Metastability in Stochastic Dynamics: Random-Field Curie-Weiss-Potts - PowerPoint PPT Presentation

05. September 2011 Martin Slowik Metastability in Stochastic Dynamics: Random-Field Curie-Weiss-Potts Model Prag summer school 1 (16)

Metastability in stochastic dynamics Metastability: A common phenomenon The paradigm. Related to the dynamics of first order phase transitions Change parameters quickly across the line of first order phase transition, the system reveals the existence of multiple time scales: Short time scales. ⊲ Existence of disjoint subsets M i , viewed as metastable sets/states ⊲ The system appears to be in a quasi-equilibrium within M i Larger time scales. ⊲ Rapid transitions between metastable sets occur induced by random fluctuations The goal. Understanding of quantitative aspects of dynamical phase transitions: ⊲ expected time of a transition from a metastable to a stable state ⊲ distribution of the exit time from a metastable state ⊲ small eigenvalues and corresponding eigenvectors of the generator Prag summer school 2 (16)

Metastability in stochastic dynamics Metastability: A common phenomenon The paradigm. Related to the dynamics of first order phase transitions Change parameters quickly across the line of first order phase transition, the system reveals the existence of multiple time scales: Short time scales. ⊲ Existence of disjoint subsets M i , viewed as metastable sets/states ⊲ The system appears to be in a quasi-equilibrium within M i Larger time scales. ⊲ Rapid transitions between metastable sets occur induced by random fluctuations The goal. Understanding of quantitative aspects of dynamical phase transitions: ⊲ expected time of a transition from a metastable to a stable state ⊲ distribution of the exit time from a metastable state ⊲ small eigenvalues and corresponding eigenvectors of the generator Prag summer school 2 (16)

Metastability in stochastic dynamics Stochastic spin models We are interested in studying the stochastic dynamics of (disordered) spin systems, i.e. Markov process with ⊲ State space S Λ = S Λ , where S finite set and e.g. Λ ⊂ Z d ⊲ Hamiltonian H Λ : S Λ → R µ Λ ,β ( σ ) = Z − 1 ⊲ Gibbs measure ` ´ Λ ,β exp − βH Λ ( σ ) ⊲ Transition rates p Λ ,β ( σ, η ) reversible with respect to µ Λ ,β and ”local”, i.e. essentially single site flips only. Prag summer school 3 (16)

Metastability in stochastic dynamics Well understood situations Low temperature limit. β → ∞ ⊲ metastable states correspond to local minima of H N ⊲ exit from metastable states occur through minimal saddle points of H N connecting one minimum to deeper ones, only a few path are relevant ⊲ the mean exit time of a metastable state is proportional to ` ´ exp β ( H N ( saddle ) − H N ( min )) ⊲ normalized metastable exit times are Exp (1) distributed Mean-field models. H N ( σ ) = E ( ̺ N ( σ )) for some mesoscopic variable ̺ N ⊲ an exact reduction to a low-dimensional model is possible; nearest-neighbor random walk in the free energy landscape, F N ⊲ metastable states correspond to local minima of F N ⊲ the mean exit time of a metastable state is proportional to ` ´ exp βN ( F N ( saddle ) − F N ( min )) ⊲ normalized metastable exit times are Exp (1) distributed Prag summer school 4 (16)

Metastability in stochastic dynamics Well understood situations Low temperature limit. β → ∞ ⊲ metastable states correspond to local minima of H N ⊲ exit from metastable states occur through minimal saddle points of H N connecting one minimum to deeper ones, only a few path are relevant ⊲ the mean exit time of a metastable state is proportional to ` ´ exp β ( H N ( saddle ) − H N ( min )) ⊲ normalized metastable exit times are Exp (1) distributed Mean-field models. H N ( σ ) = E ( ̺ N ( σ )) for some mesoscopic variable ̺ N ⊲ an exact reduction to a low-dimensional model is possible; nearest-neighbor random walk in the free energy landscape, F N ⊲ metastable states correspond to local minima of F N ⊲ the mean exit time of a metastable state is proportional to ` ´ exp βN ( F N ( saddle ) − F N ( min )) ⊲ normalized metastable exit times are Exp (1) distributed Prag summer school 4 (16)

Properties of the random field CWP model The random field Curie–Weiss–Potts Model and the dynamics Random Hamiltonian. N N q H N ( σ ) = − 1 X X X h i σ ∈ S N ≡ { 1 , . . . , q } N δ ( σ i , σ j ) − r δ ( σ i , r ) , N i,j =1 i =1 r =1 { h i } i ∈ N are i.i.d. random variables taking values in R q . 1 d H ( σ,η ) = 1 µ N ( σ ) = Z − 1 ` ´ q − N Gibbs measure. exp − βH N ( σ ) N Equilibrium properties. ⊲ J.M. Amaro de Matos, A.E. Patrick, V.A. Zagrebnov (JSP , 1992), C. Külske (JSP , 1997, 1998) ⊲ G. Iacobelli, C. Külske (JSP , 2010) Glauber dynamics. Discrete-time Markov chain { σ ( t ) } t ∈ N 0 on S N reversible w.r.t. µ N with Metropolis transition probabilities 1 ` ˆ ˜ ´ p N ( σ, η ) = qN exp − β H N ( η ) − H N ( σ ) + and p N ( σ, σ ) = P η p N ( σ, η ) . Prag summer school 5 (16)

Properties of the random field CWP model The random field Curie–Weiss–Potts Model and the dynamics Random Hamiltonian. N N q H N ( σ ) = − 1 X X X h i σ ∈ S N ≡ { 1 , . . . , q } N δ ( σ i , σ j ) − r δ ( σ i , r ) , N i,j =1 i =1 r =1 { h i } i ∈ N are i.i.d. random variables taking values in R q . 1 d H ( σ,η ) = 1 µ N ( σ ) = Z − 1 ` ´ q − N Gibbs measure. exp − βH N ( σ ) N Equilibrium properties. ⊲ J.M. Amaro de Matos, A.E. Patrick, V.A. Zagrebnov (JSP , 1992), C. Külske (JSP , 1997, 1998) ⊲ G. Iacobelli, C. Külske (JSP , 2010) Glauber dynamics. Discrete-time Markov chain { σ ( t ) } t ∈ N 0 on S N reversible w.r.t. µ N with Metropolis transition probabilities 1 ` ˆ ˜ ´ p N ( σ, η ) = qN exp − β H N ( η ) − H N ( σ ) + and p N ( σ, σ ) = P η p N ( σ, η ) . Prag summer school 5 (16)

Properties of the random field CWP model Coarse graining and mesoscopic approximation The entropic problem can be solved by passing on to Mesoscopic variables. k =1 e k ⊗ 1 X n ̺ n : S N → Γ n ⊂ R n · q , ̺ n ( σ ) = X i ∈ Λ k δ σ i N ⊲ {H k } n k =1 is a partition of support of the distribution of the random field, diam H k < ε ( n ) ¯ is a random partition of { 1 , . . . , N } ⊲ Λ k = ˘ i ∈ { 1 , . . . , N } | h i ∈ H k Q n = µ N ◦ ( ̺ n ) − 1 on the set Γ n Induced measure. △ ̺ n ` ˘ σ ( t ) ´¯ In general, t ∈ N 0 is not Markovian ! Strategy. Approximate the original dynamics by Markovian dynamics on Γ n which are reversible w.r.t. Q n with 1 r n ( x , y ) = X X µ N ( σ ) p N ( σ, η ) . Q n ( x ) σ ∈ ( ̺ n ) − 1 ( x ) η ∈ ( ̺ n ) − 1 ( y ) Prag summer school 6 (16)

Properties of the random field CWP model Coarse graining and mesoscopic approximation The entropic problem can be solved by passing on to Mesoscopic variables. k =1 e k ⊗ 1 X n ̺ n : S N → Γ n ⊂ R n · q , ̺ n ( σ ) = X i ∈ Λ k δ σ i N ⊲ {H k } n k =1 is a partition of support of the distribution of the random field, diam H k < ε ( n ) ¯ is a random partition of { 1 , . . . , N } ⊲ Λ k = ˘ i ∈ { 1 , . . . , N } | h i ∈ H k Q n = µ N ◦ ( ̺ n ) − 1 on the set Γ n Induced measure. △ ̺ n ` ˘ σ ( t ) ´¯ In general, t ∈ N 0 is not Markovian ! Strategy. Approximate the original dynamics by Markovian dynamics on Γ n which are reversible w.r.t. Q n with 1 r n ( x , y ) = X X µ N ( σ ) p N ( σ, η ) . Q n ( x ) σ ∈ ( ̺ n ) − 1 ( x ) η ∈ ( ̺ n ) − 1 ( y ) Prag summer school 6 (16)

Properties of the random field CWP model Mesoscopic free energy landscape Sharp large deviation estimates − Nβ F n ( x ) ` ´ ` ´ exp 1 + O N (1) Z N Q n ( x ) = , q − 1 q˛ ˛ det Q n ´˜˛ ˆ π k ∇ 2 U | Λ k | ` t ∗ ( x k /π k ) k =1 (2 πN ) 2 ˛ P n where π k = | Λ k | /N and F n ( x ) := E ( x ) + 1 k =1 π k I | Λ k | ( x k ) β Critical points. ⊲ Deterministic in the limit N → ∞ ⊲ explicit expression for F n ( x ) at critical points Prag summer school 7 (16)

Properties of the random field CWP model Main result Let m be a local minimum of F n and M the set of deeper local minima of F n . Theorem 1. Suppose z be a unique critical point of index 1 separating m from M and denote by A = ( ̺ n ) − 1 ( m ) and B = ( ̺ n ) − 1 ( M ) . Then, P h -a.s., s˛ ˛ det ´˛ ` I − 2 β ∇ 2 U N (2 βz ) = 2 πN ´ e βN ( F N ( z ) − F N ( m )) ` ˛ ˆ ˜ ´ τ B 1 + O N (1) E ν β | γ 1 | ` det I − 2 β ∇ 2 U N (2 βm ) where ν is a probability measure on A and 1 F N ( x ) = � x � 2 − X N “P q ´” 1 ` 2 βz r + βh i i =1 ln q exp r r =1 βN Previous and related work ⊲ F. den Hollander and P . dai Pra (JSP , 1996) large deviations, logarithmic asymptotics ⊲ P . Mathieu and P . Picco (JSP , 1998) Bernoulli distribution, up to polynomial errors ⊲ A. Bovier, M. Eckhoff, V. Gayrard and M. Klein (PTRL, 2001) discrete distribution, up to a multiplicative constant ⊲ A. Bianchi, A. Bovier and D. Ioffe (EJP , 2008) bounded continuous distribution, precise prefactor Prag summer school 8 (16)