Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
Kinetic theory and non-equilibrium phase transitions
in systems with long-range interactions
F. BOUCHET (CNRS) – ENS-Lyon and CNRS
November 2013 – Gravasco, IHP, Paris
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
Collaborators
Anomalous diffusion in the kinetic theory of systems with long
range interactions: T. Dauxois (ENS-Lyon) and Y. Yamaguchi
(Tokyo Univ.).
Non-equilibrium phase transitions for systems with long range
interactions: T. Dauxois, S. Gupta, C. Nardini, and S. Ruffo
(ENS-Lyon).
Non-equilibrium phase transitions for the 2D Navier-Stokes
equations: E. Simonnet (INLN-Nice) (ANR Statocean).
Instantons and large deviations for the 2D Navier-Stokes
equations: J. Laurie (Post-doc ANR Statocean), O.
Zaboronski (Warwick Univ.).
Kinetic theory of atmosphere turbulent jets: C. Nardini and T.
Tangarife (ENS-Lyon).
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
Random Transitions in Turbulent Flows
Magnetic Field Reversal (Turbulent Dynamo, MHD Dynamics)
Magnetic field timeseries
Zoom on reversal paths
(VKS experiment)
In turbulent flows, transitions from one attractor to another often
occur through a predictable path.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
Phase Transitions in Rotating Tank Experiments
The rotation as an ordering field (Quasi Geostrophic dynamics)
Transitions between blocked and zonal states
Y. Tian and col, J. Fluid. Mech. (2001) (groups of H. Swinney and
M. Ghil)
Compute attractors, transition pathways and probabilities.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
Non-Equilibrium Phase Transitions for Systems with Long
Range Interactions
with a theoretical prediction based on non-equilibrium kinetic theory
Time series for the order parameter for the 1D stochastic Vlasov Eq.
C. NARDINI, S. GUPTA, S. RUFFO, T. DAUXOIS, and F. BOUCHET, 2012,
J. Stat. Mech., L01002, and 2012 J. Stat. Mech., P12010.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
Outline
1
2
3
Anomalous diffusion and aging in systems with long range
interactions
The Lenard-Balescu equation
Anomalous diffusion, aging and correlations
Kinetic theory and non-equilibrium phase transitions for systems
with long range interactions
Stochastic Mean Field Hamiltonian model
The kinetic equation for the stochastic dynamics
Non-equilibrium phase transitions
Kinetic theory and non-equilibrium phase transitions for
geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging and jet formation in geostrophic
turbulence.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The Lenard-Balescu equation
Lenard-Balescu – Aging correlations
Outline
1
2
3
Anomalous diffusion and aging in systems with long range
interactions
The Lenard-Balescu equation
Anomalous diffusion, aging and correlations
Kinetic theory and non-equilibrium phase transitions for systems
with long range interactions
Stochastic Mean Field Hamiltonian model
The kinetic equation for the stochastic dynamics
Non-equilibrium phase transitions
Kinetic theory and non-equilibrium phase transitions for
geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging and jet formation in geostrophic
turbulence.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The Lenard-Balescu equation
Lenard-Balescu – Aging correlations
Systems with Long Range Interactions
Long range interactions = non-integrable potential
H=
1
1 N 2
pk +
∑
2 k=1
2N
N
∑
k,l=1
V (qk − ql ).
In the algebraic case V (r ) ∝r →∞ 1/r α ; non integrable when
α < d.
2D Euler G (r ) ∝r →∞ log(r ):
E=
1
2
Z
D
d 2 xd 2 y G (x−y)ω (x) ω (y) '
N
1
∑ Gij,kl ωij ωkl .
2N 2 i,j,k,l=1
Examples: self-gravitating stars, 2D and geophysical flows,
plasma, cold atoms, etc?
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The Lenard-Balescu equation
Lenard-Balescu – Aging correlations
Systems with Long Range Interactions
Long range interactions = non-integrable potential
H=
1
1 N 2
pk +
∑
2 k=1
2N
N
∑
k,l=1
V (qk − ql ).
In the algebraic case V (r ) ∝r →∞ 1/r α ; non integrable when
α < d.
2D Euler G (r ) ∝r →∞ log(r ):
E=
1
2
Z
D
d 2 xd 2 y G (x−y)ω (x) ω (y) '
N
1
∑ Gij,kl ωij ωkl .
2N 2 i,j,k,l=1
Examples: self-gravitating stars, 2D and geophysical flows,
plasma, cold atoms, etc?
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The Lenard-Balescu equation
Lenard-Balescu – Aging correlations
Kinetic Theory for Systems with Long Range Interactions
A classical and fascinating framework
H=
1 N 2
1
pk +
∑
2 k=1
2N
N
∑
k,l=1
V (qk − ql ).
A common framework for many systems: plasma physics, self
gravitating systems, the point vortex model.
We will assume V smooth and x ∈ D with D bounded. For
instance D a d-dimensional torus.
Classical approach: BBGKY hierarchy, small parameter,
chaotic hypothesis, derivation of kinetic equations.
Vlasov equation (40’s), Landau equation, Lenard–Balescu
equation (collisional Boltzmann equation) (60’s,70’s).
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The Lenard-Balescu equation
Lenard-Balescu – Aging correlations
Kinetic Theory for Systems with Long Range Interactions
A classical and fascinating framework
H=
1 N 2
1
pk +
∑
2 k=1
2N
N
∑
k,l=1
V (qk − ql ).
A common framework for many systems: plasma physics, self
gravitating systems, the point vortex model.
We will assume V smooth and x ∈ D with D bounded. For
instance D a d-dimensional torus.
Classical approach: BBGKY hierarchy, small parameter,
chaotic hypothesis, derivation of kinetic equations.
Vlasov equation (40’s), Landau equation, Lenard–Balescu
equation (collisional Boltzmann equation) (60’s,70’s).
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The Lenard-Balescu equation
Lenard-Balescu – Aging correlations
The Mean Field Limit and the Vlasov Equation
H=
1 N 2
1
∑ pk + 2N
2 k=1
N
∑
k,l=1
V (qk − ql ).
Dynamics of the empirical measure
fe (q, p, t) = N1 ∑k δ (q − qk (t), p − pk (t)).
The mean field limit: N → ∞ for times t C log(N). fe
remain close to the solution of the Vlasov equation :
∂f
∂f
∂ Φ[f ] ∂ f
+p
−
= 0.
∂t
∂q
∂q ∂p
With φ the mean field potential
Φ[f ](q) ≡
F. Bouchet
Z
dq1 dp1 V (q − q1 )f (q1 , p1 , t).
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The Lenard-Balescu equation
Lenard-Balescu – Aging correlations
Beyond the Mean Field Limit
H=
1
1 N 2
∑ pk + 2N
2 k=1
N
∑
k ,l=1
V (qk − ql ).
Dynamics of the empirical measure
∂ fe ∂ Φ[fe ] ∂ fe
∂ fe
+p
−
= 0.
∂t
∂q
∂q ∂p
We see the Vlasov equation as a consequence of the law of
large numbers limN→∞ fe = f .
Beyond the
Gaussian fluctuations (of
√ law of large numbers:
√
order 1/ N). fe = f + δ f / N.
What is the dynamics of those fluctuations?
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The Lenard-Balescu equation
Lenard-Balescu – Aging correlations
Rough Ideas About Corrections to the Vlasov
EquationsFluctuation Dynamics
∂ fe
∂ fe ∂ Φ[fe ] ∂ fe
+p
−
= 0.
∂t
∂q
∂q ∂p
√
Formal asymptotic expansion fe = f + δ f / N
∂f
∂f
∂ Φ[f ] ∂ f
1 ∂ Φ[δ f ] ∂ δ f
,
+p
−
=
∂t
∂q
∂q ∂p N
∂q ∂p
∂δf
+ Lf [δ f ] = 0.
∂t
At leading order the fluctuations are transported by the
linearized Vlasov equation.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The Lenard-Balescu equation
Lenard-Balescu – Aging correlations
Rough Ideas About Fluctuation Dynamics
∂δf
+ Lf [δ f ] = 0.
∂t
Second order correlation function
g (q1 , p1 , q2 , p2 , t) = hδ f (q1 , p1 , t)δ f (q2 , p2 , t)i
∂g
+ L1f [g ] + L2f [g ] = S.
∂t
Fluctuation dynamics is governed by a Lyapunov equation (the
equation for the two-points correlation of an
Ornstein-Uhlenbeck process). This is a general rule. See for
instance fluctuating hydrodynamics.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The Lenard-Balescu equation
Lenard-Balescu – Aging correlations
The Lenard-Balescu Equation
After solving the Lyapunov equations for the second order
correlation function we obtain the Lenard-Balescu equation:
∂f
1 ∂ Φ[δ f ] ∂ δ f
1
+ Vlasov [f ] =
= L B [f ] .
∂t
N
∂q ∂p
N
The Vlasov equation is an approximation of the
Lenard-Balescu equation. In plasma, the Vlasov operator is a
good approximation of the Lenard-Balescu one for scales much
larger than the Debye length.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The Lenard-Balescu equation
Lenard-Balescu – Aging correlations
Outline
1
2
3
Anomalous diffusion and aging in systems with long range
interactions
The Lenard-Balescu equation
Anomalous diffusion, aging and correlations
Kinetic theory and non-equilibrium phase transitions for systems
with long range interactions
Stochastic Mean Field Hamiltonian model
The kinetic equation for the stochastic dynamics
Non-equilibrium phase transitions
Kinetic theory and non-equilibrium phase transitions for
geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging and jet formation in geostrophic
turbulence.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The Lenard-Balescu equation
Lenard-Balescu – Aging correlations
Kinetic Theory for Systems with Long Range Interactions
A classical and fascinating framework
H=
1 N 2 1
∑ pk + N
2 k=1
N
∑
k,l=1
V (xk − xl ).
For instance a plasma in the weak coupling limit.
A common framework for many systems: plasma physics, self
gravitating systems, point vortex model.
Classical approach: BBGKY hierarchy, small parameter,
chaotic hypothesis, derivation of kinetic equations.
Vlasov equation (40’s), Landau equation, Lenard Balescu
equation (60’s,70’s).
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The Lenard-Balescu equation
Lenard-Balescu – Aging correlations
Kinetic Theory: Diluted Gases Versus Long Range Systems
Two opposite limits but strong analogies for the kinetic theories
Small parameter
Initial evolution
Late relaxation
Vanishing correlations
Boltzmann entropy
Stosszahl Ansatz
Steady states of
the initial evolution
Relaxation time scale
Long temporal correlations
and algebraic decays
Anomalous diffusion
F. Bouchet
Short-ranged (gases)
a/l = 1/ πa2 n
Collisionless Boltzmann
Boltzmann equation
Yes
Yes
Yes
Local thermal equilibrium
CNRS–ENSL
∝ l/v̄ or larger
Yes
Yes
Long-range
1/N
Vlasov equation
Lenard-Balescu
Yes
Yes
Yes
Quasistationary
states
∝ N or larger
?
?
?
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The Lenard-Balescu equation
Lenard-Balescu – Aging correlations
Algebraic Decays and Anomalous Diffusion
Long range system : the relaxation of a test particle in a bath
Using kinetic theory, we derive a Fokker-Planck equation for a
particle in a bath. Rapidly decaying diffusion coefficient.
The Fokker Planck equation has a continuous spectrum.
Algebraic decay for large times of correlation functions.
We compute analytically the exponent using matched
asymptotic expansions.
F. Bouchet, T. Dauxois (2004) Phys. Rev. E,
Y. Yamaguchi, F. Bouchet, and T. Dauxois (2007) J. Stat. Mech.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The Lenard-Balescu equation
Lenard-Balescu – Aging correlations
What are the Limits of Kinetic Theory ?
How long are the Vlasov equation valid, or the Lenard-Balescu equation valid ?
The C log(N) upper bound for the validity of the Vlasov
equation is optimal K. Jain, F. Bouchet, and D. Mukamel (2007), J.Stat.
Mech.
τ ∝ N 1.7 . This is not consistent with simple use of kinetic theory
Y.Y. Yamaguchi, J. Barré, F. Bouchet, T. Dauxois and S. Ruffo, (2004) Physica A,
F. Bouchet, T. Dauxois (2004) Phys. Rev. E
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
Stochastic Mean Field Hamiltonian model
The kinetic equation for the stochastic dynamics
Non-equilibrium phase transitions
Outline
1
2
3
Anomalous diffusion and aging in systems with long range
interactions
The Lenard-Balescu equation
Anomalous diffusion, aging and correlations
Kinetic theory and non-equilibrium phase transitions for systems
with long range interactions
Stochastic Mean Field Hamiltonian model
The kinetic equation for the stochastic dynamics
Non-equilibrium phase transitions
Kinetic theory and non-equilibrium phase transitions for
geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging and jet formation in geostrophic
turbulence.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
Stochastic Mean Field Hamiltonian model
The kinetic equation for the stochastic dynamics
Non-equilibrium phase transitions
Mean Field Hamiltonian and Stochastic Forces
A simpler framework from a mathematical point of view
H=
1 N 2
1
pk +
∑
2 k=1
2N
N
∑
k ,l=1
V (qk − ql ).
Hamiltonian dynamics plus stochastic forces
q̇i =
∂H
,
∂ pi
ṗi = −
and
√
∂H
− αpi + α F (qi , t).
∂ qi
α: friction constant. F (q, t) is a homogeneous Gaussian
process with zero mean and variance
hF (q, t)F (q 0 , t 0 )i = C (|q − q 0 |)δ (t − t 0 ).
Force spectrum: ck ≡ 21π 02π dq C (q)e −ikq > 0.
√
We expect fluctuations
of order α due to the stochastic
√
force and 1/ N due to the potential.
R
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
Stochastic Mean Field Hamiltonian model
The kinetic equation for the stochastic dynamics
Non-equilibrium phase transitions
Energy Balance
H=
q̇i =
∂H
,
∂ pi
1
1 N 2
∑ pk + 2N
2 k=1
N
∑
k ,l=1
ṗi = −
and
V (qk − ql )
√
∂H
− αpi + α F (qi , t).
∂ qi
A close equation for the average specific energy e = H/N
de
dt
= −2α hT i +
α
C (0).
2
2
T = ∑N
i=1 pi /(2N) is the kinetic energy per particle.
Average kinetic energy, for the stationary state
hT iss = C (0)/4.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
Stochastic Mean Field Hamiltonian model
The kinetic equation for the stochastic dynamics
Non-equilibrium phase transitions
N-particle Fokker-Planck Equation
Evolution of the N-particle distribution function
fN (q1 , ..., qN , p1 , ..., pN , t) (after averaging over the noise
realization)
N
∂f
1
∂ fN
+ ∑ pi N +
∂t
∂ qi
2N
i=1
... = α
N
∑
i,j=1
V 0 (qi − qj )
N
N
∑
∑
∂ (pi fN ) α
+
2
i=1 ∂ pi
i,j=1
∂
∂
−
f = ...
∂ pi ∂ pj N
C (qi − qj )
∂ 2 fN
.
∂ pi ∂ pj
With α = 0, we get the Liouville equation for the Hamiltonian
dynamics.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
Stochastic Mean Field Hamiltonian model
The kinetic equation for the stochastic dynamics
Non-equilibrium phase transitions
Outline
1
2
3
Anomalous diffusion and aging in systems with long range
interactions
The Lenard-Balescu equation
Anomalous diffusion, aging and correlations
Kinetic theory and non-equilibrium phase transitions for systems
with long range interactions
Stochastic Mean Field Hamiltonian model
The kinetic equation for the stochastic dynamics
Non-equilibrium phase transitions
Kinetic theory and non-equilibrium phase transitions for
geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging and jet formation in geostrophic
turbulence.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
Stochastic Mean Field Hamiltonian model
The kinetic equation for the stochastic dynamics
Non-equilibrium phase transitions
The BBGKY Hierarchy
The n particle distribution function
fn (q1 , p1 , ..., qn , pn , t) =
Z
N
∏
dqi dpi fN (q1 , p1 , ..., qN , pN , t).
i=n+1
We denote f (q, p, t) = f1 (q, p, t)
We anticipate that at leading order we expect loss of
correlation (Stosszahl ansatz)
N
fn (q1 , p1 , ..., qn , pn , t) = ∏ f (qi , pi , t) + αgn (q1 , p1 , ..., qn , pn , t).
i=1
Each gn is governed by an equation involving (f , g2 , ..., gn+1 ).
This is called the BBGKY hierarchy.
g2 is denoted g .
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
Stochastic Mean Field Hamiltonian model
The kinetic equation for the stochastic dynamics
Non-equilibrium phase transitions
The Two First Equations of the BBGKY Hierarchy
For pedagogical reasons, we assume an homogenous state:
f (q, p) = f (p) and g (q1 , q2 , p1 , p2 ) = g (q1 − q2 , p1 , p2 )
Z
∂f
∂
C (0) ∂ 2 f
∂
0
+Vlasov [f ] = α
(pf ) +
dq2 dp2 V (q2 )g (q2 , p, p2 , t) .
+
∂t
∂p
2 ∂ p2 ∂ p
Z
∂g
∂g
∂f
+ p1
(p1 ) dq3 dp3 V 0 (q1 − q3 )g (q3 − q2 , p3 , p2 , t) +{1 ↔ 2} = ...
−
∂t
∂ q1 ∂ p
... = C (q1 − q2 )
∂f
∂f
α
(p1 ) (p2 ) + N2 (f , g ) + αN3 (f , g , g3 ).
∂p
∂p
N
The hierarchy appears as an ordered expansion both in powers
of α and 1/N.
We discuss for instance the mean-field stochastic regime
1/N α 1 (the limit limα→0 limN →∞ ).
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
Stochastic Mean Field Hamiltonian model
The kinetic equation for the stochastic dynamics
Non-equilibrium phase transitions
Truncation at First Non Trivial Order
Z
∂f
∂
C (0) ∂ 2 f
∂
0
+ Vlasov [f ] = α
(pf ) +
+
dq2 dp2 V (q2 )g (q2 , p, p2 , t) .
∂t
∂p
2 ∂ p2 ∂ p
∂f
∂g
∂f
+ L1f [g ] + L2f [g ] = C (q1 − q2 ) (p1 ) (p2 ).
∂t
∂p
∂p
We assume that f (p) is a stable steady solution of the Vlasov
equation, in order for f to evolve over a slow time scale (for
instance 1/α ).
We assume, that for fixed f , the Lyapunov equation converge for
large times to g∞ [f ].
Bogolyubov hypothesis: the kinetic equation is
Z
∂f
∂
C (0) ∂ 2 f
∂
0
+Vlasov [f ] = α
(pf ) +
+
dq
dp
V
(q
)g
[f
](q
,
p,
p
)
.
∞
2
2
2
2
2
∂t
∂p
2 ∂ p2 ∂ p
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
Stochastic Mean Field Hamiltonian model
The kinetic equation for the stochastic dynamics
Non-equilibrium phase transitions
Kinetic Equation
All computations can be performed explicitly. Then
∂f
∂ (pf )
∂
∂f
+ Vlasov [f ] = α
+
D[f ]
∂t
∂p
∂p
∂p
∞
1
D[f ](p) = C (0)+2π ∑ Vk ck
2
k=1
Z ∗
dp1
1
1
1
∂ f +
.
|ε(k, kp)|2 |ε(k, kp1 )|2 p1 − p ∂ p p1
R∗
indicates the Cauchy principal value of the integral, and the
dielectric function ε is
ε(k, ω) = lim
η→0+
Z
1 − 2πivk k dp
1
∂f
.
−i(ω + iη) + ikp ∂ p
The equation correctly predicts the energy balance.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
Stochastic Mean Field Hamiltonian model
The kinetic equation for the stochastic dynamics
Non-equilibrium phase transitions
Kinetic Evolution from Numerical Simulation
Kinetic energy hκi as a function of
hp 4 i as a function of αt
αt
(C (0) = 1.5, and c1 = 0.75)
This is an evidence of kinetic evolution over a time scale 1/α
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
Stochastic Mean Field Hamiltonian model
The kinetic equation for the stochastic dynamics
Non-equilibrium phase transitions
Kinetic Equation Predicts the Stationary Distribution
Stationary distribution
Stationary diffusion coefficient
(α = 0.01, C (0) = 1.5, and
c1 = 0.75)
Very good agreement between kinetic theory and N-particle
numerical simulations.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
Stochastic Mean Field Hamiltonian model
The kinetic equation for the stochastic dynamics
Non-equilibrium phase transitions
Mathematical Study of the Kinetic Equation ?
All computations can be performed explicitly. Then
∂f
∂ (pf )
∂
∂f
+ Vlasov [f ] = α
+
D[f ]
∂t
∂p
∂p
∂p
∞
1
D[f ](p) = C (0)+2π ∑ Vk ck
2
k=1
Z ∗
dp1
1
1
1 ∂f
+
(p1 ).
|ε(k, kp)|2 |ε(k, kp1 )|2 p1 − p ∂ p
R∗
indicates the Cauchy principal value of the integral, and the
dielectric function ε is
ε(k, ω) = lim
η→0+
Z
1 − 2πivk k dp
1
∂f
.
−i(ω + iη) + ikp ∂ p
The equation correctly predicts the energy balance.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
Stochastic Mean Field Hamiltonian model
The kinetic equation for the stochastic dynamics
Non-equilibrium phase transitions
Outline
1
2
3
Anomalous diffusion and aging in systems with long range
interactions
The Lenard-Balescu equation
Anomalous diffusion, aging and correlations
Kinetic theory and non-equilibrium phase transitions for systems
with long range interactions
Stochastic Mean Field Hamiltonian model
The kinetic equation for the stochastic dynamics
Non-equilibrium phase transitions
Kinetic theory and non-equilibrium phase transitions for
geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging and jet formation in geostrophic
turbulence.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
Stochastic Mean Field Hamiltonian model
The kinetic equation for the stochastic dynamics
Non-equilibrium phase transitions
Mean Field Hamiltonian and Stochastic Forces
A simpler framework from a mathematical point of view
We assume now V (q) = − cos(q).
H=
1
1 N 2
∑ pk − 2N
2 k=1
N
∑
k ,l=1
cos(qk − ql )
Hamiltonian dynamics plus stochastic forces
q̇i =
∂H
,
∂ pi
ṗi = −
and
√
∂H
− αpi + α F (qi , t).
∂ qi
α: friction constant. F (q, t) is a homogeneous Gaussian
process with zero mean and variance
hF (q, t)F (q 0 , t 0 )i = C (|q − q 0 |)δ (t − t 0 ).
Force spectrum: ck ≡
F. Bouchet
1
2π
R 2π
0
CNRS–ENSL
dq C (q)e −ikq > 0.
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
Stochastic Mean Field Hamiltonian model
The kinetic equation for the stochastic dynamics
Non-equilibrium phase transitions
Equilibrium Phase Transition
The force spectrum ck is uniform for 1 ≤ k ≤ 50: close to
equilibrium dynamics (Langevin).
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
Stochastic Mean Field Hamiltonian model
The kinetic equation for the stochastic dynamics
Non-equilibrium phase transitions
Non-Equilibrium Phase Transitions
ck is constant for 1 ≤ k ≤ 7
ck is uniform for 1 ≤ k ≤ 3 and
and zero otherwise.
sero otherwise
When the force spectrum ck is not uniform, the transition is
shifted and becomes discontinuous.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
Stochastic Mean Field Hamiltonian model
The kinetic equation for the stochastic dynamics
Non-equilibrium phase transitions
Non-Equilibrium Phase Transitions and Hysteresis
α = 0.01
α = 0.005
The hysterical behavior of the non-equilibrium phase
transitions confirm they are discontinuous.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
Stochastic Mean Field Hamiltonian model
The kinetic equation for the stochastic dynamics
Non-equilibrium phase transitions
Non-Equilibrium Phase Transitions: Temperature
Dependance
The order parameter as a
PDF of the order parameter
function of time
The temperature dependance of the order paramater PDF is
consistent with a first order transition.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
Stochastic Mean Field Hamiltonian model
The kinetic equation for the stochastic dynamics
Non-equilibrium phase transitions
Non-Equilibrium Phase Transitions: Transition Rates
The order parameter as a
Waiting times as a function of
function of time
α
Even if our data quality is not very good, the transition rates
are consistent with an Arhenius law.
C. NARDINI, S. GUPTA, S. RUFFO, T. DAUXOIS, and F. BOUCHET, 2012,
J. Stat. Mech., L01002, and 2012 J. Stat. Mech., P12010.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging for geostrophic jets.
Outline
1
2
3
Anomalous diffusion and aging in systems with long range
interactions
The Lenard-Balescu equation
Anomalous diffusion, aging and correlations
Kinetic theory and non-equilibrium phase transitions for systems
with long range interactions
Stochastic Mean Field Hamiltonian model
The kinetic equation for the stochastic dynamics
Non-equilibrium phase transitions
Kinetic theory and non-equilibrium phase transitions for
geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging and jet formation in geostrophic
turbulence.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging for geostrophic jets.
The 2D Euler Equations
2D Euler equations:
∂ω
+ v [ω] .∇ω = 0,
∂t
Vorticity ω = (∇ ∧ v) .ez . Stream function ψ: v = ez × ∇ψ,
ω = ∆ψ.
Conservative dynamics - Hamiltonian (non canonical) and time
reversible.
Invariants:
Z
Z
1
1
Energy: E [ω] =
v2 dr = −
ωψ dr,
2 D
2 D
Casimir’s functionals: Cs [ω] =
Vorticity distribution:
F. Bouchet
Z
D
s(ω) dr,
dA
D (σ ) =
with A (σ ) =
dσ
CNRS–ENSL
Z
D
χ{ω(x)≤σ } dr.
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging for geostrophic jets.
Equilibrium Large Deviation: Macrostate Entropy
The most probable vorticity field
A probabilistic description of the vorticity field ω: p (x, σ ) is
the local probability to have ω (x) = σ at point x.
A measure of the number of microscopic field ω corresponding
to a probability p (Liouville and Sanov theorems):
Macrostate entropy : S [p] ≡ −
1
|D|
Z
D
drdσ p log p.
The microcanonical RSM variational problem (MVP):
S (E ) = sup {S2 [p] | E [ω] = E and D (σ ) = d (σ )} (MVP).
{p|N [p]=1}
Critical points are steady solutions of the 2D Euler equations:
ω = fd (β ψ).
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging for geostrophic jets.
Statistical Equilibria for the 2D-Euler Eq. (torus)
A second order phase transition.
Z. Yin, D. C. Montgomery, and H. J. H. Clercx, Phys. Fluids (2003)
F. Bouchet, and E. Simonnet, PRL, (2009) (Lyapunov Schmidt
reduction, normal form analysis).
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging for geostrophic jets.
Outline
1
2
3
Anomalous diffusion and aging in systems with long range
interactions
The Lenard-Balescu equation
Anomalous diffusion, aging and correlations
Kinetic theory and non-equilibrium phase transitions for systems
with long range interactions
Stochastic Mean Field Hamiltonian model
The kinetic equation for the stochastic dynamics
Non-equilibrium phase transitions
Kinetic theory and non-equilibrium phase transitions for
geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging and jet formation in geostrophic
turbulence.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging for geostrophic jets.
The 2D Stochastic-Navier-Stokes (SNS) Equations
The simplest model for two dimensional turbulence.
Navier Stokes equations with random forces
√
∂ω
+ v.∇ω = ν∆ω − αω + σ fs ,
∂t
where ω = (∇ ∧ v) .ez is the vorticity, fs is a random force, α is the
Rayleigh friction coefficient.
An academic model with experimental realizations (Sommeria
and Tabeling experiments, rotating tanks, magnetic flows, and
so on). Analogies with geophysical flows (Quasi Geostrophic
and Shallow Water layer models).
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging for geostrophic jets.
Statistical Equilibria for the 2D-Euler Eq. (torus)
A second order phase transition.
Z. Yin, D. C. Montgomery, and H. J. H. Clercx, Phys. Fluids (2003)
F. Bouchet, and E. Simonnet, PRL, (2009) (Lyapunov Schmidt
reduction, normal form analysis).
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging for geostrophic jets.
The 2D Stochastic Navier-Stokes Equations
√
∂ω
+ u.∇ω = ν∆ω − αω + 2αfs .
∂t
We would like to describe the invariant measure:
Is it concentrated close to steady solutions of the 2D Euler
(quasi-geostrophic) equations?
Can we describe the dynamics among these states?
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging for geostrophic jets.
Non-Equilibrium Phase Transition (2D Navier–Stokes Eq.)
The time series and PDF of the Order Parameter
R
Order parameter : z1 = dxdy exp(iy )ω (x, y ).
For unidirectional flows |z1 | ' 0, for dipoles |z1 | ' 0.6 − 0.7
F. Bouchet and E. Simonnet, PRL, 2009.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging for geostrophic jets.
2D Stochastic Navier-Stokes Eq. and 2D Euler Steady
States
√
∂ω
+ v.∇ω = ν∆ω − αω + 2αfs
∂t
This is no more a Langevin dynamics.
Time scale separation: magenta terms are small.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging for geostrophic jets.
Non-Equilibrium Phase Transition (2D Navier–Stokes Eq.)
The time series and PDF of the Order Parameter
R
Order parameter : z1 = dxdy exp(iy )ω (x, y ).
For unidirectional flows |z1 | ' 0, for dipoles |z1 | ' 0.6 − 0.7
F. Bouchet and E. Simonnet, PRL, 2009.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging for geostrophic jets.
Instantons: Maximum Likelihood Paths
Most trajectories that lead to a rare event follow the easiest
path.
Large deviation theory: instantons as minimum action paths.
0.8
|ω̂(0,1) |
0.6
0.4
0.2
0
2D Navier-Stokes equations
(time: 10 000) (PRL)
0
0.2
0.4
0.6
|ω̂(1,0) |
0.8
1
Numerical instanton (time of
order 1) (J. Stat. Phys.)
Goal: predict attractors, transition pathways and probabilities.
Instanton computations will predict them when it is not
possible to do that using direct numerical simulations.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging for geostrophic jets.
Bistability in a Rotating Tank Experiment
Rotating tank with a single-bump topography
Bistability (hysteresis) in rotating tank experiments
M. MATHUR, and J. SOMMERIA, to be submitted to J. Geophys. Res., M.
MATHUR, J. SOMMERIA, E. SIMONNET, and F. BOUCHET, in preparation.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging for geostrophic jets.
Numerical Computation of Rare Events and Large Deviations
Computation of least action paths (instantons) and/or multilevel splitting
0.8
|ω̂(0,1) |
0.6
0.4
0.2
0
Multilevel-splitting: Ginzburg-Landau
transitions (with E. Simonnet)
0
0.2
0.4
0.6
|ω̂(1,0) |
0.8
1
2D Navier-Stokes
instantons (with J. Laurie)
Rare events and their probability can now be computed
numerically in complex dynamical systems.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging for geostrophic jets.
Outline
1
2
3
Anomalous diffusion and aging in systems with long range
interactions
The Lenard-Balescu equation
Anomalous diffusion, aging and correlations
Kinetic theory and non-equilibrium phase transitions for systems
with long range interactions
Stochastic Mean Field Hamiltonian model
The kinetic equation for the stochastic dynamics
Non-equilibrium phase transitions
Kinetic theory and non-equilibrium phase transitions for
geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging and jet formation in geostrophic
turbulence.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging for geostrophic jets.
The Barotropic Quasi-Geostrophic Equations
The simplest model for geostrophic turbulence.
Quasi-Geostrophic equations with random forces
√
∂q
+ v.∇q = ν∆ω − αω + 2αfs ,
∂t
with q = ω + β y .
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging for geostrophic jets.
The Inertial Limit
The non-dimensional version of the barotropic QG equation.
Quasi-Geostrophic equations with random forces
√
∂q
+ v.∇q = ν∆ω − αω + 2αfs ,
∂t
with q = ω + β 0 y .
Spin up or spin down time = 1/α 1 = jet inertial time scale.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging for geostrophic jets.
Jet Formation in the Barotropic QG Model
In the inertial (weak forces and dissipation) limit
NL
U ( y , t ) , f /f c =10
6
6
0.2
5
4
0.1
3
0
2
−0.1
2
1
−0.2
1
0
0
500
NL
t
1000
4
y
y
5
0
1500
0
0.2
U
6
0.2
5
5
4
0.1
3
0
2
−0.1
2
−0.2
1
1
0
0
−0.2
U ( y , t ) , f /f c =10
6
500
t
1000
1500
4
y
y
3
3
0
−0.2
0
0.2
U
Figure by P. Ioannou (Farrell and Ioannou).
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging for geostrophic jets.
Weak Fluctuations around Jupiter’s Zonal Jets
Jupiter’s atmosphere.
Jupiter’s zonal winds (Voyager and
Cassini, from Porco et al 2003).
We will treat those weak fluctuations perturbatively (inertial limit).
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging for geostrophic jets.
Averaging out the Turbulence
√
∂q
+ v.∇q = ν∆ω − αω + 2αfs .
∂t
P [q] is the PDF for the Potential Vorticity field q (a
functional). Fokker–Planck equation:
∂P
=
∂t
Z
δ
dr
δ q(r)
Z
δ
0
0
v.∇q − ν∆ω + αω + dr C (r, r )
P .
δ q(r)
Time scale separation. We decompose into slow (zonal flows)
and fast variables (eddy turbulence)
qz (y ) = hqi ≡
1
2π
Z
D
√
dx q and q = qz + αqm .
Stochastic reduction (Van Kampen, Gardiner, ...) using the
time scale separation.
We average out the turbulent degrees of freedom.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging for geostrophic jets.
A New Fokker–Planck Equation for the Zonal Jets
R [qz ] is the PDF to observe the Zonal Potential Vorticity qz :
1 ∂R
=
α ∂t
Z
dy1
∂
ν ∂ 2 qz
Eqz vm,y qm + ωz (y1 ) −
(y1 )+
∂y
α ∂y2
Z
δ
R .
+ dy2 Cz (y1 , y2 )
δ qz (y2 )
δ
δ qz (y1 )
This new Fokker–Planck equation is equivalent to the
stochastic dynamics
∂
ν ∂ 2 qz
1 ∂ qz
= − Eqz hvm,y qm i − ωz +
+ ηz ,
α ∂t
∂y
α ∂y2
with hηz (y , t)ηz (y 0 , t 0 )i = Cz (y , y 0 )δ (t − t 0 ).
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging for geostrophic jets.
The Deterministic Part and the Quasilinear Approximation
Deterministic quasilinear dynamics
1 ∂ qz
ν ∂ 2 qz
= F [qz ] − ω z +
.
α ∂t
α ∂y2
F [qz ] = − ∂∂y Eqz hvm,y qm i. The average of the Reynolds stress
is over the statistics of the quasilinear dynamics:
∂t qm + U(y )
√
∂ qm
∂ qz
+ vm,y
= ν∆qm − αωm + 2αfs .
∂x
∂y
and
hvm,y qm i =
1
Ly
Z
dy Eqz [vm,y qm ] .
We identify SSST by Farrell and Ioannou (JAS, 2003); quasilinear
theory by Bouchet (PRE, 2004); CE2 by Marston, Conover and
Schneider (JAS, 2008); Sreenivasan and Young (JAS, 2011).
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging for geostrophic jets.
Dynamics of the Relaxation to the Averaged Zonal Flows
Deterministic quasilinear dynamics
NL
U ( y , t ) , f /f c =10
6
6
0.2
5
4
0.1
3
0
2
−0.1
2
1
−0.2
1
0
0
500
NL
t
1000
4
y
y
5
0
1500
0
0.2
U
6
0.2
5
5
4
0.1
3
0
2
−0.1
2
1
−0.2
1
0
0
−0.2
U ( y , t ) , f /f c =10
6
500
t
1000
1500
4
y
y
3
3
0
−0.2
0
0.2
U
Figure by P. Ioannou (Farrell and Ioannou).
Extremely efficient numerical simulation of the average jet
dynamics.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging for geostrophic jets.
Troposphere Dynamics and the Quasilinear Approximation
Comparison of quasilinear approximation and DNS for the primitive equations
30
30
36
36
6
−3
30
30
30
6
−9
30
20
20
0.2
0.2
15
15
21
21
10
−3
10
3
12
3
0
12
Sigma
3
0
Sigma
9
9
3
3
3
−10
−10
3
3
3
3
0.8
0.8
−3
−3
−20
−20
−3
−3
−30
−60
−30
0
30
60
−30
−60
Latitude
−30
0
30
60
Latitude
Full equations (DNS).
Quasilinear approximation.
Zonal wind and momentum convergence for the primitive equations.
Farid Ait Chaalal and Tapio Schneider (Caltech and ETH Zurich).
The qualitative structure of a fast rotating Earth troposphere
is well approximated by quasilinear dynamics.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging for geostrophic jets.
The Stochastic Dynamics of the Zonal Jet
The turbulence has been averaged out
We can now go further. What is the effect of the noise term?
1 ∂ qz
ν ∂ 2 qz
= F [qz ] − ω z +
+ηz .
α ∂t
α ∂y2
R [qz ] is the PDF to observe the Zonal Potential Vorticity qz :
1 ∂R
=
α ∂t
Z
dy1
∂
ν ∂ 2 qz
Eqz vm,y qm + ωz (y1 ) −
(y1 )+
∂y
α ∂y2
Z
δ
R .
+ dy2 Cz (y1 , y2 )
δ qz (y2 )
δ
δ qz (y1 )
This equation describes the zonal jet statistics and not only
the mean zonal flow.
This statistics can be nearly Gaussian, but can also be strongly
non-Gaussian.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging for geostrophic jets.
Rare Transitions in Real Flows?
Rotating tank experiments (Quasi Geostrophic dynamics)
Transitions between blocked and zonal states:
Y. Tian and col, J. Fluid. Mech. (2001) (groups of H. Swinney and
M. Ghil).
Can such multiple attractors and rare transitions exists for
geostrophic turbulence?
Theory based on non-equilibrium statistical mechanics?
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging for geostrophic jets.
Multiple Attractors Do Exist for the Barotropic QG Model
Two attractors for the same set of parameters
NL
U ( y , t ) , f /f c =10
6
6
0.2
5
4
0.1
3
0
2
−0.1
2
−0.2
1
1
0
0
500
NL
t
1000
4
y
y
5
0
1500
0
0.2
U
6
0.2
5
5
4
0.1
3
0
2
−0.1
2
1
−0.2
1
0
0
−0.2
U ( y , t ) , f /f c =10
6
500
t
1000
1500
4
y
y
3
3
0
−0.2
0
0.2
U
Figure by P. Ioannou (Farrell and Ioannou).
Two attractors for the mean zonal flow for one set of
parameters.
What is the dynamics for the transition? What is the rate?
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging for geostrophic jets.
Work in Progress : Zonal Flow Instantons
Onsager Machlup formalism (50’). Statistical mechanics of histories
1 ∂ qz
ν ∂ 2 qz
= F [qz ] − ω z +
+ηz .
α ∂t
α ∂y2
Path integral representation of transition probabilities:
P(qz,0 , qz,T , T ) =
S [qz ] =
1 T
dt
2 0
Z
Z
dy1 dy2
Z q(T )=q
z,T
q(0)=qz,0
D [qz ] exp (−S [qz ]) with
ν ∂ 2 qz
∂ qz
ν ∂ 2 qz
∂ qz
− F [qz ] + ω z −
(y1 )CZ (y1 , y2 )
− F [qz ] + ω z −
(y2 ).
∂t
α ∂y2
∂t
α ∂y2
Instanton (or Freidlin-Wentzel theory): the most probable path
with fixed boundary conditions
S(qz ,0 , qz ,T , T ) =
F. Bouchet
{S [qz ]} .
min
{qz |qz (0)=qz,0 and qz (T )=qz,T }
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging for geostrophic jets.
The Real Issue was to Cope with UltraViolet Divergences
We have proven that they are no such divergences
∂t qm + U(y )
√
∂ qm
∂ qz
+ vm,y
= ν∆qm − αωm + 2αfs
∂x
∂y
We need to prove that the Gaussian process has an invariant
measure which is well behaved in the limit ν → 0, and α → 0.
This is true because the linearized Quasi-Geostrophic or Euler
dynamics is non-normal.
The result is based on asymptotics of the linearized equations:
vm,x (y , t) ∼
t→∞
vm,y ,∞ (y )
vm,x,∞ (y )
exp (−ikU(y )t) and vm,y (y , t) ∼
exp (−ikU(y )t) .
t→∞
t
t2
F. Bouchet and H. Morita, 2010, Physica D.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging for geostrophic jets.
Stat. Mech. of Zonal Jets: Conclusions
Stochastic averaging for the barotropic Quasi-Geostrophic
equation leads to a non-linear Fokker-Planck equation.
This Fokker-Planck equation predicts the Reynolds stress and
jet statistics. Related to Quasilinear theory and SSST.
For some parameters, multiple attractors are observed.
Path integral, instanton and large deviation theories can
predict rare transitions between attractors.
F. Bouchet, C. Nardini and T. Tangarife, 2013 J. Stat. Phys. in press,
http://hal.archives-ouvertes.fr/hal-00819779.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.
Kinetic theory of particle systems
Non equilibrium transitions in particle dynamics
Non equilibrium phase transitions in geostrophic turbulence
The 2D Euler equations and equilibrium statistical mechanics
Non equilibrium phase transitions
Stochastic averaging for geostrophic jets.
Summary and Perspectives
Non-equilibrium statistical mechanics and large deviations can
be applied to self-gravitating systems and geophysical
turbulence.
Ongoing projects and perspectives:
Large deviations and non-equilibrium free energies for particles
with long range interactions (with K. Gawedzki).
Microcanonical measures for the Shallow Water equations
(with M. Potters and A. Venaille) and for the 3D axisymmetric
Euler equations (with S. Thalabard).
Instantons for zonal jets in the quasi-geostrophic dynamics
(with C. Nardini, T. Tangarife and O. Zaboronski).
Rare events, large deviations, and extreme heat waves in the
atmosphere (with J. Wouters).
F. Bouchet, and A. Venaille, Physics Reports, 2012, Statistical mechanics of
two-dimensional and geophysical flows.
F. Bouchet
CNRS–ENSL
Phase transitions in systems with long range interactions.