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9 Irreversibility

[216.1.1] The result in eq. (27) has provided new insight into the irreversibility paradox [21, p. 554]. [216.1.2] For \alpha\to 0^{-} one finds

h_{1}(x)=\lim _{{\alpha\to 0^{-}}}h_{\alpha}(x)=\delta(x-1) (30)

and therefore

{\rule[-2.0pt]{0.0pt}{10.0pt}_{{{G}}}\! T}_{1}^{t}\widetilde{\varrho}({\dot{s}})=\int\limits _{0}^{\infty}\widetilde{\varrho}({\dot{s}}-{\dot{t}})\delta\left(\frac{{\dot{t}}}{t}-1\right)\frac{\mathrm{d}{\dot{t}}}{t}=\widetilde{\varrho}({\dot{s}}-t) (31)

is a right translation. [216.1.3] Here {\dot{t}}\in\mathbb{R} denotes a time instant, while t\geq 0 is a time duration. [216.1.4] This shows, that induced right translations do not form a group, but only a semigroup.

[page 217, §1]    [217.1.1] These observations suggest a reformulation of the controversial irreversibility problem [49, 6]. [217.1.2] The problem of irreversibility is normally formulated as:

Definition 9.1 (The normal irreversibility problem).

[217.1.3] Assume that time is reversible. Explain how and why time irreversible equations arise in physics.

[217.1.4] The assumption that time is reversible, i.e. {\dot{t}}\in\mathbb{R}, is made in all fundamental theories of modern physics. [217.1.5] The explanation of macroscopically irreversible behaviour for macroscopic nonequilibrium states of subsystems is due to Boltzmann. [217.1.6] 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 [34].

[217.2.1] The problem with assuming {\dot{t}}\in\mathbb{R} is not the second law of thermodynamics, because the foundations of thermodynamics and statistical mechanics do not cover all dynamical systems in nature. [217.2.2] The problem with the arrow of time is that an experiment (i.e. the preparation of certain intial conditions for a dynamical system) cannot be repeated yesterday, but only tomorrow [49]. [217.2.3] While it is possible to translate the spatial position of a physical system, it is not possible to translate the temporal position of a physical system backwards in timeg (This is a footnote:) g Note, that this is not the same as reversing the momenta of all particles in a physical system.. [217.2.4] This was emphasized in [49, 6]. [217.2.5] These simple observations combined with eqs. (30) and (31) suggest to reformulate the standard irreversibility problem:

Definition 9.2 (The reversed irreversibility problem).

[217.2.6] Assume that time is irreversible. Explain how and why time reversible equations arise in physics.

[217.2.7] The reversed irreversibility problem was introduced in [49]. [217.2.8] Its solution is given by Theorem 8.1 combined with (30) and (31). [217.2.9] The impossibility of performing experiments in the past is fundamental and evident. [217.2.10] Therefore, as emphasized in [49], it must be assumed that time is irreversible. [217.2.11] The normal irreversibility problem starts from an assumption, that contradicts experiment, while the reversed problem starts from the correct assumption. [217.2.12] Theorem 8.1 combined with (30) and (31) explains why time translations, i.e. the case \alpha=1, arise in physics, and why it arises more frequently than the case \alpha<1.