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# 3 Almost invariant states

[631.2.1] Strictly stationary or invariant states [15] are an idealization. [631.2.2] In experiments stationarity is never ideal, but only approximate. [631.2.3] Expectation values are uncertain within the accuracy of the experiment. [631.2.4] Experimental accuracy depends on the response and integration times of the experimental apparatus.

[631.3.1] These experimental restrictions suggest to focus on a class of states that are stationary (invariant) only up to a given experimental accuracy . [631.3.2] To do so, recall the definition of invariant (stationary) states. [631.3.3] A state is called invariant, if

 (15)

holds for all and , i.e. if the expectation values of all observables are constant. [631.3.4] The set of invariant states over is convex and compact in the weak*-topology [6]. [631.3.5] The same holds for the set of all states . [631.3.6] Invariant states are fixed points of the adjoint time evolution as seen from eq. (8). [631.3.7] Because invariant states are fixed points of , they are of limited benefit for a proper mathematical formulation of the problems discussed above. [631.3.8] Once an orbit in state space reaches an invariant state, it remains forever in that state and cannot leave it.

[631.4.1] Almost invariant states are based on states whose expectation values are of bounded mean oscillation (BMO). [631.4.2] A state is called a BMO-state if all maps have bounded mean oscillation for all . [631.4.3] The Banach space of functions with bounded mean oscillation on is defined as the linear space

 (16)

where is the space of locally integrable functions . [631.4.4] The BMO-norm is defined as

 (17)

where denotes intervals of length and

 (18)

denotes the average of over the interval . [631.4.5] The set of all BMO-states

 (19)

is convex by linearity. [631.4.6] As a subset of a weak* compact set it is itself weak* compact. [631.4.7] Hence a decomposition theory into extremal BMO-states exists by virtue of the Krein-Milman theorem. [631.4.8] The set of invariant states is identified through

 (20)

as a subset .

[page 632, §1]    [632.1.1] A BMO-state will be called -almost invariant or almost invariant with accuracy if the expectation of all observables are stationary to within experimental accuracy . [632.1.2] More precisely, the set of all -almost invariant states is defined as

 (21)

as a family of subsets of . [632.1.3] For small these states are almost invariant. [632.1.4] The accuracy measures temporal fluctuations away from the time average.

[632.2.1] The following inclusions of classes of states used in the following are summarized for orientation and convenience

 (22)

where and the set of KMS-states at inverse temperature are defined as states such that the KMS-condition[16]

 (23)

holds for all and . [632.2.2] The KMS-states are invariant states for all , but KMS-states for different are disjoint [16]. [632.2.3] For the KMS-states are trace states, i.e.  holds for all . [632.2.4] Because KMS-states are Gibbs states they are usually interpreted as equilibrium states with extremal states corresponding to pure thermodynamic phases [16].