Invariant Measures and Heat Conduction Networks

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Invariant Measures and Heat Conduction Networks

AlainCamanes

alain.camanes@univ-nantes.fr

Laboratoire Jean Leray - Universit´e de Nantes Journ´ees ANR Evol - Rennes

03 avril 2009

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Invariant Measures and

Heat Conduction

Networks A. Camanes

Model Networks Equations Chains Quadratic Case Uniqueness &

Weak Controlability Uniqueness &

Lasalle Existence Conclusion

The oscillator networks

V

Particles :◦

Pinning potential :V Interaction potential:U Damped particles :• Excited particles :• Positions :qi

Momentum :pi

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Heat Conduction

Networks A. Camanes

Journ´ees ANR Evol - Rennes 2/25

The oscillator networks

V

Particles :◦

Pinning potential :V

Interaction potential:U Damped particles :• Excited particles :• Positions :qi

Momentum :pi

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Heat Conduction

Networks A. Camanes

The oscillator networks

U V

Particles :◦

Pinning potential :V Interaction potential:U

Damped particles :• Excited particles :• Positions :qi

Momentum :pi

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Heat Conduction

Networks A. Camanes

The oscillator networks

U V

Particles :◦

Pinning potential :V Interaction potential:U Damped particles :• Excited particles :•

Positions :qi

Momentum :pi

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Heat Conduction

Networks A. Camanes

The oscillator networks

U V

Particles :◦

Pinning potential :V Interaction potential:U Damped particles :• Excited particles :• Positions :qi

Momentum :p_i

IHamiltonian:

H(q,p) =X

i∈V

p²_i

2 +X

i∈V



V(q_i) +¹₂X

j∼i

U(q_i −q_j)





I Assumption :U,V even polynomials, U symmetric, H con- vex.

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Heat Conduction

Networks A. Camanes

The Dynamics

IThe Stochastic Differential Equations







dq_i=∂_p_iH dt

dp_i=−∂_q_iH dt+· · ·

· · · −pi1i∈Ddt+√

2Ti1i∈∂VdBi

2 4

1

3

IThe generator of semigroup (Pt) L=X

i∈V

∂p_iH∂q_i −∂q_iH∂p_i −X

i∈D

p_i∂p_i + X

i∈∂V

T_i∂_p²_i.

IThe invariant measures

L^?µ= 0.

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Heat Conduction

Networks A. Camanes

The Dynamics

IThe Stochastic Differential Equations







dq_i=∂_p_iH dt

dpi=−∂_q_iH dt+· · ·

· · ·+ n

−p_idt+√ 2T dBi

o 1i∈∂V

2 4

1

3

WhenTi =T,i ∈∂V andD=∂V. IThe generator of semigroup (P_t)

L=X

i∈V

∂_p_iH∂_q_i −∂_q_iH∂_p_i − X

i∈∂V

p_i∂_p_i + X

i∈∂V

T∂_p²_i.

ITheGibbsmeasure µ_H is invariant, where

µ_H(dz) = e^−H/T Z dz.

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Heat Conduction

Networks A. Camanes

Historical Background

IExistence and Uniqueness for chains of oscillators.

[Eckmann, Pillet, Rey-Bellet] 1999

Methods : Generators with compact resolvent.

Quadratic behavior at∞ Non-degeneracy ofU.

U&V.

[Eckmann, Hairer] 2000 Non-quadratic behavior.

[Rey-Bellet, Thomas] 2002

Methods : Lyapunov functions.

Exopnential convergence to equilibrium.

IExistence and Uniqueness for networks of oscillators.

[Maes, Neto˘cn`y, Verschuere] 2003

Methods : Non-degeneracy via propagation.

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Heat Conduction

Networks A. Camanes

Model Quadratic Case

Linear Equations Completeness Counter- Example Smoothness Uniqueness &

The Linear Equation

Potentials are quadratic, i.e.

U(x) =V(x) =x²/2.

I TheAdjacency & Degree Matrices Λ = (δi∼j)i,j,D=diag(X

j∼i

1).

Λ =





 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0







2 4

1

3

IEquations of the dynamics dZ_t=

0 I

−(I +D−Λ) −I_D

Z_tdt+

0 0 0 T∂V

dB_t.

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Heat Conduction

Networks A. Camanes

The Asymmetry Assumption

Theorem

The diffusion has a unique invariant measure iff the graph (G,∼,D) is asymmetric.

I TheAsymmetrycondition : E_M,D = Vect

n

M^ke_i,i ∈ D,k ∈N o

= Rⁿ.

Λ =





 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0







2 4

1

3

IExamples :

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Heat Conduction

Networks A. Camanes

Eigenvalues & Completeness : Sketch of proof

Theorem

If the graph(G,∼, ∂V) is asymmetric then the diffusion has a unique invariant measure.

Let us notice that for anyx,y ∈Rⁿ,

kPt^?δx−P_t^?δyk ≤ kZt^x−Z_t^yk

≤ ke^Mt(x−y)k, where

M=

„ 0 I

Λ−D−I −ID

« .

The asymmetric condition implies that the matrixMis stable and we obtain the upper bound

kPt^?δx−Pt^?δyk ≤Ce^−µ⁰^tkx−yk.

The conclusion follows from thecompletenessof the Wasserstein spaces.

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Heat Conduction

Networks A. Camanes

Symmetry & Non-Uniqueness

Theorem

If the graph isnotasymmetric and there is an invariant measureµ, there isinfinitely many invariant measures.

We find a flow invariant quantity independant of the damped particles.

L^?f = −{H,f}+X

i∈D

pi∂p_if +|D|f +X

i∈∂V

Ti∂²p_if

= −{H,f}+L^?Df. Flow-invariant quantity:

K(q,p) =hz,pi²+αhz,qi², (D+I−∆)z=αz,z∈ E_M,D^⊥

z

EM,D

Then, for any constantγ >0, the following measuresµγ are invariant µγ(dz) =e^−γK

Z µ(dz).

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Heat Conduction

Networks A. Camanes

Smoothness

Theorem

Letµbe the unique invariant measure. Suppµ=Rⁿ iff dimE_M,∂V =n.

Recall that

Zt^z =e^Mtz+ Z t

0

e^M(t−s)σdBs.

The covarianceKt of the Gaussian process (Zt^z)t satisfies the differential equation

∂tKt =σσ^?+MKt+KtM^?. We consider the solutionsQ ofLyapunov’s equation

MQ+QM^?=−σσ^?. See [Snyders et Zakai, 1970],

rank(Q) = dimEM,∂V.

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Heat Conduction

Networks A. Camanes

Model Quadratic Case Uniqueness &

Weak Controlability

Smoothness &

Support Support &

Control Support &

Recurrence Weak Controlability &

Support Uniqueness &

H¨ ormander’s condition & Uniqueness

Damped atoms are excited

D=∂V.

Theorem

IfD=∂V and H¨ormander’s condition is satisfied, there is at most one invariant measure.

ITheLie algebra : L=

∂pi,i ∈∂V+ Poisson bracket with X

j

∂pjH∂qj−∂qjH∂pj

& stability w.r.t. inner Poisson brackets

. IH¨ormander’s condition : For any z ∈Rⁿ, dimL(z) =n.

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Heat Conduction

Networks A. Camanes

Weak Controlability

Smoothness &

Support Support &

Control Support &

H¨ ormander’s condition & Uniqueness

Damped atoms are excited

D=∂V.

Theorem

IfD=∂V and H¨ormander’s condition is satisfied, there is at most one invariant measure.

I H¨ormander’s condition entails that the semigroup and the invariant measure have a smoothdensityw.r.t. Lebesgue’s measure.

WhenU,V are quadratic, the asymmetry condition is equivalent to the H¨ormander’s condition.

I Constant Temperatures : if for any i ∈ ∂V, Ti = T, Gibbs measureµ_H is invariant, where

µH(dz) = e^−H/T Z dz.

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Heat Conduction

Networks A. Camanes

Weak Controlability

Smoothness &

Support Support &

Control Support &

Control & Support

Stochastic System (Σ) Deterministic System (S) dZ_t =f(Z_t)dt+σ◦dB_t z˙_t =f(z_t) +σu(t),u∈ C^morc ISupport Theorem[Stroock-Varadhan 1972] :∀t₀ >0,x ∈Rⁿ,

Supp Pt0(x,·) = C`

z_t₀;∃u ∈ C^morc,z₀ =x,z˙_t =f(z_t) +σu(t) .

IWeak Controlability:∀z ∈Rⁿ,A⊂Rⁿ,∃T_z,A >0,u s.t.

z0 =z,z_T_z,A ∈A.

A zTz,A

u(t) z

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Heat Conduction

Networks A. Camanes

Weak Controlability

Smoothness &

Support Support &

Control Support &

Support & Recurrence

Theorem

Under H¨ormander’s condition, suppose there exists a measure µs.t.

Suppµ=Rⁿ, µ is invariant.

Then, the diffusion(Z_t^z) is recurrent.

µergodic

h(z) =P[lim sup_t1Z_t∈A] invariant ⇒ (Zt^z) recurrent.

h(Zt) convergent h(z) = 1µ-a.s.

Corollary

The deterministic system(S) is weakly controlable.

Remark :We will use Gibbs measure.

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Heat Conduction

Networks A. Camanes

Weak Controlability

Smoothness &

Support Support &

Control Support &

Controlability & Uniqueness

Stochastic System (Σ) Deterministic System (S) dZ_t =f(Z_t)dt+σ◦dB_t z˙_t =f(z_t) +σu(t),u∈ C^morc

IUniqueness of the Invariant Measure [Hairer 2005]: Suppose that H¨ormander’s condition is satisfied. If (S) is weakly controlable then (Σ) has at most one invariant measureµ.

Then,Suppµ=Rⁿ.

Remark :For any invertible matrix σ, weak controlability of the following systems is equivalent :

(S₁) z˙_t=f(z_t) +σu(t) (S2) z˙t=f(zt) +u(t)

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Heat Conduction

Networks A. Camanes

Weak Controlability

Smoothness &

Support Support &

Control Support &

In a nutshell

IfD=∂V and H¨ormander’s condition is satisfied,

Uniqueness of

the invariant measure for (Σ)!Weak Controlability for (S) dXt=f(Xt)dt+TidBt x˙t=f(xt) +Tiut

!Weak Controlability for (eS)

˙

xt=f(xt) +Tut

!Invariant measure for (eΣ) dXt=f(Xt)dt+TdBt

Gibbs measure µ(dz) =e^−H/T/Z dz

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Heat Conduction

Networks A. Camanes

Lasalle Lasalle &

Stability Existence Conclusion

Smoothness & Support

ISmoothness : (P_t) isasymptotically strong Felleratc₀∈Rⁿ if there existsdn(x,y)→1x6=y, (tn) s.t.

γ→0limlim sup

n→∞ sup

y∈B(c0,γ)

kP_t_n(c0,·)−Ptn(y,·)k_d_n = 0.

Proposition :If (Pt) is strong Feller,

γ→0lim sup

y∈B(c₀,γ)

kP2t(c0,·)−P2t(y,·)kTV= 0.

Remark :WhenU,V arequadraticand the graph isasymmetric, the semigroup isASF.

IProposition[Hairer, Mattingly 06] : If the semigroup isASF at c0 then c0 belongs to the support ofat most one ergodic invariant measures.

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Heat Conduction

Networks A. Camanes

Lasalle Lasalle &

Rigidity & Uniqueness

Letc0 be the argument of the minimum ofH.

Theorem

When the semigroup is ASF at c₀ and the network is stable, there exists at most one invariant measureµ. Then

c0∈Suppµ.

IStability condition: the only solution of the noise-free equation







˙

qi = ∂piH

˙

pi = −∂_q_iH p_i1i∈D = 0 is the constantz ≡c0.

Remark :When the potentials are quadratic, the network isstableiff the graph isasymmetric.

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Heat Conduction

Networks A. Camanes

Lasalle Lasalle &

Lasalle’s principle

ILasalle’s principle: If there exists only one solution to the equation ˙H(z) = 0 then every solution of the noise-free equation goes to this solutionc₀.

R

T 0

c

Notice that

H(z) =˙ −X

i∈D

p²_i.

IConclusion :For any invariant measureµ, c0 ∈Suppµ.

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Heat Conduction

Networks A. Camanes

Lasalle Existence

Partial Results Interaction &

Pinning Conclusion

Partial Results

Slow energy dissipation in anharmonic oscillator chains M. Hairer & J. Mattingly (2007)

IFor any chain with3 oscillators, thereexists a unique invariant measure.

IIf the interaction is quadratic and the pinning grows at least asx⁴, then 0 belongs to the essential spectrum of the

conjugated generatorLe=e^−βH/2Le^βH/2.

This result can begeneralised to the heat conduction networks.

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Heat Conduction

Networks A. Camanes

Lasalle Existence

Pinning Conclusion

Existence of the invariant measure

Theorem

If interaction is stronger than pinningandthe graph is rigid, there exists a unique invariant measure.

Requires a compactness argument [Rey Bellet & Thomas 2002] (W Lyapunov function). . .

1 LW(z)≤CW(z) for any constantC,

2 there existst0>0 andan↑+∞s.t.

n→∞lim sup

{z;W(z)>a_n}

Pt₀W(z) W(z) = 0.

. . . and a scaling argument PtW(z)

W(z) ≤e^C^βt^Pⁱ^TⁱEz

h

e^−C^R⁰^t^P^i∈D^p²ⁱ^dsi1/b

.

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Heat Conduction

Networks A. Camanes

Perspectives

IWhat about existency when pinning is stronger than interaction ?

IIs there a link between stability and H¨ormander’s condition ? IWhat about entropy production ?

Invariant Measures and Heat Conduction Networks