Overdamped Langevin dynamics The Langevin dynamics PDE methods for statistical physics Julien Roussel Cermics, ENPC Equipe-projet INRIA Matherials November 9, 2016, Cermics Julien Roussel Cermics, ENPC Equipe-projet INRIA Matherials PDE methods for statistical physics 1 / 19
Overdamped Langevin dynamics The Langevin dynamics The Overdamped Langevin dynamics N particles at positions q t = ( q 1 , · · · , q N ) ∈ D � 2 β − 1 d W t d q t = −∇ V ( q t ) d t + d W t : Brownian motion V ( q ) : potential, for example V ( q ) = � 1 ≤ i < j ≤ N v ( | q i − q j | ) β − 1 = k B T is fixed Julien Roussel Cermics, ENPC Equipe-projet INRIA Matherials PDE methods for statistical physics 2 / 19
Overdamped Langevin dynamics The Langevin dynamics Invariant measure Maxwell-Boltzmann distribution: ν ( d q ) = Z − 1 ν e − β V ( q ) d q (1) Unique invariant measure : � ∀ ϕ, E [ ϕ ( q t ) | q 0 ∼ ν ] = E ν [ ϕ ] = ϕ ( q ) d ν ( q ) (2) D Julien Roussel Cermics, ENPC Equipe-projet INRIA Matherials PDE methods for statistical physics 3 / 19
Overdamped Langevin dynamics The Langevin dynamics Generator of the dynamics Let ϕ and q ∈ D , the generator L is defined by ( L ϕ )( q ) := d d t E [ ϕ ( q t ) | q 0 = q ] (3) = β − 1 ∆ ϕ ( q ) − ∇ V · ∇ ϕ ( q ) using Itô calculus. Denoting � � ϕ, ψ � = (4) ϕ ψ d ν D the scalar product on L 2 ( ν ) and ∗ the associated adjoint � � − β − 1 ∇ ϕ · ∇ ψ d ν. � ϕ, L ψ � = ϕ L ψ d ν = (5) D D We denote L = − β − 1 ∇ ∗ ∇ , it is self-adjoint in L 2 ( ν ) . Julien Roussel Cermics, ENPC Equipe-projet INRIA Matherials PDE methods for statistical physics 4 / 19
Overdamped Langevin dynamics The Langevin dynamics Proof: Uniqueness of the invariant measure Let us prove that ν is the unique invariant measure. Let f a density of probability such that � ∀ ϕ, 0 = d d d t E [ ϕ ( q t ) | q 0 ∼ f ] = d t E [ ϕ ( q t ) | q 0 = q ] f ( q ) d q D � � � L ϕ, f = L ϕ ( q ) f ( q ) d q = ν D (6) then 0 = L ∗ � � � � f f = L ν ν � � �� = − β − 1 � � �� 2 � � f f f ie. 0 = ν , L � ∇ � ν ν so f = ν since they are normalized. Julien Roussel Cermics, ENPC Equipe-projet INRIA Matherials PDE methods for statistical physics 5 / 19
Overdamped Langevin dynamics The Langevin dynamics Convergence of empirical averages � t ϕ t = 1 Let � 0 ϕ ( q s ) d s the empirical mean of ϕ . t The dynamics can be shown to be ergodic ϕ t − − − → � t →∞ E ν [ ϕ ] a.s. (7) and Central Limit Theorem : � � √ L 0 , σ 2 t ( � ϕ t − E ν [ ϕ ]) − t →∞ N − − → (8) ϕ Julien Roussel Cermics, ENPC Equipe-projet INRIA Matherials PDE methods for statistical physics 6 / 19
Overdamped Langevin dynamics The Langevin dynamics Convergence of empirical averages � t ϕ t = 1 Let � 0 ϕ ( q s ) d s the empirical mean of ϕ . t The dynamics can be shown to be ergodic ϕ t − � t →∞ E ν [ ϕ ] a.s. − − → (7) and Central Limit Theorem : � � √ L 0 , σ 2 t ( � ϕ t − E ν [ ϕ ]) − t →∞ N − − → (8) ϕ But what is σ ϕ ? Julien Roussel Cermics, ENPC Equipe-projet INRIA Matherials PDE methods for statistical physics 6 / 19
Overdamped Langevin dynamics The Langevin dynamics Calculus: Asymptotic variance Suppose E ν [ ϕ ] = 0, then the asymptotical variance of ϕ is σ 2 ϕ t 2 ] . ϕ = lim t →∞ t E [ � (9) where � t � t � t � � 2 ] = 1 1 − s E [ ϕ ( q s ) ϕ ( q s ′ )] d s d s ′ = 2 t E [ � ϕ t E [ ϕ ( q 0 ) ϕ ( q s )] d s t t 0 0 0 � ∞ � t E [ ϕ ( q 0 ) e s L ϕ ( q 0 )] d s − t →∞ 2 − − → E [ ϕ ( q 0 ) ϕ ( q s )] d s = 2 0 0 � t � � ϕ ( q 0 ) , e s L ϕ ( q 0 ) = 2 d s 0 so � � σ 2 ϕ, L − 1 ϕ ϕ = − 2 (10) . Julien Roussel Cermics, ENPC Equipe-projet INRIA Matherials PDE methods for statistical physics 7 / 19
Overdamped Langevin dynamics The Langevin dynamics Application 1.6 2 V(q) q 1.4 1.5 nu(q) mean of q 95% 1 1.2 energy saddle 0.5 1 Position q 0 0.8 -0.5 0.6 energy well energy well -1 0.4 -1.5 0.2 -2 0 20 40 60 80 100 0 -1.5 -1 -0.5 0 0.5 1 1.5 Time t 2 q 1.5 mean of q 95% 1 0.5 Position q 0 -0.5 -1 -1.5 -2 0 200 400 600 800 1000 Time t Julien Roussel Cermics, ENPC Equipe-projet INRIA Matherials PDE methods for statistical physics 8 / 19
Overdamped Langevin dynamics The Langevin dynamics Coercivity of the generator Decompose the function space into an orthogonal direct sum � � L 2 ( ν ) = L 2 with L 2 ϕ ∈ L 2 ( ν ) | E ν [ ϕ ] = 0 0 ( ν ) ⊕ R 1 , 0 ( ν ) = (11) Reminder : L ( 1 ) = 0 Poincaré-Wirtinger for ϕ ∈ L 2 0 ( ν ) : 1 − � ϕ, L ϕ � = β − 1 �∇ ϕ � 2 ≥ C β � ϕ � 2 (12) −L is coercive so invertible on L 2 0 ( ν ) . Therefore σ 2 ϕ ≤ C β � ϕ � 2 . Julien Roussel Cermics, ENPC Equipe-projet INRIA Matherials PDE methods for statistical physics 9 / 19
Overdamped Langevin dynamics The Langevin dynamics Exponential decay and inversibility Take ϕ 0 ∈ L 2 0 ( µ ) and ϕ ( t ) = e t L ϕ 0 , and define H ( t ) = 1 2 � ϕ ( t ) � 2 H ′ ( t ) = �L ϕ ( t ) , ϕ ( t ) � ≤ − 1 C β � ϕ ( t ) � 2 = − 2 C β H ( t ) . (13) By Gronwall H ( t ) ≤ e − 2 t / C β H ( 0 ) 0 ( µ )) ≤ e − t / C β , � e t L � B ( L 2 ie. (14) so on L 2 0 ( µ ) � ∞ L − 1 = − e t L �L − 1 � B ( L 2 and 0 ( µ )) ≤ C β. (15) 0 Julien Roussel Cermics, ENPC Equipe-projet INRIA Matherials PDE methods for statistical physics 10 / 19
Overdamped Langevin dynamics The Langevin dynamics The Langevin dynamics � positions q t = ( q 1 , · · · , q N ) ∈ D N particles at momenta p t = ( p 1 , · · · , p N ) ∈ R D � d q t = p t d t � 2 γβ − 1 d W t = −∇ V ( q t ) d t − γ p t d t + d p t d W t : Brownian motion V ( q ) : potential, for example V ( q ) = � 1 ≤ i < j ≤ N v ( | q i − q j | ) β − 1 = k B T is fixed γ friction fixed Julien Roussel Cermics, ENPC Equipe-projet INRIA Matherials PDE methods for statistical physics 11 / 19
Overdamped Langevin dynamics The Langevin dynamics Invariant measure Maxwell-Boltzmann distribution µ ( d q , d p ) = Z − 1 µ e − β H ( q , p ) d q is the unique invariant measure µ ( d q , d p ) = ν ( d q ) κ ( d p ) . Hamiltonian : H ( q , p ) = V ( q ) + 1 2 | p | 2 (16) Limit γ → 0 : Hamiltonian system Limit γ → ∞ + time scaling : Overdamped Julien Roussel Cermics, ENPC Equipe-projet INRIA Matherials PDE methods for statistical physics 12 / 19
Overdamped Langevin dynamics The Langevin dynamics Generator of the dynamics Let ϕ and q ∈ D , the generator L is defined by ( L ϕ )( q ) := d d t E [ ϕ ( q t ) | q 0 = q ] (17) = p ∇ q − ∇ V ⊤ ∇ p − γβ − 1 ∇ ∗ p ∇ p = L ham + γ L FD using Itô calculus. Antisymetric transport part L ∗ ham = −L ham Symmetric fluctuation-dissipation part L FD ∗ = L FD Julien Roussel Cermics, ENPC Equipe-projet INRIA Matherials PDE methods for statistical physics 13 / 19
Overdamped Langevin dynamics The Langevin dynamics Kernel of the generator Let ϕ such that L ϕ = 0. Then 0 = � ϕ, L ϕ � = � ϕ, L FD ϕ � = − β − 1 �∇ p ϕ � 2 ⇒ ϕ = ϕ ( q ) (18) so 0 = L ϕ = p ⊤ ∇ q ϕ ⇒ ϕ ∈ R 1 . Therefore Ker ( L ) = Ker ( L ∗ ) = R 1 . But not coercive � ϕ, −L ϕ � = β − 1 �∇ p ϕ � 2 � � � ϕ � 2 (19) Julien Roussel Cermics, ENPC Equipe-projet INRIA Matherials PDE methods for statistical physics 14 / 19
Overdamped Langevin dynamics The Langevin dynamics Kernel of the generator Let ϕ such that L ϕ = 0. Then 0 = � ϕ, L ϕ � = � ϕ, L FD ϕ � = − β − 1 �∇ p ϕ � 2 ⇒ ϕ = ϕ ( q ) (18) so 0 = L ϕ = p ⊤ ∇ q ϕ ⇒ ϕ ∈ R 1 . Therefore Ker ( L ) = Ker ( L ∗ ) = R 1 . But not coercive � ϕ, −L ϕ � = β − 1 �∇ p ϕ � 2 � � � ϕ � 2 (19) So is it invertible ? Julien Roussel Cermics, ENPC Equipe-projet INRIA Matherials PDE methods for statistical physics 14 / 19
Overdamped Langevin dynamics The Langevin dynamics Hypocoercivity It suffices to show ∀ ϕ ∈ L 2 0 ( µ ) , ∀ t ≥ 0 , � e t L ϕ � ≤ C e − α t � ϕ � (20) to have on L 2 0 ( µ ) � ∞ �L − 1 � ≤ C L − 1 = − e t L d t , and α . (21) 0 Julien Roussel Cermics, ENPC Equipe-projet INRIA Matherials PDE methods for statistical physics 15 / 19
Overdamped Langevin dynamics The Langevin dynamics Proof: Hypocoercivity 2 � ϕ � 2 − ε � ϕ, A ϕ � where Define H [ ϕ ] = 1 � A = ( 1 + ( L ham Π p ) ∗ ( L ham Π p )) − 1 ( L ham Π p ) ∗ (22) � Π p ϕ = R D ϕ ( q , p ) d κ ( p ) Check � · � 2 ∼ H [ · ] d t H [ e t L ϕ ] ≤ α ′ � ϕ � 2 ≤ 2 α ′ d Prove 1 + ε H [ ϕ ] Conclude using the Gronwall lemma and the norm equivalence Julien Roussel Cermics, ENPC Equipe-projet INRIA Matherials PDE methods for statistical physics 16 / 19
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