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05/11/2011

SEMINAIRE : Spectral analysis of Fokker-Planck and Kramers-Fokker-Planck operators

Spectral analysis of Fokker-Planck

and Kramers-Fokker-Planck operators

Francis Nier

IRMAR, UMR-CNRS 6625, Univ. Rennes 1

INRIA-MICMAC project-team

In the last ten years, many advances have be done in the spectral analysis of

the Fokker-Planck operator

2∇V (x).∇x −

∆x

and the Kramers-Fokker-Planck operators

v.∂x −

(∂xV (x)).∂b +

mβ

(−∂v + mβv).∂v .

The main questions which were studied were about estimating the rate of

return to the equilibrium, w.r.t the mass m, the friction coeﬃcient γ and the

inverse temperature β =

kBT

, and more generally the life time of metastable

states in the small temperature asymptotics (large β). These operators can

be considered on the whole space, on riemannian manifolds or domains with

boundary.

Mathematically this has connection with hypoellipticity, ﬁne spectral analysis of non self-adjoint operators, accurate estimate of the tunnel eﬀect for

Schr¨odinger type operators, diﬀerential geometry, and even algebraic topology or Lie algebras.

In this short course, I will present the main ideas, illustrated by simple examples and give some general results which can be useful in some other

frameworks. Many references to recent (and less recent) works will be given

and brieﬂy explained.

The outline of the course will be:

• Introduction: Smoluchowski, Langevin processes and their semigroup.

• Semigroups, contour integrals and hypoellipticity.

• Results about the Kramers-Fokker-Planck equation and its relation

with the Fokker-Planck equation.

• About the constant Cα for semigroup decay estimates ke

−tL

k ≤ Cαe

−αt

• Additional structures: a) PT-symmetry; b) supersymmetry; c) nilpotent Lie algebras and maximal hypoellipticity.

Francis Nier (Université de Rennes)

lundi 7 novembre 2011 à 14h00 (B413)
mardi 15 novembre 2011 à 14h00 (la salle sera annoncée ultérieurement)
lundi 21 novembre 2011 à 14h00 (B413)
lundi 5 décembre 2011 à 14h00 (B413)

Sources :

http://cermics.enpc.fr/seminaires/cs/Seminaire11-12/nier....

http://cermics.enpc.fr/seminaires/cs/seminaire.html

Séminaire le 28 novembre 2011 : Unication of Dynamic Density Functional Theory for Colloids

Unication of Dynamic Density Functional Theory for Colloids

Adeline Augier, Université Paris-Sud

Lundi 28 novembre 2011 à 11h00

We study the acoustic properties of a microstructured material such as glass wool or foam. In our model, the

solid matrix is governed by linear elasticity and the surrounding uid obeys Stokes equations. The microstructure

is assumed to be periodic at some small scale and the viscosity coecient of the uid is assumed to be of order

. We consider the time-harmonic regime forced by vibrations applied on a part of the boundary. We use the

two-scale convergence theory to prove the convergence of the displacements to the solution of a homogeneous

problem as the size of the microstructure shrinks to 0. We conclude with some numerical results. The talk is based

on joint work with François Alouges, Benjamin Graille and Benoît Merlet.

Source :

http://cermics.enpc.fr/seminaires/cs/seminaire.html