Mean Field Limits for Coulomb Type Dynamics Sylvia Serfaty Courant - PowerPoint PPT Presentation

Mean Field Limits for Coulomb Type Dynamics Sylvia Serfaty Courant Institute, NYU In celebration of Alessio Figalli, Fields Institute, October 20, 2020

The discrete coupled ODE system Consider H N ( x 1 , . . . , x N ) = 1 � x i ∈ R d w ( x i − x j ) , 2 1 ≤ i � = j ≤ N w ( x ) = − log | x | d = 1 , 2 log case 1 w ( x ) = max ( d − 2 , 0 ) ≤ s < d Riesz case | x | s Evolution equation   x i = − 1 � ˙  ∇ i H N ( x 1 , . . . , x N ) + v ( x i − x j ) gradient flow  N j � = i x i = − 1 ( J T = − J ) ˙ N J ∇ i H N ( x 1 , . . . , x N ) conservative flow x i = − 1 ¨ N ∇ i H N ( x 1 , . . . , x N ) Newton’s law √ θ dW t possibly with added noise i , N independent Brownian motions, θ =temperature, v smooth force.

The discrete coupled ODE system Consider H N ( x 1 , . . . , x N ) = 1 � x i ∈ R d w ( x i − x j ) , 2 1 ≤ i � = j ≤ N w ( x ) = − log | x | d = 1 , 2 log case 1 w ( x ) = max ( d − 2 , 0 ) ≤ s < d Riesz case | x | s Evolution equation   x i = − 1 � ˙  ∇ i H N ( x 1 , . . . , x N ) + v ( x i − x j ) gradient flow  N j � = i x i = − 1 ( J T = − J ) ˙ N J ∇ i H N ( x 1 , . . . , x N ) conservative flow x i = − 1 ¨ N ∇ i H N ( x 1 , . . . , x N ) Newton’s law √ θ dW t possibly with added noise i , N independent Brownian motions, θ =temperature, v smooth force.

Questions For a general system √ x i = 1 � θ dW t ˙ K ( x i − x j ) + i N j � = i ◮ What is the limit of the empirical measure? Is there µ t such that for each t N 1 � i ⇀ µ t δ x t (1) N i = 1 ◮ if f 0 N ( x 1 , . . . , x N ) is the probability density of position of the system at time 0, what is the limit behavior of f t N ? ◮ propagation of chaos (Boltzmann, Kac, Dobrushin): if f 0 N ( x 1 , . . . , x N ) ≃ µ 0 ( x 1 ) . . . µ 0 ( x N ) is it true that f t N ( x 1 , . . . , x N ) ≃ µ t ( x 1 ) . . . µ t ( x N )? in the sense of convergence of the k -point marginal f N , k . ◮ In fact (1) is equivalent to propagation of chaos. Check : ϕ ∈ C ∞ c � � N � � 1 � � ( f N , 1 − µ t ) ϕ = f N ( x 1 , . . . , x N ) ϕ ( x i ) − µϕ dx 1 . . . dx N N i = 1

Questions For a general system √ x i = 1 � θ dW t ˙ K ( x i − x j ) + i N j � = i ◮ What is the limit of the empirical measure? Is there µ t such that for each t N 1 � i ⇀ µ t δ x t (1) N i = 1 ◮ if f 0 N ( x 1 , . . . , x N ) is the probability density of position of the system at time 0, what is the limit behavior of f t N ? ◮ propagation of chaos (Boltzmann, Kac, Dobrushin): if f 0 N ( x 1 , . . . , x N ) ≃ µ 0 ( x 1 ) . . . µ 0 ( x N ) is it true that f t N ( x 1 , . . . , x N ) ≃ µ t ( x 1 ) . . . µ t ( x N )? in the sense of convergence of the k -point marginal f N , k . ◮ In fact (1) is equivalent to propagation of chaos. Check : ϕ ∈ C ∞ c � � N � � 1 � � ( f N , 1 − µ t ) ϕ = f N ( x 1 , . . . , x N ) ϕ ( x i ) − µϕ dx 1 . . . dx N N i = 1

Questions For a general system √ x i = 1 � θ dW t ˙ K ( x i − x j ) + i N j � = i ◮ What is the limit of the empirical measure? Is there µ t such that for each t N 1 � i ⇀ µ t δ x t (1) N i = 1 ◮ if f 0 N ( x 1 , . . . , x N ) is the probability density of position of the system at time 0, what is the limit behavior of f t N ? ◮ propagation of chaos (Boltzmann, Kac, Dobrushin): if f 0 N ( x 1 , . . . , x N ) ≃ µ 0 ( x 1 ) . . . µ 0 ( x N ) is it true that f t N ( x 1 , . . . , x N ) ≃ µ t ( x 1 ) . . . µ t ( x N )? in the sense of convergence of the k -point marginal f N , k . ◮ In fact (1) is equivalent to propagation of chaos. Check : ϕ ∈ C ∞ c � � N � � 1 � � ( f N , 1 − µ t ) ϕ = f N ( x 1 , . . . , x N ) ϕ ( x i ) − µϕ dx 1 . . . dx N N i = 1

Questions For a general system √ x i = 1 � θ dW t ˙ K ( x i − x j ) + i N j � = i ◮ What is the limit of the empirical measure? Is there µ t such that for each t N 1 � i ⇀ µ t δ x t (1) N i = 1 ◮ if f 0 N ( x 1 , . . . , x N ) is the probability density of position of the system at time 0, what is the limit behavior of f t N ? ◮ propagation of chaos (Boltzmann, Kac, Dobrushin): if f 0 N ( x 1 , . . . , x N ) ≃ µ 0 ( x 1 ) . . . µ 0 ( x N ) is it true that f t N ( x 1 , . . . , x N ) ≃ µ t ( x 1 ) . . . µ t ( x N )? in the sense of convergence of the k -point marginal f N , k . ◮ In fact (1) is equivalent to propagation of chaos. Check : ϕ ∈ C ∞ c � � N � � 1 � � ( f N , 1 − µ t ) ϕ = f N ( x 1 , . . . , x N ) ϕ ( x i ) − µϕ dx 1 . . . dx N N i = 1

Questions For a general system √ x i = 1 � θ dW t ˙ K ( x i − x j ) + i N j � = i ◮ What is the limit of the empirical measure? Is there µ t such that for each t N 1 � i ⇀ µ t δ x t (1) N i = 1 ◮ if f 0 N ( x 1 , . . . , x N ) is the probability density of position of the system at time 0, what is the limit behavior of f t N ? ◮ propagation of chaos (Boltzmann, Kac, Dobrushin): if f 0 N ( x 1 , . . . , x N ) ≃ µ 0 ( x 1 ) . . . µ 0 ( x N ) is it true that f t N ( x 1 , . . . , x N ) ≃ µ t ( x 1 ) . . . µ t ( x N )? in the sense of convergence of the k -point marginal f N , k . ◮ In fact (1) is equivalent to propagation of chaos. Check : ϕ ∈ C ∞ c � � N � � 1 � � ( f N , 1 − µ t ) ϕ = f N ( x 1 , . . . , x N ) ϕ ( x i ) − µϕ dx 1 . . . dx N N i = 1

Formal limit Use ∂ t δ x ( t ) + div ( ˙ x δ x ( t ) ) = 0 or Liouville equation + BBGKY hierarchy N � N � 1 � � ∂ t f N + ∇ x i f N K ( x i − x j ) = 0 N i = 1 i = 1 i ⇀ µ t where µ t solves the � N We formally expect µ t N := 1 i = 1 δ x t N mean-field equation ∂ t µ = div (( K ∗ µ ) µ ) + 1 2 θ ∆ µ or in the second order case the Vlasov equation ∂ t ρ + v · ∇ x ρ + ( K ∗ µ ) · ∇ v ρ + 1 � 2 θ ∆ ρ = 0 µ = R d ρ ( x , v ) dv

How to prove convergence? ◮ Classical method ([Sznitman]...): compare true trajectories of points to trajectories following characteristics for the limit equation, show they remain close. OK for K Lipschitz. If K not regular, try to control the minimal distance between points in order to control K ( x i − x j ) [Hauray-Jabin] ◮ Find a good metric, typically Wasserstein W 1 such that ∂ t W 1 ( µ 1 ( t ) , µ 2 ( t )) ≤ CW 1 ( µ 1 ( t ) , µ 2 ( t )) for two solutions of the mean-field evolution. Apply to µ t N and µ t . [Braun-Hepp, Dobrushin, Neunzert-Wick] ◮ Use a relative entropy method : show a Gronwall relation for 0 ≤ H N ( f N | ρ ⊗ N ) := 1 � f N log f N ρ ⊗ N dx 1 . . . dx N . N [Jabin-Wang ’16] for θ > 0, K not too irregular.

How to prove convergence? ◮ Classical method ([Sznitman]...): compare true trajectories of points to trajectories following characteristics for the limit equation, show they remain close. OK for K Lipschitz. If K not regular, try to control the minimal distance between points in order to control K ( x i − x j ) [Hauray-Jabin] ◮ Find a good metric, typically Wasserstein W 1 such that ∂ t W 1 ( µ 1 ( t ) , µ 2 ( t )) ≤ CW 1 ( µ 1 ( t ) , µ 2 ( t )) for two solutions of the mean-field evolution. Apply to µ t N and µ t . [Braun-Hepp, Dobrushin, Neunzert-Wick] ◮ Use a relative entropy method : show a Gronwall relation for 0 ≤ H N ( f N | ρ ⊗ N ) := 1 � f N log f N ρ ⊗ N dx 1 . . . dx N . N [Jabin-Wang ’16] for θ > 0, K not too irregular.

How to prove convergence? ◮ Classical method ([Sznitman]...): compare true trajectories of points to trajectories following characteristics for the limit equation, show they remain close. OK for K Lipschitz. If K not regular, try to control the minimal distance between points in order to control K ( x i − x j ) [Hauray-Jabin] ◮ Find a good metric, typically Wasserstein W 1 such that ∂ t W 1 ( µ 1 ( t ) , µ 2 ( t )) ≤ CW 1 ( µ 1 ( t ) , µ 2 ( t )) for two solutions of the mean-field evolution. Apply to µ t N and µ t . [Braun-Hepp, Dobrushin, Neunzert-Wick] ◮ Use a relative entropy method : show a Gronwall relation for 0 ≤ H N ( f N | ρ ⊗ N ) := 1 � f N log f N ρ ⊗ N dx 1 . . . dx N . N [Jabin-Wang ’16] for θ > 0, K not too irregular.

Specialization to the Coulomb or Riesz interaction Limiting equations ∂ t µ = div ( ∇ ( w ∗ µ ) µ ) ( MFD ) ∂ t µ = div ( J ∇ ( w ∗ µ ) µ ) ( MFC ) or Vlasov-Poisson � ∂ t ρ + v · ∇ x ρ + ( ∇ w ∗ µ ) · ∇ v ρ = 0 µ = R d ρ ( x , v ) dv

Previous results ◮ d = 2 log, point vortex system → 2D incompressible Euler in vorticity form [Goodman-Hou-Lowengrub ’90, Schochet ’96] with noise → 2D Navier-Stokes [Osada ’87, Fournier-Hauray-Mischler ’14] ◮ [Hauray’ 09, Carrillo-Choi-Hauray ’14] ( s < d − 2) stability in Wasserstein W ∞ ◮ [Carrillo-Ferreira-Precioso ’12, Berman-Onnheim ’15] ( d = 1) Wasserstein gradient flow, use convexity of the interaction in 1D ◮ [Duerinckx ’15] ( d ≤ 2, s < 1) modulated energy method ◮ for convergence to Vlasov-Poisson [Hauray-Jabin ’15, Jabin-Wang ’17] s < d − 2, relative entropy method. Coulomb interaction (or more singular) remains open. ◮ [Boers-Pickl ’16, Lazarovici ’16, Lazarovici-Pickl ’17] with N -dependent cut-off of the interaction kernel