Random-Field Curie-Weiss-Potts Model: From average to pointwise - PowerPoint PPT Presentation

10. February 2011 Martin Slowik Random-Field Curie-Weiss-Potts Model: From average to pointwise estimates of metastable times Warwick Statistical Mechanics Seminar 1 (21)

Metastability in stochastic dynamics Metastability: A common phenomenon The paradigm. Metastability is a phenomenon related to the dynamics of first order phase transitions. Changing the parameters of a system quickly across the line of a first order phase transition, the behaviour of the system reveals the existence of multiple, well separated time-scales: ⊲ Short time-scales. The system reaches quickly a quasi-equilibrium and remains for a long time in a confined subset of the phase space, called the metastable state. ⊲ Larger time-scales. Rapid transitions between metastable states occur which are influenced 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, Warwick Statistical Mechanics Seminar 2 (21)

Metastability in stochastic dynamics Stochastic spin models We are interested in studying the stochastic dynamics of (disordered) spin systems, i.e. Markov chain with S Λ = S Λ , where Λ ⊂ Z d and S finite set, ⊲ State space ⊲ Hamiltonian H Λ : S Λ → R , µ Λ ,β ( σ ) = Z − 1 ⊲ Gibbs measure ` ´ Λ ,β exp − βH Λ ( σ ) , ⊲ Order parameter e.g. ̺ Λ ( σ ) = 1 P i ∈ Λ δ σ i , | Λ | ⊲ Transition rates p Λ ,β ( σ, η ) reversible with respect to µ Λ ,β and ”local”, i.e. essentially single site flips only. Warwick Statistical Mechanics Seminar 3 (21)

Metastability in stochastic dynamics Well understood situations Low temperature limit. If β ↑ ∞ and the state space S N is finite, then ⊲ 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 )) . Mean-field models. If H N ( σ ) = E ( ̺ N ( σ )) for some mesoscopic variable ̺ N and ̺ N ( σ ( t )) is itself a Markov chain, then ⊲ an exact reduction to a low-dimensional model is possible; nearest-neighbor random walk in the free energy landscape. Warwick Statistical Mechanics Seminar 4 (21)

Properties of the random field CWP model The random field Curie–Weiss–Potts Model and the dynamics Random Hamiltonian. q N N H N ( σ ) = − 1 i δ ( σ i , r ) , X X X σ ∈ S N ≡ { 1 , . . . , q } N δ ( σ i , σ j ) − h r N i,j =1 i =1 r =1 { h i } i ∈ N are (bounded) i.i.d. random variables taking values in R q . 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. Consider a 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 ) = N ( q − 1) exp − β H N ( η ) − H N ( σ ) + if σ, η ∈ S N differ on a single coordinate, and zero else. Warwick Statistical Mechanics Seminar 5 (21)

Properties of the random field CWP model Free energy landscape N N ⊂ R q , 1 P Macroscopic variable. ̺ N : S N → Γ ̺ N ( σ ) = δ σ i N i =1 Q N = µ N ◦ ̺ − 1 Induced measure. on the set Γ N N Using sharp large deviation estimates, one gets in the limit when N ↑ ∞ ` ´ ` ´ Z N Q N ( x ) = exp − Nβ F N ( x ) 1 + O N (1) , q˛ q − 1 ˛ det ˆ ˜˛ (2 πN ) ∇ 2 U N ( t ∗ ) 2 ˛ where F N ( x ) := −� x � 2 + 1 β I N ( x ) and I N ( x ) is the Legendre–Fenchel transform of 1 1 X N X q β h i ` ´ U N ( t ) = i =1 ln q exp r + t r . N r =1 Critical points. Solutions of β (2 z ∗ ` l + h i ´ exp l ) 1 X N z ∗ = i =1 ln ´ , l = 1 , . . . , q. l P q ` N r =1 exp β (2 z ∗ r + h i r ) Moreover, at any critical point z ∗ one gets an explicit expression for F N ( z ∗ ) . Warwick Statistical Mechanics Seminar 6 (21)

Properties of the random field CWP model Main question Let m ∗ be a local minimum and consider the set of ’deeper’ local minima m | F N ( m ) ≤ F N ( m ∗ ) ˘ ¯ M = . We will be interested in the metastable exit time ˛ σ ( t ) ∈ S [ M ] , σ (0) ∈ S [ m ∗ ] ˛ τ S [ M ] = inf ˘ t > 0 ¯ , where S [ I ] := { σ ∈ S N | ̺ N ( σ ) ∈ I } . for all σ ∈ S [ m ∗ ] ? ˆ ˜ Question: What can we say about E σ τ S [ M ] Question: What can we say about the distribution of τ S [ M ] ? Warwick Statistical Mechanics Seminar 7 (21)

Properties of the random field CWP model Coarse graining The entropic problem can be solved by passing on to n Mesoscopic variables. ̺ n : S N → Γ n ⊂ R n · q , e k ⊗ ̺ n ( σ ) = 1 P P δ σ i N k =1 i ∈ Λ k ⊲ {H k } n k =1 is a partition of support of the distribution of the random field. i ∈ { 1 , . . . , N } | h i ∈ H k ⊲ Λ k = ˘ ¯ is a random partition of { 1 , . . . , N } . Q n = µ N ◦ ( ̺ n ) − 1 on the set Γ n Induced measure. This allows to rewrite the Hamiltonian as X n ̺ n ( σ ) X i ∈ Λ k � ˜ h i , e σ � ` ´ H N ( σ ) = − NE − k =1 ⊲ ˜ h i = h i − ¯ h k for i ∈ Λ k and ‚ ˜ ‚ ≤ c/n ≡ ε ( n ) ‚ h i ‚ Strategy. Approximate the original dynamics by effective mesoscopic dynamics on Γ n which are reversible w.r.t. Q n with 1 X X r N ( x , y ) = µ N ( σ ) p N ( σ, η ) . Q n ( x ) σ ∈S n [ x ] η ∈S n [ y ] Warwick Statistical Mechanics Seminar 8 (21)

Potential theoretic approach Potential theory Consider two disjoint sets A, B ⊂ S N and let L N = P N − I be the discrete generator. Equilibrium potential. Solution of the equation 8 ` ´ ` ´ L N h A,B ( σ ) = 0 , σ ∈ S N \ A ∪ B > < h A,B ( σ ) = 1 , σ ∈ A > h A,B ( σ ) = 0 , σ ∈ B : ` ´ Equilibrium measure. e A,B ( σ ) = − L N h A,B ( σ ) cap( A, B ) = P Capacity. σ ∈ B µ N ( σ ) e A,B ( σ ) Dirichlet form. E ( h ) = 1 ´ 2 X ` µ N ( σ ) p N ( σ, η ) h ( σ ) − h ( η ) 2 σ,η ∈S N First hitting probability. 8 h A,B ( σ ) , σ �∈ A ∪ B > < ˆ ˜ τ A < τ B = 1 − e A,B ( σ ) , σ ∈ A P σ > e B,A ( σ ) , σ ∈ B : Warwick Statistical Mechanics Seminar 9 (21)

Potential theoretic approach Probabilistic interpretation Mean hitting time. 1 X ˆ ˜ E ν A,B τ B = µ N ( σ ) h A,B ( σ ) , cap( A, B ) σ ∈S N where ν A,B is a measure on A ˆ ˜ µ N ( σ ) P τ B < τ A ν A,B ( σ ) = µ N ( σ ) e A,B ( σ ) σ = ˜ . cap( A, B ) ˆ P η ∈ A µ N ( σ ) P τ B < τ A η The full beauty. To obtain sharp estimates for the mean hitting time, we need: ⊲ precise control on capacities. ⊲ some rough bounds on the equilibrium potential. To obtain pointwise estimates it would be much simpler, if we were aware of a reasonable △ ! quantitative version of an elliptic Harnack inequality for such processes. Warwick Statistical Mechanics Seminar 10 (21)

Potential theoretic approach Probabilistic interpretation Mean hitting time. 1 X ˆ ˜ E ν A,B τ B = µ N ( σ ) h A,B ( σ ) , cap( A, B ) σ ∈S N where ν A,B is a measure on A ˆ ˜ µ N ( σ ) P τ B < τ A ν A,B ( σ ) = µ N ( σ ) e A,B ( σ ) σ = ˜ . cap( A, B ) ˆ P η ∈ A µ N ( σ ) P τ B < τ A η The full beauty. To obtain sharp estimates for the mean hitting time, we need: ⊲ precise control on capacities. ⊲ some rough bounds on the equilibrium potential. To obtain pointwise estimates it would be much simpler, if we were aware of a reasonable △ ! quantitative version of an elliptic Harnack inequality for such processes. Warwick Statistical Mechanics Seminar 10 (21)

Potential theoretic approach Computation of capacities Variational principles for capacities offers two convenient options for upper and lower bounds: Dirichlet principle. 1 ´ 2 X ` cap( A, B ) = inf µ ( σ ) p ( σ, η ) h ( σ ) − h ( η ) 2 h ∈H A,B σ,η H A,B is the space of functions with boundary constraints; minimizer harmonic function Berman-Konsowa principle. # − 1 " f ( σ, η ) E f X cap( A, B ) = sup µ ( σ ) p ( σ, η ) f ∈U A,B ( σ,η ) ∈X U A,B is the space of unit flows; maximizer harmonic flow. E f denotes the law of a directed Markov chain with transition probabilities proportional to the flow. Warwick Statistical Mechanics Seminar 11 (21)

Metastable exit times in the random-field CWP model Averaged result Theorem 1. Let m ∗ be a local minimum of F N and let z ∗ be the critical point of index 1 separating m ∗ from M . Then, P h -a.s., = C ( β, q, m ∗ , z ∗ ) N e βN ( F N ( z ∗ ) − F N ( m ∗ )) ` ˆ ˜ ´ E ν τ S [ M ] 1 + O N (1) where ν is a probability measure on S [ m ∗ ] and C ( β, q, m ∗ , z ∗ ) is the prefactor (explicit formulae). 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 Warwick Statistical Mechanics Seminar 12 (21)