Metastability for the Ginzburg Landau equation with space time white noise

Transcription

1 Barbara Gentz gentz Metastability for the Ginzburg Landau equation with space time white noise Barbara Gentz University of Bielefeld, Germany Joint work with Florent Barret (Ecole Polytechnique, Palaiseau, France) Nils Berglund (Université d Orléans, France) SAMSI, 10 February 2010

2 Metastability for the Ginzburg Landau equation SAMSI 10 February / 26

3 Metastability for the Ginzburg Landau equation SAMSI 10 February / 26 Examples Supercooled liquid Supersaturated gas Wrongly magnetized ferromagnet Metastability in the real world Free energy Order parameter Near first-order phase transitions Nucleation implies crossing of energy barrier

4 Metastability for the Ginzburg Landau equation SAMSI 10 February / 26 Ingredients Lattice: Λ Z d Metastability in stochastic lattice models Configuration space: X = S Λ, S finite set (e.g. { 1, 1}) Hamiltonian: H : X R (e.g. Ising model or lattice gas) Gibbs measure: µ β (x) = e βh(x) /Z β Dynamics: Markov chain with invariant measure µ β (e.g. Metropolis such as Glauber or Kawasaki dynamics)

5 Metastability for the Ginzburg Landau equation SAMSI 10 February / 26 Ingredients Lattice: Λ Z d Metastability in stochastic lattice models Configuration space: X = S Λ, S finite set (e.g. { 1, 1}) Hamiltonian: H : X R (e.g. Ising model or lattice gas) Gibbs measure: µ β (x) = e βh(x) /Z β Dynamics: Markov chain with invariant measure µ β (e.g. Metropolis such as Glauber or Kawasaki dynamics) Results (for β 1) Transition time from empty to full configuration Typical transition paths Shape of critical droplet References Frank den Hollander, Metastability under stochastic dynamics, Stochastic Process. Appl. 114 (2004), 1 26 Enzo Olivieri & Maria Eulália Vares, Large deviations and metastability, Cambridge University Press, Cambridge, 2005

6 Metastability for the Ginzburg Landau equation SAMSI 10 February / 26 Reversible diffusions Gradient dynamics (ODE) ẋ det t = V (x det t ) Random perturbation by Gaussian white noise (SDE) z dxt ε (ω) = V (xt ε (ω)) dt + 2ε db t (ω) with V : R d R : confining potential, growth condition at infinity {B t (ω)} t 0 : d-dimensional Brownian motion x x + Invariant measure or equilibrium distribution (for gradient systems) µ ε (dx) = 1 e V (x)/ε dx with Z ε = e V (x)/ε dx Z ε R d Dynamics reversible w.r.t. invariant measure µ ε (detailed balance)

7 Metastability for the Ginzburg Landau equation SAMSI 10 February / 26 Transition times between potential wells First-hitting time of a small ball B δ (x +) around minimum x + τ + = τ ε x + (ω) = inf{t 0: x ε t (ω) B δ (x +)} Eyring Kramers Law [Eyring 35, Kramers 40] d = 1: E x τ + 2π V (x ) V (z ) e[v (z ) V (x )]/ε d 2: E x τ + 2π det 2 V (z ) (z ) V (x λ 1 (z ) det 2 V (x ) e[v )]/ε where λ 1 (z ) is the unique negative eigenvalue of 2 V at saddle z

8 Metastability for the Ginzburg Landau equation SAMSI 10 February / 26 Proving Kramers Law Exponential asymptotics and optimal transition paths via large deviations approach [Wentzell & Freidlin 69 72] lim ε 0 ε log E x τ + = V (z ) V (x ) Only 1-saddles are relevant for transitions between wells Low-lying spectrum of generator of the diffusion (analytic approach) [Helffer & Sjöstrand 85, Miclo 95, Mathieu 95, Kolokoltsov 96,... ] Potential theoretic approach [Bovier, Eckhoff, Gayrard & Klein 04] E x τ + = 2π det 2 V (z ) (z ) V (x λ 1 (z ) det 2 V (x ) e[v )]/ε [1 + O ( (ε log ε ) 1/2) ] Full asymptotic expansion of prefactor [Helffer, Klein & Nier 04] Asymptotic distribution of τ + [Day 83, Bovier, Gayrard & Klein 05] lim P x {τ ε 0 + > t E x τ + } = e t

9 Metastability for the Ginzburg Landau equation SAMSI 10 February / 26 Ginzburg Landau equation t u(x, t) = xx u(x, t) + u(x, t) u(x, t) 3 + noise On finite interval x [0, L] u(x, t) R (one-dimensional, representing e.g. magnetization) Boundary conditions Periodic b.c. u(0, t) = u(l, t) and xu(0, t) = xu(l, t) Neumann b.c. with zero flux xu(0, t) = xu(l, t) = 0 Weak space time white noise Deterministic dynamics minimizes energy functional as V (u) = L 0 [ 1 2 u (x) u(x) u(x)4 ] dx xx u(x, t) + u(x, t) u(x, t) 3 = δv δu

10 Metastability for the Ginzburg Landau equation SAMSI 10 February / 26 Stationary states for the deterministic system d 2 dx 2 u(x) = u(x) + u(x)3 = d [ ] du Uniform stationary states u±(x) ±1 (stable; global minima of V ) u0(x) 0 (unstable when is u 0 a transition state?) Periodic b.c.: For k = 1, 2,... and L > 2πk Continuous one-parameter family of stationary states 2m u k,ϕ (x) = m + 1 sn( kx + ϕ, m ) where 4k m + 1K(m) = L m + 1 Neumann b.c.: For k = 1, 2,... and L > πk Two stationary states 2m u k,± (x) = ± m + 1 sn( kx + K(m), m ) where 2k m + 1K(m) = L m + 1

11 Metastability for the Ginzburg Landau equation SAMSI 10 February / 26 Stationary states: Neumann b.c. For k = 1, 2,... and L > πk: 2m u k,± (x) = ± m + 1 sn( kx + K(m), m ) m + 1 where 2k m + 1K(m) = L u ,+ u 2,+ u 3,+ u 4,+ 0 π 2π 3π L u0 0 u 4, u 3, u 2, u 1, u 1

12 Metastability for the Ginzburg Landau equation SAMSI 10 February / 26 Stability of the stationary states: Neumann b.c. Consider linearization of GL equation at stationary solution u : [0, L] R t v = A[u]v where A[u] = d2 dx u2 Stability is determined by the eigenvalues of A[u] u ± (x) ±1: A[u ± ] has eigenvalues (2 + (πk/l) 2 ), k = 0, 1, 2,... u 0 (x) 0: A[u 0 ] has eigenvalues 1 (πk/l) 2, k = 0, 1, 2,... Counting the number of positive eigenvalues: None for u ± and... 0 π 2π 3π L u u 4,± u 3,± u 2,± u 1,±

13 Metastability for the Ginzburg Landau equation SAMSI 10 February / 26 Stability of the stationary states: Neumann b.c. For L < π: u±(x) ±1 are stable; global minima u0(x) 0 is unstable; transition state Activation energy V (u0) V (u ±) = L/4 For L > π: u±(x) ±1 remain stable; global minima u0(x) 0 remains unstable; but no longer forms the transition state u1,±(x) are the new transition states (of instanton shape) Pitchfork bifurcation as L increases through π: Uniform transition state u 0 bifurcates into pair of instanton states u 1,± Subsequent bifurcations at L = kπ for k = 2, 3,... do not affect transition states

14 Metastability for the Ginzburg Landau equation SAMSI 10 February / 26 Ginzburg Landau equation with noise t u(x, t) = xx u(x, t) + u(x, t) u(x, t) 3 + 2εξ(t, x) u(, 0) = ϕ( ) x u(0, t) = x u(l, t) = 0 (Neumann b.c.) Space time white noise ξ(t, x) as formal derivative of Brownian sheet Mild / evolution formulation, following [Walsh 86]: u(x, t) = L 0 + 2ε t G t (x, z)ϕ(z) dz + t L L G t s (x, z)w (ds, dz) 0 G t s (x, z) [ u(s, z) u(s, z) 3] dz ds where G is the fundamental solution of the deterministic equation W is the Brownian sheet Existence and a.s. uniqueness [Faris & Jona-Lasinio 82]

15 Metastability for the Ginzburg Landau equation SAMSI 10 February / 26 Question How long does a noise-induced transition from the global minimum u (x) 1 to (a neighbourhood of) u + (x) 1 take? τ u+ = first hitting time of such a neighbourhood Metastability: We expect E u τ u+ e const/ε We seek Activation energy W Transition rate prefactor Γ 1 0 Exponent α of error term such that E u τ u+ = Γ 1 0 e W /ε [1 + O(ε α )]

16 Metastability for the Ginzburg Landau equation SAMSI 10 February / 26 Large deviations for the Ginzburg Landau equation Large deviation principle [Faris & Jona Lasinio 82]: For L π: For L > π: W = V (u 1,± ) V (u ) = W = V (u 0 ) V (u ) = L/4 1 [ 3 8E(m) 1 + m (1 m)(3m + 5) ] K(m) 1 + m

17 Metastability for the Ginzburg Landau equation SAMSI 10 February / 26 Formal computation of the prefactor for the GL equation Consider L < π Transition state: u 0 (x) 0, V [u 0 ] = 0 Activation energy: W = V [u 0 ] V [u ] = L/4 Eigenvalues at stable state u (x) 1: µ k = 2 + (πk/l) 2 Eigenvalues at transition state u 0 0: λ k = 1 + (πk/l) 2 Thus formally [Maier & Stein 01, 03] Γ 0 λ 0 µ k 2π λ k = 1 sinh( 2L) 2 3/4 π sin L k=0 For L > π: Spectral determinant computed by Gelfand s method Problems What happens when L π? (Approaching bifurcation) Is the formal computation correct in infinite dimension?

18 Metastability for the Ginzburg Landau equation SAMSI 10 February / 26 Ginzburg Landau equation: Introducing Fourier variables Fourier series u(x, t) = 1 y 0 (t) + 2 y k (t) cos(πkx/l) = 1 ỹ k (t) e i kπx/l L L L k=1 k Z Rewrite energy functional V in Fourier variables V (y) = 1 λ k yk 2 + V 4 (y), λ k = 1 + (πk/l) 2 2 where k=0 Resulting system of SDEs V 4 (y) = 1 4L ẏ k = λ k y k 1 L with i.i.d. Brownian motions W (k) t k 1+k 2+k 3+k 4=0 k 1+k 2+k 3=k ỹ k1 ỹ k2 ỹ k3 ỹ k4 ỹ k1 ỹ k2 ỹ k3 + 2εẆ (k) t

19 Metastability for the Ginzburg Landau equation SAMSI 10 February / 26 Truncating the Fourier series Truncate Fourier series (projected equation) u d (x, t) = 1 L y 0 (t) + 2 L d y k (t) cos(πkx/l) k=1 Retain only modes k d in the energy functional V where V (d) (y) = 1 2 V (d) 4 (y) = 1 4L d k=0 λ k y 2 k + V (d) 4 (y) k 1+k 2+k 3+k 4=0 k i { d,...,0,...,+d} Resulting d-dimensional system of SDEs ẏ k = λ k y k 1 L k 1+k 2+k 3=k k i { d,...,0,...,+d} ỹ k1 ỹ k2 ỹ k3 ỹ k4 ỹ k1 ỹ k2 ỹ k3 + 2εẆ (k) t

20 Metastability for the Ginzburg Landau equation SAMSI 10 February / 26 Reduction to finite-dimensional system Show the following result for the projected finite-dimensional systems ε γ C(d) e W (d) /ε [1 R d (ε)] E τ u (d) (d) u ε γ C(d) e W (d) /ε [1 + R + d (ε)] + (The contribution ε γ is only present at bifurcation points / non-quadratic saddles) The following limits exist and are finite lim C(d) =: C( ) and lim W (d) =: W ( ) d d Important: Uniform control of error terms (uniform in d): R ± (ε) := sup R ± d (ε) 0 as ε 0 d Away from bifurcation points, c.f. [Barret, Bovier & Méleard 09]

21 Metastability for the Ginzburg Landau equation SAMSI 10 February / 26 Taking the limit d For any ε, distance between u(x, t) and solution u (d) (x, t) of the projected equation becomes small [Liu 03] on any finite time interval [0, T ] Uniform error bounds and large deviation results allow to decouple limits of small ε and large d Yielding ε γ C( ) e W ( ) /ε [1 R (ε)] E u τ u+ ε γ C( ) e W ( ) /ε [1 + R + (ε)]

22 Metastability for the Ginzburg Landau equation SAMSI 10 February / 26 Result for the Ginzburg Landau equation Theorem [Barret, Berglund & G., in preparation] For the Ginzburg Landau equation with Neumann b.c., L < π (Similar expression for L > π) Γ 0 E u τ u+ = 1 Γ 0 (L) el/4ε [1+O((ε log ε ) 1/4 )] where the rate prefactor satisfies (recall: λ 1 = 1 + (π/l) 2 ) L/π Γ 0 (L) = 1 2 3/4 π sinh( 2L) sin L λ 1 λ 1 + 3ε/4L Ψ + Γ(1/4) sinh( 2π) ε 1/4 2(3π 7 ) 1/4 ( λ ) 1 3ε/4L as L π

23 Metastability for the Ginzburg Landau equation SAMSI 10 February / 26 Towards a proof in the finite-dimensional case: Potential theory for Brownian motion I First-hitting time τ A = inf{t > 0: B t A} of A R d Fact I: The expected first-hitting time w A (x) = E x τ A is a solution to the Dirichlet problem { w A (x) = 1 for x A c w A (x) = 0 for x A and can be expressed with the help of the Green function G A c (x, y) as w A (x) = G A c (x, y) dy A c

24 Metastability for the Ginzburg Landau equation SAMSI 10 February / 26 Potential theory for Brownian motion II The equilibrium potential (or capacitor) h A,B is a solution to the Dirichlet problem h A,B (x) = 0 for x (A B) c h A,B (x) = 1 for x A h A,B (x) = 0 for x B Fact II: h A,B (x) = P x [τ A < τ B ] The equilibrium measure (or surface charge density) is the unique measure ρ A,B on A s.t. h A,B (x) = G B c (x, y) ρ A,B (dy) A

25 Metastability for the Ginzburg Landau equation SAMSI 10 February / 26 Capacities Key observation: For a small ball C = B δ (x), h C,A (y) dy = G A c (y, z) ρ C,A (dz) dy A c A c C = w A (z) ρ C,A (dz) w A (x) cap C (A) C where cap C (A) = ρ C,A (dy) denotes the capacity C 1 E x τ A = w A (x) h Bδ (x),a(y) dy cap Bδ (x)(a) A c Variational representation via Dirichlet form cap C (A) = h C,A (x) 2 dx = inf h(x) 2 dx (C A) c h H C,A (C A) c where H C,A = set of sufficiently smooth functions h satisfying b.c.

26 Metastability for the Ginzburg Landau equation SAMSI 10 February / 26 General case dx ε t = V (x ε t ) dt + 2ε db t What changes as the generator is replaced by ε V? cap C (A) = ε inf h(x) 2 e V (x)/ε dx h H C,A (C A) c 1 E x τ A = w A (x) h Bδ (x),a(y) e V (y)/ε dy cap Bδ (x)(a) A c It remains to investigate capacity and integral. Assume, x = x is a quadratic minimum. Use rough a priori bounds on h Ac h Bδ(x ),A (y) e V (y)/ε dy (2πε)d/2 V (x e )/ε det 2 V (x )

27 Metastability for the Ginzburg Landau equation SAMSI 10 February / 26 For the truncated energy functional where V (d) (y) = 1 2 d k=0 Estimating the capacity λ k y 2 k + V (d) 4 (y) = 1 2 y u 1 (y 1 ) d λ k yk k=2 u 1 (y 1 ) = 1 2 λ 1y y 4 1 To show cap C (A) = ε e u1(y1)/ε dy 2 2πε d j=2 2πε λ j [ 1 + O(R(ε)) ] where R(ε) = O((ε log ε ) 1/4 ) is uniformly bounded in d

28 Metastability for the Ginzburg Landau equation SAMSI 10 February / 26 Sketch of the proof Proof follows along the lines of [Bovier, Eckhoff, Gayrard & Klein 04] Upper bound: Use Dirichlet form representation of capacity cap = inf Φ(h) Φ(h +) = Φ(h + ) = ε h + (y) 2 e V (y)/ε dy h Choose δ = cε log ε and 1 for y 0 < δ h + (z) = f (y 0 ) for δ < y 0 < δ 0 for y 0 > δ where εf (y 0 ) + y0 V (y 0, 0)f y 0 ) = 0 with b.c. f (±δ) = 0 or 1, resp. Lower bound: Bound Dirichlet form for capacity from below by restricting domain taking only 1st component of h using b.c. derived from a priori bound on hc,a