Effect of long range interactions on the growth of compact clusters under deposition

In the paper the role of long range interactions on the growth of a volume conserving surface is studied using the Nonlocal Conserved Kardar-Parisi-Zhang (NCKPZ) equation. It is shown that previous theoretical predictions are inconsistent with an exact one-dimensional result. This serves as a motivation for construction of a Self-Consistent Expansion (SCE) that recovers the exact one-dimensional result, and gives the scaling exponents in higher dimensions as well. A possible application of this result to colloidal systems is discussed.     

On some natural and model 2D bimodal random cellular structures

The topological and metric properties of a few natural 2D random cellular structures, namely an armadillo shell structure and young soap froths, which are formed from two classes of cells, large and small, have been characterized. The topological properties of a model generated from a Kagome tiling, which mimics such random binary structures, have also been exactly calculated. The distribution of the number of cell sides is bimodal for the structures investigated. In contrast to the classical Aboav-Weaire law for homogeneous 2D random cellular structures, nm(n), the mean total number of edges of neighbouring cells of cells with n sides does not vary linearly with n. Only the nm(i, n) (i=1,2) determined separately for every class of cells are linear in n for all investigated structures. Topological properties and correlations between metric and topological properties are finally compared with the predictions of various literature models. Received: 24 December 1997 / Revised: 7 April 1998 / Accepted: 20 April 1998     

Simulation study of random sequential adsorption of mixtures on a triangular lattice

Random sequential adsorption of binary mixtures of extended objects on a two-dimensional triangular lattice is studied numerically by means of Monte Carlo simulations. The depositing objects are formed by self-avoiding random walks on the lattice. We concentrate here on the influence of the symmetry properties of the shapes on the kinetics of the deposition processes in two-component mixtures. Approach to the jamming limit in the case of mixtures is found to be exponential, of the form: θ(t) ∼ θ_jam - Δθ exp(- t/σ), and the values of the parameter σ are determined by the order of symmetry of the less symmetric object in the mixture. Depending on the local geometry of the objects making the mixture, jamming coverage of a mixture can be either greater than both single-component jamming coverages or it can be in between these values. Results of the simulations for various fractional concentrations of the objects in the mixture are also presented.     

Non-commutative geometry and irreversibility

A kinetics built upon q-calculus, the calculus of discrete dilatations, is shown to describe diffusion on a hierarchical lattice. The only observable on this ultrametric space is the “quasi-position” whose eigenvalues are the levels of the hierarchy, corresponding to the volume of phase space available to the system at any given time. Motion along the lattice of quasi-positions is irreversible. Received: 24 June 1997 / Revised: 15 September 1997 / Accepted: 6 October 1997     

Scale-dependent functions in statistical hydrodynamics: a functional analysis point of view

Most of dynamic systems which exhibit chaotic behavior are also known to posses self-similarity and manifest strong fluctuations of all possible scales. The meaning of this terms is not always same. In present note we make an attempt to formulate the problem in the framework of functional analysis. The statistical hydrodynamics is taken as a vivid physical example. The links to wavelet analysis are presented. Received 22 August 1997     

When do finite sample effects significantly affect entropy estimates?

An expression is proposed for determining the error made by neglecting finite sample effects in entropy estimates. It is based on the Ansatz that the ranked distribution of probabilities tends to follow a Zipf scaling. Received 17 August 1998 and Received in final form 8 February 1999     

Propagation and ignition of fast gasless detonation waves of phase or chemical transformation in condensed matter

Fast self sustained waves of chemical or phase transformations, observed in several contexts in condensed matter effectively result in “gasless detonation". The phenomenon is modelled by coupling the reaction diffusion equation, describing chemical or phase transformations, and the wave equation, describing elastic perturbations. The coupling considered in this work involves (i) a dependence of the sound velocity on the chemical (phase) field, and (ii) the destruction of the initial chemical equilibrium when the strain exceeds a critical value (strain induced phase transition). Both the case of an initially unstable state (first order kinetics) and metastable state (second order kinetics) are considered. An exhaustive analytic and numerical study of travelling waves reveals the existence of supersonic modes of transformations. The practically important problem of ignition of fast waves by mechanical perturbation is investigated. With the present model, the critical strain necessary to ignite gasless detonation by local perturbations is determined. Received 18 November 1999     

Detonation type waves in phase (chemical) transformation processes in condensed matter

Fast self sustained waves (autowaves) associated with chemical or phase transformations are observed in many situations in condensed matter. They are governed neither by diffusion of matter or heat (as in combustion processes) nor by a travelling shock wave (as in gaseous detonation). Instead, they result from a coupling between phase transformation and the stress field, and may be classified as gasless detonation autowaves in solids. We propose a simple model to describe these regimes. The model rests on the classical equations of elastic deformations in a 1-dimensional solid bar, with the extra assumption that the phase (chemical) transformation induces a change of the sound velocity. The transformations are assumed to occur through a chain branched mechanism, which starts when the mechanical stress exceeds a given threshold. Our investigation shows that supersonic autowaves exist in this model. In the absence of diffusion (dissipation factor, losses), a continuum of travelling wave solutions is found. In the presence of diffusion, a steady state supersonic wave solution is found, along with a slower wave controlled by diffusion. Received 15 October 1998     

The short-time critical behaviour of the Ginzburg-Landau model with long-range interaction

The renormalisation group approach is applied to the study of the short-time critical behaviour of the d-dimensional Ginzburg-Landau model with long-range interaction of the form in momentum space. Firstly the system is quenched from a high temperature to the critical temperature and then relaxes to equilibrium within the model A dynamics. The asymptotic scaling laws and the initial slip exponents and of the order parameter and the response function respectively, are calculated to the second order in . Received 9 June 2000 and Received in final form 2 August 2000     

Power, Lévy, exponential and Gaussian-like regimes in autocatalytic financial systems

We study by theoretical analysis and by direct numerical simulation the dynamics of a wide class of asynchronous stochastic systems composed of many autocatalytic degrees of freedom. We describe the generic emergence of truncated power laws in the size distribution of their individual elements. The exponents α of these power laws are time independent and depend only on the way the elements with very small values are treated. These truncated power laws determine the collective time evolution of the system. In particular the global stochastic fluctuations of the system differ from the normal Gaussian noise according to the time and size scales at which these fluctuations are considered. We describe the ranges in which these fluctuations are parameterized respectively by: the Lévy regime α < 2, the power law decay with large exponent ( α > 2), and the exponential decay. Finally we relate these results to the large exponent power laws found in the actual behavior of the stock markets and to the exponential cut-off detected in certain recent measurement. Received 29 July 2000 and Received in final form 25 September 2000     

A definition of metastability for Markov processes with detailed balance

A definition of metastable states applicable to arbitrary finite state Markov processes satisfying detailed balance is discussed. In particular, we identify a crucial condition that distinguishes genuine metastable states from other types of slowly decaying modes and which leads to properties similar to those postulated in the restricted ensemble approach [1]. The intuitive physical meaning of this condition is simply that the total equilibrium probability of finding the system in the metastable state is negligible. As a concrete application of our formalism we present preliminary results on a 2D kinetic Ising model.     

Slow coarsening in a class of driven systems

The coarsening process in a class of driven systems is studied. These systems have previously been shown to exhibit phase separation and slow coarsening in one dimension. We consider generalizations of this class of models to higher dimensions. In particular we study a system of three types of particles that diffuse under local conserving dynamics in two dimensions. Arguments and numerical studies are presented indicating that the coarsening process in any number of dimensions is logarithmically slow in time. A key feature of this behavior is that the interfaces separating the various growing domains are macroscopically smooth (well approximated by a Fermi function). This implies that the coarsening mechanism in one dimension is readily extendible to higher dimensions. Received 3 April 2000     

Anomalous scaling in the Zhang model

We apply the moment analysis technique to analyze large scale simulations of the Zhang sandpile model. We find that this model shows different scaling behavior depending on the update mechanism used. With the standard parallel updating, the Zhang model violates the finite-size scaling hypothesis, and it also appears to be incompatible with the more general multifractal scaling form. This makes impossible its affiliation to any one of the known universality classes of sandpile models. With sequential updating, it shows scaling for the size and area distribution. The introduction of stochasticity into the toppling rules of the parallel Zhang model leads to a scaling behavior compatible with the Manna universality class. Received 8 August 2000 and Received in final form 4 October 2000     

Wealth distributions in asset exchange models

A model for the evolution of the wealth distribution in an economically interacting population is introduced, in which a specified amount of assets are exchanged between two individuals when they interact. The resulting wealth distributions are determined for a variety of exchange rules. For “random” exchange, either individual is equally likely to gain in a trade, while “greedy” exchange, the richer individual gains. When the amount of asset traded is fixed, random exchange leads to a Gaussian wealth distribution, while greedy exchange gives a Fermi-like scaled wealth distribution in the long-time limit. Multiplicative processes are also investigated, where the amount of asset exchanged is a finite fraction of the wealth of one of the traders. For random multiplicative exchange, a steady state occurs, while in greedy multiplicative exchange a continuously evolving power law wealth distribution arises. Received: 13 August 1997 / Revised: 31 December 1997 / Accepted: 26 January 1998     

The extremal independence problem

Consider a finite sequence of independent-though not, necessarily, identically distributed-real-valued random scores. If the scores are absolutely continuous random variables, the sequence possesses a unique maximum (minimum). We say that “maximal (minimal) independence” holds if the value and the identity of the sequence’s unique maximal (minimal) score are independent random variables. In this research we study the class of statistics for which maximal (minimal) independence holds, and: (i) establish explicit characterizations of this class; (ii) connect this class with the class of Lévy processes; (iii) unveil the underlying spatial Poissonian structure of this class.     

Revealing of the simplest mechanisms of a structure formation

In this paper, results of investigations of the simplest mechanisms of a structure formation are presented. In frameworks of the suggested model the main attention was focused on such characteristics as wiring of the system, clusters formation, dynamics of the wiring. The idea to take into account an influence of the environment factor is employed in the proposed model. Investigations of systems with such principle of a structure formation reveal that the system's dynamics has typical features of self-organized criticality phenomenon. In the avalanche-like processes, which occur in the wiring dynamics, a power law was found with the index close to 1.4. It is independent on the environment factor (which in a sense can be considered as system parameter). The system wiring is approximated pretty well by the Gaussian distribution. The size of the system does not play any role in the dynamics of the model. Received 10 March 1999 and Received in final form 24 May 1999     

Diffusion and electromigration on disordered surfaces

We study a one-dimensional disordered solid-on-solid model in which neighboring columns are shifted by quenched random phases. The static height-difference correlation function displays a minimum at a nonzero temperature. The model is equipped with volume-conserving surface diffusion dynamics, including a possible bias due to an electromigration force. In the case of Arrhenius jump rates a continuum equation for the evolution of macroscopic profiles is derived and confirmed by direct simulation. Dynamic surface fluctuations are investigated using simulations and phenomenological Langevin equations. In these equations the quenched disorder appears in the form of time-independent random forces. The disorder does not qualitatively change the roughening dynamics of near-equilibrium surfaces, but in the case of biased surface diffusion with Metropolis rates it induces a new roughening mechanism, which leads to an increase of the surface width as . Received 7 February 2000     

Truncated Lévy laws and 2D turbulence

We study the probability distribution functions and scaling properties of truncated Lévy processes with sharp cut-offs. We find that they display features analog to those observed in some 2D numerical simulations of turbulence. Received: 29 October 1997 / Revised: 12 February 1998 / Accepted: 10 April 1998