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 
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 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].
