[212.4.1] Consider a subset with small but positive measure of a measure preserving many body system . [page 213, §0] [213.0.1] Because of the subset becomes a probability measure space with induced probability measure and being the trace of in .
[213.1.1] The measure preserving continuous time evolution is discretized by setting
with and the discretization time step. [213.1.2] A character is called recurrent, if there exists an integer such that . [213.1.3] If and is invariant under , then almost every character in is recurrent by virtue of the Poincarè recurrence theorem. [213.1.4] A subset is called recurrent, if -almost every point is recurrent. [213.1.5] By Poincarè’s recurrence theorem the recurrence time of the character , defined as
is positive and finite for almost every . [213.1.6] For every let
denote the set of characters with recurrence time . [213.1.7] Then the number
is the probability to find a recurrence time . [213.1.8] The numbers define a discrete probability density on the arithmetic progression . [213.1.9] Every probability measure on at time instant is then defined on the same arithmetic progression through
where . [213.1.11] For characters , one recovers the first step in the discretized microscopic time evolution as expected. [213.1.12] For mixed states 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.