Classical Motion in a Field of Soft Obstacles

icon

25

pages

icon

English

icon

Documents

Écrit par

Publié par

Lire un extrait
Lire un extrait

Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus

Découvre YouScribe en t'inscrivant gratuitement

Je m'inscris

Découvre YouScribe en t'inscrivant gratuitement

Je m'inscris
icon

25

pages

icon

English

icon

Ebook

Lire un extrait
Lire un extrait

Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus

Classical Motion in a Field of Soft Obstacles B. Aguer S. De Bievre, P. Lafitte- Godillon, P. Parris Introduction Numerical Results A Random Walk Approximation Derive the random walk Analysis of R Predict the dynamics Remarks Classical Motion in a Field of Soft Obstacles B. Aguer S. De Bievre, P. Lafitte-Godillon, P. Parris INRIA Lille Nord Europe , Equipe projet SIMPAF Lab. Paul Painleve, Lille Journees Systemes Ouverts - Grenoble, March 2009 1 / 25

  • random walk

  • rd ?r ?

  • walk approximation

  • fokker-planck equation


Voir icon arrow

Publié par

Nombre de lectures

20

Langue

English

Classical
Motion in a
Field of Soft
Obstacles
B. Aguer
S. De Bi`evre,
P. Lafitte- Classical Motion in a Field of Soft Obstacles
Godillon, P.
Parris
Introduction B. Aguer
Numerical
S. De Bi`evre, P. Lafitte-Godillon, P. ParrisResults
A Random
Walk INRIA Lille Nord Europe , Equipe projet SIMPAF
Approximation
Lab. Paul Painlev´e, Lille
Derive the
random walk
Analysis of R
Predict the
dynamics Journ´ees Syst`emes Ouverts - Grenoble, March 2009
Remarks
1/25Classical
Motion in a
Field of Soft Equations
Obstacles
B. Aguer We study the motion of a fast particle in a force field F with
S. De Bi`evre,
short spatial correlation:P. Lafitte-
Godillon, P.
Parris
q¨(t) = F(q(t),t), q(t ) = q , p(t ) = p0 0 0 0
Introduction
d d dwith q∈R and F :R ×R→R .Numerical
Results 2 cases are studied:
A Random
•Walk F is a random force field satisfying
Approximation
Derive the ′ ′ ′ ′
random walk < F(q,t) >= 0, < F(q,t)F(q ,t ) >= C q−q ,t−t ,
Analysis of R
Predict the
dynamics where C is a function of rapid decay in its space variable;
Remarks
• F is defined by
X
F(q,t) = f (q−x ,t),N N
N
where f are random smooth functions of compactN
support in B(0,1/2) and inf kx −x k≥ 1.N,M N M
2/25Classical
Motion in a
Field of Soft Litterature
Obstacles
B. Aguer • In mathematics:
S. De Bi`evre,
P. Lafitte- • F =−∇V; Kesten-Papanicolaou (1981) and
Godillon, P.
2 2/3
Parris Dolgopyat-Koralov (2008): p (t)∼ t .
• Rescaling - derivation of a Fokker-Planck equation:
Introduction
Poupaud-Vasseur (2003) (F =−∇V), Goudon-Rousset
Numerical
(2008) (F =−∇V).Results
A Random • In physics: 90’s. Random process F =−∇V with
Walk
Approximation
′ ′ ′ ′Derive the hV(q,t)i = 0, hV(q,t)V(q ,t )i = C(q−q ,t−t ).random walk
Analysis of R
Predict the ′ ′ ′ ′
dynamics • C(q−q ,t−t ) = C (q−q )δ(t−t ):0
Remarks
2 1/2 2 3hp (t)i∼ t ,hq (t)i = t (d ≥ 1).
′ ′• C(q−q ,t−t ) smooth, rapid decay. Then
2 2/5 1/2hp (t)i∼ t (d ≥ 1) or t (d > 1),
2 12/5 9/4hq (t)i∼ t (d = 1) and t (d > 1).
3/2566Classical
Motion in a
Field of Soft Hypotheses
Obstacles
B. Aguer
S. De Bi`evre,
P. Lafitte-
Godillon, P. q¨(t) = F(q(t),t),
Parris
with XIntroduction
F(q,t) = c f (q−x ,t +φ ),N N NNumerical
Results N
A Random
Walk • x chosen randomly or on a regular lattice;Approximation N
Derive the
random walk • c : coupling constants,φ : phases. iidrv according to aN N
Analysis of R
Predict the fixed probability measure on [−1,1];
dynamics
Remarks • finite horizon;
• f is spherical symmetric in its space variable
• f =−∇V, where V is smooth, compactly supported in
B(0,1/2) wrt its space variable and quasi-periodic wrt its
time variable.
4/25Classical
Motion in a
Field of Soft Results
Obstacles
B. Aguer
S. De Bi`evre,
P. Lafitte-
Godillon, P.
Parris
Numerically, we get:
Introduction
Numerical 2 2/5hp (t)i∼ t ,
Results
A Random
2 12/5
Walk hq (t)i∼ t (d = 1),
Approximation
Derive the 2 2
random walk hq (t)i∼ t (d > 1).
Analysis of R
Predict the
dynamics We derived the same asymptotics by the analysis of a simple
Remarks
and intuitive random walk describing the motion of the
particles.
5/25Classical
Motion in a
Field of Soft Outline
Obstacles
B. Aguer
S. De Bi`evre,
P. Lafitte-
Godillon, P.
Parris
Introduction
1 Numerical Results
Numerical
Results
A Random
Walk
Approximation
2 A Random Walk Approximation
Derive the
random walk Derive the random walkAnalysis of R
Predict the
dynamics Analysis of R
Remarks Predict the dynamics
6/25Classical
Motion in a
Field of Soft Outline
Obstacles
B. Aguer
S. De Bi`evre,
P. Lafitte-
Godillon, P.
Parris
Introduction
1 Numerical Results
Numerical
Results
A Random
Walk
Approximation
2 A Random Walk Approximation
Derive the
random walk Derive the random walkAnalysis of R
Predict the
dynamics Analysis of R
Remarks Predict the dynamics
7/25Classical
Motion in a
Field of Soft The Model
Obstacles
B. Aguer
S. De Bi`evre,
P. Lafitte- F(q,t) =−∇V(q,t)
Godillon, P.
Parris X
V(q,t) =λ c ρ(q−x )cos(t +φ ),N N N
Introduction
dN∈Z
Numerical
Results with
A Random
Walk • d = 1,2, λ = 1/4;
Approximation
Derive the • x on a square or hexagonal lattice;Nrandom walk
Analysis of R
Predict the • ρ is the characteristic function of
dynamics √
B(0,r), 3/4≤ r ≤ 1/2;Remarks
• c (resp. φ ) iidrv, uniformly distributed on [−1,1] (resp.N N
[0,2π]);
• Initial conditions:
• q : uniformly distributed on S(0,r);0
• kp k constant, in [0.5,1.5], p outgoing.0 0
8/25Classical
Motion in a
Field of Soft Trajectories
Obstacles
B. Aguer
S. De Bi`evre,
P. Lafitte-
Godillon, P.
Parris
200
Introduction
Numerical
Results 0
A Random
Walk
Approximation −200
Derive the
random walk
Analysis of R
Predict the
−400dynamics
Remarks
−600
−800
−1000
−400 −200 0 200 400 600 800 1000 1200
9/25Classical
Motion in a
Field of Soft Trajectories
Obstacles
B. Aguer
S. De Bi`evre,
P. Lafitte-
Godillon, P.
Parris
Introduction
Numerical
Results
A Random
Walk
Approximation
Derive the
random walk
Analysis of R
Predict the
dynamics
Remarks
10/25

Voir icon more
Alternate Text