On critical phenomena in interacting growth systems. Part II: Bounded growth

This paper completes the classification of some infinite and finite growth systems which was started in Part I. Components whose states are integer numbers interact in a local deterministic way, in addition to which every component's state grows by a positive integerk with a probability ^k (1-) at every moment of the discrete time. Proposition 1 says that in the infinite system which starts from the state all zeros, percentages of elements whose states exceed a given valuek0 never exceed (C) ^k , whereC=const. Proposition 2 refers to finite systems. It states that the same inequalities hold during a time which depends exponentially on the system size.     

Scaling and critical phenomena in a cellular automaton slider-block model for earthquakes

The dynamics of a general class of two-dimensional cellular automaton slider-block models of earthquake faults is studied as a function of the failure rules that determine slip and the nature of the failure threshold. Scaling properties of clusters of failed sites imply the existence of a mean-field spinodal line in systems with spatially random failure thresholds, whereas spatially uniform failure thresholds produce behavior reminiscent of self-organized critical behavior. This model can describe several classes of faults, ranging from those that only exhibit creep to those that produce large events.     

Universal multifractal properties of circle maps from the point of view of critical phenomena I. Phenomenology

The strange attractor for maps of the circle at criticality has been shown to be characterized by a remarkable infinite set of exponents. This characterization by an infinite set of exponents has become known as the multifractal approach. The present paper reformulates the multifractal properties of the strange attractor in a way more akin to critical phenomena. This new approach allows one to study the universal properties of both the critical point and of its vicinity within the same framework, and it allows universal properties to be extracted from experimental data in a straightforward manner. Obtaining Feigenbaum's scaling function from the experimental data is, by contrast, much more difficult. In addition to the infinite set of exponents, universal amplitude ratios here appear naturally. To study the crossover region near criticality, a correlation time, which plays a role analogous to the correlation length in critical phenomena, is introduced. This new approach is based on the introduction of a joint probability distribution for the positive integer moments of the closest-return distances. This joint probability distribution is physically motivated by the large fluctuations of the multifractal moments with respect to the choice of origin. The joint probability distribution has scaling properties analogous to those of the free energy close to a critical point.     

Phase transitions in a probabilistic cellular automaton: Growth kinetics and critical properties

We investigate a discrete-time kinetic model without detailed balance which simulates the phase segregation of a quenched binary alloy. The model is a variation on the Rothman-Keller cellular automaton in which particles of type A (B) move toward domains of greater concentration of A (B). Modifications include a fully occupied lattice and the introduction of a temperature-like parameter which endows the system with a stochastic evolution. Using computer simulations, we examine domain growth kinetics in the two-dimensional model. For long times after a quench from disorder, we find that the average domain sizeR(t) t ^1/3, in agreement with the prediction of Lifshitz-Slyozov-Wagner theory. Using a variety of methods, we analyze the critical properties of the associated second-order transition. Our analysis indicates that this model does not fall within either the Ising or mean-field classes.     

Intermediate processes and critical phenomena: Theory, method and progress of fractional operators and their applications to modern mechanics

From point of view of physics, especially of mechanics, we briefly introduce fractional operators (with emphasis on fractional calculus and fractional differential equations) used for describing intermediate processes and critical phenomena in physics and mechanics, their progress in theory and methods and their applications to modern mechanics. Some authors’ researches in this area in recent years are included. Finally, prospects and evaluation for this subject are made.     

Part I. The 2D classical Coulomb gas near the zero-density Kosterlitz-Thouless critical point: Correlations and critical line

The 2D classical Coulomb gas undergoes the famous Kosterlitz-Thouless (KT) transition between a high-temperature conducting phase and a low-temperature insulating phase. We present various studies of the correlations in the insulating phase near the zero-density critical point. First, we briefly recall the phenomenological approach of Kosterlitz and Thouless. This theory predicts that the decay of the charge correlation is entirely controlled by the bare Coulomb potential between opposite charges only renormalized by the dielectric constante. Then, we present an analysis of the low-fugacity expansions of the correlations. The particle correlations are found to decay as 1/r⁴. The large-distance decay of the charge correlation is shown to be tightly related to the behavior of l/s in the regime of interest. Systematic resummations allow one to recover the algebraic decay predicted by the heuristic KT model. This settles on a rigorous basis various assumptions of this model. In particular, the nested pair mechanism naturally arises in the resummation scheme. Finally, we describe the phase diagram of the system according to the most recent calculations which include finite-density effects.     

On the theory of transport phenomena in ferrofluids. Effect of chain-like aggregates

The paper deals with the theoretical study of the effect of chain-like aggregates on diffusion and magnetophoretic transport in ferrofluids. Analysis shows that the appearance of the chains leads to a strong anisotropy of the diffusion transport–the coefficient of diffusion in the direction of applied magnetic field is significantly more than that in the perpendicular direction. The presence of the chains in a ferrofluid strongly affects the coefficient of the particle magnetophoresis.     

Bethe lattice spin glass: The effects of a ferromagnetic bias and external fields. I. Bifurcation analysis

We present a rigorous analysis of the ±J Ising spin-glass model on the Bethe lattice with fixed uncorrelated boundary conditions. Phase diagrams are derived as a function of temperature vs. concentration of ferromagnetic bonds and, for a symmetric distribution of bonds, external field vs. temperature. In this part we characterize the bulk ordered phases using bifurcation theory: we prove the existence of a distribution of single-site magnetizations far inside the lattice which is stable with respect to changes in the boundary conditions.     

Core percolation in random graphs: a critical phenomena analysis

We study both numerically and analytically what happens to a random graph of average connectivity α when its leaves and their neighbors are removed iteratively up to the point when no leaf remains. The remnant is made of isolated vertices plus an induced subgraph we call the core. In the thermodynamic limit of an infinite random graph, we compute analytically the dynamics of leaf removal, the number of isolated vertices and the number of vertices and edges in the core. We show that a second order phase transition occurs at α = e = 2.718 ... : below the transition, the core is small but above the transition, it occupies a finite fraction of the initial graph. The finite size scaling properties are then studied numerically in detail in the critical region, and we propose a consistent set of critical exponents, which does not coincide with the set of standard percolation exponents for this model. We clarify several aspects in combinatorial optimization and spectral properties of the adjacency matrix of random graphs. Received 31 January 2001 and Received in final form 26 June 2001     

Difference equations in statistical mechanics. I. Cluster statistics models

A review and some new results are presented for several cluster statistics models, solutions of which can be reduced to difference equations. Mathematical techniques suitable for solving these equations are surveyed.     

Macrodynamics: Large-scale structures in turbulent media

We develop a method to derive the macroscopic equations governing the evolution of the mean field in continuous turbulent media. The approach is based on the concept of local equilibrium, which enables one to evaluate averages of nonlinear terms and to close the averaged equation. Examples include the Kuramoto-Sivashinsky equation and its modifications.     

Theory of irreversible processes in nonequilibrium quantum systems. I. General formalism

The theory of Van Hove for nonequilibrium quantum statistical mechanics is extensively reformulated in terms of a superspace (a kind of operator space). This reformulation enables us to introduce a diagrammatic method which makes it convenient to deal with practical problems in physical systems. In our formalism, quantum statistical effects are considered on the basis of a systematic rule for the contraction technique. A complicated statistical effect in boson or fermion systems can be treated by starting with a simple unsymmetrized formalism in the Boltzmann statistics.     

Liquid–gas critical phenomena under confinement: small-angle neutron scattering studies of CO2 in aerogel

Small angle neutron scattering (SANS) is a well-established technique for investigating the behavior of confined binary liquid solutions, as it can probe the correlation length and susceptibility in pores on length scales 1 – 100 nm. We applied SANS to explore the influence of confinement on critical behavior of an individual fluid carbon dioxide (CO₂) in a highly porous aerogel. The results demonstrate that quenched disorder induced by aerogel significantly depresses density fluctuations. Despite the negligible volume occupied by aerogel (< 4%), the macroscopic phase separation of confined CO₂ into coexisting liquid and gaseous phases is suppressed and below the critical temperature of the bulk fluid frozen methastable microdomains are formed. Experimental data show that critical adsorption is as important as the effect of confinement in defining the behavior of confined fluids.     

On the assumption of initial factorization in the master equation for weakly coupled systems I: General framework

We analyze the dynamics of a quantum mechanical system in interaction with a reservoir when the initial state is not factorized. In the weak-coupling (van Hove) limit, the dynamics can be properly described in terms of a master equation, but a consistent application of Nakajima-Zwanzig’s projection method requires that the reference (not necessarily equilibrium) state of the reservoir be endowed with the mixing property.     

Population growth in random media. I. Variational formula and phase diagram

We consider an infinite system of particles on the integer lattice Z that: (1) migrate to the right with a random delay, (2) branch along the way according to a random law depending on their position (random medium). In Part I, the first part of a two-part presentation, the initial configuration has one particle at each site. The long-time limit exponential growth rate of the expected number of particles at site 0 (local particle density) does not depend on the realization of the random medium, but only on the law. It is computed in the form of a variational formula that can be solved explicitly. The result reveals two phase transitions associated with localization vs. delocalization and survival vs. extinction. In earlier work the exponential growth rate of the Cesaro limit of the number of particles per site (global particle density) was studied and a different variational formula was found, but with similar structure, solution, and phases. Combination of the two results reveals an intermediate phase where the population globally survives but locally becomes extinct (i.e., dies out on any fixed finite set of sites).     

On the direct determination of constrained pure state one-electron density matrices: Part I. A new theoretical model for closed-shell systems

A new method based on the penalty-function way of satisfying equality constraints is proposed for the determination of constrained pure state one-electron density matrices for closed-shell many-electron systems. The algorithm suggested can handle many constraints simultaneously. Certain interesting features of the proposed algorithm are discussed with numerical examples.     

Stationary states of crystal growth in three dimensions

A new Markov process describing crystal growth in three dimensions is introduced. States of the process are configurations of the crystal surface, which has a terrace-edge-kink structure. The states are continuous along edges but discrete across edges, in accordance with the very different rates for the two types of captures of particles. Stationary distributions, describing steady crystal growth, are found in general. To our knowledge, these are the first examples of stationary distributions for layered crystal growth in three dimensions. The steady growth rate and other quantities are obtained explicitly for two interacting edges. For many interacting edges, growth behavior is determined (a) in various asymptotic regimes including thermodynamic limits, (b) via simulations, and (c) using series (cluster) expansions in the slope of the surface, the first three coefficients being computed. The theoretical growth rates show a marked dependence on surface dimensions. This may contribute to the size dependence and dispersion in the observed growth rate of small crystals.     

Phase transition in tumor growth: I avascular development

We propose a mechanism for avascular tumor growth based on a simple chemical network. This model presents a logistic behavior and shows a “second order” phase transition. We prove the fractal origin of the empirical logistics and Gompertz constant and its relation to mitosis and apoptosis rate. Finally, the thermodynamics framework developed demonstrates the entropy production rate as a Lyapunov function during avascular tumor growth.