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7 Induced Measure Preserving Transformations

[212.4.1] Consider a subset {G}\subset\Gamma with small but positive measure \mu({G})>0 of a measure preserving many body system (\Gamma,\mathfrak{G},\mu,{\rule[-2.0pt]{0.0pt}{10.0pt}_{{\Gamma}}\! T}^{{\dot{t}}}). [page 213, §0]    [213.0.1] Because of \mu({G})>0 the subset {G} becomes a probability measure space ({G},\mathfrak{S},\nu) with induced probability measure \nu=\mu/\mu({G}) and \mathfrak{S}=\mathfrak{G}\cap{G} being the trace of \mathfrak{G} in {G} [41].

[213.1.1] The measure preserving continuous time evolution {\rule[-2.0pt]{0.0pt}{10.0pt}_{{\Gamma}}\! T}^{{\dot{t}}} is discretized by setting

{\dot{t}}=k\tau (11)

with k\in\mathbb{Z} and \tau>0 the discretization time step. [213.1.2] A character x\in{G} is called recurrent, if there exists an integer k\geq 1 such that {\rule[-2.0pt]{0.0pt}{10.0pt}_{{\Gamma}}\! T}^{{k\tau}}x\in{G}. [213.1.3] If {G}\in\mathfrak{G} and \mu is invariant under {\rule[-2.0pt]{0.0pt}{10.0pt}_{{\Gamma}}\! 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 \mu-almost every point x\in{G} is recurrent. [213.1.5] By Poincarè’s recurrence theorem the recurrence time {t_{{G}}}(x) of the character x\in{G}, defined as

{t_{{G}}}(x)=\tau\min\{ k\geq 1:{\rule[-2.0pt]{0.0pt}{10.0pt}_{{\Gamma}}\! T}^{{k\tau}}(x)\in{G}\}, (12)

is positive and finite for almost every x\in{G}. [213.1.6] For every k\geq 1 let

{G}_{k}=\{ x\in{G}:{t_{{G}}}(x)=k\tau\} (13)

denote the set of characters with recurrence time k\tau. [213.1.7] Then the number

p(k)=\nu({G}_{k}) (14)

is the probability to find a recurrence time k\tau. [213.1.8] The numbers p(k) define a discrete probability density p(k)\delta({\dot{s}}-k\tau) on the arithmetic progression {\dot{s}}-k\tau,k\in\mathbb{N},{\dot{s}}\in\mathbb{R}. [213.1.9] Every probability measure \varrho({\dot{s}}) on ({G},\mathfrak{S}) at time instant {\dot{s}} is then defined on the same arithmetic progression through

\varrho(B,{\dot{s}}-k\tau)=\varrho(B\cap{G}_{k},{\dot{s}}) (15)

for all B\in\mathfrak{S} and {\dot{s}}\in\mathbb{R}. [213.1.10] The induced time evolution {\rule[-2.0pt]{0.0pt}{10.0pt}_{{{G}}}\! T} on the subset {G} is defined for every B\in\mathfrak{S} as the average [1, 21]

{\rule[-2.0pt]{0.0pt}{10.0pt}_{{{G}}}\! T}\varrho(B,{\dot{s}})=\sum _{{k=1}}^{\infty}p(k)\varrho(B,{\dot{s}}-k\tau) (16)

where {\dot{s}}\in\mathbb{R}. [213.1.11] For characters \varrho=x\in{G}, one recovers the first step in the discretized microscopic time evolution {\rule[-2.0pt]{0.0pt}{10.0pt}_{{{G}}}\! T}x({\dot{s}})=x({\dot{s}}-{t_{{G}}}(x)) as expected. [213.1.12] For mixed states \varrho 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.