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FreddyBouchet9CesareNardlnl9andTomasTangarlfe
/
D
F[q](r)
δq(r)
δG
[q]
d
r=0j
(32)
forany
q
lThosehypothesisareVerified,forinstanceifthedynamicalsystemis
anHamiltoniansystem
F(q)={qjH}j
where
{.j.}
isaPoissonbracket,and
G
oneoftheconserVedquantityofthe
Hamiltoniansystem,forinstance
G=H
lWestresshoweVerthat
G
doesnotneed
tobe
H
l
IftheLiouVillehypothesisisVerifiedand
G
isaconserVedquantity,wecalla
LangeVindynamicsforthepotential
G
thestochasticdynamics
∂q
∂t
=F[q](r)−o/
D
C(rjr!)
δq(r!)
δG
[q]
d
r!+d2o,nj
(33)
wherewehaVeintroducedastochasticforce
n
,whichweassumetobeaGaussian
process,whiteintime,andcorrelatedas
E[n(rjt)n(r!jt!)]=C(rjr!)δ(t−t!)
lAs
itisacorrelationfunction,
C
hastobeasymmetricpositiVefunctionoforany
function
0
oVer
D
/
D/
D
0(r)C(rjr!)0(r!)
d
r
d
r!≥0j
(34)
and
C(rjr!)=C(r!jr)
lForsimplicity,weassumeinthefollowingthat
C
ispositiVe
definiteandhasaninVerse
C11
suchthat
/
D
C(rjr1)C
11(r1jr!)
d
r1=δ(r−r
!).
ThemajorpropertyofaLangeVindynamicsisthatthestationaryprobability
densityfunctionalisknowna-priorilItis
Ps[q]=
Z
1
exp(−
G[q]
,)j
where
Z
isanormalizationconstantlAtaformalleVel,thiscanbecheckedeas-
ilybywritingtheFokker-PlanckequationfortheeVolutionoftheprobability
functionalslThenthefactthat
Ps
isstationaryreadilyfollowsfromtheLiouVille
theoremandthepropertythat
G
isaconserVedquantityforthedeterministic
dynamicsl
40202ReVersedLangeVindynamics
Weconsider
I
alinearinVolutiononthespaceoffields
q
(
I
isalinearfunctional
with
I2=
Id)lWedefinethereVersedLangeVindynamicswithrespectto
I
as
∂q
∂t
=Fr[q](r)−o/
D
Cr(rjr
!)
δq(r!)
δGr
[q]
d
r!+d2o,nj
(35)
