Invariant eigenvalue

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]. 

For B = BiU B2, both measures are eigenmeasures of P associated with the eigenvalue A = 1. Hence, there is no unique invariant measure /t. In fact, any liii( ar combination... 

In the perturbed case, assume now a small change in 7, which induces a small intersection of Bi and B2. Let B = Bi U B2 remain to be invariant. Then, we have a unique eigenmeasure /2 with eigenvalue Ai = 1 and another eigenvector P associated with A2 1. Under some continuity assumption A2 should be close to 1 and thus we have f (B) = 0. In view of (6), continuity for 7 = 7o then requires a = 1/2. Therefore (5) implies that... 

As it turns out, there exists a relationship between those probabilities, by which sets are almost invariant, and associated eigenvalues X (cf. [6]). 

For more than two almost invariant sets one has to consider all eigenmea-sures corresponding to eigenmodes for eigenvalues close to one. In this case, the following lemma will be helpful. 

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

If one uses the well-known properties of Y00(j 0), namely that this operator has five eigenfunctions with zero eigenvalues corresponding to the collision invariants lil. liPi-... 

Since the trace of a matrix is invariant upon rotation, we get the sum of the n eigenvalues 8j as the trace of the correlation matrix. The dimensionless total variance is therefore... 

The last problem of general interest in algebraic theory is the evaluation of the eigenvalues of the invariant operators in the basis discussed in Section 2.4. As mentioned before, the invariant operators commute with all the Xs. As a result, they are diagonal in the basis [A,], A, ..., A.v],... 

The eigenvalue problem for the Hamiltonian H [Eq. (2.92)] can be solved in closed form whenever H does not contain all the elements but only a subset of them, the invariant or Casimir operators. For three-dimensional problems there are two such situations corresponding to the two chains discussed in the preceding sections. We begin with chain (I). Restricting oneself only to terms up to quadratic in the elements of the algebra, one can write the most general Hamiltonian with dynamic symmetry (I) as... 

We have stated several times that whenever the Hamiltonian can be written in terms of invariant (Casimir) operators of a chain, its eigenvalue problem can be solved analytically. This method can be applied to the construction of both local and normal Hamiltonians. For local Hamiltonians, one writes H in terms of Casimir invariants of Eq. (4.43). 

To an energy eigenvalue or state of 3C, there will generally correspond several independent eigenvectors or state functions i,. . . , n is the degeneracy of the state. These functions must form a basis for a representation of the group G if is invariant under G. If i2 is an element of G... 

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