Sie sind hier: ICP » R. Hilfer » Publikationen

5 Conservative Systems

[211.2.1] In classical mechanics the commutative algebra of observables \mathcal{A}=C_{0}(\Gamma) is the algebra of continuous functions on phase space \Gamma, that vanish at infinitye (This is a footnote:) e This means that for each a\in C_{0}(\Gamma) and \varepsilon>0 there is a compact subset K\subseteq\Gamma such that |a(x)|<\varepsilon for all x\in\Gamma\setminus K.. [211.2.2] The characters (pure states) are point measures on phase space \Gamma, and one has the isomorphism \Gamma\equiv X(\mathcal{A}). [211.2.3] By the Riesz representation theorem the states \mu\in\mathcal{A}^{*}\equiv C_{0}(X(\mathcal{A}))^{*}\equiv C_{0}(\Gamma)^{*} in classical mechanics are probability measures on phase space \Gamma\equiv X(\mathcal{A}). [211.2.4] Every state \mu\in C_{0}(\Gamma)^{*} gives rise to a probability measure space (\Gamma,\mathfrak{G},\mu) where \mathfrak{G} is the \sigma-algebra of measurable subsets of phase space \Gamma.

[211.3.1] Let S\colon\Gamma\to\Gamma be an invertible map such that S and S^{{-1}} are both measurable, i.e. such that S^{{-1}}\mathfrak{G}=S\mathfrak{G}=\mathfrak{G} where S{G}:=\{ Sx:x\in{G}\} for {G}\in\mathfrak{G}. [211.3.2] The map S is called a measure preserving transformation and the measure \mu on \Gamma is called invariant under S, if \mu({G})=\mu(S{G})=\mu(S^{{-1}}{G}) for all {G}\in\mathfrak{G}. [page 212, §0]    [212.0.1] An invariant measure is called ergodic with respect to S, if it cannot be decomposed into a convex combination of S-invariant measures.

[212.1.1] Here and in the following the measure preserving transformation is the adjoint time evolution {\rule[-2.0pt]{0.0pt}{10.0pt}_{{\mathcal{A}^{*}}}\! T}^{{\dot{t}}} which is denoted more briefly as {\rule[-2.0pt]{0.0pt}{10.0pt}_{{\Gamma}}\! T}^{{\dot{t}}}={\rule[-2.0pt]{0.0pt}{10.0pt}_{{\mathcal{A}^{*}}}\! T}^{{\dot{t}}}. [212.1.2] Pure states (characters) are not invariant under {\rule[-2.0pt]{0.0pt}{10.0pt}_{{\Gamma}}\! T}^{{\dot{t}}}. [212.1.3] Examples of invariant probability measures are furnished by the set of equilibrium states of a conservative system with Hamiltonian dynamics. [212.1.4] If \mu is an equilibrium state of a conservative system then (\Gamma,\mathfrak{G},\mu,{\rule[-2.0pt]{0.0pt}{10.0pt}_{{\Gamma}}\! T}^{{\dot{t}}}) is a measure preserving system.