7 Induced Measure Preserving Transformations

[212.4.1] Consider a subset G⊂Γ with small but positive measure μ⁢G>0 of a measure preserving many body system Γ,G,μ,Γ⁢Tt˙. [page 213, §0]    [213.0.1] Because of μ⁢G>0 the subset G becomes a probability measure space G,S,ν with induced probability measure ν=μ/μ⁢G and S=G∩G being the trace of G in G [41].

[213.1.1] The measure preserving continuous time evolution Γ⁢Tt˙ is discretized by setting

with k∈Z and τ>0 the discretization time step. [213.1.2] A character x∈G is called recurrent, if there exists an integer k≥1 such that Γ⁢Tk⁢τ⁢x∈G. [213.1.3] If G∈G and μ is invariant under Γ⁢T, then almost every character in G is recurrent by virtue of the Poincarè recurrence theorem. [213.1.4] A subset G is called recurrent, if μ-almost every point x∈G is recurrent. [213.1.5] By Poincarè’s recurrence theorem the recurrence time tG⁢x of the character x∈G, defined as

is positive and finite for almost every x∈G. [213.1.6] For every k≥1 let

denote the set of characters with recurrence time k⁢τ. [213.1.7] Then the number

is the probability to find a recurrence time k⁢τ. [213.1.8] The numbers p⁢k define a discrete probability density p⁢k⁢δ⁢s˙-k⁢τ on the arithmetic progression s˙-k⁢τ,k∈N,s˙∈R. [213.1.9] Every probability measure ϱ⁢s˙ on G,S at time instant s˙ is then defined on the same arithmetic progression through

for all B∈S and s˙∈R. [213.1.10] The induced time evolution G⁢T on the subset G is defined for every B∈S as the average [1, 21]

where s˙∈R. [213.1.11] For characters ϱ=x∈G, one recovers the first step in the discretized microscopic time evolution G⁢T⁢x⁢s˙=x⁢s˙-tG⁢x as expected. [213.1.12] For mixed states ϱ this formula allows the transition from the microscopic to the macroscopic time evolution. [213.1.13] It assigns an averaged translation to the first step in the induced time evolution of mixed states.