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# 2 Time Flow and Induced Transformations

[550.4.1]Let be the phase or state space of a dynamical system, let be a -algebra of measurable subsets of , and a measure on such that .[550.4.2]The triple forms a probability measure space.[550.4.3]In general the time evolution of the system is given as a flow (or semiflow) on , defined as a one-parameter family of maps such that is the identity, for all and such that for every measurable function the function is measurable on the direct product .[550.4.4]For every also holds.[550.4.5]The measure is called invariant under the flow if for all .[550.4.6]An invariant measure is called ergodic if it cannot be decomposed into a convex combination of invariant measures, i.e. if with invariant and implies or .

[550.5.1]The flow defines the time evolution of measures through as a map on the space of measures on .[550.5.2]Defining as usual [22, 23] shows that

 (1)

and thus the flow acts on measures as a right translation in time.[550.5.3]The existence of the inverse for a flow expresses microscopic reversibility.[550.5.4]The infinitesimal [page 551, §0]   generator of is defined (assuming all the necessary structure for and ) as the strong limit

 (2)

where denotes the identity, and one has for right translations.[551.0.1]The invariance of the measure can be expressed as and it implies that for given

 (3)

for all .

[551.1.1]The continuous time evolution with may be discretized into the discrete time evolution with generated by the map with discretization time step .[551.1.2]Consider an arbitrary subset corresponding to a physically interesting reduced or coarse grained description of the original dynamical system.[551.1.3]Not all choices of correspond to a physically interesting situation, and the choice of reflects physical modeling or insight.[551.1.4]A point is called recurrent with respect to if there exists a for which .[551.1.5]The Poincarè recurrence theorem asserts that if is invariant under and then almost every point of is recurrent with respect to .[551.1.6]A set is called a -recurrent set if -almost every is recurrent with respect to .[551.1.7]By virtue of Poincarè’s recurrence theorem the transformation defines an induced transformation on subsets of positive measure, , through

 (4)

for almost every .[551.1.8]The recurrence time of the point , defined as

 (5)

is positive and finite for almost every point .[551.1.9]Because has positive measure it becomes a probability measure space with the induced measure .[551.1.10]If was invariant under then is invariant under , and ergodicity of implies ergodicity also for [22].

[551.2.1]The induced transformation exists for -almost every with by virtue of the Poincare recurrence theorem.[551.2.2]To extend the definition to the case let denote a subspace of measure with -algebra contained in , , in the sense that for all .[551.2.3] for all while for all sets with .[551.2.4]Let .[551.2.5]If is -recurrent under in the sense that -almost every point (rather than ) is recurrent with respect to then the recurrence time and the map are defined for -almost every point .[551.2.6]Throughout the following it will be assumed that is -recurrent under , and that .[551.2.7]An example is given by solidification where represents the high temperature phase space, while corresponds to the phase space at low temperatures when a large number of nuclear translational degrees of freedom is frozen out.

[551.3.1]The pointwise definition of can be extended to a transformation on measures by averaging over the recurrence times.[551.3.2]This extension was first given in [21].[551.3.3]Let

 (6)

be the set of points whose recurrence time is .[551.3.4]The number

 (7)

[page 552, §0]   is the probability to find a recurrence time with .[552.0.1]The numbers define a discrete (lattice) probability density concentrated on the arithmetic progression .[552.0.2]The induced transformation acting on a measure on is defined as the mathematical expectation

 (8)

where , and was given in (1).[552.0.3]This defines a transformation on the space of measures on .[552.0.4]The next section discusses the iterated transformation and the long time limit .