Metastability for interacting particles in double-well potentials and Allen Cahn SPDEs

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Nils Berglund nils.berglund@univ-orleans.fr http://www.

fr/mapmo/membres/berglund/ SPA 2017 Contributed

interacting particles in double-well potentials and

Orléans, France Moskva / Moscow, July 28 2017 Joint

1 Nils Berglund SPA 2017 Contributed Session: Interacting particle systems Metastability for interacting particles in double-well potentials and Allen Cahn SPDEs Nils Berglund MAPMO, Université d Orléans, France Moskva / Moscow, July Joint works with Giacomo Di Gesù (Vienna), Bastien Fernandez (Paris), Barbara Gentz (Bielefeld) and Hendrik Weber (Warwick)

2 Metastability for Allen Cahn SPDEs July 28, /12 Examples of particle systems Example 1 [B, Fernandez, Gentz, Nonlinearity 2007] N particles on a circle Z/NZ Bistable local dynamics Ferromagnetic nearest neighbour coupling Independent noise on each site x 6 x 7 x 5 x 0 x 1 x 2 x 3 x 4 dxt i =[xt i (xt) i 3 ] dt + γ i+1 [xt 2 2xt i + xt i 1 ] dt + 2ε dwt i

3 Metastability for Allen Cahn SPDEs July 28, /12 Examples of particle systems Example 1 [B, Fernandez, Gentz, Nonlinearity 2007] N particles on a circle Z/NZ Bistable local dynamics Ferromagnetic nearest neighbour coupling Independent noise on each site x 6 x 7 x 5 x 0 x 1 x 2 x 3 x 4 Gradient system dxt i =[xt i (xt) i 3 ] dt + γ i+1 [xt 2 potential V (x) = U(x i ) + γ i 4 i dx t = V (x t ) dt + 2ε dw t 2xt i + xt i 1 ] dt + 2ε dwt i (x i+1 x i ) 2 U(ξ) = 1 4 ξ4 1 2 ξ2

4 Metastability for Allen Cahn SPDEs July 28, /12 Examples of particle systems Example 1 [B, Fernandez, Gentz, Nonlinearity 2007] N particles on a circle Z/NZ Bistable local dynamics Ferromagnetic nearest neighbour coupling Independent noise on each site x 6 x 7 x 5 x 0 x 1 x 2 x 3 x 4 Gradient system dxt i =[xt i (xt) i 3 ] dt + γ i+1 [xt 2 potential V (x) = U(x i ) + γ i 4 i dx t = V (x t ) dt + 2ε dw t Example 2 Replace circle by 2D torus (Z/NZ) 2 2xt i + xt i 1 ] dt + 2ε dwt i (x i+1 x i ) 2 U(ξ) = 1 4 ξ4 1 2 ξ2 Example 3 [B, Dutercq, J Stat Phys 2016]: Same V + constraint i x i = 0

5 Metastability for Allen Cahn SPDEs July 28, /12 Coarsening dynamics with noise (Link to simulation)

6 General gradient systems with noise Metastability for Allen Cahn SPDEs July 28, /12 dx t = V (x t ) dt + 2ε dw t V R N R: confining potential, class C 2

7 Metastability for Allen Cahn SPDEs July 28, /12 General gradient systems with noise dx t = V (x t ) dt + 2ε dw t V R N R: confining potential, class C 2 Stationary points: X = {x V (x) = 0} Local minima: X 0 = {x X all ev of Hessian 2 V (x) are > 0} Saddles of Morse index 1: X 1 = {x X 2 V (x) has 1 negative ev }

8 Metastability for Allen Cahn SPDEs July 28, /12 General gradient systems with noise dx t = V (x t ) dt + 2ε dw t V R N R: confining potential, class C 2 Stationary points: X = {x V (x) = 0} Local minima: X 0 = {x X all ev of Hessian 2 V (x) are > 0} Saddles of Morse index 1: X 1 = {x X 2 V (x) has 1 negative ev } Dynamics markovian jump process on G = (X 0, E), E X 1 [Fre@idlin & Ventcel' / Freidlin & Wentzell 1970], [Bovier et al 2004]

9 Metastability for Allen Cahn SPDEs July 28, /12 Eyring Kramers law Definition: Communication height H(x i, x j ) = inf γ x i x j = V (z ij ) V (x i ) sup V (γ t ) V (xi ) t Definition: Metastable hierarchy x 1 x 2 x n θ > 0: k H k = H(x k, {x 1,..., x k 1 }) min i<k H(x i, {x 1,..., x i 1, x i+1,..., x k }) θ z2 = z 21 z3 = z 32 H 3 =H(x3, x 2 ) H 2 =H(x2, x 1 ) x3 x2 x 1

10 Eyring Kramers law Metastability for Allen Cahn SPDEs July 28, /12 Definition: Communication height H(x i, x j ) = inf γ x i x j = V (z ij ) V (x i ) sup V (γ t ) V (xi ) t Definition: Metastable hierarchy x 1 x 2 x n θ > 0: k H k = H(x k, {x 1,..., x k 1 }) min i<k H(x i, {x 1,..., x i 1, x i+1,..., x k }) θ z2 = z 21 z3 = z 32 H 3 =H(x3, x 2 ) H 2 =H(x2, x 1 ) x3 x2 x 1 Theorem: Eyring Kramers law [Bovier,Eckhoff,Gayrard,Klein 2004] τ k = first-hitting time of nbh of {x 1,..., x k } 2π E x k [τ k 1 ] = λ (zk ) λ k = k th ev of generator det 2 V (z k ) det 2 V (x k ) eh k/ε [1 + Oε(1)] λ k 1

11 Potential landscape for Example 1 Metastability for Allen Cahn SPDEs July 28, /12 V (x) = U(x i ) + γ (x i+1 x i ) 2 U(ξ) = 1 i Z/NZ 4 4 ξ4 1 2 ξ2 i Z/NZ γ = 0: X = { 1, 0, 1} N, X 0 = { 1, 1} N, X 1 = {x X one x i = 0} γ = 0: G = hypercube

12 Metastability for Allen Cahn SPDEs July 28, /12 Potential landscape for Example 1 V (x) = U(x i ) + γ (x i+1 x i ) 2 U(ξ) = 1 i Z/NZ 4 4 ξ4 1 2 ξ2 i Z/NZ γ = 0: X = { 1, 0, 1} N, X 0 = { 1, 1} N, X 1 = {x X one x i = 0} γ = 0: G = hypercube Theorem [BFG, Nonlinearity 2007] No bifurcation for 0 γ γ (N) where γ (N) > 1 4 N 2 V γ (zγ ) = V 0 (z0 ) + γ(# interfaces) + Ising-like dynamics

13 Metastability for Allen Cahn SPDEs July 28, /12 Potential landscape for Example 1 V (x) = U(x i ) + γ (x i+1 x i ) 2 U(ξ) = 1 i Z/NZ 4 4 ξ4 1 2 ξ2 i Z/NZ γ = 0: X = { 1, 0, 1} N, X 0 = { 1, 1} N, X 1 = {x X one x i = 0} γ = 0: G = hypercube Theorem [BFG, Nonlinearity 2007] No bifurcation for 0 γ γ (N) where γ (N) > 1 4 N 2 V γ (zγ ) = V 0 (z0 ) + γ(# interfaces) + Ising-like dynamics Theorem [BFG, Nonlinearity 2007] γ > 1 2 sin 2 (π/n) N2 X 0 = {±(1,..., 1)}, X 1 = {0} Synchronization

14 Metastability for Allen Cahn SPDEs July 28, /12 Limitations of the standard Eyring Kramers law Question 1: What happens when V is invariant under a group of symmetries? Representation theory of finite groups clustering of eigenvalues [B, Dutercq, JoTP 2015]

15 Metastability for Allen Cahn SPDEs July 28, /12 Limitations of the standard Eyring Kramers law Question 1: What happens when V is invariant under a group of symmetries? Representation theory of finite groups clustering of eigenvalues [B, Dutercq, JoTP 2015] Question 2: What happens when V has saddles with zero eigenvalues? (det 2 V (z ) = 0 at bifurcations) Eyring Kramers law with ε-dependent prefactor [B, Gentz, MPRF 2010]

16 Metastability for Allen Cahn SPDEs July 28, /12 Limitations of the standard Eyring Kramers law Question 1: What happens when V is invariant under a group of symmetries? Representation theory of finite groups clustering of eigenvalues [B, Dutercq, JoTP 2015] Question 2: What happens when V has saddles with zero eigenvalues? (det 2 V (z ) = 0 at bifurcations) Eyring Kramers law with ε-dependent prefactor [B, Gentz, MPRF 2010] Question 3: What happens when γ N 2 and N in Example 1? One expects convergence to Allen Cahn SPDE t u(t, x) = γ 2N 2 u(t, x) + u(t, x) u(t, x)3 + 2εξ(t, x) where ξ is space-time white noise Is there an Eyring Kramers law for such SPDEs?

17 Metastability for Allen Cahn SPDEs July 28, /12 Deterministic Allen Cahn PDE [Chafee & Infante 74, Allen & Cahn 75] t u(x, t) = xx u(x, t) + f (u(x, t)) x [0, L], L: bifurcation parameter u(x, t) R Either periodic or zero-flux Neumann boundary conditions In this talk: f (u) = u u 3 (results more general)

18 Deterministic Allen Cahn PDE Metastability for Allen Cahn SPDEs July 28, /12 [Chafee & Infante 74, Allen & Cahn 75] t u(x, t) = xx u(x, t) + f (u(x, t)) x [0, L], L: bifurcation parameter u(x, t) R Either periodic or zero-flux Neumann boundary conditions In this talk: f (u) = u u 3 (results more general) Energy function: L V [u] = [1 0 2 u (x) u(x) u(x)4 ] dx min Scaling limit of particle system of Example 1 with γ = 2 N2 L 2 Stationary solutions: u 0 (x) = u 0(x) + u 0 (x) 3 critical points of V Stability: Sturm Liouville problem t v t (x) = v t (x) + [1 3u 0 (x) 2 ]v t (x)

19 Metastability for Allen Cahn SPDEs July 28, /12 Stationary solutions u 0 (x) = f (u 0(x)) = u 0 (x) + u 0 (x) 3 u ± (x) ±1 u 0 (x) 0 Nonconstant solutions satisfying b.c. (expressible in terms of Jacobi elliptic fcts) H = 1 2 (u ) u2 1 4 u4 1 1 u H = 1 4 u

20 Metastability for Allen Cahn SPDEs July 28, /12 Stationary solutions u 0 (x) = f (u 0(x)) = u 0 (x) + u 0 (x) 3 u ± (x) ±1 u 0 (x) 0 Nonconstant solutions satisfying b.c. (expressible in terms of Jacobi elliptic fcts) H = 1 2 (u ) u2 1 4 u4 1 1 u H = 1 4 u Neumann b.c: 2k nonconstant solutions when L > kπ 0 Number of positive eigenvalues (= unstable directions) Transition state π 2π 3π 4π L Periodic b.c: k families when L > 2kπ

21 Eyring Kramers law for 1D SPDEs: heuristics Metastability for Allen Cahn SPDEs July 28, /12 u t (x) = u t (x) + f (u t (x)) + 2ε ξ(t, x) ( xx, f (u) = u u 3 ) Initial condition: u in near u 1 Target: u + 1, τ + = inf{t > 0 u t u + L < ρ} with eigenvalues ν k = ( βkπ L )2 + 2 Transition state: (β = 1 for Neumann b.c., β = 2 for periodic b.c.) u 0 (x) 0 if L βπ with ev λ k =( βkπ L u ts (x) = u 1 (x) β-kink stationary sol. if L > βπ with ev λ k )2 1

22 Eyring Kramers law for 1D SPDEs: heuristics Metastability for Allen Cahn SPDEs July 28, /12 u t (x) = u t (x) + f (u t (x)) + 2ε ξ(t, x) ( xx, f (u) = u u 3 ) Initial condition: u in near u 1 Target: u + 1, τ + = inf{t > 0 u t u + L < ρ} with eigenvalues ν k = ( βkπ L )2 + 2 Transition state: (β = 1 for Neumann b.c., β = 2 for periodic b.c.) u 0 (x) 0 if L βπ with ev λ k =( βkπ L u ts (x) = u 1 (x) β-kink stationary sol. if L > βπ with ev λ k [Faris & Jona-Lasinio 82]: large-deviation principle Arrhenius law: E u in (V [uts] V [u ])/ε [τ + ] e [Maier & Stein 01]: formal computation; for Neumann b.c. E u in [τ + ] 2π 1 λ 0 ν 0 λ k (V [uts] V [u ])/ε k=1 ν e k )2 1

23 Eyring Kramers law for 1D SPDEs: main result Metastability for Allen Cahn SPDEs July 28, /12 Theorem: Neumann b.c. [B & Gentz, Elec J Proba 2013] If L < π c with c > 0, then E u in [τ + ] = 2π 1 λ k e (V [uts] V [u ])/ε [1 + O(ε 1/2 log ε 3/2 )] λ 0 ν 0 k=1 ν k If L > π + c, then same formula with extra factor 1 2 (since 2 saddles) and λ k instead of λ k

24 Eyring Kramers law for 1D SPDEs: main result Metastability for Allen Cahn SPDEs July 28, /12 Theorem: Neumann b.c. [B & Gentz, Elec J Proba 2013] If L < π c with c > 0, then E u in [τ + ] = 2π 1 λ k e (V [uts] V [u ])/ε [1 + O(ε 1/2 log ε 3/2 )] λ 0 ν 0 k=1 ν k If L > π + c, then same formula with extra factor 1 2 (since 2 saddles) and λ k instead of λ k Results also for L near π and periodic b.c.

25 Eyring Kramers law for 1D SPDEs: main result Metastability for Allen Cahn SPDEs July 28, /12 Theorem: Neumann b.c. [B & Gentz, Elec J Proba 2013] If L < π c with c > 0, then E u in [τ + ] = 2π 1 λ k e (V [uts] V [u ])/ε [1 + O(ε 1/2 log ε 3/2 )] λ 0 ν 0 k=1 ν k If L > π + c, then same formula with extra factor 1 2 (since 2 saddles) and λ k instead of λ k Results also for L near π and periodic b.c. Prefactor involves a Fredholm determinant: λ k k=1 = det[( 1)( + 2) 1 ] = det[1 3( + 2) 1 ] ν k converges because ( + 2) 1 is trace class (limit = 2 sin(l) sinh( 2L) )

26 Eyring Kramers law for 1D SPDEs: main result Metastability for Allen Cahn SPDEs July 28, /12 Theorem: Neumann b.c. [B & Gentz, Elec J Proba 2013] If L < π c with c > 0, then E u in [τ + ] = 2π 1 λ k e (V [uts] V [u ])/ε [1 + O(ε 1/2 log ε 3/2 )] λ 0 ν 0 k=1 ν k If L > π + c, then same formula with extra factor 1 2 (since 2 saddles) and λ k instead of λ k Results also for L near π and periodic b.c. Prefactor involves a Fredholm determinant: λ k k=1 = det[( 1)( + 2) 1 ] = det[1 3( + 2) 1 ] ν k converges because ( + 2) 1 is trace class (limit = 2 sin(l) sinh( 2L) ) Proof uses spectral Galerkin approx. and potential-theoretic formula E ν A,B [τ B ] = 1 cap A (B) B c P y {τ A < τ B } e V (y)/ε dy

27 Metastability for Allen Cahn SPDEs July 28, /12 The two-dimensional case (Link to simulation)

28 Metastability for Allen Cahn SPDEs July 28, /12 The two-dimensional case Large-deviation principle: [Hairer & Weber, Ann. Fac. Sc. Toulouse, 2015] Naive computation of prefactor fails (( + 2) 1 is not trace class): log det[1 3( + 2) 1 ] log(1 3L2 k (N k 2 π ) ) k (N 2 0 ) 3L2 k 2 π 2 3L2 π 2 1 r dr r 2 =

29 Metastability for Allen Cahn SPDEs July 28, /12 The two-dimensional case Large-deviation principle: [Hairer & Weber, Ann. Fac. Sc. Toulouse, 2015] Naive computation of prefactor fails (( + 2) 1 is not trace class): log det[1 3( + 2) 1 ] log(1 3L2 k (N k 2 π ) ) k (N 2 0 ) 3L2 k 2 π 2 3L2 π 2 In fact, the equation needs to be renormalised 1 r dr r 2 = Theorem: [Da Prato & Debussche 2003] Let ξ δ be a mollification on scale δ of white noise. Then t u = u + [1 + 3εC(δ)]u u 3 + 2εξ δ with C(δ) = Tr(( P δ 1 ) 1 ) log(δ 1 ) admits local solution cv as δ 0 (Global version: [Mourrat & Weber 2015]) [Mourrat & Weber 2014]: Renormalised eq = scaling limit of Ising Kac model

30 Main result in dimension 2 Metastability for Allen Cahn SPDEs July 28, /12 Theorem: [B, Di Gesù, Weber, Elec J Proba 2016] For L < 2π, appropriate A u, B u +, µ N probability measures on A: lim sup E µ N [τ B ] 2π N λ 0 lim inf N Eµ N [τ B ] 2π λ 0 λ k k Z ν 2 k λ k k Z ν 2 k ν k λ k λ e k e (V [uts] V [u ])/ε [1 + c + ε ] ν k λ k e λ k e (V [uts] V [u ])/ε [1 c ε]

31 Main result in dimension 2 Metastability for Allen Cahn SPDEs July 28, /12 Theorem: [B, Di Gesù, Weber, Elec J Proba 2016] For L < 2π, appropriate A u, B u +, µ N probability measures on A: lim sup E µ N [τ B ] 2π N λ 0 lim inf N Eµ N [τ B ] 2π λ 0 λ k k Z ν 2 k λ k k Z ν 2 k ν k λ k λ e k e (V [uts] V [u ])/ε [1 + c + ε ] ν k λ k e λ k e (V [uts] V [u ])/ε [1 c ε] Prefactor involves Carleman Fredholm determinant: det 2 (Id + T ) = det(id + T )e Tr T Defined whenever T is only Hilbert Schmidt

32 Main result in dimension 2 Metastability for Allen Cahn SPDEs July 28, /12 Theorem: [B, Di Gesù, Weber, Elec J Proba 2016] For L < 2π, appropriate A u, B u +, µ N probability measures on A: lim sup E µ N [τ B ] 2π N λ 0 lim inf N Eµ N [τ B ] 2π λ 0 λ k k Z ν 2 k λ k k Z ν 2 k ν k λ k λ e k e (V [uts] V [u ])/ε [1 + c + ε ] ν k λ k e λ k e (V [uts] V [u ])/ε [1 c ε] Prefactor involves Carleman Fredholm determinant: det 2 (Id + T ) = det(id + T )e Tr T Defined whenever T is only Hilbert Schmidt Proof uses absolute continuity of e V /ε wrt Gaussian free field and Nelson argument for Wick powers: if X n th Wiener chaos then E[X 2p ] 1/2p C n (2p 1) n/2 E[X 2 ] 1/2

33 Metastability for Allen Cahn SPDEs July 28, /12 Outlook Dim d = 3 (in progress): more difficult because 2 renormalisation constants needed, no Nelson argument (e V /ε singular wrt GFF)

34 Metastability for Allen Cahn SPDEs July 28, /12 Outlook Dim d = 3 (in progress): more difficult because 2 renormalisation constants needed, no Nelson argument (e V /ε singular wrt GFF) References: Particle system: [1]; nonquadratic: [2,3]; SPDEs: [4,5]; symmetries: [6,7]; 1. N. B., Bastien Fernandez & Barbara Gentz, Metastability in interacting nonlinear stochastic differential equations I: From weak coupling to synchronisation & II: Large-N behaviour, Nonlinearity 20, ; (2007) 2. N. B. & Barbara Gentz, Anomalous behavior of the Kramers rate at bifurcations in classical field theories, J. Phys. A: Math. Theor. 42, (2009) 3., The Eyring Kramers law for potentials with nonquadratic saddles, Markov Processes Relat. Fields 16, (2010) 4., Sharp estimates for metastable lifetimes in parabolic SPDEs: Kramers law and beyond, Electronic J. Probability 18, (24):1 58 (2013) 5. N. B., Giacomo Di Gesù & Hendrik Weber, An Eyring Kramers law for the stochastic Allen Cahn equation in dimension two, Electronic J. Probability 22, (41):1 27 (2017) 6. N. B. & Sébastien Dutercq, The Eyring Kramers law for Markovian jump processes with symmetries, J. Theoretical Probability, 29, (4): (2016) 7., Interface dynamics of a metastable mass-conserving spatially extended diffusion, J. Statist. Phys. 162, (2016)

Sharp estimates on metastable lifetimes for one- and two-dimensional Allen Cahn SPDEs

Nils Berglund nils.berglund@univ-orleans.fr http://www.univ-orleans.fr/mapmo/membres/berglund/ COSMOS Workshop, École des Ponts ParisTech Sharp estimates on metastable lifetimes for one- and two-dimensional