[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

[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 ^{i} (This is a footnote:) ^{i}This 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].