Sie sind hier: ICP » R. Hilfer » Publikationen

1 Introduction and Statement of the Problem

[page 1, §1]   
[1.2.1.1] A recent classification theory [1, 2, 3] has derived fractional equations of motion from abstract ergodic theory. [1.2.1.2] Fractional equations of motion contain fractional rather than integer order time derivatives as generators of the time evolution. [1.2.1.3] Fractional equations of motion arise at anequilibrium phase transitions [1, 2] or whenever a dynamical system is restricted to subsets of measure zero of its state space [3].

[1.2.2.1] Master equations in which the time derivative is replaced with a derivative of fractional order form the subject of the present paper. [1.2.2.2] Such fractional master equations arise as special cases of the more general fractional Liouville equation introduced in [1, 2, 3], and they contain the fractional diffusion equation as a special case. [1.2.2.3] A fractional master equation for a translationally invariant d-dimensional system may be written formally, but in suggestive notation, as

\frac{\partial^{\omega}}{\partial t^{\omega}}p(\bm{\vec{r}},t)=\sum _{{\bm{\vec{r}}^{\prime}}}w(\bm{\vec{r}}-\bm{\vec{r}}^{\prime})p(\bm{\vec{r}}^{\prime},t) (1.1)

where p(\bm{\vec{r}},t) denotes the probability density to find the diffusing entity at the position \bm{\vec{r}}\in{\mathbb{R}}^{d} at time t if it was at the origin \bm{\vec{r}}=0 at time t=0. [1.2.2.4] The positions \bm{\vec{r}}\in{\mathbb{R}}^{d} may be discrete or continuous. [1.2.2.5] The fractional transition rates w(\bm{\vec{r}}) measure the propensity for a displacement \bm{\vec{r}} in units of (1/time)^{\omega}, and obey the relation \sum _{{\bm{\vec{r}}}}w(\bm{\vec{r}})=0. [1.2.2.6] The fractional order \omega plays the role of a dynamical critical exponent. [1.2.2.7] Equation (1.1) can be made precise by applying the fractional Riemann-Liouville integral as

p(\bm{\vec{r}},t)=\delta _{{\bm{\vec{r}}0}}+\frac{1}{\Gamma(\omega)}\int _{0}^{t}(t-t^{\prime})^{{\omega-1}}\sum _{{\bm{\vec{r}}^{\prime}}}w(\bm{\vec{r}}-\bm{\vec{r}}^{\prime})p(\bm{\vec{r}}^{\prime},t^{\prime})\, dt^{\prime} (1.2)

where the initial condition p(\bm{\vec{r}},0)=\delta _{{\bm{\vec{r}}0}} has been incorporated.

[1.3.1.1] Diffusion in a d-dimensional euclidean space is contained in the fractional master equations (1.1) or (1.2) as the special case in which \omega=1 and w(\bm{\vec{r}}) is the discretized Laplacian on a d-dimensional regular lattice. [1.3.1.2] The integral form (1.2) suggests a relation with the well known theory of continuous time random walks [4, 5, 6, 7, 8, 9, 10]. [1.3.1.3] It is the purpose and objective of the present paper to show that there exists a precise and rigorous relation between the fractional master equation and the theory of continuous time random walks. [1.3.1.4] It will be shown that the fractional master equation describes a fractal time process [11, 10]. [1.3.1.5] Fractal time processes (see [10] for a review) are defined here as continuous time random walks whose waiting time density has an infinite first moment [12, 13, 14, 15, 16].

[1.3.2.1] Given the existence of an exact relation between fractional master equations and fractal time random walks, it might seem that (1.1) or (1.2) describe also diffusion on fractals. [1.3.2.2] Dimensional analysis suggests anomalous subdiffusive behaviour of the form\langle\bm{\vec{r}}^{2}(t)\rangle\propto t^{{\widetilde{d}/\overline{d}}} where \overline{d} is the fractal dimension, and \widetilde{d} is the spectral or fracton dimension [17, 18, 19], and indeed some authors have suggested that \omega=\widetilde{d}/\overline{d}. [1.3.2.3] It must be clear however, that while the relation between fractional master equations and fractal time random walks established in this paper is exact, the relation with diffusion on fractals is not. [1.3.2.4] It appears doubtful that the latter relation can exist beyond superficial scaling similarities because exactly solvable cases show that the spectral properties as well as the eigenfunctions for fractal time walks and walks on fractals are radically different [20, 21, 22, 23].