On different notions of timescales in molecular dynamics Origin of Scaling Cascades in Protein Dynamics June 8, 217 IHP Trimester Stochastic Dynamics out of Equilibrium
Overview 1. Motivation 2. Definition of the timescales of interest: convergence rates and mean first exit times 3. Review of results for overdamped Langevin equation 4. Approach for linear but irreversible systems: entropy production
Origin of the scaling cascades in protein dynamics Proteinstructure (Source: MaxPlanckForschung 4/23) Observation: single point mutations large non-local effects Trying to understand the observations: study parameter sensitivies of timescales study relations between timescales
A model to simulate molecular dynamics Overdamped Langevin equation dx t = V (X t )dt + 2β 1 db t where X R 3N : positions of the atoms V : R 3N R is the interaction potential B t is standard N dimensional Brownian motion β 1 R + is the temperature.
Illustration of the model Overdamped Langevin equation dx t = V (X t )dt + 2β 1 db t Associated probability density ρ t and equilibrium distribution ρ e V β
Quantities of interest Convergence to equilibrium: ρ t ρ c γ(t) Mean first exit times (MFET): E(τ x ( D)) = E(inf {t > : X t / D})
Decay towards equilibrium in L 2 µ 1 Generator: L = β 1 2 V dx t = V (X t )dt + 2β 1 db t describes evolution of expectation values and probability densities is self-adjoint wrt dµ = 1 Z e βv (x) dx, i.e. Lf, g µ = f, Lg µ spec( L) {} [λ, ) this implies convergence towards equilibrium in L 2, i.e. µ 1 ρ t ρ 2 L 2 µ 1 Problems: = λ is not known ρ L 2 µ 1 ρ t (x) ρ (x) 2 ρ 1 (x)dx e 2λt ρ ρ 2 L 2 µ 1 ρ (x) 2 e βv (x) dx <
Mean first exit times and eigenvalues of L Theorem [Bovier et al. 24]: Assumptions V has n minima x 1,..., x n ordering of minima according to energy barrier height possible Define τ xk (S k 1 ) = inf {t : X t S k 1, X = x k }, S k 1 = k 1 B 1 (x j). β Then L has n eigenvalues = Λ 1 > Λ 2 >... > Λ n and δ > such that 1 ( Λ k = 1 + O(1 + e βδ ) ) E(τ xk (S k 1 )) = c exp ( β (x k, {x 1,..., x k 1 })) (1 + O(1 + β 1 ) log β 1 ), c >. j=1
What we would like to do Convergence in terms of relative entropy instead of L 2 : µ 1 Relative entropy H(ρ t ρ ) = ( ρt (x) ρ t (x) log ρ (x) ) dx Relation to L 1 norm via Csizsàr-Kullback/Pinsker inequality: ρ t ρ L 1 2H(ρ t ρ ) L 1 is the natural norm for probability densities relation to measurable quantities applicable for any process that admits a pdf, not only reversible processes H is computable from simulation data
Linear but possibly irreversible processes Conditions dx t = AX t dt + σ 2β 1 db t, X R n, A R n n, (i) spec(a) C = {λ C : R(λ) < } (ii) A and σ fulfill the Kalman rank condition rank( [ σ, Aσ,..., A n 1 σ ] ) = n σ R n m, B t R m, m n. = existence of unique positive invariant measure ρ = N (, Σ). Theorem[Arnold, Erb 214] H(ρ t ρ ) c H(ρ ρ )e 2λ A t, λ A = min { R(λ) : λ spec(a)}, c 1.
Mean first exit times and eigenvalues of the covariance dx t = AX t dt + σ 2β 1 db t Interested in: τ x ( D) = inf {t > : X t / D}, D = {x : x < 1}. Theorem [Zabczyk 85] Assume that conditions (i) and (ii) are fulfilled s. th. ρ = N (, Σ) exists. Let λ Σ = max {λ : λ spec(σ)} >, E = {v : Σv = λ Σ v}. Then for large β, i.e. small temperature lim β β 1 log E(τ x ( D)) = 1 2λ exit time Σ and for any η > lim P(dist(X τ x ( D), E) η) = 1 exit path β We can show that λ A (2λ Σ) 1 λ + σ, λ + σ = min{λ > : λ spec(σσ T )}.
Analysis of relaxation behaviour Splitting up: H(ρ t ρ ) = ( ρt (x) log ρ (x) =a(t) ) ρ t (x) dx = 1 [ Tr(Σt Σ 1 ) n Tr(log(Σ t Σ 1 )) + µ T t Σ 1 ] µ t. 2 }{{}}{{} =b(t) } {{ } Covariance Same structure in all terms: z T e AT t Σ 1 e At z. For a(t) and b(t) : z = (Σ Σ ) 1 2, for c(t) : z = x. } {{ } } =c(t) {{ } Mean
Some examples for different relaxation behaviour high temperature low temperature, x = EVec(A) H/H 1.4 1.2 1.8.6.4 relative entropy H()*e -2 t a(t) b(t) c(t) H/H 1.8.6.4 relative entropy H()*e -2 t a(t) b(t) c(t).2.2 -.2 -.4 1 2 3 4 5 time -.2 1 2 3 4 5 time low temperature, x EVec(A) low temperature, A = A T H/H 2 1.5 1.5 relative entropy H()*e -2 t c*h()*e -2 t a(t) b(t) c(t) H/H 1.8.6.4.2 relative entropy H()*e -2 t a(t) b(t) c(t) 1 2 3 4 5 time -.2 1 2 3 4 5 time
Understanding plateaus in the entropy decay Necessary and sufficient condition for the existence of a plateau: degeneracy of the noise, i.e. det σσ T =. For c(t) this translates to: ċ(t) = ż p = z moves along contour lines of the potential p Here z i (t) = e λ i t (Sx ) i, p(z) = z T S T Σ 1 S 1 z with S such that SAS 1 = diag(λ 1,..., λ n ). det(σσ T ) = H/H 1.8.6.4 a b c c at discrete times S 2 -.5-1 -1.5 4 35 3 25 2.2.1.2.3.4.5.6.7.8 time -2-2.5 z at discrete times 1 1 1.5 2 2.5 3 3.5 4 S 1 15
From degenerate to isotropic noise c(t) = z T S T Σ 1 S 1 z, z i = e λ i t z i (), λ 1 = 1, λ 2 = 1. det(σσ T ) = H/H 1.8.6.4 a b c c at discrete times S 2 -.5-1 -1.5 4 35 3 25 2.2.1.2.3.4.5.6.7.8 time -2-2.5 z at discrete times 1 1 1.5 2 2.5 3 3.5 4 S 1 15 det(σσ T ) = 1 H/H 1.8.6.4.2 a b c c at discrete times.1.2.3.4.5.6.7.8 time S 2 -.5-1 -1.5-2 -2.5 z at discrete times 1 1 1.5 2 2.5 3 3.5 4 S 1 35 3 25 2 15
Identification of slow and fast? Can we identify slow and fast dof? Can we get estimates on the marginals? Can we get hierarchichal order of timescales?