# 8 Discussion

[639.2.1] The introduction of the sets Bε
of ε-almost invariant
BMO-states with 0≤ε≤∞
has provided a mathematical framework in which
questions concerning the abundance of time-invariant states and
their embedding in the set of all states can be posed
mathematically in a proper way.
[639.2.2] The class of BMO-states reflects in its definition the
experimental reality that observations
are always performed by integration of experimental
data over time intervals.
[639.2.3] BMO-states allow for singular expectation values, thereby
establishing a general framework to discuss
Problems 1 and 2 above.

[page 640, §1]
[640.1.1] There exists a direct relation between Theorem 3
and the BMO-states.
[640.1.2] It is given by Equation (32), which directly determines
the values of α and D, as well as the function Λ
in Theorem 3 and Equation (70).

[640.2.1] The result in Equation (70)
shows that the left-hand side
in a coarse-grained or rescaled version of
Equation (1) may not always be a time translation
along the orbits of the original unscaled dynamics.
[640.2.2] Instead, the left-hand side is in general the infinitesimal
generator of a convolution along time rescaled orbits of
ε-almost invariant states.
[640.2.3] The orbits of ε-almost invariant
states can approach the manifold of invariant states
of the physical system or subsystem of
interest at every point for any length of time without
being trapped.

[640.3.1] As discussed above, the result in Equation (70) implies
a general concept of time flow and, hence, provides a new
perspective on the issue of irreversibility [22, 24, 25].
[640.3.2] It suggest a reformulation [25, 26]
of the much discussed irreversibility problem.
[640.3.3] The normal problem can be stated as:

###### Problem 4 (The normal irreversibility problem).

[640.3.4] Assume that time is reversible.

[640.3.5] Explain how and why time irreversible equations arise in physics.

[640.3.6] The assumption that time is reversible, *i.e.*, t∈R,
is made in all fundamental theories of modern physics.
[640.3.7] The explanation of macroscopically irreversible
behavior for macroscopic nonequilibrium states of subsystems
is due to Boltzmann.
[640.3.8] It is based on the applicability of statistical mechanics
and thermodynamics, the large separation of scales, the importance of
low entropy initial conditions and probabilistic reasoning [27].

[640.4.1] The problem with with assuming t∈R is that an experiment
(*i.e.*, the preparation of an initial state within an infinity of
η-indistinguishable initial states for a dynamical system)
cannot be repeated yesterday, but only tomorrow [25].
[640.4.2] While it is possible to translate the spatial position of a physical
system forward and backward in space, it is not possible to translate
the temporal position of a physical system backwards in time.
[640.4.3] Translating an experiment backward in time is not
the same as reversing the momenta of all particles in a physical
system, as emphasized in [25, 26].
[640.4.4] These observations combined with
Equations (71) and (72)
suggest to reformulate the normal irreversibility problem above as:

###### Problem 5 (The reversed irreversibility problem).

[640.4.5] Assume that time evolution is always irreversible.

[640.4.6] Explain why time reversible equations are more frequent in physics.

[640.4.7] The reversed irreversibility problem was
introduced in [25].
[640.4.8] Its solution is given by
Theorem 3 combined with
two additional facts.
[640.4.9] Firstly, ultra-long-time evolutions with 0<α<1
are always irreversible, while those with
α=1 may be irreversible or reversible,
depending on the operator on the right-hand side
of Equation (1).
[640.4.10] Secondly, the set of recurrence time
distributions pk in the domain of attraction for the
case α=1 comprises all distributions whose
first moment ∑kkpk exists,
independent of their tail behavior.
[640.4.11] Contrary to this, the domain of attraction for the
case 0<α<1 is restricted to those pk
with the correct tail behavior.
[640.4.12] Thus, the domain of attraction is much larger for
α=1 than for 0<α<1.
[640.4.13] This explains why equations of motion with
time reversal symmetry arise more frequently.

[page 641, §1]
[641.1.1] Because anomalous time evolutions from Equation (70)
with 0<α<1 must be expected on theoretical grounds, they
are attracting increasing experimental interest [15, 28].
[641.1.2] For the example of broadband dielectric spectroscopy in
glasses, generalized relaxation functions and susceptibilities
based on Equation (70)
have already been successfully compared
to experiments [29, 30, 23, 31, 32].
[641.1.3] Theoretical, mathematical and experimental studies are encouraged
to further explore the consequences of the generalized concept.