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 α 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 α 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].