Metastability Reversible diffusions Timescales Why non-quadratic saddles? Transition times Metastable Lifetimes in Coupled Random Dynamical Systems Barbara Gentz University of Bielefeld, Germany Joint work with Nils Berglund, Universit´ e d’Orl´ eans, France Bastien Fernandez, CPT–CNRS Luminy, France SAMSI, 31 August 2009 Barbara Gentz gentz@math.uni-bielefeld.de http://www.math.uni-bielefeld.de/ ˜ gentz
Metastability Reversible diffusions Timescales Why non-quadratic saddles? Transition times Metastability: A common phenomenon ⊲ Observed in the dynamical behaviour of complex systems ⊲ Related to first-order phase transitions in nonlinear dynamics Characterization of metastability ⊲ Existence of quasi-invariant subspaces Ω i , i ∈ I ⊲ Multiple timescales ⊲ A short timescale on which local equilibrium is reached within the Ω i ⊲ A longer metastable timescale governing the transitions between the Ω i Important feature ⊲ High free-energy barriers to overcome Consequence ⊲ Generally very slow approach to the (global) equilibrium distribution Metastable Lifetimes in Coupled Random Dynamical Systems SAMSI 31 August 2009 1 / 25
Metastability Reversible diffusions Timescales Why non-quadratic saddles? Transition times Example: Liquid–cristal transition through nucleation Change parameters quickly across the line of a first-order phase transition: ⊲ System remains in metastable equilibrium for long time before undergoing a rapid transition to the new equilibrium state due to (random) perturbations Example: Supercooled liquid ⊲ Pure water freezes at about − 44 ◦ F rather than at its freezing temperature of 32 ◦ F if no crystal nuclei are present Metastable Lifetimes in Coupled Random Dynamical Systems SAMSI 31 August 2009 2 / 25
Metastability Reversible diffusions Timescales Why non-quadratic saddles? Transition times Example: Liquid–cristal transition through nucleation Change parameters quickly across the line of a first-order phase transition: ⊲ System remains in metastable equilibrium for long time before undergoing a rapid transition to the new equilibrium state due to (random) perturbations Example: Supercooled liquid ⊲ Pure water freezes at about − 44 ◦ F rather than at its freezing temperature of 32 ◦ F if no crystal nuclei are present Supercooled water Metastable Lifetimes in Coupled Random Dynamical Systems SAMSI 31 August 2009 2 / 25
Metastability Reversible diffusions Timescales Why non-quadratic saddles? Transition times Reversible diffusions Gradient dynamics (ODE) x det = −∇ V ( x det ˙ ) z ⋆ t t Random perturbation by Gaussian white noise (SDE) √ d x ε t ( ω ) = −∇ V ( x ε t ( ω )) d t + 2 ε d B t ( ω ) x ⋆ − with x ⋆ + ⊲ V : R d → R : confining potential, growth condition at infinity ⊲ { B t ( ω ) } t ≥ 0 : d -dimensional Brownian motion Invariant measure or equilibrium distribution (for gradient systems) � µ ε ( dx ) = 1 e − V ( x ) /ε dx R d e − V ( x ) /ε d x with Z ε = Z ε µ ε concentrates in the minima of V Metastable Lifetimes in Coupled Random Dynamical Systems SAMSI 31 August 2009 3 / 25
Metastability Reversible diffusions Timescales Why non-quadratic saddles? Transition times Metastability in reversible diffusions: Timescales Let V double-well potential as before, start in x ε 0 = x ⋆ − = left-hand well How long does it take until x ε t is well described by its invariant distribution? ⊲ For ε small, paths will stay in the left-hand well for a long time ⊲ x ε t first approaches a Gaussian distribution, centered in x ⋆ − , 1 1 T relax = − ) = ( d =1) V ′′ ( x ⋆ curvature at the bottom of the well ⊲ With overwhelming probability, paths will remain inside left-hand well, for all times significantly shorter than Kramers’ time T Kramers = e H /ε , where H = V ( z ⋆ ) − V ( x ⋆ − ) = barrier height ⊲ Only for t ≫ T Kramers , the distribution of x ε t approaches p 0 The dynamics is thus very different on the different timescales Metastable Lifetimes in Coupled Random Dynamical Systems SAMSI 31 August 2009 4 / 25
Metastability Reversible diffusions Timescales Why non-quadratic saddles? Transition times Transition times between potential wells First-hitting time of a small ball B δ ( x ⋆ + ) around minimum x ⋆ + τ + = τ ε + ( ω ) = inf { t ≥ 0: x ε t ( ω ) ∈ B δ ( x ⋆ + ) } x ⋆ Eyring–Kramers Law [Eyring 35, Kramers 40] 2 π − ) | V ′′ ( z ⋆ ) | e [ V ( z ⋆ ) − V ( x ⋆ − )] /ε ⊲ d = 1: − τ + ≃ E x ⋆ � V ′′ ( x ⋆ Metastable Lifetimes in Coupled Random Dynamical Systems SAMSI 31 August 2009 5 / 25
Metastability Reversible diffusions Timescales Why non-quadratic saddles? Transition times Transition times between potential wells First-hitting time of a small ball B δ ( x ⋆ + ) around minimum x ⋆ + τ + = τ ε + ( ω ) = inf { t ≥ 0: x ε t ( ω ) ∈ B δ ( x ⋆ + ) } x ⋆ Eyring–Kramers Law [Eyring 35, Kramers 40] 2 π − ) | V ′′ ( z ⋆ ) | e [ V ( z ⋆ ) − V ( x ⋆ − )] /ε ⊲ d = 1: − τ + ≃ E x ⋆ � V ′′ ( x ⋆ � 2 π | det ∇ 2 V ( z ⋆ ) | − ) e [ V ( z ⋆ ) − V ( x ⋆ − )] /ε ⊲ d ≥ 2: − τ + ≃ E x ⋆ | λ 1 ( z ⋆ ) | det ∇ 2 V ( x ⋆ where λ 1 ( z ⋆ ) is the unique negative eigenvalue of ∇ 2 V at saddle z ⋆ Metastable Lifetimes in Coupled Random Dynamical Systems SAMSI 31 August 2009 5 / 25
Metastability Reversible diffusions Timescales Why non-quadratic saddles? Transition times Proving Kramers Law I ⊲ Exponential asymptotics and optimal transition paths via large deviations approach [Wentzell & Freidlin 69–72] ⊲ Probability of observing sample paths being close to a function ϕ : [0 , T ] → R d behaves like ∼ exp {− 2 I ( ϕ ) /ε } ⊲ Large-deviation rate function R T ϕ s − ( −∇ V ( ϕ s )) � 2 d s ( 1 0 � ˙ for ϕ ∈ H 1 2 I [0 , T ] ( ϕ ) = + ∞ otherwise ⊲ Domain D with unique asymptotically stable equilibrium point x ⋆ − Quasipotential with respect to x ⋆ − = Cost to reach z against the flow V ( x ⋆ t > 0 inf { I [0 , t ] ( ϕ ): ϕ ∈ C ([0 , t ] , D ) , ϕ 0 = x ⋆ − , z ) = inf − , ϕ t = z } ⊲ Gradient case (reversible diffusion) ⊲ Cost for leaving potential well: V := min z ∈ ∂ D V ( x ⋆ − , z ) = 2[ V ( z ⋆ ) − V ( x ⋆ − )] ⊲ Attained for paths going against the flow: ˙ ϕ t = + ∇ V ( ϕ t ) Metastable Lifetimes in Coupled Random Dynamical Systems SAMSI 31 August 2009 6 / 25
Metastability Reversible diffusions Timescales Why non-quadratic saddles? Transition times Proving Kramers Law II ⊲ Exponential asymptotics depends only on barrier height − τ + = V ( z ⋆ ) − V ( x ⋆ ε → 0 ε log E x ⋆ lim − ) Only 1-saddles are relevant for transitions between wells ⊲ Low-lying spectrum of generator of the diffusion (analytic approach) [Helffer & Sj¨ ostrand 85, Miclo 95, Mathieu 95, Kolokoltsov 96, . . . ] ⊲ Potential theoretic approach [Bovier, Eckhoff, Gayrard & Klein 04] � | det ∇ 2 V ( z ⋆ ) | � � 2 π − )] /ε [1 + O − ) e [ V ( z ⋆ ) − V ( x ⋆ ε 1 / 2 | log ε | E x ⋆ − τ + = ] det ∇ 2 V ( x ⋆ | λ 1 ( z ⋆ ) | ⊲ Full asymptotic expansion of prefactor [Helffer, Klein & Nier 04] ⊲ Asymptotic distribution of τ + [Day 83, Bovier, Gayrard & Klein 05] − τ + } = e − t lim − { τ + > t · E x ⋆ ε → 0 P x ⋆ Metastable Lifetimes in Coupled Random Dynamical Systems SAMSI 31 August 2009 7 / 25
Metastability Reversible diffusions Timescales Why non-quadratic saddles? Transition times Non-quadratic saddles What happens if det ∇ 2 V ( z ⋆ ) = 0 ? det ∇ 2 V ( z ⋆ ) = 0 At least one vanishing eigenvalue at saddle z ⋆ ⇒ ⇒ Saddle has at least one non-quadratic direction ⇒ Kramers Law not applicable Why do we care about this non-generic situation? Parameter-dependent systems may undergo bifurcations Quartic unstable direction Quartic stable direction Metastable Lifetimes in Coupled Random Dynamical Systems SAMSI 31 August 2009 8 / 25
Metastability Reversible diffusions Timescales Why non-quadratic saddles? Transition times Example: Two harmonically coupled particles V γ ( x 1 , x 2 ) = U ( x 1 ) + U ( x 2 ) + γ 2 ( x 1 − x 2 ) 2 Change of variable: Rotation by π/ 4 yields U ( x ) = x 4 4 − x 2 2 � � V γ ( y 1 , y 2 ) = − 1 1 − 1 − 2 γ 2 + 1 � 2 y 2 y 2 y 4 1 + 6 y 2 1 y 2 2 + y 4 2 2 8 Note: det ∇ 2 � V γ (0 , 0) = 1 − 2 γ ⇒ Pitchfork bifurcation at γ = 1 / 2 γ > 1 1 2 > γ > 1 1 3 > γ > 0 2 3 Metastable Lifetimes in Coupled Random Dynamical Systems SAMSI 31 August 2009 9 / 25
Metastability Reversible diffusions Timescales Why non-quadratic saddles? Transition times Further examples: More particles N particles with nearest-neighbour coupling : i ∈ Λ = Z / N Z � � U ( x i ) + γ ( x i +1 − x i ) 2 V γ ( x ) = 4 i ∈ Λ i ∈ Λ Results [Berglund, G. & Fernandez 07] ⊲ Bifurcation diagram ⊲ Optimal transition paths ⊲ Exponential asymptotics of transition times Metastable Lifetimes in Coupled Random Dynamical Systems SAMSI 31 August 2009 10 / 25
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