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6 Statement of the Problem

[212.2.1] Let Γ,G,μ,ΓTt˙ be a measure preserving system for a many body system. [212.2.2] The detailed microscopic time evolution ΓTt˙:ΓΓ 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 GTt˙:GG on subsets GΓ of phase space.

[212.3.1] It is not possible to define GTt˙=ΓTt˙G as the restriction of ΓTt˙ to G, because for fixed initial state x0GΓ the time evolution ΓTt˙ produces states ΓTt˙x0=xt˙G. [212.3.2] Equivalently, for fixed time t˙ the map ΓTt˙ maps states xG to states not in G. [212.3.3] The restriction ΓTt˙G is not defined for all t˙R. [212.3.4] This seems to preclude a sensible definition of GTt˙. [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 ΓTt˙, averaging Kakutani’s induced measure preserving transformation [40, 26] and Kac’s theorem for recurrence times [21, 1].