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12 Dissipative Systems

[221.2.1] The concept of time is the same for conservative and dissipative systems. [221.2.2] For conservative dynamical systems a mathematically rigorous derivation of fractional dynamics from an underlying nonfractional dynamical system has remained elusive, although some authors have tried to relate \alpha to invariant tori, strange attractors or other phase space structures [63, 64]. [221.2.3] For dissipative systems the rigorous derivation has been possible for Bochner-Levy diffusion [65, 47, 7, 44] and Montroll-Weiss diffusion [66, 67, 68, 69, 70]. [221.2.4] Due to restrictions on page number and preparation time only the latter case will be considered very briefly.

[221.3.1] For diffusive dynamical systems a mathematically rigorous relation of fractional dynamics with microscopic Montroll-Weiss continuous time random walks was discovered in [71, 72]. [221.3.2] It was shown that a diffusion (or master) equation with fractional time derivatives (i.e. a dissipative fractional dynamical system) can be related rigorously to the microscopic model of Montroll-Weiss continuous time random walks (CTRW’s) [66, 70] in the same way as ordinary diffusion is related to random walks [44]. [221.3.3] This discovery became decoupled from its source in the widely cited review [19], and was later incorrectly attributed in [73]h (This is a footnote:) h[221.3.4] Contrary to [73, p. 51] fractional derivatives are never mentioned in [74]. .

[221.4.1] The fractional order \alpha can be identified and has a physical meaning related to the statistics of waiting times in the Montroll-Weiss theory. [221.4.2] The relation was established in two steps. [221.4.3] First, it was shown in [71] that Montroll-Weiss continuous time random walks with a Mittag-Leffler waiting time density are rigorously equivalent to a fractional master equation. [221.4.4] Then, in [72] this underlying random walk model was connected to the fractional time diffusion equation in the usual asymptotic sense [75] of long times and large distancesi (This is a footnote:) iThis is emphasized in eqs. (1.8) and (2.1) in [72] that are, of course, asymptotic.. [page 222, §0]    [222.0.1] For additional results see also [76, 50, 77, 78].

[222.0.2] The relation between fractional diffusion and continuous time random walks, established in [71, 72] and elaborated in [76, 50, 77, 78], has initiated many subsequent investigations of fractional dissipative systems, particularly into fractional Fokker-Planck equations with drift [79, 80, 81, 82, 83, 84, 85, 17, 18, 19, 86, 73].