arXiv subjects

Zbigniew Koza

Publications and source records attributed to Zbigniew Koza.

28 records · Page 2Linked to original sources

arXiv2006

The Ising model in a Bak-Tang-Wiesenfeld sandpile

We study the spin-1 Ising model with non-local constraints imposed by the Bak-Tang-Wiesenfeld sandpile model of self-organized criticality (SOC). The model is constructed as if the sandpile is being built on a (honeycomb) lattice with Ising interactions. In this way we combine two models that exhibit power-law decay of correlation functions characterized by different exponents. We discuss the model properties through an order parameter and the mean energy per node, as well as the temperature dependence of their fourth-order Binder cumulants. We find (i) a thermodynamic phase transition at a finite T_c between paramagnetic and antiferromagnetic phases, and (ii) that above T_c the correlation functions decay in a way typical of SOC. The usual thermodynamic criticality of the two-dimensional Ising model is not affected by SOC constraints (the specific heat critical exponent α\approx 0), nor are SOC-induced correlations affected by the interactions of the Ising model. Even though the constraints imposed by the SOC model induce long-range correlations, as if at standard (thermodynamic) criticality, these SOC-induced correlations have no impact on the thermodynamic functions.

cond-mat.stat-mech↗

arXiv2004

Fractal dimension of transport coefficients in a deterministic dynamical system

In many low-dimensional dynamical systems transport coefficients are very irregular, perhaps even fractal functions of control parameters. To analyse this phenomenon we study a dynamical system defined by a piece-wise linear map and investigate the dependence of transport coefficients on the slope of the map. We present analytical arguments, supported by numerical calculations, showing that both the Minkowski-Bouligand and Hausdorff fractal dimension of the graphs of these functions is 1 with a logarithmic correction, and find that the exponent $γ$ controlling this correction is bounded from above by 1 or 2, depending on some detailed properties of the system. Using numerical techniques we show local self-similarity of the graphs. The local self-similarity scaling transformations turn out to depend (irregularly) on the values of the system control parameters.

nlin.CD↗

arXiv2004

Fractal properties of the diffusion coefficient in a simple deterministic dynamical system: a numerical study

Using a numerical library for arbitrary precision arithmetic I study the irregular dependence of the diffusion coefficient on the slope of a piecewise linear map defining a dynamical system. I find that the graph of the diffusion coefficient as a function of the slope has the fractal dimension 1, but the convergence to this limit is slowed down by logarithmic corrections. The exponent controlling this correction depends on the slope and is either 1 or 2 depending on existence and properties of a Markov partition.

nlin.CD↗

arXiv2003

Numerical analysis of reversible A + B <-> C reaction-diffusion systems

We develop an effective numerical method of studying large-time properties of reversible reaction-diffusion systems of type A + B <-> C with initially separated reactants. Using it we find that there are three types of asymptotic reaction zones. In particular we show that the reaction rate can be locally negative and concentrations of species A and B can be nonmonotonic functions of the space coordinate x, locally significantly exceeding their initial values.

cond-mat.stat-mech↗

arXiv2002

General relation between drift velocity and dispersion of a molecular motor

We model a processive linear molecular motor as a particle diffusing in a one-dimensional periodic lattice with arbitrary transition rates between its sites. We present a relatively simple proof of a theorem which states that the ratio of the drift velocity V to the diffusion coefficient D has the upper bound 2N/d, where N is the number of nodes in an elementary cell and d denotes its length. This relation can be used to estimate the minimal value of internal states of the motor and the maximal value of the so called Einstein force, which approximately equals to the maximal force exerted by a molecular motor.

cond-mat.stat-mech↗

arXiv2002

Asymptotic expansion for reversible A + B <-> C reaction-diffusion process

We study long-time properties of reversible reaction-diffusion systems of type A + B <-> C by means of perturbation expansion in powers of 1/t (inverse of time). For the case of equal diffusion coefficients we present exact formulas for the asymptotic forms of reactant concentrations and a complete, recursive expression for an arbitrary term of the expansions. Taking an appropriate limit we show that by studying reversible reactions one can obtain "singular" solutions typical of irreversible reactions.

cond-mat.stat-mech↗

arXiv1999

General technique of calculating drift velocity and diffusion coefficient in arbitrary periodic systems

We develop a practical method of computing the stationary drift velocity V and the diffusion coefficient D of a particle (or a few particles) in a periodic system with arbitrary transition rates. We solve this problem both in a physically relevant continuous-time approach as well as for models with discrete-time kinetics, which are often used in computer simulations. We show that both approaches yield the same value of the drift, but the difference between the diffusion coefficients obtained in each of them equals V*V/2. Generalization to spaces of arbitrary dimension and several applications of the method are also presented.

cond-mat.stat-mech↗

arXiv1996

The asymptotic behaviour of the initially separated A + B(static) -> 0 reaction-diffusion systems

We examine the long-time behaviour of A+B \to 0 reaction-diffusion systems with initially separated species A and B. All of our analysis is carried out for arbitrary (positive) values of the diffusion constant D_A of particles A and initial concentrations a_0 and b_0 of A's and B's. We derive general formulae for the location of the reaction zone centre, the total reaction rate, and the concentration profile of species A outside the reaction zone. The general properties of the reaction zone are studied with a help of the scaling ansatz. Using the mean-field approximation we find the functional forms of `tails' of the reaction rate R and the dependence of the width of the reaction zone on the external parameters of the system. We also study the change in the kinetics of the system with D_B > 0 in the limit D_B \to 0. Our results are supported by numerical solutions of the mean-field reaction-diffusion equation.

cond-mat.stat-mech↗

arXiv1996

The long time behavior of initially separated A+B->0 reaction-diffusion systems with arbitrary diffusion constants

We examine the long time behaviour of A+B->0 reaction diffusion systems with initially segregated species A and B. All of our analysis is carried out for arbitrary (positive) values of the diffusion constants $D_A$, $D_B$, and initial concentrations $a_0$ and $b_0$ of A's and B's. We divide the domain of the partial differential equations describing the problem into several regions in which they can be reduced to simpler, solvable equations, and we merge the solutions. Thus we derive general formulae for the concentration profiles outside the reaction zone, the location of the reaction zone center, and the total reaction rate. An asymptotic condition for the reaction front to be stationary is also derived. The properties of the reaction layer are studied in the mean-field approximation, and we show that not only the scaling exponents, but also the scaling functions are independent of $D_A$, $D_B$, $a_0$ and $b_0$.

cond-mat↗

arXiv1994

Dynamic Multiscaling of the Reaction-Diffusion Front for mA+nB ->0

We consider the reaction zone that grows between separated regions of diffusing species $A$ and $B$ that react according to $mA+nB\to 0$, within the framework of the mean-fieldlike reaction-diffusion equations. For distances from the centre of the reaction zone much smaller than the diffusion length $X_D\equiv \sqrt{Dt}$, the particle density profiles are described by the scaling forms predicted by a quasistatic approximation, whereas they have a diffusive cutoff at a distance of order $X_D$. This cutoff, and the power-law decay of the quasistatic profiles, give rise to multiscaling behaviour, with anomalous values for the exponents describing the moments of the density and reaction profiles. Numerical solutions of the reaction-diffusion equations are in good quantitative agreement with the predictions of this theory.

cond-mat↗