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     

Non-linear evolution of step meander during growth of a vicinal surface with no desorption

Step meandering due to a deterministic morphological instability on vicinal surfaces during growth is studied. We investigate nonlinear dynamics of a step model with asymmetric step kinetics, terrace and line diffusion, by means of a multiscale analysis. We give the detailed derivation of the highly nonlinear evolution equation on which a brief account has been given [6]. Decomposing the model into driving and relaxational contributions, we give a profound explanation to the origin of the unusual divergent scaling of step meander (where F is the incoming atom flux). A careful numerical analysis indicates that a cellular structure arises where plateaus form, as opposed to spike-like structures reported erroneously in reference [6]. As a robust feature, the amplitude of these cells scales as t ^1/2, regardless of the strength of the Ehrlich-Schwoebel effect, or the presence of line diffusion. A simple ansatz allows to describe analytically the asymptotic regime quantitatively. We show also how sub-dominant terms from multiscale analysis account for the loss of up-down symmetry of the cellular structure. Received 4 May 2000 and Received in final form 8 September 2000     

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     

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     

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 .     

Exactly solvable reaction diffusion models on a Cayley tree

The most general reaction-diffusion model on a Cayley tree with nearest-neighbor interactions is introduced, which can be solved exactly through the empty-interval method. The stationary solutions of such models, as well as their dynamics, are discussed. Concerning the dynamics, the spectrum of the evolution Hamiltonian is found and shown to be discrete, hence there is a finite relaxation time in the evolution of the system towards its stationary state.     

A quasi-elastic regime for vibrated granular gases

Using simple scaling arguments and two-dimensional numerical simulations of a granular gas excited by vibrating one of the container boundaries, we study a double limit of small 1-r and large L, where r is the restitution coefficient and L the size of the container. We show that if the particle density n₀ and (1-r²)(n₀ Ld) where d is the particle diameter, are kept constant and small enough, the granular temperature, i.e. the mean value of the kinetic energy per particle, 〈E 〉/N, tends to a constant whereas the mean dissipated power per particle, 〈D 〉/N, decreases like when N increases, provided that (1-r²)(n₀ Ld)² < 1. The relative fluctuations of E, D and the power injected by the moving boundary, I, have simple properties in that regime. In addition, the granular temperature can be determined from the fluctuations of the power I(t) injected by the moving boundary.     

Phase transition in annihilation-limited processes

A system of particles is studied in which the stochastic processes are one-particle type-change (or one-particle diffusion) and multi-particle annihilation. It is shown that, if the annihilation rate tends to zero but the initial values of the average number of the particles tend to infinity, so that the annihilation rate times a certain power of the initial values of the average number of the particles remain constant (the double scaling) then if the initial state of the system is a multi-Poisson distribution, the system always remains in a state of multi-Poisson distribution, but with evolving parameters. The large time behavior of the system is also investigated. The system exhibits a dynamical phase transition. It is seen that for a k-particle annihilation, if k is larger than a critical value k_c, which is determined by the type-change rates, then annihilation does not enter the relaxation exponent of the system; while for k < k_c, it is the annihilation (in fact k itself) which determines the relaxation exponent.     

Relationships between a microscopic parameter and the stochastic equations for interface's evolution of two growth models

The relationship between a microscopic parameter p, that is related to the probability of choosing a mechanism of deposition, and the stochastic equation for the interface's evolution is studied for two different models. It is found that in one model, that is similar to ballistic deposition, the corresponding stochastic equation can be represented by a Kardar-Parisi-Zhang (KPZ) equation where both λ and ν depend on p in the following way: ν(p) = νp and λ(p) = λp ^3/2. Furthermore, in the other studied model, which is similar to random deposition with relaxation, the stochastic equation can be represented by an Edwards-Wilkinson (EW) equation where ν depends on p according to ν(p) = νp ². It is expected that these results will help to find a framework for the development of stochastic equations starting from microscopic details of growth models. Received 26 August 2002 / Received in final form 20 November 2002 Published online 6 March 2003 RID="a" ID="a"e-mail: ealbano@inifta.unlp.edu.ar     

Exactly solvable models through the generalized empty interval method,for multi-species interactions

Multi-species reaction-diffusion systems, with nearest-neighbor interaction on a one-dimensional lattice are considered. Necessary and sufficient constraints on the interaction rates are obtained, that guarantee the closedness of the time evolution equation for E _n(t)'s, the expectation value of the product of certain linear combination of the number operators on n consecutive sites at time t. The constraints are solved for the single-species left-right-symmetric systems. Also, examples of multi-species system for which the evolution equations of E _n(t)'s are closed, are given. Received 25 September 2002 / Received in final form 3 December 2002 Published online 14 February 2003 RID="a" ID="a"e-mail: mamwad@iasbs.ac.ir     

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     

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     

Dynamic renormalization-group analysis of the d+1 dimensional Kuramoto-Sivashinsky equation with both conservative and nonconservative noises

The long-wavelength properties of the (d + 1)-dimensional Kuramoto-Sivashinsky (KS) equation with both conservative and nonconservative noises are investigated by use of the dynamic renormalization-group (DRG) theory. The dynamic exponent z and roughness exponent α are calculated for substrate dimensions d = 1 and d = 2, respectively. In the case of d = 1, we arrive at the critical exponents z = 1.5 and α = 0.5 , which are consistent with the results obtained by Ueno et al. in the discussion of the same noisy KS equation in 1+1 dimensions [Phys. Rev. E 71, 046138 (2005)] and are believed to be identical with the dynamic scaling of the Kardar-Parisi-Zhang (KPZ) in 1+1 dimensions. In the case of d = 2, we find a fixed point with the dynamic exponents z = 2.866 and α = -0.866 , which show that, as in the 1 + 1 dimensions situation, the existence of the conservative noise in 2 + 1 or higher dimensional KS equation can also lead to new fixed points with different dynamic scaling exponents. In addition, since a higher order approximation is adopted, our calculations in this paper have improved the results obtained previously by Cuerno and Lauritsen [Phys. Rev. E 52, 4853 (1995)] in the DRG analysis of the noisy KS equation, where the conservative noise is not taken into account.     

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     

The underdamped Josephson junction subjected to colored noises

The properties of the underdamped Josephson junction subjected to colored noises were investigated with large and small phase difference (φ). For the case of the large φ, we found numerically that: (i) the probability distribution function of φ exhibits monostability → bistability → monostability transitions as the autocorrelation rate (λ) of a colored noise increases; (ii) in the bistability region the multiplicative noise drives the phase difference to turn over periodically; (iii) the slope K of the linear response of the junction potential difference (〈V 〉) can be somewhat reduced by means of tuning an optimal λ; (iv) the amplitude of φ in response to external sinusoidal signals changes with λ. For the case of small φ, after deriving the analytical expressions of the potential difference amplitude (〈V 〉_max) and the K in the presence of a dichotomous noise, we found nonmonotonic behavior of 〈V 〉_max and the slope K as a function of λ.     

Measuring a colloidal particle's interaction with a flat surface under nonequilibrium conditions

A new and general approach is proposed to analyze the dynamics of a colloidal particle interacting with a nearby wall. This analysis can be used to determine the acting forces even when the system is non-stationary. As an illustration, we use total internal reflection microscopy to investigate the forces acting on a polystyrene sulfate latex particle as it is receding from a charged glass surface. Received 10 October 2002 Published online: 16 April 2003 RID="a" ID="a"Present address: Department of Polymer Physics, BASF Aktiengesellschaft, 67056 Ludwigshafen, Germany RID="b" ID="b"Present address: Arryx. Inc., Chicago, IL 60601, USA     

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     

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     

Dynamics and scaling of noise-induced domain growth

The domain growth processes originating from noise-induced nonequilibrium phase transitions are analyzed, both for non-conserved and conserved dynamics. The existence of a dynamical scaling regime is established in the two cases, and the corresponding growth laws are determined. The resulting universal dynamical scaling scenarios are those of Allen-Cahn and Lifshitz-Slyozov, respectively. Additionally, the effect of noise sources on the behaviour of the pair correlation function at short distances is studied. Received 28 June 2000 and Received in final form 29 September 2000