The Kramers law

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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


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The Kramers law :
Validity, derivations and generalizations
Nils Berglund MAPMO,Universite´dOrl´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
nigatinnity
. 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,growingtanintiy
. 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 ) d x . System is reversible w.r.t. µ (detailed balance) p ( y, t | x, 0) e V ( x ) = p ( x, t | y, 0) e V ( y )
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