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Applications of Fractional Calculus

The idea of generalizing the concepts of differentiation and integration to noninteger (fractional) orders has a long mathematical history. In mathematics one learns very early that the $ n$-th derivative of $ f(x)=x^m$ is $ f^{(n)}(x)=n!x^{m-n}/(m-n)!$ for $ m\geq n$. It is little known, however, that this rule can be naturally generalized to noninteger orders $ \beta$ such that $ f^{(\beta)}\sim x^{\alpha-\beta}$ for arbitrary real (and complex) $ \beta$. Over the centuries many mathematicians have built up a large body of mathematical knowledge on fractional integrals and derivatives. Although fractional calculus is a natural generalization of calculus, and although its mathematical history is equally long, it has, until recently, played a negligible role in physics. This situation is beginning to change and there has been some interest in Applications of Fractional Calculus in Physics. Our Institute has published a book concerning this topic which was well received by the scientific public (see section 7).

Many applications of fractional calculus in physics amount to replacing the time derivative in an evolution equation with a derivative of fractional order. This is not merely a phenomenological procedure providing an additional fit parameter. Rather fractional derivatives seem to arise generally and universally, and for deep mathematical reasons. One central result of the work at our institute has been the fact that fractional derivatives arise as the infinitesimal generators of a class of translation invariant convolution semigroups. These semigroups appear universally as attractors for coarse graining procedures or scale changes. They are parametrized by a number in the unit interval corresponding to the order of the fractional derivative. Ongoing work concentrates on applications of fractional time evolutions in particular to diffusion phenomena and glassy relaxation.




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Next: Fractional Dynamics in the Up: Scientific Report Previous: The Broad Histogram Method:   Contents
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