## 1. Introduction

## 2. Problems and Objective

- (A)
- Open quantum systems: For open quantum systems with infinite reservoirs, $\mathfrak{A}=\mathfrak{B}\left(\mathfrak{H}\right)$ is the C${}^{*}$-algebra of bounded operators on the Hilbert space $\mathfrak{H}$ of the system $S\cup R$ in a suitably-chosen GNS-representation. The states $\langle \rho ,A\rangle =Tr\rho A$ can be identified with trace class operators on $\mathfrak{H}$ if Tr denotes the trace operation. The unitary time-automorphisms ${T}^{t}:\mathfrak{B}\left(\mathfrak{H}\right)\to \mathfrak{B}\left(\mathfrak{H}\right)$ of $S\cup R$ are generated by a Hamiltonian H of $S\cup R$$$H={H}_{S}\otimes \mathbf{1}+\mathbf{1}\otimes {H}_{R}+{H}_{SR}=-\mathrm{i}\text{\mathcal{L}}$$$$\begin{array}{cc}\hfill {T}_{S}^{t}A& =\underset{R}{Tr}(\mathbf{1}\otimes {\rho}_{R}){T}^{-t}(A\otimes \mathbf{1}){T}^{t}\hfill \end{array}$$$$\begin{array}{cc}\hfill {T}_{S}^{*\phantom{\rule{0.277778em}{0ex}}t}{\rho}_{S}& =\underset{R}{Tr}\phantom{\rule{0.277778em}{0ex}}{T}^{t}({\rho}_{S}\otimes {\rho}_{R}){T}^{-t}\hfill \end{array}$$$$\begin{array}{cc}& {T}_{S}^{{h}_{1}}{T}_{S}^{{h}_{2}}\ne {T}_{S}^{{h}_{1}+{h}_{2}}\hfill \end{array}$$$$\begin{array}{cc}& {T}_{S}^{*\phantom{\rule{0.277778em}{0ex}}{h}_{1}}{T}_{S}^{*\phantom{\rule{0.277778em}{0ex}}{h}_{2}}\ne {T}_{S}^{*\phantom{\rule{0.277778em}{0ex}}{h}_{1}+{h}_{2}}\hfill \end{array}$$
- (B)
- Classical dynamical systems: In classical systems, it is well known [14] that the orbits in the abelian algebra $\mathfrak{A}$ of functions on phase space cannot always be defined for all $t\in \mathbb{R}$ and for all initial conditions ${A}_{0}$ in the thermodynamics limit. The integration of Equation (1) does not generally give a dynamical flow of time for all initial conditions, and the problem is to find sufficiently large subsets of $\mathfrak{A}$, such that catastrophic behavior is absent and a unique orbit exists for all $t\in \mathbb{R}$.
- (C)
- Quantum field theory: For quantum field theories or infinite systems, the Stone–von Neumann uniqueness breaks down. Haag’s theorem shows that the determination of a suitable representation of the canonical commutation relations becomes a dynamical problem, if the vacuum states for different couplings are different. Non-normal states arise that yield representations assigning different values to global observables, like densities. Due to the problem of inequivalent representations, it is not possible to represent the time evolution as a group of unitary transformations within a single representation, because the representation algebra may change into an inequivalent representation as time evolves.

**Problem 1.**Are there global solutions of equation (1), i.e., solutions for all $t\in \mathbb{R}$?

**Problem 2.**If global solutions of Equation (1) exist, how can invariant solutions still change with time?

## 3. Almost Invariant States

## 4. Indistinguishability of States

## 5. Invariant Measures on BMO-states

- $P\left(\varnothing \right)=0$, $P\left(\mathsf{B}\right)=\mathbf{1}$
- Each $P\left(\mathsf{G}\right)$ is a self-adjoint projector.
- $P(\mathsf{G}\cap {\mathsf{G}}^{\prime})=P\left(\mathsf{G}\right)P\left({\mathsf{G}}^{\prime}\right)$
- If $\mathsf{G}\cap {\mathsf{G}}^{\prime}=\varnothing $, then $P(\mathsf{G}\cup {\mathsf{G}}^{\prime})=P\left(\mathsf{G}\right)+P\left({\mathsf{G}}^{\prime}\right)$
- For every $\psi \in {\mathfrak{H}}_{\mathsf{z}}$ and $\varphi \in {\mathfrak{H}}_{\mathsf{z}}$, the set function ${P}_{\psi ,\varphi}:\text{\mathcal{B}}\to \mathbb{C}$ defined by$$\begin{array}{c}\hfill {P}_{\psi ,\varphi}\left(\mathsf{G}\right)=\left(P\left(\mathsf{G}\right)\psi ,\varphi \right)\end{array}$$

## 6. Almost Invariance and Recurrence

## 7. Results

**Theorem 3**Let ${p}_{N}\left(k\right)$ be the probability density of ${W}_{N}$ specified above in (41). If the distributions of ${W}_{N}/{D}_{N}$ converge to a limit as $N\to \infty $ for suitable norming constants ${D}_{N}\ge 0$, then there exist constants $D\ge 0$ and $0<\alpha \le 1$, such that

**Proof.**The existence of a limiting distribution for ${W}_{N}/{D}_{N}>0$ is known to be equivalent to the stability of the limit [18]. If the limit distribution is nondegenerate, this implies that the rescaling constants ${D}_{N}$ have the form

## 8. Discussion

**Problem 4**(The normal irreversibility problem). Assume that time is reversible. Explain how and why time irreversible equations arise in physics.

**Problem 5**(The reversed irreversibility problem). Assume that time evolution is always irreversible. Explain why time reversible equations are more frequent in physics.

## Acknowledgments

## Publication Notice

## References

- Levy, M. On the Description of Unstable Particles in Quantum Field Theory. Il Nuovo Cimento
**1959**, 13, 115–143. [Google Scholar] [CrossRef] - Thirring, W. Quantum Mathematical Physics: Atoms, Molecules and Large Systems; Springer: Berlin, Germany, 2002. [Google Scholar]
- Allahverdyan, A.; Balian, R.; Nieuwenhuizen, T. Understanding quantum measurement from the solution of dynamical models. Phys. Rep.
**2013**, 525, 1–166. [Google Scholar] [CrossRef] - Balian, R. From Microphysics to Macrophysics I+II; Springer Verlag: Berlin, Germany, 1991. [Google Scholar]
- Spohn, H. Large Scale Dynamics of Interacting Particles; Springer: Berlin, Germany, 1991. [Google Scholar]
- Bratteli, O.; Robinson, D. Operator Algebras and Quantum Statistical Mechanics I; Springer: Berlin, Germany, 1979. [Google Scholar]
- Haag, R. Local Quantum Physics; Springer Verlag: Berlin, Germany, 1992. [Google Scholar]
- Hille, E.; Phillips, R. Functional Analysis and Semi-Groups; American Mathematical Society: Providence, RI, USA, 1957. [Google Scholar]
- Yosida, K. Functional Analysis; Springer: Berlin, Germany, 1965. [Google Scholar]
- Pazy, A. Semigroups of Linear Operators and Applications to Partial Differential Equations; Springer: Berlin, Germany, 1983. [Google Scholar]
- Neerven, J. The Adjoint of a Semigroup of Linear Operators; Springer: Berlin, Germany, 1992. [Google Scholar]
- Phillips, R. On the Generation of Semigroups of Linear Operators. Pacific J. Math.
**1952**, 2, 343–369. [Google Scholar] [CrossRef] - Rudin, W. Functional Analysis; McGraw-Hill: New York, NY, USA, 1973. [Google Scholar]
- Bunimovich, L.; Dani, S.; Dobrushin, R.; Kornfeld, I.; Maslova, N.; Pesin, Y.; Sinai, Y.; Smillie, J.; Shukov, Y.; Vershik, A. Dynamical Systems, Ergodic Theory and Applications; Springer: Berlin, Germany, 2000. [Google Scholar]
- Hilfer, R. Applications of Fractional Calculus in Physics; World Scientific Publ. Co.: Singapore, Singapore, 2000. [Google Scholar]
- Bratteli, O.; Robinson, D. Operator Algebras and Quantum Statistical Mechanics II; Springer: Berlin, Germany, 1981. [Google Scholar]
- Haag, R.; Kastler, D. An Algebraic Approach to Quantum Field Theory. J. Math. Phys.
**1964**, 5. [Google Scholar] [CrossRef] - Gnedenko, B.; Kolmogorov, A. Limit Distributions for Sums of Independent Random Variables; Addison-Wesley: Cambridge, UK, 1954. [Google Scholar]
- Seneta, E. Regularly Varying Functions; Springer Verlag: Berlin, Germany, 1976. [Google Scholar]
- Ibragimov, I.; Linnik, Y. Independent and Stationary Sequences of Random Variables; Wolters-Nordhoff Publishing: Groningen, The Netherlands, 1971. [Google Scholar]
- Bingham, N.; Goldie, C.; Teugels, J. Regular Variation; Cambridge University Press: Cambridge, UK, 1987. [Google Scholar]
- Hilfer, R. Foundations of Fractional Dynamics. Fractals
**1995**, 3. [Google Scholar] [CrossRef] - Hilfer, R. Foundations of Fractional Dynamics: A Short Account. In Fractional Dynamics: Recent Advances; Klafter, J., Lim, S., Metzler, R., Eds.; World Scientific: Singapore, Singapore, 2011; p. 207. [Google Scholar]
- Hilfer, R. An Extension of the Dynamical Foundation for the Statistical Equilibrium Concept. Phys. A
**1995**, 221, 89–96. [Google Scholar] [CrossRef] - Hilfer, R. Remarks on Fractional Time. In Time, Quantum and Information; Castell, L., Ischebeck, O., Eds.; Springer: Berlin, Germany, 2003; p. 235. [Google Scholar]
- Hilfer, R. Threefold Introduction to Fractional Derivatives. In Anomalous Transport: Foundations and Applications; Klages, R., Radons, G., Sokolov, I., Eds.; Wiley-VCH: Weinheim, Germany, 2008; pp. 17–74. [Google Scholar]
- Lebowitz, J. Statistical Mechanics: A Selective Review of Two Central Issues. Rev. Mod. Phys.
**1999**, 71. [Google Scholar] [CrossRef] - Hilfer, R.; Metzler, R.; Blumen, A.; Klafter, J. Strange Kinetics. Chem. Phys.
**2002**, 284. [Google Scholar] - Hilfer, R. Fitting the excess wing in the dielectric α-relaxation of propylene carbonate. J. Phys.: Condens. Matter
**2002**, 14, 2297. [Google Scholar] [CrossRef] - Hilfer, R. Experimental Evidence for Fractional Time Evolution in Glass Forming Materials. Chem. Phys.
**2002**, 284, 399–408. [Google Scholar] [CrossRef] - Hilfer, R. Applications and Implications of Fractional Dynamics for Dielectric Relaxation. In Recent Advances in Broadband Dielectric Spectroscopy; Kalmykov, Y., Ed.; Springer: Berlin, Germany, 2012; p. 123. [Google Scholar]
- Candelaresi, S.; Hilfer, R. Excess Wings in Broadband Dielectric Spectroscopy. In Proceedings of the American Institute of Physics (AIP) Conference Proceedings, Narvik, Norway, 15 July 2014; Volume 1637, pp. 1283–1290.

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