The Kramers law

Nils Berglund

Description

Niveau: Supérieur
The Kramers law : Validity, derivations and generalizations Nils Berglund MAPMO, Universite d'Orleans CNRS, UMR 6628 et Federation Denis Poisson Partly based on joint work with Florent Barret, Ecole Polytechnique, Palaiseau Bastien Fernandez, CPT, CNRS, Marseille Barbara Gentz, University of Bielefeld Institut Henri Poincare, Paris, 26 January 2011

• langevin equation


• stochastic differential


• equivalent notation


• invariant probability


• system need


• local minima


• gaussian white


• arrhenius law

Sujets

• Barret
• Poisson
• Berglund
• École polytechnique
• Poincaré
• Fernández
• Langevin equation
• Stochastic differential equation
• Maxima and minima

Informations

• Publié par: chaeh
• Nombre de lectures: 33
• Langue: English

Extrait

The Kramers law :

Validity, derivations and generalizations

Nils Berglund MAPMO,Universite´d’Orl´eans CNRS,UMR6628etFe´d´erationDenisPoisson www.univ-orleans.fr/mapmo/membres/berglund

Partly based on joint work with Florent Barret , Ecole Polytechnique, Palaiseau Bastien Fernandez , CPT, CNRS, Marseille Barbara Gentz , University of Bielefeld

InstitutHenriPoincare´,Paris,26January2011

Stochastic differential equation a.k.a. (overdamped) Langevin equation d x t = −r V ( x t ) d t + √ 2 ε d W t

:poetntial,rgow

. V : R d → R . W t : d -dim Brownian motion

Equivalent notation : x ˙ = −r V ( x ) + √ 2 ε ξ t

nigatinfinity

. ξ t : Gaussian white noise, h ξ t i = 0 , h ξ t ξ s i = δ ( t − s )

1

Stochastic differential equation a.k.a. (overdamped) Langevin equation d x t = −r V ( x t ) d t + √ 2 ε d W t . V : R d → R . W t : d -dim Brownian motion

:poten

Equivalent notation : x ˙ = −r V ( x ) + √ 2 ε ξ t

tial,growingtaninfitiy

. ξ t : Gaussian white noise, h ξ t i = 0 , h ξ t ξ s i = δ ( t − s )

Some properties :

. Invariant probability measure : µ (d x ) = Z 1 e − V ( x ) /ε d x . System is reversible w.r.t. µ (detailed balance) p ( y, t | x, 0) e − V ( x ) /ε = p ( x, t | y, 0) e − V ( y ) /ε

1-a

Main question

Assume V ( x ) has several (= at least

two) local minima.

How long does the system need

average

another

to go from one minimum

one

(for

ε



1 )?

on

to

2