How to Measure Subdiffusion Parameters

We propose a method to measure the subdiffusion parameter $\alpha$ and subdiffusion coefficient $D_{\alpha}$ which are defined by means of the relation $=\frac{2D_\alpha} {\Gamma(1+\alpha)} t^\alpha$ where $$ denotes a mean square displacement of a random walker starting from $x=0$ at the initial time $t=0$. The method exploits a membrane system where a substance of interest is transported in a solvent from one vessel to another across a thin membrane which plays here only an auxiliary role. We experimentally study a diffusion of glucose and sucrose in a gel solvent, and we precisely determine the parameters $\alpha$ and $D_{\alpha}$, using a fully analytic solution of the fractional subdiffusion equation.Comment: short version of cond-mat/0309072, to appear in Phys. Rev. Let

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 $x_{f} \sim t^{\alpha/2}$, with $\alpha$ 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.Comment: 18 pages, 3 figure

Measuring subdiffusion parameters

We propose a method to extract from experimental data the subdiffusion parameter $\alpha$ and subdiffusion coefficient $D_\alpha$ which are defined by means of the relation $=2D_\alpha/\Gamma(1+\alpha) t^\alpha$ where $$ denotes a mean square displacement of a random walker starting from $x=0$ at the initial time $t=0$. The method exploits a membrane system where a substance of interest is transported in a solvent from one vessel to another across a thin membrane which plays here only an auxiliary role. Using such a system, we experimentally study a diffusion of glucose and sucrose in a gel solvent. We find a fully analytic solution of the fractional subdiffusion equation with the initial and boundary conditions representing the system under study. Confronting the experimental data with the derived formulas, we show a subdiffusive character of the sugar transport in gel solvent. We precisely determine the parameter $\alpha$, which is smaller than 1, and the subdiffusion coefficient $D_\alpha$.Comment: 17 pages, 9 figures, revised, to appear in Phys. Rev.

First passage time for subdiffusion: The nonextensive entropy approach versus the fractional model

We study the similarities and differences between different models concerning subdiffusion. More particularly, we calculate first passage time (FPT) distributions for subdiffusion, derived from Greens' functions of nonlinear equations obtained from Sharma-Mittal's, Tsallis's and Gauss's nonadditive entropies. Then we compare these with FPT distributions calculated from a fractional model using a subdiffusion equation with a fractional time derivative. All of Greens' functions give us exactly the same standard relation $=2 D_\alpha t^\alpha$ which characterizes subdiffusion ($0<\alpha<1$), but generally FPT's are not equivalent to one another. We will show here that the FPT distribution for the fractional model is asymptotically equal to the Sharma--Mittal model over the long time limit only if in the latter case one of the three parameters describing Sharma--Mittal entropy $r$ depends on $\alpha$, and satisfies the specific equation derived in this paper, whereas the other two models mentioned above give different FTPs with the fractional model. Greens' functions obtained from the Sharma-Mittal and fractional models - for $r$ obtained from this particular equation - are very similar to each other. We will also discuss the interpretation of subdiffusion models based on nonadditive entropies and the possibilities of experimental measurement of subdiffusion models parameters.Comment: 12 pages, 8 figure

Hyperbolic subdiffusive impedance

We use the hyperbolic subdiffusion equation with fractional time derivatives (the generalized Cattaneo equation) to study the transport process of electrolytes in media where subdiffusion occurs. In this model the flux is delayed in a non-zero time with respect to the concentration gradient. In particular, we obtain the formula of electrochemical subdiffusive impedance of a spatially limited sample in the limit of large and of small pulsation of the electric field. The boundary condition at the external wall of the sample are taken in the general form as a linear combination of subdiffusive flux and concentration of the transported particles. We also discuss the influence of the equation parameters (the subdiffusion parameter and the delay time) on the Nyquist impedance plots.Comment: 10 pages, 5 figure

Ból i cierpienie. Materiały konferencyjne

Ze wstępu: "Coroczne spotkania lekarzy w klasztorze Sióstr Duehaczek, czyli poprawnie Sióstr Kanoniczck Ducha Świętego przy kościele Św. Tomasza w Krakowie przy ulicy Szpitalnej, w tradycyjnym dla tego zakonu terminie - tj. w drugą niedzielę po święcie Trzech Króli - przekształciły się w 1994 roku w sympozja naukowe, poświęcone stałemu tematowi: „Ból i cierpienie”. W tym roku spotkaliśmy się w dniach 17 i 18 stycznia 2004, tradycyjnie już, w Domu Towarzystwa Lekarskiego Krakowskiego przy ulicy Radziwiłłowskiej 4. W skład Komitetu Organizacyjnego Konferencji, wzorem lat ubiegłych weszli: prof, dr hab. med. Andrzej Środka, Kierownik Katedry Historii Medycyny CM UJ, prof, dr hab. med. Janusz Andres, Kierownik Katedry Anestezjologii i Intensywnej Terapii CM UJ, dr hab. med. Zdzisław Gajda, prof. UJ, Przewodniczący Stowarzyszenia Absolwentów Wydziałów Medycznych UJ, dr med. Alicja Macheta, Przewodnicząca Podkarpackiego Oddziału Towarzystwa Anestezjologii i Intensywnej Terapii, dr Maria Dorota Schmidt-Pospuła, Przewodnicząca Krakowskiego Towarzystwa Miłośników Historii Medycyny."(...

Subdiffusion–Absorption Process in a System with a Thin Membrane

We study a subdiffusion–absorption process which takes place in a system with a thin membrane. We present the method of deriving the Green’s functions (probability densities) describing the process. Within this method we consider a particle’s random walk in a system with both a discrete time and space variable. Then, we move from a discrete system to a continuous system by means of the procedures which are presented in this paper

How to identify absorption in a subdiffusive medium

We consider subdiffusion in a system which consists of two different media separated by a thin membrane. In one of the media particles’ absorption can occur. Such systems can be studied experimentally but, due to technical reasons, it is not always possible to measure concentration profiles in the medium in which absorption can be present. We show the method which allows one to recognize whether absorption is present in such a medium knowing concentration profiles of diffusing substance in the other medium