Temperature scaling, glassiness and stationarity in the Bak-Sneppen model

We show that the emergence of criticality in the locally-defined Bak-Sneppen model corresponds to separation over a hierarchy of timescales. Near to the critical point the model obeys scaling relations, with exponents which we derive numerically for a one-dimensional system. We further describe how the model can be related to the glass model of Bouchaud (J. Phys. I France 2, 1705 (1992)), and we use this insight to comment on the usual assumption of stationarity in the Bak-Sneppen model. Finally, we propose a general definition of self-organised criticality which is in partial agreement with other recent definitions. Received 14 January 2000 and Received in final form 18 April 2000     

A deterministic sandpile automaton revisited

The Bak-Tang-Wiesenfeld (BTW) sandpile model is a cellular automaton which has been intensively studied during the last years as a paradigm for self-organized criticality. In this paper, we reconsider a deterministic version of the BTW model introduced by Wiesenfeld, Theiler and McNamara, where sand grains are added always to one fixed site on the square lattice. Using the Abelian sandpile formalism we discuss the static properties of the system. We present numerical evidence that the deterministic model is only in the BTW universality class if the initial conditions and the geometric form of the boundaries do not respect the full symmetry of the square lattice. Received 19 August 1999     

Stochastic resonance in a model of opinion formation on small-world networks

We analyze the phenomenon of stochastic resonance in an Ising-like system on a small-world network. The system, which is subject to the combined action of noise and an external modulation, can be interpreted as a stylized model of opinion formation by imitation under the effects of a “fashion wave”. Both the amplitude threshold for the detection of the external modulation and the width of the stochastic-resonance peak show considerable variation as the randomness of the underlying small-world network is changed. Received 19 December 2001     

Brownian particles far from equilibrium

We study a model of Brownian particles which are pumped with energy by means of a non-linear friction function, for which different types are discussed. A suitable expression for a non-linear, velocity-dependent friction function is derived by considering an internal energy depot of the Brownian particles. In this case, the friction function describes the pumping of energy in the range of small velocities, while in the range of large velocities the known limit of dissipative friction is reached. In order to investigate the influence of additional energy supply, we discuss the velocity distribution function for different cases. Analytical solutions of the corresponding Fokker-Planck equation in 2d are presented and compared with computer simulations. Different to the case of passive Brownian motion, we find several new features of the dynamics, such as the formation of limit cycles in the four-dimensional phase-space, a large mean squared displacement which increases quadratically with the energy supply, or non-equilibrium velocity distributions with crater-like form. Further, we point to some generalizations and possible applications of the model. Received 24 November 1999     

Escape of Brownian particles in a bistable sawtooth potential driven by correlated white noises

An exact analytic expression of the relative escape rate (RER) for Brownian particles in a bistable sawtooth potential driven by correlated white noises is obtained. It is found that the RER vs. R (the multiplicative to the additive noise intensities ratio) exhibits a suppression platform for positive correlation, whereas the resonant activation and suppression platform appear successively for negative correlation. The mechanism of the present phenomena is explained. The effects of a nonlinear potential on the RER are studied. We have numerically calculated the RER of the system under a parabolic potential and a quartic potential and have compared the differences of the RER in the case of the linear potential and the one of the nonlinear potential. Received 16 September 2000 and Received in final form 15 January 2001     

Visualizing the log-periodic pattern before crashes

We present a method for visualizing the pattern which we believe to be a precursor signature of financial crashes (or ruptures). The log-periodicity of the pattern is investigated through the envelope function technique. Three periods of the Dow Jones Industrial Average (DJIA) are investigated: 1982-1987, 1992-1997 and 1993-1998. The presence of a rupture in the end of 1998 is outlined from data taken before the end of August 1998. Received 15 October 1998 and Received in final form 19 November 1998     

Multifractality of some random and disordered systems as a thermodynamics in fractal phase space

It is shown that multifractal properties of some random and disordered systems can be simulated using thermodynamics of a generalized ideal monoatomic gas in a fractal phase space. Received 25 November 1998 and Received in final form 16 December 1998     

Escape of Brownian particles in an asymmetric bistable sawtooth potential driven by cross-correlated noises

The relative escape rate (RER) for Brownian particles in an asymmetric bistable sawtooth potential driven by cross correlations between multiplicative white noise and additive white noise is studied. A new expression of the mean first-passage time is derived under the condition of piecewise linear potentials and discontinuous diffusion function. Based on the results of RER numerically calculated, it is found that (i) under positively correlated noises action (i.e. λ > 0, and λ is the correlation strength), the escape rate exhibits the suppression platform as the intensity of multiplicative noise varies. The effect of suppression becomes more pronounced with the growth of height of the deterministic potential barrier for transition, and with the increase of λ. However, for the case of uncorrelated noises (λ = 0) and of negatively correlated noises (λ < 0), the suppression platform disappears. (ii) The positive correlation between noises amplifies the change of the escape rate with the height of barrier for transition, while the negative correlation between noises suppresses this change. Received 20 November 2002 / Received in final form 19 October 2003 Published online 23 May 2003 RID="a" ID="a"e-mail: kmdcmei@public.km.yn.cn     

Extremal coupled map lattices

Multifractal critical phenomena with infinite-temperature critical point and with complex coexistence of the infinite and finite temperature critical points are considered and it is shown that strange attractors generated by cascades of period-doubling bifurcations (Feigenbaum scenario) as well as fields of velocity differences in fluid turbulence belong to the former subclass of the multifractal critical phenomena, while the real traffic processes and real currency exchange processes belong to the last (complex) subclass of the multifractal critical phenomena. Data obtained by different authors are used for this purpose. Received 5 February 1999     

Activated drift motion of a classical particle with a dynamical pinning effect

A one dimensional trap model for a thermally activated classical particle is introduced to simulate driven dynamics in presence of “ageing” effects. The depth of each trap increases with the time elapsed since the particle has fallen into it. The consequences of this dynamical pinning are studied, and velocity-force characteristics are numerically obtained. A special attention is paid to the situation where the particle is pulled with a spring to ensure a finite average velocity. In the low velocity regime, the presence of a broad distribution of trapping times leads to suppression of linear response, replaced by a threshold or by sublinear dynamics. A regime of strong fluctuations is obtained when the particle is driven at intermediate velocities. Received: 12 December 1997 / Accepted: 25 February 1998     

Steady-states and kinetics of ordering in bus-route models: connection with the Nagel-Schreckenberg model

A Bus Route Model (BRM) can be defined on a one-dimensional lattice, where buses are represented by “particles” that are driven forward from one site to the next with each site representing a bus stop. We replace the random sequential updating rules in an earlier BRM by parallel updating rules. In order to elucidate the connection between the BRM with parallel updating (BRMPU) and the Nagel-Schreckenberg (NaSch) model, we propose two alternative extensions of the NaSch model with space-/time-dependent hopping rates. Approximating the BRMPU as a generalization of the NaSch model, we calculate analytically the steady-state distribution of the time headways (TH) which are defined as the time intervals between the departures (or arrivals) of two successive particles (i.e., buses) recorded by a detector placed at a fixed site (i.e., bus stop) on the model route. We compare these TH distributions with the corresponding results of our computer simulations of the BRMPU, as well as with the data from the simulation of the two extended NaSch models. We also investigate interesting kinetic properties exhibited by the BRMPU during its time evolution from random initial states towards its steady-states. Received 16 December 1999     

Cellular oscillator with a small number of particles

To illustrate complex spatio-temporal effects which are possible in cellular reactions with a small number of particles, we present Monte Carlo simulations of the formation of oscillatory spark-like patterns in a model completely stochastic Ca⁺² oscillator. Our analysis shows that in order to observe such patterns the minimum average number of Ca⁺² ions in the cytosol may be as low as about 50. Received 21 April 2002 / Received in final form 19 June 2002 Published online 14 October 2002 RID="a" ID="a"e-mail: zhdanov@catalysis.nsk.su     

Noise-induced bifurcations and chaos in the average motion of globally-coupled oscillators

A system of coupled master equations simplified from a model of noise-driven globally coupled bistable oscillators under periodic forcing is investigated. In the thermodynamic limit, the system is reduced to a set of two coupled differential equations. Rich bifurcations to subharmonics and chaotic motions are found. This behavior can be found only for certain intermediate noise intensities. Noise with intensities which are too small or too large will certainly spoil the bifurcations. In a system with large though finite size, the bifurcations to chaos induced by noise can still be detected to a certain degree. Received 6 April 1999 and Received in final form 1 November 1999     

Quantum-mechanical tunneling in associative neural networks

We investigate the quantum-mechanical tunneling between the “patterns" of the, so-called, associative neural networks. Being the relatively stable minima of the “configuration-energy" space of the networks, the “patterns" represent the macroscopically distinguishable states of the neural nets. Therefore, the tunneling represents a macroscopic quantum effect, but with some special characteristics. Particularly, we investigate the tunneling between the minima of approximately equal depth, thus requiring no energy exchange. If there are at least a few such minima, the tunneling represents a sort of the “random walk" process, which implies the quantum fluctuations in the system, and therefore “malfunctioning" in the information processing of the nets. Due to the finite number of the minima, the “random walk" reduces to a dynamics modeled by the, so-called, Pauli master equation. With some plausible assumptions, the set(s) of the Pauli master equations can be analytically solved. This way comes the main result of this paper: the quantum fluctuations due to the quantum-mechanical tunneling can be “minimized" if the “pattern"-formation is such that there are mutually “distant" groups of the “patterns", thus providing the “zone" structure of the “pattern" formation. This qualitative result can be considered as a basis of the efficient deterministic functioning of the associative neural nets. Received 15 July 1999     

On the structure of competitive societies

We model the dynamics of social structure by a simple interacting particle system. The social standing of an individual agent is represented by an integer-valued fitness that changes via two offsetting processes. When two agents interact one advances: the fitter with probability p and the less fit with probability 1-p. The fitness of an agent may also decline with rate r. From a scaling analysis of the underlying master equations for the fitness distribution of the population, we find four distinct social structures as a function of the governing parameters p and r. These include: (i) a static lower-class society where all agents have finite fitness; (ii) an upwardly-mobile middle-class society; (iii) a hierarchical society where a finite fraction of the population belongs to a middle class and a complementary fraction to the lower class; (iv) an egalitarian society where all agents are upwardly mobile and have nearly the same fitness. We determine the basic features of the fitness distributions in these four phases.     

On the 1/D expansion for directed polymers

We present a variational approach for directed polymers in D transversal dimensions which is used to compute the correction to the mean field theory predictions with broken replica symmetry. The trial function is taken to be a symmetrized version of the mean-field solution, which is known to be exact for . We compute the free energy corresponding to that function and show that the finite-D corrections behave like D ^-4/3 . It means that the expansion in powers of 1/D should be used with great care here. We hope that the techniques developed in this note will be useful also in the study of spin glasses. Receveid 19 May 1998     

Noisy covariance matrices and portfolio optimization

According to recent findings [#!bouchaud!#,#!stanley!#], empirical covariance matrices deduced from financial return series contain such a high amount of noise that, apart from a few large eigenvalues and the corresponding eigenvectors, their structure can essentially be regarded as random. In [#!bouchaud!#], e.g., it is reported that about 94% of the spectrum of these matrices can be fitted by that of a random matrix drawn from an appropriately chosen ensemble. In view of the fundamental role of covariance matrices in the theory of portfolio optimization as well as in industry-wide risk management practices, we analyze the possible implications of this effect. Simulation experiments with matrices having a structure such as described in [#!bouchaud!#,#!stanley!#] lead us to the conclusion that in the context of the classical portfolio problem (minimizing the portfolio variance under linear constraints) noise has relatively little effect. To leading order the solutions are determined by the stable, large eigenvalues, and the displacement of the solution (measured in variance) due to noise is rather small: depending on the size of the portfolio and on the length of the time series, it is of the order of 5 to 15%. The picture is completely different, however, if we attempt to minimize the variance under non-linear constraints, like those that arise e.g. in the problem of margin accounts or in international capital adequacy regulation. In these problems the presence of noise leads to a serious instability and a high degree of degeneracy of the solutions. Received 31 December 2001     

Noise effects on one-Pauli channels

The possibility of stochastic resonance of a quantum channel and hence the noise enhanced quantum channel capacity is explored by considering one-Pauli channels which are more classical like. The fidelity of the channel is also considered. Received 4 June 1999     

Quantum Brownian motion and the second law of thermodynamics

We consider a single harmonic oscillator coupled to a bath at zero temperature. As is well-known, the oscillator then has a higher average energy than that given by its ground state. Here we show analytically that for a damping model with arbitrarily discrete distribution of bath modes and damping models with continuous distributions of bath modes with cut-off frequencies, this excess energy is less than the work needed to couple the system to the bath, therefore, the quantum second law is not violated. On the other hand, the second law may be violated for bath modes without cut-off frequencies, which are, however, physically unrealistic models. An erratum to this article is available at .     

Symmetry alteration of ensemble return distribution in crash and rally days of financial markets

We select the n stocks traded in the New York Stock Exchange and we form a statistical ensemble of daily stock returns for each of the k trading days of our database from the stock price time series. We study the ensemble return distribution for each trading day and we find that the symmetry properties of the ensemble return distribution drastically change in crash and rally days of the market. In crash and rally days, the distribution becomes asymmetric. In particular for crashes the positive tail is steeper than the negative one whereas the reverse is observed in rally days. Received 25 February 2000