Perrone

A key observation for our purposes here is that the numerical computation of invariant measures is equivalent to the solution of an eigenvalue problem for the so-called Frobenius-Perron operator P M - M defined on the set M. of probability measures on F by virtue of... 

From a mathematical point of view, conformations are special subsets of phase space a) invariant sets of MD systems, which correspond to infinite durations of stay (or relaxation times) and contain all subsets associated with different conformations, b) almost invariant sets, which correspond to finite relaxation times and consist of conformational subsets. In order to characterize the dynamics of a system, these subsets are the interesting objects. As already mentioned above, invariant measures are fixed points of the Frobenius-Perron operator or, equivalently, eigenmodes of the Frobenius-Perron operator associated with eigenvalue exactly 1. In view of this property, almost invariant sets will be understood to be connected with eigenmodes associated with (real) eigenvalues close (but not equal) to 1 - an idea recently developed in [6]. 

The literature on ergodic theory contains an interesting theorem concerning the spectrum of the Frobenius-Perron operator P. In order to state this result, we have to reformulate P as an operator on the Hilbert space L P) of all square integrable functions on the phase space P. Since and, therefore, / are volume preserving, this operator P L P) —+ L r) is unitary (cf. [20], Thm. 1.25). As a consequence, its spectrum lies on the unit circle. 

Setting up the Frobenius-Perron operator with respect to this subset. 

Then, the discretized Frobenius-Perron operator v = PdU can be written componentwise as... 

Almost Invariant Sets Recall that the relevant almost invariant sets correspond to eigenvalues X 1 with A < 1 of the associated Frobenius-Perron operator. 

Fig. 7. Eigenmeasure V2 of the Frobenius-Perron operator to the second largest eigenvalue A2 = 0.9963 for the test system (15) with 7 = 3. iV2 was computed via our new subdivision algorithm (cf. Section 4).

Lipson, C., Sheth, N. and Disney, R. L. 1967 Reliability Prediction - Mechanical Stessj Strength Interference Models (Perrons), RADC-TR-66-710, March (AD/813574). 

N. Fried, Degradation of Composite Materials The Effect of Water on Glass-Reinforced Plastic, in Mechanics of Composite Materials, Proceedings of the 5th Symposium on Naval Structural Mechanics, Philadelphia, Pennsylvania, 8-10 May 1967, F. W. Wendt, H. Liebowitz, and N. Perrone (Editors), Pergamon, Now York, 1970, pp. 813-837. 

J. C. Perrone, A. lachan, and L. A. Moreira Cameiro, Anais Acad. Brasil... 

While there is, at present, no known CA analogue of a Froebenius-Perron construction, a systematic n -order approximation to the invariant probability distributions for CA systems is readily obtainable from the local structure theory (LST), developed by Gutowitz, et.al. [guto87a] LST is discussed in some detail in section 5.3. 

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