Variational Principles in Conformation Dynamics - FU Berlin, FB MI

14 II. The variational principleFor a symmetric operator like T (τ), theRayleighcoefficientisalsocalledmatrix element orsimply expectation value. Thenexttworesultswillrevealtousthesignificanceofautocorrelationfunctions when it comes to approximating eigenvalues and -functions, [Noé, 2011,sec. 2.3]:Lemma II.5: The transfer operator’s eigenfunctions ψ i satisfy the relation:acf(ψ i ,ψ i ; τ) =λ i .(II.34)Proof. All we have to do is recognize the action of T (τ) on its eigenfunctions and use theorthogonality relation between those:acf(ψ i ,ψ i ; τ) =T (τ)ψ i | ψ i µ= λ i ψ i | ψ i µ= λ i .(II.35)Theorem II.6 (Variational principle): For any function ψ ∈ L 2 µ(Ω) which is orthogonalto ψ 1 ,wehave:acf(ψ, ψ; τ) ≤ λ 2 .(II.36)Proof. First, expand ψ in terms of the orthonormal basis {ψ i }.Notethatthereisnooverlapwith ψ 1 :acf(ψ, ψ; τ) =T (τ)ψ | ψ µ=∞c i c j T (τ)ψ i | ψ j µ,i,j=2(II.37)where c i = ψ | ψ i µ.Likeinthepreviousproof,weusetheactionofthetransferoperatoron its eigenfunctions:acf(ψ, ψ; τ) =∞c i c j λ i ψ i | ψ j µ=i,j=2∞c 2 i λ i .i=2(II.38)Finally, we recall that the eigenvalues are sorted in decreasing order, and that in an orthonormalbasis expansion, the squares of the coefficients sum up to unity:acf(ψ, ψ; τ) ≤ λ 2∞i=2c 2 i = λ 2 .(II.39)