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2 Time Flow and Induced Transformations

[550.4.1]Let \Gamma be the phase or state space of a dynamical system, let \mathcal{G} be a \sigma-algebra of measurable subsets of \Gamma, and \mu a measure on \mathcal{G} such that \mu(\Gamma)=1.[550.4.2]The triple (\Gamma,\mathcal{G},\mu) forms a probability measure space.[550.4.3]In general the time evolution of the system is given as a flow (or semiflow) on (\Gamma,\mathcal{G},\mu), defined as a one-parameter family of maps \widetilde{T}^{t}:\Gamma\rightarrow\Gamma such that \widetilde{T}^{0}=I is the identity, \widetilde{T}^{{s+t}}=\widetilde{T}^{s}\widetilde{T}^{t} for all t,s\in\mathbb{R} and such that for every measurable function f the function f(\widetilde{T}^{t}x) is measurable on the direct product \Gamma\times\mathbb{R}.[550.4.4]For every G\in\mathcal{G} also \widetilde{T}G,\widetilde{T}^{{-1}}G\in\mathcal{G} holds.[550.4.5]The measure \mu is called invariant under the flow \widetilde{T}^{t} if \mu(G)=\mu(\widetilde{T}^{t}G)=\mu((\widetilde{T}^{t})^{{-1}}G) for all t\in\mathbb{R},G\in\mathcal{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 \mu=\lambda\mu _{1}+(1-\lambda)\mu _{2} with \mu _{1},\mu _{2} invariant and 0\leq\lambda\leq 1 implies \lambda=1,\mu _{1}=\mu or \lambda=0,\mu _{2}=\mu.

[550.5.1]The flow \widetilde{T}^{t} defines the time evolution of measures through T^{t}\mu(G)=\mu(\widetilde{T}^{t}G) as a map T^{t}:\Gamma^{\prime}\rightarrow\Gamma^{\prime} on the space \Gamma^{\prime} of measures on \Gamma.[550.5.2]Defining as usual [22, 23] \mu(G,t)=\mu((\widetilde{T}^{t})^{{-1}}G) shows that

T^{t}\mu(G,t_{0})=\mu(G,t_{0}-t) (1)

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

A=\lim _{{t\rightarrow 0^{+}}}\frac{T^{t}-I}{t} (2)

where I=T^{0} denotes the identity, and one has A=-d/dt for right translations.[551.0.1]The invariance of the measure \mu can be expressed as A\mu=-d\mu/dt=0 and it implies that for given t_{0}\in\mathbb{R}

T^{t}\mu(G,t_{0})=\mu(G,t_{0}) (3)

for all G\in\mathcal{G},t\in\mathbb{R}.

[551.1.1]The continuous time evolution \widetilde{T}^{t} with t\in\mathbb{R} may be discretized into the discrete time evolution \widetilde{T}^{k} with k\in\mathbb{Z} generated by the map \widetilde{T}=\widetilde{T}^{{\Delta t}} with discretization time step \Delta t.[551.1.2]Consider an arbitrary subset G\subset\Gamma 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\in G is called recurrent with respect to G if there exists a k\geq 1 for which \widetilde{T}^{k}x\in G.[551.1.5]The Poincarè recurrence theorem asserts that if \mu is invariant under \widetilde{T} and G\in\mathcal{G} then almost every point of G is recurrent with respect to G.[551.1.6]A set G\in\mathcal{G} is called a \mu-recurrent set if \mu-almost every x\in G is recurrent with respect to G.[551.1.7]By virtue of Poincarè’s recurrence theorem the transformation \widetilde{T} defines an induced transformation \widetilde{S}_{G} on subsets G of positive measure, \mu(G)>0, through

\widetilde{S}_{G}x(t_{0})=\widetilde{T}^{{\tau _{G}(x)}}x(t_{0})=x(t_{0}+\tau _{G}(x)) (4)

for almost every x\in G.[551.1.8]The recurrence time \tau _{G}(x) of the point x, defined as

\tau _{G}(x)=\Delta t\min\{ k\geq 1:\widetilde{T}^{k}x\in G\}, (5)

is positive and finite for almost every point x\in G.[551.1.9]Because G has positive measure it becomes a probability measure space with the induced measure \nu=\mu/\mu(G).[551.1.10]If \mu was invariant under \widetilde{T} then \nu is invariant under \widetilde{S}_{G}, and ergodicity of \mu implies ergodicity also for \nu [22].

[551.2.1]The induced transformation \widetilde{S}_{G}:G\rightarrow G exists for \mu-almost every x\in G with \mu(G)>0 by virtue of the Poincare recurrence theorem.[551.2.2]To extend the definition to the case \mu(G)=0 let (G,\mathfrak{G},\nu) denote a subspace G\subset\Gamma of measure \mu(G)=0 with \sigma-algebra \mathfrak{G} contained in \mathcal{G}, \mathfrak{G}\subset\mathcal{G}, in the sense that B\in\mathfrak{G} for all B\in\mathcal{G}.[551.2.3]\mu(B)=0 for all B\in\mathfrak{G} while \nu(B)=\infty for all sets B\in\mathcal{G} with \mu(B)>0.[551.2.4]Let 0<\nu(G)<\infty.[551.2.5]If G is \nu-recurrent under \widetilde{T} in the sense that \nu-almost every point (rather than \mu) is recurrent with respect to G then the recurrence time \tau _{G}(x) and the map \widetilde{S}_{G} are defined for \nu-almost every point x\in G.[551.2.6]Throughout the following it will be assumed that G is \nu-recurrent under \widetilde{T}, and that \nu(G\setminus\widetilde{S}_{G}G)=0.[551.2.7]An example is given by solidification where \Gamma 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 \widetilde{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

G_{k}=\{ x\in G:\tau(x)=k\Delta t\} (6)

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

p(k)=\frac{\nu(G_{k})}{\nu(G)} (7)

[page 552, §0]   is the probability to find a recurrence time k\Delta t with k\in\mathbb{N}.[552.0.1]The numbers p(k) define a discrete (lattice) probability density p(k)\delta(t-k\Delta t) concentrated on the arithmetic progression k\Delta t,k\in\mathbb{N}.[552.0.2]The induced transformation S_{G} acting on a measure \varrho on G is defined as the mathematical expectation

S_{G}\varrho(B,t_{0})=\langle T^{{\tau _{G}}}\varrho(B,t_{0})\rangle=\sum _{{k=1}}^{\infty}\varrho(B,t_{0}-k\Delta t)p(k) (8)

where B\subset G, and T^{t} was given in (1).[552.0.3]This defines a transformation S_{G}:G^{\prime}\rightarrow G^{\prime} on the space G^{\prime} of measures on G.[552.0.4]The next section discusses the iterated transformation S_{G}^{N} and the long time limit N\rightarrow\infty.