Searcharxiv⌕ Search

arXiv subjects

Tadeusz Kosztołowicz

Publications and source records attributed to Tadeusz Kosztołowicz.

At least 19 recordsLinked to original sources
arXiv2026

Subdiffusion equation with Cattaneo effect

The ordinary subdiffusion equation, with a fractional time derivative of at most first order, describes a process in which the propagation velocity of diffusing molecules is unlimited. To avoid this non-physical property different forms of the Cattaneo subdiffusion equation have been proposed. We define the Cattaneo effect as a delay of the ordinary subdiffusion flux activation by a random time. By incorporating this effect into the flux equation we get a Cattaneo--type subdiffusion equation (CTSE). We consider a subdiffusion process in which the Cattaneo effect is generated by the time-delay probability distribution controlled by the Mittag-Leffler function. Then, CTSE differs from the ordinary subdiffusion equation by a term with a fractional time derivative, whose order can be independent of the subdiffusion exponent. The influence of the Cattaneo effect on the solutions to the CTSE is discussed. We show that the process described by CTSE is subdiffusion in the entire time domain even though the temporal evolution of the mean square displacement of diffusing particle in the short-time limit is typical for superdiffusion. The delay in the flux activation in the subdiffusion equation should also cause a flux delay in a boundary condition. As an example, we study subdiffusion with the Cattaneo effect in a system with a partially absorbing wall at which the Robin boundary condition is assumed. We also propose a method for experimentally identifying the Cattaneo effect in a subdiffusive system.

cond-mat.stat-mech↗
arXiv2025

Cattaneo--type subdiffusion equation

The ordinary subdiffusion equation, with a fractional time derivative of at most first order, describes a process in which the propagation velocity of diffusing molecules is unlimited. To avoid this non-physical property different forms of the Cattaneo subdiffusion equation have been proposed. We define the Cattaneo effect as a delay of the ordinary subdiffusion flux activation by a random time. By incorporating this effect into the flux equation we get a Cattaneo--type subdiffusion equation (CTSE). We study the CTSE that differs from the ordinary subdiffusion equation by an additional integro--differential operator (AO) controlled by a time delay probability distribution. A method for deriving CTSE within the standard continuous time random walk model is also shown. As examples, we consider the CTSE with AO being the Caputo fractional time derivative of the order independent of the subdiffusion exponent and with the AO with a kernel that is a slowly varying function. In the first case the Cattaneo effect disappears over time much faster than in the second one. Based on the Green's functions, the time evolutions of the first passage time distribution and of the mean square displacement of a diffusing molecule, we discuss whether the influence of the Cattaneo effect is significant. In the considered examples, this influence seems to be small. However, a relative probability of finding a molecule at a long distance from the starting point for the CTSE equation with respect to the ordinary subdiffusion equation increases rapidly with distance. Even small changes caused by the Cattaneo effect can lead to different results in modeling processes where a faster appearance of a diffusing object changes the nature of the process. The effect may be important, for example, in modeling the spread of an epidemic when a diffusing object is a source of infection.

cond-mat.stat-mech↗
arXiv2023

Subdiffusion equation with fractional Caputo time derivative with respect to another function in modeling transition from ordinary subdiffusion to superdiffusion

We use a subdiffusion equation with fractional Caputo time derivative with respect to another function $g$ ($g$--subdiffusion equation) to describe a smooth transition from ordinary subdiffusion to superdiffusion. Ordinary subdiffusion is described by the equation with the ``ordinary'' fractional Caputo time derivative, superdiffusion is described by the equation with a fractional Riesz type spatial derivative. We find the function $g$ for which the solution (Green's function, GF) to the $g$--subdiffusion equation takes the form of GF for ordinary subdiffusion in the limit of small time and GF for superdiffusion in the limit of long time. To solve the $g$--subdiffusion equation we use the $g$--Laplace transform method. It is shown that the scaling properties of the GF for $g$--subdiffusion and the GF for superdiffusion are the same in the long time limit. We conclude that for a sufficiently long time the $g$--subdiffusion equation describes superdiffusion well, despite a different stochastic interpretation of the processes. Then, paradoxically, a subdiffusion equation with a fractional time derivative describes superdiffusion. The superdiffusive effect is achieved here not by making anomalously long jumps by a diffusing particle, but by greatly increasing the particle jump frequency which is derived by means of the $g$--continuous time random walk model. The $g$--subdiffusion equation is shown to be quite general, it can be used in modeling of processes in which a kind of diffusion change continuously over time. In addition, some methods used in modeling of ordinary subdiffusion processes, such as the derivation of local boundary conditions at a thin partially permeable membrane, can be used to model $g$--subdiffusion processes, even if this process is interpreted as superdiffusion.

cond-mat.stat-mech↗
arXiv2023

Subdiffusion with particle immobilization process described by differential equation with Riemann--Liouville type fractional time derivative

An equation describing subdiffusion with possible immobilization of particles is derived by means of the continuous time random walk model. The equation contains a fractional time derivative of Riemann--Liouville type which is a differential-integral operator with the kernel defined by the Laplace transform. We propose the method for calculating the inverse Laplace transform providing the kernel in the time domain. In the long time limit the subdiffusion--immobilization process reaches a stationary state in which the probability density of a particle distribution is an exponential function.

cond-mat.stat-mech↗
arXiv2022

Subdiffusion equation with Caputo fractional derivative with respect to another function in modelling diffusion in a complex system consisting of matrix and channels

Diffusion equation with a fractional Caputo time derivative with respect to another function $g$, which defines new time scale of the process, is applied to describe anomalous diffusion of antibiotic (colistin) in a system consisting of packed gel (alginate) beads immersed in water and impregnated with the antibiotic. We show that ultraslow diffusion of colistin in the system occurs. The equation, which is more general than the "ordinary" fractional subdiffusion equation, can be widely used in various fields of science to model diffusion in a matrix with channels system when subdiffusion parameters and even a type of diffusion evolves over time.

cond-mat.stat-mech↗
arXiv2022

First passage time for $g$--subdiffusion process of vanishing particles

Subdiffusion equation and molecule survival equation, both with Caputo fractional time derivatives with respect to another functions $g_1$ and $g_2$, respectively, are used to describe diffusion of a molecule that can disappear at any time with a constant probability. The process can be interpreted as ``ordinary'' subdiffusion and ``ordinary'' molecule survival process in which timescales are changed by the functions $g_1$ and $g_2$. We derive the first-passage time distribution for the process. The mutual influence of subdiffusion and molecule vanishing processes can be included in the model when the functions $g_1$ and $g_2$ are related to each other. As an example, we consider the processes in which subdiffusion and molecule survival are highly related, which corresponds to the case of $g_1\equiv g_2$.

cond-mat.stat-mech↗
arXiv2022

$G$-subdiffusion equation that describes transient subdiffusion

A $g$--subdiffusion equation with fractional Caputo time derivative with respect to another function $g$ is used to describe a process of a continuous transition from subdiffusion with parameters $α$ and $D_α$ to subdiffusion with parameters $β$ and $D_β$. The parameters are defined by the time evolution of the mean square displacement of diffusing particle $σ^2(t)=2D_i t^i/Γ(1+i)$, $i=α,β$. The function $g$ controls the process at "intermediate" times. The $g$--subdiffusion equation is more general than the "ordinary" fractional subdiffusion equation with constant parameters, it has potentially wide application in modelling diffusion processes with changing parameters.

cond-mat.stat-mech↗
arXiv2021

Stochastic foundations of $g$-subdiffusion process

Recently, in the paper: T. Kosztołowicz and A. Dutkiewicz, Phys. Rev. E \textbf{104}, 014118 (2021) the $g$--subdiffusion equation with fractional Caputo time derivative with respect to another function $g$ has been considered. This equation offers new possibilities for modelling diffusion such as a process in which a type of diffusion evolves continuously over time. However, the equation has not been derived from a stochastic model and the stochastic interpretation of $g$--subdiffusion has been unknown. In this paper we show stochastic foundations of this process. We derive the equation by means of a modified Continuous Time Random Walk model. Interpretation of the $g$--subdiffusion process is also discussed.

cond-mat.stat-mech↗
arXiv2021

Subdiffusion equation with Caputo fractional derivative with respect to another function

We show an application of a subdiffusion equation with Caputo fractional time derivative with respect to another function $g$ to describe subdiffusion in a medium having a structure evolving over time. In this case a continuous transition from subdiffusion to other type of diffusion may occur. The process can be interpreted as "ordinary" subdiffusion with fixed subdiffusion parameter (subdiffusion exponent) $α$ in which time scale is changed by the function $g$. As example, we consider the transition from "ordinary" subdiffusion to ultraslow diffusion. The function $g$ generates the additional aging process superimposed on the "standard" aging generated by "ordinary" subdiffusion. The aging process is analyzed using coefficient of relative aging of $g$--subdiffusion with respect to "ordinary" subdiffusion. The method of solving the $g$-subdiffusion equation is also presented.

cond-mat.stat-mech↗
arXiv2021

Boundary conditions at a thin membrane for normal diffusion equation which generate subdiffusion

We consider a particle transport process in a one-dimensional system with a thin membrane, described by a normal diffusion equation. We consider two boundary conditions at the membrane that are linear combinations of integral operators, with time dependent kernels, which act on the functions and their spatial derivatives define on both membrane surfaces. We show how boundary conditions at the membrane change the temporal evolution of the first and second moments of particle position distribution (the Green's function) which is a solution to normal diffusion equation. As these moments define the kind of diffusion, an appropriate choice of boundary conditions generates the moments characteristic for subdiffusion. The interpretation of the process is based on a particle random walk model in which the subdiffusion effect is caused by anomalously long stays of the particle in the membrane.

cond-mat.stat-mech↗
arXiv2020

Subdiffusion in a system with a partially permeable partially absorbing wall

We consider subdiffusion of a particle in a one-dimensional system with a thin partially permeable wall. Passing through the wall, the particle can be absorbed with a certain probability. We call such a wall partially permeable partially absorbing wall (PPAW). Using the diffusion model in a system with discrete time and spatial variable, probability densities (Green's functions) describing subdiffusion in the system have been derived. Knowing the Green's functions we derive boundary conditions at the wall. The boundary conditions take a specific form in which time derivatives of the fractional order controlled by the subdiffusion parameter are involved. We assume that the absorption of a particle can occur only when the particle jumps through the wall. It is not possible to temporarily retain a particle inside a thin wall. The wall can represent a thin membrane. If a system with a thick membrane inside which particles may diffuse is considered, it can be treated as a three-part with a thick membrane as the middle part. The boundary conditions at membrane surfaces can be assumed as for PPAW. The system with PPAW can be used to filter diffusing particles. The temporal evolution of the probability that the diffusing molecule has not been absorbed is considered. This function shows the efficiency of the filtering process. The impact of subdiffusion and wall parameters on this function is discussed.

cond-mat.stat-mech↗
arXiv2020

Boundary conditions at a thin membrane that generate non--Markovian normal diffusion

We show that some boundary conditions assumed at a thin membrane may result in normal diffusion not being the stochastic Markov process. We consider boundary conditions defined in terms of the Laplace transform in which there is a linear combination of probabilities and probability fluxes defined on both membrane surfaces. The coefficients of the combination may depend on the Laplace transform parameter. Such boundary conditions are most commonly used when considering diffusion in a membrane system unless collective or non-local processes in particles diffusion occur. We find Bachelier-Smoluchowski-Chapmann-Kolmogorov (BSCK) equation in terms of the Laplace transform and we derive the criterion to check whether the boundary conditions lead to fundamental solutions of diffusion equation satisfying this equation. If the BSCK equation is not met, the Markov property is broken. When a probability flux is continuous at the membrane, the general forms of the boundary conditions for which the fundamental solutions meet the BSCK equation are derived. A measure of broken of semi-group property is also proposed. The relation of this measure to the non-Markovian property measure is discussed.

cond-mat.stat-mech↗
arXiv2020

Diffusion of antibiotics through a biofilm in the presence of diffusion and absorption barriers

We propose a model of antibiotic diffusion through a bacterial biofilm when diffusion and/or absorption barriers develop in the biofilm. The idea of this model is: We deduce details of the diffusion process in a medium in which direct experimental study is difficult, based on probing diffusion in external regions. Since a biofilm has a gel-like consistency, we suppose that subdiffusion of particles in the biofilm may occur. To describe this process we use a fractional subdiffusion-absorption equation with an adjustable anomalous diffusion exponent. The boundary conditions at the boundaries of the biofilm are derived by means of a particle random walk model on a discrete lattice leading to an expression involving a fractional time derivative. We show that the temporal evolution of the total amount of substance that has diffused through the biofilm explicitly depends on whether there is antibiotic absorption in the biofilm. This fact is used to experimentally check for antibiotic absorption in the biofilm and if the biofilm parameters change over time. We propose a four-stage model of antibiotic diffusion in biofilm based on the mentioned above physical characteristics. The biological interpretation of the stages, in particular their relation with the bacterial defence mechanisms, is discussed. Theoretical results are compared with empirical results of ciprofloxacin diffusion through Pseudomonas aeruginosa biofilm, and ciprofloxacin and gentamicin diffusion through Proteus mirabilis biofilm.

physics.bio-ph↗
arXiv2019

Model of anomalous diffusion--absorption process in a system consisting of two different media separated by a thin membrane

We present the model of a diffusion-absorption process in a system which consists of two media separated by a thin partially permeable membrane. The kind of diffusion as well as the parameters of the process may be different in both media. Based on a simply model of particle's random walk in a membrane system we derive the Green's functions, then we find the boundary conditions at the membrane. One of the boundary conditions are rather complicated and takes a relatively simple form in terms of the Laplace transform. Assuming that particles diffuse independently of one another, the obtained boundary conditions can be used to solve to differential or differential-integral equations describing the processes in multilayered systems for any initial condition. We consider normal diffusion, subdiffusion and slow subdiffusion processes, and we also suggest how superdiffusion could be included in this model. The presented method provides the functions in terms of the Laplace transform and some useful methods of calculation the inverse Laplace transform are shown.

cond-mat.stat-mech↗
arXiv2018

Model of subdiffusion--absorption process in a membrane system consisting of two different media

We consider the subdiffusion--absorption process in a system which consists of two different media separated by a thin membrane. The process is described by subdiffusion--absorption equations with fractional Riemann--Liouville time derivative. We present the method of deriving the probabilities (the Green's functions) described particle's random walk in the system. Within the method we firstly consider the random walk of a particle in a system with both discrete time and space variables, and then we pass from discrete to continuous variables by means of the procedure presented in this paper. Using the Green's functions we derive boundary conditions at the membrane.

cond-mat.stat-mech↗
arXiv2017

How to determine a boundary condition at a thin membrane for diffusion from experimental data

We present a new method of deriving a boundary condition at a thin membrane for diffusion from experimental data. Based on experimental results obtained for normal diffusion of ethanol in water, we show that the derived boundary condition at a membrane contains a term with the Riemann--Liouville fractional time derivative of the $1/2$ order. Such a form of the boundary condition shows that a transfer of particles through a thin membrane is a `long memory process'. Presented method is an example that an important part of mathematical model of physical process may be derived directly from experimental data.

cond-mat.stat-mech↗
arXiv2016

Subdiffusion in a system consisting of two different media separated by a thin membrane

We consider subdiffusion in a system which consists of two media separated by a thin membrane. The subdiffusion parameters may be different in each of the medium. Using the new method presented in this paper we derive the probabilities (the Green's functions) describing a particle's random walk in the system. Within this method we firstly consider the particle's random walk in a system with both discrete time and space variables in which a particle can vanish due to reactions with constant probabilities $R_1$ and $R_2$, defined separately for each medium. Then, we move from discrete to continuous variables. The reactions included in the model play a supporting role. We link the reaction probabilities with the other subdiffusion parameters which characterize the media by means of the formulae presented in this paper. Calculating the generating functions for the difference equations describing the random walk in the composite membrane system with reactions, which depend explicitly on $R_1$ and $R_2$, we are able to correctly incorporate the subdiffusion parameters of both the media into the Green's functions. Finally, placing $R_1=R_2=0$ into the obtained functions we get the Green's functions for the composite membrane system without any reactions. From the obtained Green's functions, we derive the boundary conditions at the thin membrane. One of the boundary conditions contains the Riemann--Liouville fractional time derivative, which shows that the additional `memory effect' is created in the system. As is discussed in this paper, the `memory effect' can be created both by the membrane and by the discontinuity of the medium at the point at which the various media are joined.

cond-mat.stat-mech↗