Classical Motion in a Field of Soft Obstacles

Paul Painleve

Description

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

Sujets

• Painlevé
• Random walk
• Fokker?Planck equation

Informations

• Publié par: pefav
• Nombre de lectures: 20
• Langue: English

Extrait

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