Hamiltonian formulation of reduced Vlasov-Maxwell equations Cristel CHANDRE Centre de Physique Théorique – CNRS, Marseille, France
Contact:
[email protected]
¾ importance of stability vs instability in devices involving a large number of charged particles interacting with fields: plasma physics (tokamaks), free electron lasers ¾ Here: reduced models of such systems (easier simulation, better understanding of the dynamics)
Outline - Hamiltonian description of charges particles and electromagnetic fields - Reduction of Vlasov-Maxwell equations using Lie transforms Alain J. BRIZARD (Saint Michael’s College, Vermont, USA) - Reduced Hamiltonian model for the Free Electron Laser Romain BACHELARD (Synchrotron Soleil, Paris) Michel VITTOT (CPT, Marseille)
Motion of a charged particle in electromagnetic fields > in canonical form 2
H (p, q, t ) =
⎛ ⎞ ⎜⎜p − e A (q, t )⎟⎟ ⎟ ⎜⎝ c ⎠⎟ 2m
+ eV (q, t )
with
{ f , g} =
⎧ ⎪ ∂H ⎪ , p H = p = − { } ⎪ ⎪ ∂q equations of motion : ⎪ ⎨ ⎪ ∂H ⎪ q = {q, H } = ⎪ ⎪ ∂p ⎪ ⎩
> in non-canonical form: physical variables
∂ f ∂g ∂ f ∂ g ⋅ − ⋅ ∂q ∂ p ∂ p ∂ q
1⎛ e ⎞ ⎜⎜p − A (q, t )⎟⎟ ⎠ m⎝ c x=q v=
1 1 ⎛ ∂f ∂g ∂f ∂g ⎞⎟ eB ⎛⎜ ∂f ∂g ⎞⎟ ⎟⎟ + 2 ⋅ ⎜ × ⎟⎟ − ⋅ mv 2 + eV (x, t ) with { f , g } = ⎜⎜⎜ ⋅ 2 m ⎝ ∂x ∂v ∂v ∂x ⎠⎟ m c ⎜⎝ ∂v ∂v ⎠⎟ ⎧⎪ gyroscopic bracket x = {x, H } = v ⎪⎪ equations of motion : ⎪⎨ e⎛ v × B ⎞⎟ ⎪⎪v = {v, H } = ⎜⎜E + ⎟⎟ ⎜ ⎪⎪⎩ m⎝ c ⎠⎟ H (v, x, t ) =
Definition: Hamiltonian system - a scalar function H , the Hamiltonian - a Poisson bracket {F ,G } with the properties
{F ,G } = − {G, F } Leibnitz law {F ,GK } = {F ,G } K + G {F , K } Jacobi identity {{F ,G }, K } + {{K , F },G } + {{G, K }, F } = 0 antisymmetric
- equations of motion dF = {F , H } dt - a conserved quantity
{F , H } = 0
Eulerian version: case of a density of charged particles - density of particles in phase space f (x, v, t ) 1 example: f (x, v, t ) = ∑ δ x − xi (t ) δ v − vi (t ) Klimontovitch distribution N i
(
)(
)
- evolution given by the Vlasov equation ⎞⎟ ∂f ∂f e ⎛⎜ v = −v ⋅ ∇f − ⎜⎜E + × B⎟⎟ ⋅ m⎝ c ∂t ⎠⎟ ∂v - Eulerian, not Lagrangian: dF ∂F for any observable F ⎡⎢⎣ f ⎤⎦⎥ , we have = = ⎡⎢⎣ F , H⎤⎦⎥ dt ∂t - still a Hamiltonian system Hamiltonian : H ⎡⎢⎣ f ⎤⎥⎦ =
⎛m 2 ⎞⎟ ⎜ ∫∫ d xd v f ⎜⎜⎝ 2 v + eV ⎠⎟⎟⎟ 3
3
with ⎡⎢⎣ F , G ⎤⎦⎥ =
∫∫
⎧⎪ δF δG ⎫⎪ d xd v f ⎪⎨ , ⎪⎬ ⎪⎩⎪ δf δf ⎪⎪⎭ 3
3
Eulerian version: case of a density of charged particles - an example: ρ ⎡⎢⎣ f ⎤⎦⎥ (x 0 ) = e ∫ d 3v f (x 0 , v) ⎧ ⎪ ∂ρ ⎡ δρ δH ⎫⎪⎪ 3 3 ⎪ ⎤ = ⎢⎣ρ, H⎥⎦ = ∫ d xd v f ⎨ , ⎬ ⎪ ∂t ⎪ δf δf ⎪⎭⎪ ⎩ - functional derivatives F
⎡ f + φ ⎤ =F ⎢⎣ ⎥⎦
⎡ f ⎤ + d 3xd 3v δF φ + O φ2 ⎢⎣ ⎥⎦ ∫ δf
δρ δH 1 - here: = e δ (x − x 0 ) and = mv2 + eV 2 δf δf - therefore:
∂ ∂ρ =− ⋅ J with J ⎣⎡⎢ f ⎦⎤⎥ = e ∫ d 3v f v ∂x ∂t
( )
Vlasov-Maxwell equations: self-consistent dynamics > description of the dynamics of a collisionless plasma (low density)
Variables: particle density f(x,v,t), electric field E(x,t), magnetic field B(x,t)
⎞⎟ ∂f ∂f e ⎛⎜ v = −v ⋅ ∇f − ⎜E + × B⎟⎟ ⋅ ∂t m ⎜⎝ c ⎠⎟ ∂v ∂B = −c∇ × E ∂t ∂E = c∇ × B − 4πJ ∂t where ∇ ⋅ E = 4πρ and
∇⋅ B = 0
Vlasov-Maxwell equations... still a Hamiltonian system 2
E +B m 2 3 ∫∫ d xd v f 2 v + ∫ d x 8π ⎧ ⎫ ⎪ ⎪ F G δ δ 3 3 ⎪ ⎡ ⎤ with ⎢⎣ F , G ⎥⎦ = ∫∫ d xd v f ⎨ , ⎪ ⎬ ⎪ ⎪ ⎩⎪ δf δf ⎭⎪ ⎡ 4πe δF δG ⎤⎥ 3 3 ∂f ⎢ δ F δ G d xd v + ⋅ − ∫∫ ⎢ m ∂v ⎣ δ E δ f δf δE ⎥⎦ ⎡ δG δF δG ⎤⎥ 3 ⎢ δF +4πc ∫ d x ⋅ ∇× − ∇× ⋅ ⎢ δE B B δ δ δE ⎥⎦ ⎣
Hamiltonian : H ⎡⎣⎢ E, B, f ⎤⎦⎥ =
3
2
3
⎡ E, B, f ⎤ : d F = ∂F = ⎡ F , H⎤ ⎢⎣ ⎥⎦ dt ⎢⎣ ⎥⎦ ∂t Remark: div B et div E − ∫ d 3 p f are conserved quantities Equation of motion for F
antisymmetry, Leibnitz, Jacobi Morrison, PLA (1980) Marsden, Weinstein, Physica D (1982)
From microscopic to macroscopic Vlasov-Maxwell equations - Elimination (or decoupling) of fast time and small spatial scales for a better understanding of complex plasma phenomena
- reduced Maxwell equations in terms of D and H ∇ ⋅ D = 4πρR ∂D = c∇ × H − 4πJR ∂t D = E + 4πP, H = B − 4πM where
∂P ρR = ρ + ∇ ⋅ P, JR = J − c∇ × M − ∂t
reduced polarization density / magnetization current / polarization current density
- Can we represent the reduced Vlasov-Maxwell equations as a Hamiltonian system? Hint: use of Lie transforms - Deliverables: Expressions of the polarization P and magnetization M vectors
Reduced fields as Lie transforms of f, E and B Given a functional S ⎡⎢E (x, t ), B (x, t ), f (x, v, t )⎤⎥ , we define some new fields as ⎣ ⎦ ⎛ ⎞ ⎜⎜ E − ⎡ S, E⎤ + 1 ⎡ S, ⎡ S, E⎤ ⎤ + "⎟⎟ ⎟⎟ ⎢⎣ ⎥⎦ 2 ⎢⎣ ⎢⎣ ⎥⎦ ⎥⎦ ⎛ D⎞⎟ ⎛E⎞⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟ ⎜⎜ ⎟ ⎜ ⎜ ⎟ ⎟ ⎟ 1 ⎜⎜⎜H⎟⎟ = e−LS ⎜⎜⎜B⎟⎟ = ⎜⎜B − ⎢⎡ S, B⎥⎤ + ⎢⎡ S, ⎢⎡ S, B⎥⎤ ⎥⎤ + "⎟⎟ ⎟⎟ ⎣ ⎦ 2⎣ ⎣ ⎦⎦ ⎜⎜ ⎟⎟⎟ ⎜⎜ ⎟⎟⎟ ⎜⎜ ⎟⎟ ⎜⎝F ⎠⎟ 1 ⎝⎜ f ⎠⎟ ⎜⎜⎜ ⎡ S, ⎡ S, f ⎤ ⎤ + " ⎟⎟ ⎡ ⎤ − + f S , f ⎟ ⎢⎣ ⎥⎦ 2 ⎢⎣ ⎢⎣ ⎥⎦ ⎥⎦ ⎜⎝⎜ ⎠⎟ Remark: If the variable χ is only a function of x −LS
then e
χ is only a function of x −L
The functionals transforms into F = e S F , resulting in a new Hamiltonian and a new Poisson bracket...
Polarization, magnetization, reduced density, etc… 1 −LS δS ∂ ⎛⎜ δS ⎞⎟ e 3 e − 1 E = c∇ × − ∫ d vf P= ⎜⎜ ⎟⎟ + " δB m ∂v ⎝ δf ⎠⎟ 4π δS 1 −L +" 1 − e S B = c∇ × M= δE 4π
(
)
(
)
⎧ ⎪ D = E + 4πP ⎪ so that ⎨ ⎪ H = B − 4πM ⎪ ⎩ ⎛ −L ∂ L ⎞⎟ ∂F ≡ ⎜⎜⎜e S Reduced evolution operator e S ⎟⎟ F ∂t ∂t ⎝ ⎠⎟ −L L L = e S ⎡⎢ e S F , e S H⎥⎤ = ⎡⎢⎣ F , H⎤⎥⎦ ⎣ ⎦
Reduced Vlasov-Maxwell equations ∂D = c∇ × H − 4πJ ∂t ∂H = −c∇ × D ∂t ∂D ∂ D ⎡ = + ⎢⎣D, H − H⎤⎥⎦ = c∇ × H − 4πJR ∂t ∂t ∂ H ∂H ⎡ ∂M = + ⎢⎣ H, H − H⎤⎥⎦ = −c∇ × D − 4π + 4πc∇ × P ∂t ∂t ∂t
⎞⎟ ∂F v ∂F e ⎜⎛ = −v ⋅ ∇F − ⎜⎜D + × H⎟⎟ ⋅ Reduced Vlasov equation ∂t m⎝ c ⎠⎟ ∂v ⎧ ⎪ ⎪ 4πe ∂f δS δS ⎫ ⎪ ⋅ +" F = f − ⎨f , ⎪ ⎬− ⎪ m ∂v δE ⎪ δf ⎪ ⎪ ⎩ ⎭
guiding center theory / gyrokinetics
What S ?
⎛D⎞⎟ ⎛E⎞⎟ ⎜⎜ ⎟ ⎜⎜ ⎟ ⎟ ⎜⎜H⎟ = e−LS ⎜⎜B⎟⎟ ⎜⎜ ⎟⎟⎟ ⎜⎜ ⎟⎟⎟ ⎜⎜F ⎟⎟ ⎜ ⎟ ⎝ ⎠ ⎝⎜ f ⎠⎟
- Elimination of small spatial and fast time (averaging) scales of f(x,v,t) : guiding-center ⎫⎪⎪ ⎬ collisionless plasmas (low frequency phenomena) ⎪⎪ gyrokinetics ⎭ reduced models for free electron lasers -Use of KAM algorithms (at least one step process)
(
)
For F = f + δf , homological equation P δf + ⎡⎢⎣ S, f ⎤⎥⎦ = 0 - Advantages: preserve the structure of the equations, invertible, symbolic calculus
Brizard, Hahm, Rev. Mod. Phys. (2007)
Strategies to reduce Vlasov-Maxwell equations −LS
> rigorous: H = e
H
> non-rigorous: truncate the Hamiltonian system - the equations of motion - the Hamiltonian and the Poisson bracket > the canonical version provides a way out...
Reduced model for the Free Electron Laser From: Vlasov-Maxwell Hamiltonian 2
H ⎡⎢⎣E, B, f ⎤⎥⎦ =
∫∫ d xd 3
3
p f (x, p) 1 + p + ∫ d x 2
3
E +B
2
2
To: Bonifacio’s reduced FEL Hamiltonian model
⎛ p2 ⎞⎟ ⎜ ⎡ ⎤ H ⎢⎣ f , I , ϕ⎥⎦ = ∫∫ d θdp f (θ, p)⎜⎜ + 2 I sin (θ − ϕ)⎟⎟ ⎜⎝ 2 ⎠⎟ with (I , ϕ) intensity and phase of the electromagnetic wave
… in a Hamiltonian way
A Free Electron… what?
λLEL
2 λu = 2 (1 + K rms ) 2γ
Vlasov-Maxwell: canonical version c Change of variables: ( f , E, B) 6 ( fm , Y, A)
(
)
f (x, p) = fm x, p + A (x)
E = −Y B = ∇×A Bracket: canonical ⎡ ⎤ ⎣⎢ F , G ⎥⎦ =
⎡ δF ∂ δ G ⎡ δG δF δG ⎤⎥ ∂ δF δG ⎤⎥ 3 3 3 ⎢ δF ⎢∇ x ⋅ − ⋅ ∇ + ⋅ − ⋅ d xd p f d ∫∫ m ⎢ ∫ ⎥ ⎢ ⎥ δfm ⎦⎥ ⎣ δA δY δY δA ⎦ ⎣⎢ δfm ∂p δfm ∂p δfm 2
Hamiltonian: H =
3 3 3 ∫∫ d xd p fm 1 + (p − A) + ∫ d x 2
Y + ∇×A
2
2
d Translation of A by a constant function (external field-undulator): A (x, t ) 6 Aw (x) + A (x, t ) Bracket: canonical (canonical transformation) 2
Hamiltonian: H =
∫∫ d xd 3
3
p fm 1 + (p − Aw − A) + ∫ d 3x
Aw helicoidal undulator: Aw =
2
aw
e ( 2
−ikw z
ikw z
ˆe + e
ˆe*
)
Y + 2∇ × Aw ⋅ ∇ × A + ∇ × A 2
2
One mode for the radiated field ˆ y
ˆ x z
L: interaction length V: interaction volume e Paraxial approximation and circularly polarized radiated field ˆ + iy ˆ i x A=− a eikz ˆe − a *e−ikz ˆe* where ˆe = 2 2
(
Y=
k 2
(a e
)
ikz
ˆe + a *e−ikz ˆe*
)
Remark: a does not depend on x and y but depends on time (dynamical variable) Bracket: ⎡⎣⎢ F , G ⎤⎥⎦ =
∫∫
Hamiltonian: H =
⎡ δF ∂ δG ∂ δF δG ⎤⎥ ik ⎛⎜ ∂F ∂G ∂F ∂G ⎞⎟ ⎢ + − d xd p fm ⎢∇ ⋅ − ⋅∇ ⎥ V ⎜⎜⎝ ∂a ∂a * ∂a * ∂a ⎠⎟⎟⎟ f f f f δ ∂ δ ∂ δ δ p p ⎢⎣ m m m m ⎥⎦
∫∫
3
3
(
)
d 3xd 3 p fm 1 + p2 + aa * − i 2 a eikz ˆe − a * e−ikz ˆe* ⋅ Aw + Aw
+ k 2Vaa * −
ikS
dz (a e ∫ 2
ikz
)
ˆe − a * e−ikz ˆe* ⋅ ∇ × Aw
2
Dimensional reduction f The fields do not depend on x and y ⇒ no transverse velocity dispersion f (x, p) = f x, p& δ (p⊥ ) if p⊥ (t = 0) = 0
(
)
(
)
If p⊥ (t ) = 0 then no modification of the (x , y ) distribution f (x, p) = f z, p& δ (p⊥ ) δ (x ) δ (y ) if x (t = 0) = y (t = 0) = 0 (injection at the center)
⎡ ∂ δF ∂ δ G ⎤ ik ⎛ ∂F ∂G ∂F ∂G ⎞⎟ ∂ δ ∂ δ F G ⎢ ⎥ ⎜ ⎡ ⎤ ⎟ − + − Bracket: ⎢⎣ F , G ⎥⎦ = ∫∫ dzdp& f ⎢ ∂z δf ∂z δf ∂p δf ⎥ V ⎜⎜⎝ ∂a ∂a * ∂a * ∂a ⎠⎟⎟ p f ∂ δ ⎢⎣ & ⎥⎦ & Hamiltonian: H =
(
)
2 * * −ikz * ikz ∫∫ dzdp& f 1 + p& + aa − i 2 a e ˆe − a e ˆe ⋅ Aw + Aw
+ k 2Vaa * −
ikS
dz (a e ∫ 2
ikz
)
ˆe − a * e−ikz ˆe* ⋅ ∇ × Aw
∂F ∂G ∂F ∂G g Autonomization: ⎡⎣⎢ F , G ⎤⎦⎥ = ⎡⎣⎢ F , G ⎤⎦⎥ + − a ∂t ∂E ∂E ∂t Time dependent transformation (canonical): fˆ θ, p = f z, p with θ = (k + k ) z − kt
( ) &
(
&
)
2
with Ha = H + E
vanishing
w
aˆ = a eikt Eˆ = E + Vk 2aa * and tˆ = t Bracket: canonical Hamiltonian: H =
∫∫
⎛ ⎞⎟ k 2 * iθ * −i θ 2 ⎜ ˆ ˆˆ − iaw aˆ e − aˆ e d θdp& f ⎜⎜ 1 + p& + aa p ⎟⎟ + aw − ⎜⎝ k + kw & ⎠⎟
(
)
Bonifacio’s FEL model
h Resonance condition: p& = pR + p with p pR weak radiated field: aˆ γ R ≡ 1 + pR 2 ⎡ ∂ δF ∂ δ G ∂ δF ∂ δG ⎤⎥ i ⎛⎜ ∂F ∂G ∂F ∂G ⎞⎟ ˆ ⎡ ⎤ ⎢ ⎟ + − − Bracket: ⎢⎣ F , G ⎥⎦ = (k + kw ) ∫∫ d θdp f ⎢ ∂θ δfˆ ∂p δfˆ ∂p δfˆ ∂θ δfˆ ⎥ kV ⎝⎜⎜ ∂aˆ* ∂aˆ ∂aˆ ∂aˆ* ⎟⎟⎠ ⎦ ⎣ ⎛ 1 + a 2 p 2 ia ⎞⎟ ⎜ i θ − i θ * w w Hamiltonian: H = ∫∫ d θdp fˆ(θ, p )⎜⎜ − aˆ e − aˆ e ⎟⎟ 3 ⎟ ⎜⎝ γ R 2 γR ⎠⎟ i Normalization
(
)
j Transformation (canonical) into intensity/phase a = i I e−iϕ ⎡ ∂ δF ∂ δ G ∂ δF ∂ δG ⎤⎥ ∂F ∂G ∂F ∂G ⎡ ⎤ ⎢ − Bracket: canonical ⎢⎣ F , G ⎥⎦ = ∫∫ d θdp f + − ⎢ ∂θ δf ∂p δf ⎥ ∂ϕ ∂I δ f ∂ δ ∂ θ p f ∂I ∂ϕ ⎣ ⎦ p2 Hamiltonian: H = ∫∫ d θdp f (θ, p ) + 2 I ∫∫ d θdpf (θ, p ) cos (θ − ϕ ) 2
Outlook:on the use of reduced Hamiltonian models 1.5
HN = ∑ j =1
p j2
I +2 2 N
N
∑ cos(θ j =1
j
− ϕ)
1
I/N
N
0.5 0 0
100
200
z Long-range interacting systems : QSS, transition to equilibrium,… Gyrokinetics: understand plasma disruption, control,…
Contact:
[email protected] References: Bachelard, Chandre, Vittot, PRE (2008) Chandre, Brizard, in preparation.
