coefficient have been determined from the results with an imprecision of no worse than 0.11 cm 3 /mol. This estimation of imprecision includes all known sources of systematic and random errors at the level of one standard deviation. The results are compared with values calculated from two recently proposed intermolecular potential-energy functions for this system both of which were based partially on second virial coefficient data. Although neither of these functions proved able to predict the acoustic virial coefficients to within the high precision of the present measurements, the potential of Ling and Rigby (Mol. Phys. 51 (1984) 855) gives results that deviate by less than 1 cm 3 /mol. We have also obtained very precise values of the dynamic perfect-gas heat capacity of nitrogen from the zero-density limit of the sound speed measured in the frequency range 3 − 22 kHz . These resolve the effects of centrifugal distortion and are in good agreement with statistical mechanical calculations which include that effect. We report new calibration measurements in argon over the temperature range 90 to 373 K from which the mean radius of the resonator was determined as a function of temperature. In addition, these results provide values of the second acoustic virial coefficient for argon which deviate from those calculated from the interatomic potential energy function for this system by only 0.8 cm 3 /mol at 90 K and by less than 0.1 cm 3 /mol above 300 K. Ordinary ( p, V, T ) second virial coefficients have been calculated from the acoustic results for both gases. These are believed to be of superior accuracy to directly measured values at low temperatures.

Chemical spiral rotation is controlled by light-induced artificial cores

Abstract: Dynamic features of spiral-shaped excitation waves rotating around unexcitable disks are investigated experimentally. The spiral patterns are observed in the ruthenium-catalyzed Belousov-Zhabotinsky reaction (bromination of malonic acid), in which the excitability depends on the intensity of applied illumination. Measurements are performed with a novel technique, which uses an argon laser beam for manipulation of wave propagation and creation of unexcitable disks, serving as artificial spiral cores. Rotation period, wavelength and velocity of spirals increase monotonically when the core radius is enlarged by expanding the diameter of the laser beam (0.1–3.0 mm). Rapid change of the core size is followed by a continuous relaxation process into a new dynamic state. The transient wavelengths and velocities scanned during this process provide data for a fast calculation of the dispersion relation for the investigated medium.

Direct observation of laser induced freezing

Abstract: The effect of an externally applied field on the microstructure of aqueous suspensions of charge stabilized polystyrene spheres is studied by direct observation. This work complements the earlier light diffraction work of Chowdhury et al., which demonstrated that radiation pressure forces can induce crystal-like microstructures in suspensions which have equilibrium liquid-like microstructures in the absence of these external fields. Specifically, a spatially periodic radiation pressure field is produced by intersecting two mutually coherent laser beams in the sample. The microstructure is observed as a function of the input intensity and crossing angle of the two laser beams, and the data is presented in terms of one and two dimensional particle distribution functions. We find that both single- and multilayered systems exhibit a transition to crystal-like order. This crystal order is more pronounced at large input power and for a periodic external field commensurate with the lattice spacing for a final undistorted two dimensional hexagonal crystal structure. Monolayer suspensions undergo this recording with no change in density, while multilayer systems restructure to a monolayer with increased layer density.

Kinetics of clusters in Ising models

Abstract: Recent progress, partly confirming old predictions, is reviewed on the problem of how to describe collective phenomena in fluids and Ising models by a droplet picture, where clusters of all sizes conspire to produce phase transitions. Our particular emphasis is on time-dependent simulations and their partial description by the 1935 Becker-Doring equation.

Dynamic computer simulation of concentrated hard sphere suspensions: II. Re-analysis of mean square displacement data

Abstract: In view of recent theoretical results by Cichocki and Felderhof we reevaluate previously published simulation data for the mean square displacement of a particle in a hard sphere suspension without hydrodynamic interactions. This new evaluation is consistent with the theoretically predicted algebraic long-time tails. We observe better agreement for dense systems and obtain values for the relaxation time almost twice as large as previously.

Self-organized criticality in a forest-fire model

Abstract: A forest-fire model is introduced which contains a lightning probability f. This leads to a self-organized critical state in the limit f→0 provided that the time scales of free growth and burning down of forest clusters are separated. We derive scaling laws and calculate all critical exponents. The values of the critical exponents are confirmed by computer simulations. For a two-dimensional system, there is evidence that the forest density in the critical state assumes its minimum possible value, i.e. that energy dissipation is maximum.

Resistance of one-dimensional quasicrystals

Abstract: One-dimensional quasiperiodic systems are often based on an inflation rule giving rise to a recursion formula on the transfer matrices. It is shown in a general setting, that the norm growth of these matrices can be evaluated. A consequence is, for instance, a purely critical regime (namely polynomial bounds in function of the sample size) for the electrical resistance, when the energy is in the spectrum. This solves a conjecture by Sutherland and Kohmoto.