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6 Statement of the Problem

[212.2.1] Let (\Gamma,\mathfrak{G},\mu,{\rule[-2.0pt]{0.0pt}{10.0pt}_{{\Gamma}}\! T}^{{\dot{t}}}) be a measure preserving system for a many body system. [212.2.2] The detailed microscopic time evolution {\rule[-2.0pt]{0.0pt}{10.0pt}_{{\Gamma}}\! T}^{{\dot{t}}}\colon\Gamma\to\Gamma is frequently not of interest in applications, because it is much too detailed to be computable. [212.2.3] Instead one is interested in a reduced, coarse grained or averaged time evolution of macroscopic states where the system is locally or globally in equilibrium. [212.2.4] Examples are isolated systems at phase coexistence or in metastable statesf (This is a footnote:) f This differs from relaxation to equilibrium discussed in [34].. [212.2.5] This gives rise to the problem of finding the time evolution {\rule[-2.0pt]{0.0pt}{10.0pt}_{{{G}}}\! T}^{{\dot{t}}}\colon{G}\to{G} on subsets {G}\subset\Gamma of phase space.

[212.3.1] It is not possible to define {\rule[-2.0pt]{0.0pt}{10.0pt}_{{{G}}}\! T}^{{\dot{t}}}={\rule[-2.0pt]{0.0pt}{10.0pt}_{{\Gamma}}\! T}^{{\dot{t}}}|_{{G}} as the restriction of {\rule[-2.0pt]{0.0pt}{10.0pt}_{{\Gamma}}\! T}^{{\dot{t}}} to {G}, because for fixed initial state x(0)\in{G}\subset\Gamma the time evolution {\rule[-2.0pt]{0.0pt}{10.0pt}_{{\Gamma}}\! T}^{{\dot{t}}} produces states {\rule[-2.0pt]{0.0pt}{10.0pt}_{{\Gamma}}\! T}^{{\dot{t}}}x(0)=x({\dot{t}})\notin{G}. [212.3.2] Equivalently, for fixed time {\dot{t}} the map {\rule[-2.0pt]{0.0pt}{10.0pt}_{{\Gamma}}\! T}^{{\dot{t}}} maps states x\in{G} to states not in {G}. [212.3.3] The restriction {\rule[-2.0pt]{0.0pt}{10.0pt}_{{\Gamma}}\! T}^{{\dot{t}}}|_{{G}} is not defined for all {\dot{t}}\in\mathbb{R}. [212.3.4] This seems to preclude a sensible definition of {\rule[-2.0pt]{0.0pt}{10.0pt}_{{{G}}}\! T}^{{\dot{t}}}. [212.3.5] The problem of defining an induced continuous time evolution for mixed states on subsets of small measure was introduced and solved in [1, 21]. [212.3.6] It originated from the general classification theory for phase transitions [35, 36, 37, 38, 39]. [212.3.7] The solution involves discretization of {\rule[-2.0pt]{0.0pt}{10.0pt}_{{\Gamma}}\! T}^{{\dot{t}}}, averaging Kakutani’s induced measure preserving transformation [40, 26] and Kac’s theorem for recurrence times [21, 1].