Abstract
Up to now we have defined path entropy of non-random functions and we have constructed a new model of complex-valued fractional Brownian motion of order n based on a random walk in the complex plane. One of the frequent features of these results is that they can be thought of as some consequences of the maximum entropy principle, therefore having direct relations with information thermodynamics. Our purpose in the present chapter is to go a step further and to look for other characteristics common between these two concepts, and in this way we shall put in evidence some connections between these models and fractals, and more especially with Hausdorff dimension and Liapunov exponent. In addition we shall show that there is a striking identity between the generalized heat equation of order 2n and the fractional Fokker-Planck equation of order 1/n, in the special case when the mean value of the process is zero. We shall use this property to infer some results on the Fokker-Planck equation of fractal processes (or processes of fractional order) with non-zero mean values.. But before doing so, for the convenience of the reader, we shall call to mind some preliminary backgrounds.
This long road which returns backward takes for ever, and this long road which goes forward also takes for ever.
Nietzsche (Zarathustra(III, 2))
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© 2000 Springer Science+Business Media Dordrecht
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Jumarie, G. (2000). Fractals, Path Entropy, and Fractional Fokker-Planck Equation. In: Maximum Entropy, Information Without Probability and Complex Fractals. Fundamental Theories of Physics, vol 112. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9496-7_9
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DOI: https://doi.org/10.1007/978-94-015-9496-7_9
Publisher Name: Springer, Dordrecht
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