Variational Principles in Conformation Dynamics - FU Berlin, FB MI

10 II. The variational principleThe spectral decomposition, the Chapman-Kolmogorov equation Lemma II.1 and the assumptionP(τ)φ → φ for all φ ∈ L 2 µ −1 (Ω) show us that the i-th eigenvalue λ i (τ) is acontinuous function of τ and satisfies λ i (τ 1 +τ 2 )=λ i (τ 1 )λ i (τ 2 ) as well as λ i (0) = 1. Therefore,it can be written as λ i (τ) =e −κ iτ for some κ i > 0, thatis,ithasanexponentialdecayrate. Clearly, κ 1 =0and κ 2 ,...,κ m are close to zero. Looking at the action of P(τ) onsome function ρ once more shows thatP(τ)ρ =∞ρ | φ i µ −1 P(τ)φ i =i=1∞e −κiτ ρ | φ i µ −1 φ i .i=1(II.27)If the time lag τ is very close to zero, almost all terms in the above sum will contribute. Ifτ becomes larger, however, there is a range of time lags τ where, for i>m,allexponentialdecay terms e −κiτ have essentially vanished, whereas all terms with i ≤ m still contribute. Ifτ increases even further, namely if τ 1 κ 2,therearenocontributionsexcepttheveryfirstterm. For a probability distribution ρ, wehaveρ | φ 1 µ −1 = ρ | µ µ −1 =1,soP(τ)ρ ≈ µ.The system is then distributed according to its equilibrium distribution and has basically“forgotten“ the initial deviation from equilibrium expressed by ρ. Eachdominanteigenvaluedefines a time scale 1 κ iand a corresponding slow process which equilibrates only if one waitsmuch longer than this time scale. The slow processes are the link between eigenvalues andmetastable states. A slow process usually corresponds to an equilibration process between twometastable regions, and the implied time scale 1 κ i= − τlog λ idefines an average transition(τ)time. For this reason, the approximation of dominant eigenvalues is of such importance.Example 2: Consider a four state system with stochastic transition matrix⎛⎞0.7571 0.2429 0 0P= ⎜ 0.1609 0.8306 0.0085 0⎟⎝ 0 0.0084 0.8294 0.1622 ⎠ .0 0 0.2413 0.7587(II.28)Clealy, most of the transitions will occur either between states x 1 and x 2 or between x 3 andx 4 ,whereasatransitionfromx 2 to x 3 will be a very rare event. Grouped together, x 1 , x 2 andx 3 , x 4 form two metastable states. The eigenvalues are λ 1 =1, λ 2 =0.9900, λ 3 =0.5964and λ 4 =0.5894. Clearly, there is one dominant eigenvalue related to the slow transitionprocess between the two metastable states.Example 3: In chapter 2, we will discuss the diffusion process of a one-dimensional particlein an energy landscape V .TheparticleissubjecttoaforceF ,whichequalsthederivativeofthe energy function, but is additionally perturbed by random fluctuations. For the potentialfunction V shown in Figure II.1a, theparticlewillspendmostofthetimearoundoneofthetwo minima of V ,andwilljustoccasionallycrossthebarrierseparatingthem. Figure II.1bshows a trajectory of the process, where the metastable behaviour is clearly visible. This isanother typical example of metastable states, we will investigate it in more detail in the next