38 research outputs found
Percolation-dependent Reaction Rates in the Etching of Disordered Solids
A prototype statistical model for the etching of a random solid is
investigated in order to assess the influence of disorder and temperature on
the dissolution kinetics. At low temperature, the kinetics is dominated by
percolation phenomena, and the percolation threshold determines the global
reaction time. At high temperature, the fluctuations of the reaction rate are
Gaussian, whereas at low temperature they exhibit a power law tail due to
chemical avalanches. This is an example where microscopic disorder directly
induces non-classical chemical kinetics.Comment: Revtex, 4 pages, 5 figure
Fractional differentiability of nowhere differentiable functions and dimensions
Weierstrass's everywhere continuous but nowhere differentiable function is
shown to be locally continuously fractionally differentiable everywhere for all
orders below the `critical order' 2-s and not so for orders between 2-s and 1,
where s, 1<s<2 is the box dimension of the graph of the function. This
observation is consolidated in the general result showing a direct connection
between local fractional differentiability and the box dimension/ local Holder
exponent. Levy index for one dimensional Levy flights is shown to be the
critical order of its characteristic function. Local fractional derivatives of
multifractal signals (non-random functions) are shown to provide the local
Holder exponent. It is argued that Local fractional derivatives provide a
powerful tool to analyze pointwise behavior of irregular signals.Comment: minor changes, 19 pages, Late
Holder exponents of irregular signals and local fractional derivatives
It has been recognized recently that fractional calculus is useful for
handling scaling structures and processes. We begin this survey by pointing out
the relevance of the subject to physical situations. Then the essential
definitions and formulae from fractional calculus are summarized and their
immediate use in the study of scaling in physical systems is given. This is
followed by a brief summary of classical results. The main theme of the review
rests on the notion of local fractional derivatives. There is a direct
connection between local fractional differentiability properties and the
dimensions/ local Holder exponents of nowhere differentiable functions. It is
argued that local fractional derivatives provide a powerful tool to analyse the
pointwise behaviour of irregular signals and functions.Comment: 20 pages, Late