2 Time Flow and Induced Transformations

[550.4.1]Let Γ be the phase or state space of a dynamical system, let G be a σ-algebra of measurable subsets of Γ, and μ a measure on G such that μ⁢Γ=1.[550.4.2]The triple Γ,G,μ forms a probability measure space.[550.4.3]In general the time evolution of the system is given as a flow (or semiflow) on Γ,G,μ, defined as a one-parameter family of maps T~t:Γ→Γ such that T~0=I is the identity, T~s+t=T~s⁢T~t for all t,s∈R and such that for every measurable function f the function f⁢T~t⁢x is measurable on the direct product Γ×R.[550.4.4]For every G∈G also T~⁢G,T~-1⁢G∈G holds.[550.4.5]The measure μ is called invariant under the flow T~t if μ⁢G=μ⁢T~t⁢G=μ⁢T~t-1⁢G for all t∈R,G∈G.[550.4.6]An invariant measure is called ergodic if it cannot be decomposed into a convex combination of invariant measures, i.e. if μ=λ⁢μ1+1-λ⁢μ2 with μ1,μ2 invariant and 0≤λ≤1 implies λ=1,μ1=μ or λ=0,μ2=μ.

[550.5.1]The flow T~t defines the time evolution of measures through Tt⁢μ⁢G=μ⁢T~t⁢G as a map Tt:Γ′→Γ′ on the space Γ′ of measures on Γ.[550.5.2]Defining as usual [22, 23] μ⁢G,t=μ⁢T~t-1⁢G shows that

and thus the flow Tt acts on measures as a right translation in time.[550.5.3]The existence of the inverse Tt-1=T-t for a flow expresses microscopic reversibility.[550.5.4]The infinitesimal [page 551, §0]   generator of Tt is defined (assuming all the necessary structure for Γ′ and Tt) as the strong limit

where I=T0 denotes the identity, and one has A=-d/d⁢t for right translations.[551.0.1]The invariance of the measure μ can be expressed as A⁢μ=-d⁢μ/d⁢t=0 and it implies that for given t0∈R

for all G∈G,t∈R.

[551.1.1]The continuous time evolution T~t with t∈R may be discretized into the discrete time evolution T~k with k∈Z generated by the map T~=T~Δ⁢t with discretization time step Δ⁢t.[551.1.2]Consider an arbitrary subset G⊂Γ corresponding to a physically interesting reduced or coarse grained description of the original dynamical system.[551.1.3]Not all choices of G correspond to a physically interesting situation, and the choice of G reflects physical modeling or insight.[551.1.4]A point x∈G is called recurrent with respect to G if there exists a k≥1 for which T~k⁢x∈G.[551.1.5]The Poincarè recurrence theorem asserts that if μ is invariant under T~ and G∈G then almost every point of G is recurrent with respect to G.[551.1.6]A set G∈G is called a μ-recurrent set if μ-almost every x∈G is recurrent with respect to G.[551.1.7]By virtue of Poincarè’s recurrence theorem the transformation T~ defines an induced transformation S~G on subsets G of positive measure, μ⁢G>0, through

for almost every x∈G.[551.1.8]The recurrence time τG⁢x of the point x, defined as

is positive and finite for almost every point x∈G.[551.1.9]Because G has positive measure it becomes a probability measure space with the induced measure ν=μ/μ⁢G.[551.1.10]If μ was invariant under T~ then ν is invariant under S~G, and ergodicity of μ implies ergodicity also for ν [22].

[551.2.1]The induced transformation S~G:G→G exists for μ-almost every x∈G with μ⁢G>0 by virtue of the Poincare recurrence theorem.[551.2.2]To extend the definition to the case μ⁢G=0 let G,G,ν denote a subspace G⊂Γ of measure μ⁢G=0 with σ-algebra G contained in G, G⊂G, in the sense that B∈G for all B∈G.[551.2.3]μ⁢B=0 for all B∈G while ν⁢B=∞ for all sets B∈G with μ⁢B>0.[551.2.4]Let 0<ν⁢G<∞.[551.2.5]If G is ν-recurrent under T~ in the sense that ν-almost every point (rather than μ) is recurrent with respect to G then the recurrence time τG⁢x and the map S~G are defined for ν-almost every point x∈G.[551.2.6]Throughout the following it will be assumed that G is ν-recurrent under T~, and that ν⁢G∖S~G⁢G=0.[551.2.7]An example is given by solidification where Γ represents the high temperature phase space, while G 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 S~G 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

be the set of points whose recurrence time is k⁢Δ⁢t.[551.3.4]The number

[page 552, §0]   is the probability to find a recurrence time k⁢Δ⁢t with k∈N.[552.0.1]The numbers p⁢k define a discrete (lattice) probability density p⁢k⁢δ⁢t-k⁢Δ⁢t concentrated on the arithmetic progression k⁢Δ⁢t,k∈N.[552.0.2]The induced transformation SG acting on a measure ϱ on G is defined as the mathematical expectation

where B⊂G, and Tt was given in (1).[552.0.3]This defines a transformation SG:G′→G′ on the space G′ of measures on G.[552.0.4]The next section discusses the iterated transformation SGN and the long time limit N→∞.