arXiv subjects

Tadeusz Kosztołowicz

Publications and source records attributed to Tadeusz Kosztołowicz.

28 records · Page 2Linked to original sources

arXiv2016

Subdiffusion--absorption process in a system consisting of two different media

Subdiffusion with reaction $A+B\rightarrow B$ is considered in a system which consists of two homogeneous media joined together; the $A$ particles are mobile whereas $B$ are static. Subdiffusion and reaction parameters, which are assumed to be independent of time and space variable, can be different in both media. Particles $A$ move freely across the border between the media. In each part of the system the process is described by the subdiffusion--reaction equations with fractional time derivative. By means of the method presented in this paper we derive both the fundamental solutions (the Green's functions) $P(x,t)$ to the subdiffusion--reaction equations and the boundary conditions at the border between the media. One of the conditions demands the continuity of a flux and the other one contains the Riemann--Liouville fractional time derivatives $\partial^{α_1}P(0^+,t)/\partial t^{α_1}=(D_1/D_2)\partial^{α_2}P(0^-,t)/\partial t^{α_2}$, where the subdiffusion parameters $α_1$, $D_1$ and $α_2$, $D_2$ are defined in the regions $x<0$ and $x>0$, respectively.

cond-mat.stat-mech↗

arXiv2015

Subdiffusive random walk in a membrane system. The generalized method of images approach

In this paper we study subdiffusion in a system with a thin membrane. At the beginning, the random walk of a particle is considered in a system with a discrete time and space variable and then the probability describing the evolution of the particle's position (Green's function) is transformed into a continuous system. Two models are considered differing here from each other regarding the assumptions about how the particle is stopped or reflected by the membrane when the particle attempts to pass through the membrane fails. We show that for a system in which a membrane is partially permeable with respect to both its sides the Green's functions obtained for both models within the {\it continuous time random walk formalism} are equivalent to each other and expressed by the functions presented in the paper: T. Kosztołowicz, Phys. Rev. E \textbf{91}, 022102 (2015), except the values defined at the membrane's surfaces. We also show that for a system with a one--sidedly fully permeable membrane, the Green's functions obtained within both models are not equivalent to each other. We also present the generalized method of images which provides the Green's functions for the membrane system, obtained in this paper. This method, which has a simple physical interpretation, is of a general nature and, in our opinion, can be used to find the Green's functions for a system with a thin membrane in which various models of subdiffusion can be applied. As an example, we find the Green's functions for the particular case of a `slow--subdiffusion' process in a system with a thin membrane.

cond-mat.stat-mech↗

arXiv2013

Cattaneo--type subdiffusion--reaction equation

Subdiffusion in a system in which mobile particles $A$ can chemically react with static particles $B$ according to the rule $A+B\rightarrow B$ is considered within a persistent random walk model. This model, which assumes a correlation between successive steps of particles, provides hyperbolic Cattaneo normal diffusion or fractional subdiffusion equations for a system without chemical reactions. Starting with the difference equation, which describes a persistent random walk in a system with chemical reactions, using the generating function method and the continuous time random walk formalism, we will derive the Cattaneo--type subdiffusion differential equation with fractional time derivatives in which the chemical reactions mentioned above are taken into account. We will also find its solution over a long time limit. Based on the obtained results, we will find the Cattaneo--type subdiffusion--reaction equation in the case in which mobile particles of species $A$ and $B$ can chemically react according to a more complicated rule.

cond-mat.stat-mech↗

arXiv2013

A comparison study of slow--subdiffusion and subdiffusion

We study slow-subdiffusion in comparison to subdiffusion. Both of the processes are treated as random walks and can be described within continuous time random walk formalism. However, the probability density of the waiting time of a random walker to take its next step $ω(t)$ is assumed over a long time limit in the form $ω(t)\sim 1/t^{α+1}$ for subdiffusion, and in the form $ω(t)\sim h(t)/t$ for slow-subdiffusion [$h(t)$ is a slowly varying function]. We show that Green functions for slow-subdiffusion and subdiffusion can be very similar when the subdiffusion coefficient $D_α$ depends on time. This creates the possibility of describing slow-subdiffusion by means of subdiffusion with a small value of the subdiffusion parameter $α$, and the subdiffusion coefficient $D_α$ varying over time.

cond-mat.stat-mech↗

arXiv2012

The perturbation method to solve subdiffusion-reaction equations

We use the perturbation method to approximately solve subdiffusion-reaction equations. Within this method we obtain the solutions of the zeroth and the first order. The comparison our analytical solutions with the numerical results shown that the perturbation method can be useful to find approximate solutions of nonlinear subdiffusion--reaction equations.

cond-mat.stat-mech↗

arXiv2012

The nonextensive entropy approach versus the stochastic in describing subdiffusion

We have proposed a new stochastic interpretation of the sudiffusion described by the Sharma-Mittal entropy formalism which generates a nonlinear subdiffusion equation with natural order derivatives. We have shown that the solution to the diffusion equation generated by Gauss entropy (which is the particular case of Sharma-Mittal entropy) is the same as the solution of the Fokker-Planck (FP) equation generated by the Langevin generalised equation where the `long memory effect' is taken into account. The external noise which pertubates the subdiffusion coefficient (occuring in the solution of FP equation) according to the formula $D_α\rightarrow D_α/u$ where $u$ is a random variable described by the Gamma distribution, provides us with solutions of equations obtained from Sharma-Mittal entropy. We have also shown that the parameters $q$ and $r$ occuring in Sharma-Mittal entropy are controlled by the parameters $α$ and $ $, respectively.

cond-mat.stat-mech↗

arXiv2008

Subdiffusive model of substance releasing from a thick membrane

We study both theoretically and experimentally the process of subdiffusive substance releasing from a thick membrane. The theoretical model uses the subdiffusion equation with fractional time derivative and specific boundary conditions at the membrane surfaces. Using a special \textit{ansatz} we find analytical formulas describing the time evolution of concentration profiles and an amount of the substance remains in the membrane. Fitting the theoretical functions to the experimental results, we estimate the subdiffusion coefficient of polyethylene glycol 2000 in agarose hydrogel.

cond-mat.soft↗

arXiv2008

Time evolution of the reaction front in a subdiffusive system

Using the quasistatic approximation, we show that in a subdiffusion--reaction system the reaction front $x_{f}$ evolves in time according to the formula $x_{f} \sim t^{α/2}$, with $α$ being the subdiffusion parameter. The result is derived for the system where the subdiffusion coefficients of reactants differ from each other. It includes the case of one static reactant. As an application of our results, we compare the time evolution of reaction front extracted from experimental data with the theoretical formula and we find that the transport process of organic acid particles in the tooth enamel is subdiffusive.

cond-mat.stat-mech↗

arXiv2006

Scaling form of concentration profiles in a subdiffusive membrane system

We show that the concentration profiles in the subdiffusive system with a membrane, which separates a homogeneous solution from a pure solvent at an initial moment, has a general scaling form in the long time limit $C\sim t^λF(δ/t^ρ)$, where $δ$ is a distance from the membrane surface. There is also derived the relation involving the subdiffusion parameters of the medium and the membrane with the scaling parameters $λ$ and $ρ$, which are measured experimentally. The relation allows one to extract the subdiffusion parameters of the system from experimental data.

cond-mat.other↗

arXiv2006

Numerical study of subdiffusion equation

We present a numerical procedure of solving the subdiffusion equation with Caputo fractional time derivative. On the basis of few examples we show that the subdiffusion is a 'long time memory' process and the short memory principle should not be used in this case.

cond-mat.other↗