Langevin dynamics with space-time periodic nonequilibrium forcing

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Langevin dynamics with

space-time periodic nonequilibrium forcing

Gabriel STOLTZ [email protected]

(CERMICS, Ecole des Ponts & MATHERIALS team, INRIA Rocquencourt)

“Progress in Nonequilibrium Statistical Physics”, Nice, 12 June 2015

Gabriel Stoltz (ENPC/INRIA) Nice, June 2015 1 / 22

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Definition of the dynamics

• Periodic boundary conditions: positionq ∈ M= (LT)^d

• Nonequilibrium Langevin dynamics (M ∈R^d×d positive definite,γ >0)









dq^η_t =M⁻¹p^η_t dt, dp^η_t =

− ∇V(q_t^η) +ηF(t,q_t^η)

dt−γM⁻¹p^η_t dt+ s2γ

β dW_t.

• Smooth potentialV and forceF, withF periodic in time with periodT

• F = 0: inv. measureµ(dq dp)∝exp

−β

V(q) +1

2p^TM⁻¹p

dq dp

Questions

what is the steady-stateof the system?

behavior under hyperbolic space-time scaling? (average velocity) fluctuations around the average velocity: longtime effectivediffusion

P. Collet and S. Mart´ınez,J. Math. Biol.,56(6) (2008) 765–792

G. Pavliotis, R. Joubaud and G. Stoltz,J. Stat. Phys158(1) (2015) 1–36

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Convergence

to the equilibrium state

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Exponential convergence to a limiting cycle

• Lyapunov functions Kⁿ(q,p) = 1 +|p|²ⁿ (for n>1) and corresponding L^∞ norms on functions f(θ,q,p)

kfkL^∞(L^∞_K

n) = sup

θ∈TT

f(θ) Kⁿ

L^∞

Uniform convergence result for η∈[−η_∗, η_∗]

Unique probability measure ψη(θ,q,p) on E =TT× M ×R^d such that

E

f([t],q_t^η,p_t^η)

−f_η([t])6C_ne^−λⁿ^tkfkL^∞(L^∞_K_n), with time-dependent spatial average f_η(θ) =

ˆ

M×R^d

f(θ,q,p)ψ_η(θ,q,p)dq dp

• In addition, convergence of the trajectorialaverage for any (q₀,p₀) 1

t ˆ t

0

f([s],q_s^η,p_s^η)ds −−−−→_t→+∞

ˆ

E

f ψη a.s.

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Properties of the limiting cycle

• Time dependent generatorA⁰+ηA¹, adjoints on L²(E) A0 =M⁻¹p·∇q−∇V·∇p+γ

−M⁻¹p· ∇p+ 1 β∆_p

, A1 =F(θ,q)·∇p

Fokker-Planck equation for ψ_η

The invariant distribution is smooth, positive and satisfies −∂_t+A^†₀+ηA^†₁

ψ_η = 0, ˆ

E

ψ_η = 1.

• Finite moments of order 2n uniformly in the time variable

∀θ∈TT, ˆ

M×R^dKn(q,p)ψ_η(θ,q,p)dq dp6R_n<+∞

• Uniform marginals in time ψ_η(θ) = ˆ

E

ψ_η(θ,q,p)dq dp= 1 T

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Elements of proof

• Exponential convergence of the sampled chain¹ (Qn^η,P_n^η) = (q^η_nT,p_nT^η ) (U_T,ηf) (q,p) =E

f(Q_n+1^η ,P_n+1^η )(Q_n^η,P_n^η) = (q,p) .

Uniform Lyapunov condition

For any n >1 and η_∗ >0, there existb >0 and a∈[0,1) such that

∀η ∈[−η_∗, η_∗], U_T_,ηKn6aKn+b.

Uniform minorization condition

Fix any pmax>0. There exists a probability measureν onM ×R^d and a constant κ >0 such that, for all η∈[−η_∗, η_∗],

∀B ∈B(M ×R^d), P

Q_k^η₊₁,P_k^η₊₁

∈B P_k^η6pmax

>κ ν(B).

• Patch invariant measures for sampled chains (q_θ+nT^η ,p_θ+nT^η )n>0

1M. Hairer and J. Mattingly,Progr. Probab.,63(2011) 109–117

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Perturbative expansion for small forcings

• Writeψη(t,q,p) =ρη(t,q,p)µ(q,p) : adjoints on H=L²(E, µ)∩ {1}^⊥ (−∂t+A^∗0+ηA^∗1)ρη = 0, ψη =ρηµ,

ˆ

E

ρηµ= 1

• Use the invertibility of∂t+A0 onH(Fourier series in time) and the relative boundedness of A1 with respect to ∂_t+A0

Series expansion of the invariant measure There exists C,r >0 such that, for |η|<r,

ρ_η = 1 +η̺₁+η²̺₂+. . . with

ˆ

E|̺_m(t,q,p)|²µ(q,p)dq dp dt6 C r^m and

ˆ

E

̺_mµdq dp dt= 0

• Functions̺_m not explicitely known: solutions of Poisson equations

• The leading order correction ̺₁ governs thelinear response

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Linear response of the velocity

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General result on the mobility

• Linear response of the time dependent spatially averaged velocity V(t) = lim

η→0

v_η(t)

η , v_η(t) = ˆ

M

ˆ

R^d

M⁻¹pψ_η(t,q,p)dq dp

• Fourier series in time: e_n(t) =eⁱ^nωt (ω = 2π/T) F(t,q) =F₀(q) + 2X

n>1

Re

F_n(q)en(t) Space-time decomposition of the mobility

V(t) =βX

n∈Z

en(t) ˆ

M

Dn(q)Fn(q)µ(q)e dq, µ(q) =e Ze⁻¹e^−βV^(q) with the position-dependent diffusion matrix

Dn(q) = ˆ +∞

0

E

M⁻¹ps

⊗ M⁻¹p₀ q₀ =q eⁱ^nωsds

• In particular, the average (time-independent) velocity dependsonly onF₀

1 ˆ T ˆ

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Numerical illustration (1)

• Timestep ∆t >0 such thatI∆t=T

• Approximation (qⁿ,pⁿ) of (q_n∆t,p_n∆t) (omitη for simplicity)

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







p^n+1/2=α∆tpⁿ+∆t 2

− ∇V(qⁿ) +ηF(tⁿ,qⁿ) +

s 1−α²_∆t

β Gⁿ, qⁿ⁺¹ =qⁿ+ ∆t M⁻¹p^n+1/2,

pⁿ⁺¹=α∆tp^n+1/2+∆t 2

− ∇V(qⁿ⁺¹) +ηF(tⁿ⁺¹,qⁿ⁺¹) +

s 1−α²_∆t

β G^n+1/2,

with α_∆t =e^{−γ∆t M}⁻¹^/2 (Strang splitting)

• Empirical averagesv_η(i∆t)≃ I N

XN/I j=1

M⁻¹pⁱ^+jI andV ≃ 1 Iη

XI i=1

v_η(i∆t)

• Numerical analysis² of the bias with respect to ∆t

2B. Leimkuhler, Ch. Matthews, and G. Stoltz,IMA J. Numer. Anal. (2015)

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Numerical illustration (2)

• 2-dimensional case V(q) = 2 cos(2x) + cos(y) + cos(x−y)

• Static, non-gradient forcesF_0,n(q) =e^βV^(q)

cos(nx) 0

• Parameters β=γ = 1,M =Id, ∆t = 0.01, 4.5×10⁹ timesteps

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

−0.035

−0.030

−0.025

−0.020

−0.015

−0.010

−0.005

−0.000

η

linear fit computed

V_x

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.00

0.01 0.02 0.03 0.04 0.05 0.06 0.07

η

linear fit

computed

V_x

n= 1, V_x =−3.04×10⁻² n= 2 V_x = 6.88×10⁻²

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Negative mobility

• Decomposition of the real,M-periodic and symmetric matrix D₀(q) = X

K∈L^∗

D_0,Ke⁻ⁱ^K·q= X

K∈L^∗

a_0,Kcos(K ·q) +b_0,Ksin(K ·q),

• Related spatial decomposition of the external force F₀(q) = 1

e

µ(q)|M|

X

K∈L^∗

F_0,Ke⁻ⁱ^K·q

!

, F_0,K =F_0,−K ∈C^d×d.

• Normalization such thatF_0,0 = canonical equilibrium average of F₀ Spatial decomposition of the mobility

The space-time averaged linear response of the velocity reads V =D_0,0F_0,0+

ˆ

M

D₀(q)−D_0,0

F₀(q)eµ(q)−F_0,0 dq

• Non-zero velocity produced either by constant forcing or as a result of some spatial resonance

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Numerical illustration

• Time-independent forceF₀(q) =e^βV^(q)

−1 + 3 cos(2x) 0

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.00

0.01 0.02 0.03 0.04 0.05 0.06

η

linear fit computed

V_x

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

−0.18

−0.16

−0.14

−0.12

−0.10

−0.08

−0.06

−0.04

−0.02

−0.00

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

−0.18

−0.16

−0.14

−0.12

−0.10

−0.08

−0.06

−0.04

−0.02

−0.00

η F_0,x

Left: Average observedvelocityin the x direction.

Right: Average experienced forcein the x direction.

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Resonance of the frequency-dependent mobility

• Dependence on the periodT of the forcing: fix F₁(q) and consider F(t,q) = 2Re F₁(q)eⁱ^ωt

• Linear response result: V(t) = 2βRe b

V(ω)eⁱ^ωt with b

V(ω) =−2β ˆ

M

ˆ

R^d

(iω+A0)⁻¹ M⁻¹p

⊗ M⁻¹p F₁µ.

High-frequency decay of the linear response

For any n>2, there exists a constantC_n>0 and ν₁, . . . , ν_n−1 ∈C^d, such that, for allω >1,

Vb(ω)−

n−1X

m=1

νm

ω^m 6 C_n

ωⁿ, ν1 = 2iβM⁻¹ ˆ

M

F₁(q)µ(q)e dq

• Existence of alocal/global maximumof bV(ω)?

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Numerical illustration

• ForceF(t,q) = 1

0

cos(ωt) andγ = 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

ω b

V(ω)x

0

10

1

10

−3

10

−2

10

−1

10

0

10

ω

× ^computed

Cω⁻¹

b V(ω)x

• Asymptotic behavior Vb(ω)_x ∼ω⁻¹ since ν₁ 6= 0

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Longtime diffusive behavior

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Convergence to an effective Brownian motion

• Diffusive rescalingon the dynamics not reprojected intoM introduce Q^η_t =q₀^η+

ˆ t

0

M⁻¹p_s^ηds remove average drift Vη =

ˆ

E

M⁻¹pψ_η(t,q,p)dt dq dp define Q_t^η =Q^η

t −tVη and rescale as Q_t^η,ε =εQ_t/ε^η 2

• Stationary initial conditions (q₀^η,p₀^η)∼ψ_η(0,q,p)dq dp Weak convergence over finite time intervals

Limiting Brownian motion d Q_t =√

2D_η^1/2dBt with Q₀ ∼ψe_η(q)dq and symmetric, positive definite covariance matrix

∀ξ ∈R^d, ξ^TD_ηξ= γ β

ˆ

E

∇p

ξ^TΦ_η²ψ_η.

• Covariance matrix determined by the solution of the Poisson equation (∂_t+A0+ηA1)Φ_η(t,q,p) =M⁻¹p− Vη,

ˆ

E

Φ_ηψ_ηdt dq dp = 0.

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Elements of proof

• Rewriteξ^T

Q_t^η,ε−Q₀^η,ε

=ε ˆ t/ε²

0

ξ^T M⁻¹p_s^η− Vη ds as εξ^T

Φ_ηh t ε²

i, ...

−Φ_η(0, ...)

−ε s2γ

β ˆ t/ε²

0 ∇p

ξ^TΦ_η

([θ],q^η_θ,p_θ^η)·dW_θ and use Martingale CLT (cv. finite dimensional laws) + thightness

Functional estimates (following Talay (2002) and Kopec (2013)) For a smooth function f with derivatives growing at most polynomially in p, the solution to the Poisson equation

(∂t+A0+ηA1)Φ_η(t,q,p) =f(t,q,p)− ˆ

E

fψ_η, ˆ

E

Φ_ηψ_η = 0 is unique. For any k >1 and η_∗ >0, there exists a real constant C >0 and integers n,m,N>1 such that, for all η∈[−η_∗, η_∗] and|l|6k,

∂^lΦη(t,q,p)6CKⁿ(q,p) sup

|r|6Nk∂^rfkL^∞(L^∞_K_m)

D. Talay,Gabriel Stoltz (ENPC/INRIA)Markov Proc. Rel. Fields,8(2002) 163–198 Nice, June 2015 18 / 22

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Further properties of the covariance matrix

Perturbative expansion of the covariance matrix ξ^TD_ηξ =ξ^TD₀ξ+ηξ^TD1ξ+η²Deη,ξ

with Deη,ξ uniformly bounded for |ξ|61.

• Equilibrium covarianceD₀= ˆ

M

D₀(q)µ(q)e dq

• When the external force has time average 0 for all configurations

∀q∈ M, ˆ

TT

F(t,q)dt = 0, the first order correction vanishes: D1 = 0

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Numerical illustration

• Simulations usingD_η = lim

t→∞

E

[Q_t^η−E(Q^η_t)]⊗[Q_t^η−E(Q_t^η)]

2t

• External forcingF(t,q) =e^βV^(q)

−1 + 3 cos(2x) 0

cos(ωt)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0 1 2 3 4 5 6 7 8

η

D_xx(η)

Dyy(η)

Dxy(η)^0.00_0.0 _0.5 _1.0 _1.5 _2.0 _2.5 _3.0 0.05

0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55

η

Dxx(η)

Dyy(η)

Dxy(η)

ω= 0 ω= 2π

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Conclusion and perspectives

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Conclusion and perspectives

• Various resonancephenomena

“spatial” resonance leading to negative mobility

“frequency” resonance leading to enhanced synchronization

observed at the linear response level (already known in the nonlinear regime³)

• Extension to thermal transport?

→ Some results for time-dependent drivings at the boundaries⁴

3Machura, Kostur, Talkner, Luczka, H¨anggi,Phys. Rev. Lett.(2007)

4A. Dhar, O. Narayan, A. Kundu, K. Saito,Phys. Rev. E(2011)