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8 Discussion

[639.2.1] The introduction of the sets \mathsf{B}_{\varepsilon} of \varepsilon-almost invariant BMO-states with 0\leq\varepsilon\leq\infty 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 \alpha and D, as well as the function \Lambda 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 \varepsilon-almost invariant states. [640.2.3] The orbits of \varepsilon-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\in\mathbb{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\in\mathbb{R} is that an experiment (i.e., the preparation of an initial state within an infinity of \eta-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<\alpha<1 are always irreversible, while those with \alpha=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 p(k) in the domain of attraction for the case \alpha=1 comprises all distributions whose first moment \sum _{k}kp(k) exists, independent of their tail behavior. [640.4.11] Contrary to this, the domain of attraction for the case 0<\alpha<1 is restricted to those p(k) with the correct tail behavior. [640.4.12] Thus, the domain of attraction is much larger for \alpha=1 than for 0<\alpha<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<\alpha<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.