Evolution & Actions. Unifying Approaches to the Darwin Approximation. Todd B. Krause.

Evolution & Actions Unifying Approaches to the Darwin Approximation, PhD bobtodd@math.utexas.edu Institute for Fusion Studies The University of Texas at Austin April 3, 2008 Tech-X Corporation

Collaborators Amit S. Apte Centre for Applied Mathematics, Tata Institute of Fundamental Research, Bangalore, India apte@math.tifrbng. res.in Philip J. Morrison Institute for Fusion Studies, The University of Texas at Austin, Austin, Texas morrison@physics. utexas.edu

Outline Approaches to the Darwin Approximation 1 Approaches to the Darwin Approximation 2 Darwin Integral Equivalent Potentials 3 Darwin Field Action Darwin Particle-Field & Particle Actions Vlasov-Darwin Action

The Character of Physical Law Ever since there have been laws... Thou shalt not kill (Exodus 20:13) If a man seizes a woman on a mountain, it is the man s sin, and he is to die (Hittite Laws) G = 8πT (Einstein)... there have been approximations to those laws Thou shalt not suffer a witch to live (Ex. 22:18) If [a man seizes a woman] in her house, it is the woman s sin, and she is to die (Hitt. Laws) F = GMmˆr/r 2 (Newton) C. G. Darwin (1920) The order-(v/c) 2 approximation to relativistic interaction of classical charged particles

Basics of the Darwin Approximation The order-(v/c) 2 approximation to the relativistic interaction of classical charged particles: L = n a=1 + 1 2 ( ma q 2 a 2 a b + m a q a 4 ) 8c 2 1 e a e b 2 r ab a b e a e b 2c 2 r ab [ q a q b + ( q a ˆr ab )( q b ˆr ab )] The interaction stems from the coupling with the fields of other particles via A µ = (φ, A), where φ(x, t) = b e b x q b, A(x, t) = b e b [ q b + ( q b ˆr b )ˆr b ] 2c x q b

So What Good Is It? Grandson of the Charles Darwin Evidently a fervent eugenist! But that s another talk... Darwin s original motivation Bohr-Sommerfeld atom, hydrogen spectrum (Darwin 1920)

Uses in Plasma Physics Mirrortron simulations Plasma column with stationary ions, counterstreaming electrons Shear Alfvén waves

Coulomb Gauge We start with Maxwell s equations for φ and A 2 A 1 c 2 2 A t 2 2 φ + 1 c t ( A + 1 c ( A) = 4πρ, ) φ = 4π t c J Employ Coulomb condition A = 0, yielding 2 A 1 c 2 2 A t 2 2 φ = 4πρ, = 4π c J + 1 φ c t

Coulomb Solution The solution for the scalar potential is the friendly φ C (x, t) = d 3 x ρ(x, t) r The solution for the vector potential is more monstrous A C (x, t) = 1 d 3 x 1 [ J(x, t ) ˆr (ˆr J(x, t ) )] c r ret + c d 3 x 1 r/c dττ [ 3ˆr (ˆr J(x, t τ) ) r 0 J(x, t τ) ] where r := x x and ˆr = (x x )/r, ret denotes that the quantities in brackets are evaluated at the retarded time t = t r/c

Quasistatic Coulomb Solution The instantaneous, or quasistatic, form of the vector potential Substitute J(x, t τ) J(x, t) in the second integral Remove ret from the first integral, i.e. evaluate the current at the time t A qs C (x, t) = 1 2c d 3 x 1 r Clearly the continuum analogue of [ J(x, t) + ˆr (ˆr J(x, t) )] A(x, t) = b e b [ q b + ( q b ˆr b )ˆr b ] 2c x q b in Darwin s original particle action

Turning the Tables What Equations Do φ C & A qs C Satisfy? Clearly φ C satisfies Poisson s equation 2 φ C = 4πρ What about A qs C? Compute: 2 A qs C = 4π c J + 1 c d 3 x 1 r 3 [J 3(J ˆr)ˆr] What s that mess on the right?

Turning the Tables Vector Decomposition We can decompose any vector into longitudinal and transverse components In particular, decompose the current J T (x, t) = 1 4π J L (x, t) = 1 4π d 3 x J(x, t), r d 3 x J(x, t) r Re-express the longitudinal component J L (x, t) = 1 d 3 x 1 4π r 3 [J 3(J ˆr)ˆr]

Turning the Tables Equation for A qs C Just What We Need 2 A qs C (x, t) = 4π c J T (x, t) This is Poisson s equation in vector form The source is not the full current, but only the transverse current J T

Do Over Approaches to the Darwin Approximation Rather than approximate the solution, approximate the governing equations directly Argue the form of the differential operator The term (1/c 2 ) 2 A/ t 2 leads to field retardation The Darwin approximation neglects retardation So drop the time derivative of vector potential Maxwell s equations take the approximate form 2 A 2 φ + 1 c t ( A + 1 c ( A) = 4πρ, ) φ = 4π t c J

You Don t Appreciate What You Have until... Continuity Equation We now have the constraint Now you ve done it! You just lost conservation! You can t drop terms from Maxwell s equations without losing the continuity equation ρ t + J = 1 2 4πc t 2 A For sources that conserve charge, we have a natural ansatz: A = 0

AnSatz is Good Satz We make the ansatz A = 0, yielding 2 A 1 c 2 φ = 4πρ, φ t = 4π c J and see if the solution A OA satisfies A OA = 0

Problem Solved! Almost... The first equation gives the Coulomb scalar potential, and the second equation gives A OA (x, t) = 1 c d 3 x J(x, t) r 1 4πc t d 3 x φ C (x, t) r Check the ansatz A OA = 1 c d 3 x 1 r [ ] J(x, t) + ρ(x, t) t A OA is divergence-free when (ρ, J) satisfy charge conservation Consistent with the ansatz!

Making It Work for You Employing Charge Conservation Given that Coulomb gauge and charge conservation are equivalent, we can now use that to our advantage Use the continuity equation in J L J L (x, t) = 1 4π d 3 x J(x, t) r = 1 φ C (x, t) 4π t Then A OA satisfies 2 A OA = 4π c (J J L) = 4π c J T

Speak My Language! In Terms of Fields Let E = E T + E L, with E T = 0 and E L = 0. Then the preceding is equivalent to 1 E L c t 1 B c t B = 4π c J, E L = 4πρ, + E T = 0, B = 0 for E L = φ, E T = (1/c) A/ t, and B = A Changing gauge by function χ 4π c J = B + 1 φ c t = B + 1 c φ t + 1 c 2 2 χ t 2 so gauge invariant only when the function χ is of the same order in 1/c as A

Pros & Cons Approaches to the Darwin Approximation Pros Eliminates high-frequency, transverse electromagnetic effects 1 2 A c 2 t 2 0 field time-variation 0 speed of light Retains electrostatic and low-frequency inductive effects Cons Numerical instabilities Difficulties with boundary conditions

Approaches to the Darwin Approximation Loch Lommond The High Road vs. the Low Road L[A] = J solve approx. A C = L 1 [J] L[A] = J approx. A qs? = solve C = L 1 [J] A OA = L 1 [J]

Loch Lommond Toy Model Approaches to the Darwin Approximation t = solve dx dt = 1 1 x approx. x1(1 x)dx t = x2 dx 1 + x approx. solve x 1 (t) = 1 1 2t t x 2 (t) = e t 1? =

Outline Approaches to the Darwin Approximation Darwin Integral Equivalent Potentials 1 Approaches to the Darwin Approximation 2 Darwin Integral Equivalent Potentials 3 Darwin Field Action Darwin Particle-Field & Particle Actions Vlasov-Darwin Action

The Heart of the Matter Darwin Integral Equivalent Potentials We have the Darwin field equations 2 φ OA (x, t) = 4πρ(x, t), 2 A OA (x, t) = 4π c J T (x, t) A direct solution for the vector potential is d 3 x A OA (x, t) = 1 4πc r [ ] d 3 x J(x, t) r x x But Darwin s original potential is A qs C (x, t) = 1 d 3 x 1 [ J(x, t) + ˆr (ˆr J(x, t) )] 2c r Are they equal? If not, then A OA isn t really Darwin

Setting Up Approaches to the Darwin Approximation Darwin Integral Equivalent Potentials Using vector identities, we bring A OA to the form A OA = 1 d 3 x J(x, t) c x x + 1 d 3 x ( J ) d 3 x 4πc x x x x The first term has the form we need for A qs C Everything hinges on the second term We need to perform the integral I D := d 3 x x x x x = letting y := x x and z := x x d 3 y y y z

The Darwin Integral What s a Little Infinity among Friends? Darwin Integral Equivalent Potentials The integral is simple if we use Legendre polynomials 1 y z = l=0 r< l r> l+1 P l (cos γ), 1 1 P k (ξ)p l (ξ)dξ = 2δ kl 2l + 1 Then we just calculate I D = d 3 y y y z = 2π l=0 y z = 4π dy = 4π 0 r > 0 = 4π 2πz r l 1 < 0 r> l+1 ydy 1 y dy + 4π z z y y dy P 0 (ξ)p l (ξ)dξ

Regularize Approaches to the Darwin Approximation Darwin Integral Equivalent Potentials The infinity is actually hidden behind a derivative We are free to subtract a term canceling the x divergence A OA = 1 d 3 x J(x, t) c x x + 1 d 3 x ( J ) [ d 3 x 4πc x x x x d 3 x ] x 2 = 1 d 3 x J(x, t) c x x 1 d 3 x ( J ) x x 2c

There Can Be Only One! Darwin Integral Equivalent Potentials This gives us exactly what we need A OA = 1 d 3 x J(x, t) + ˆr (ˆr J(x, t)) 2c x x = A qs C The two potentials are in fact equal!

Outline Approaches to the Darwin Approximation Darwin Field Action Darwin Particle-Field & Particle Actions Vlasov-Darwin Action 1 Approaches to the Darwin Approximation 2 Darwin Integral Equivalent Potentials 3 Darwin Field Action Darwin Particle-Field & Particle Actions Vlasov-Darwin Action

Got Milk? Approaches to the Darwin Approximation Darwin Field Action Darwin Particle-Field & Particle Actions Vlasov-Darwin Action Got Action?

Wanna Li l Action... Darwin Field Action Darwin Particle-Field & Particle Actions Vlasov-Darwin Action Joe Monaghan (2004), talk entitled SPH and Simulation Conservation of general properties may be more important than high order integration Continuing... Lagrangians, when they can be used, are good because the physics can be added consistently and invariants can be satisfied more easily

Putting Words into Action Maxwell Field Action Darwin Field Action Darwin Particle-Field & Particle Actions Vlasov-Darwin Action The standard Maxwell field action is { S Mf [φ, A] = dtd 3 x ρφ + 1 c J A + 1 [E 2 B 2]} 8π where E = φ (1/c) A/ t and B = A are just shorthand notation Variation gives Maxwell s equations 2 A 1 c 2 2 A t 2 2 φ + 1 c t ( A + 1 c ( A) = 4πρ, ) φ = 4π t c J

Putting Words into Action Darwin Field Action Darwin Field Action Darwin Particle-Field & Particle Actions Vlasov-Darwin Action Similarly, we write a Darwin field action { S Df [φ, A] = dtd 3 x ρφ + 1 c J A + 1 [E 2 ET 2 8π B2]} where E L = φ and E T = (1/c) A/ t Variation gives the equations 2 A 2 φ + 1 c t ( A + 1 c ( A) = 4πρ, ) φ = 4π t c J

Putting Words into Action Darwin Field Action with Constraint Darwin Field Action Darwin Particle-Field & Particle Actions Vlasov-Darwin Action Setting A = 0 is equivalent to imposing charge conservation, and so 2 φ = 4πρ, 2 φ = 4πρ, 2 A 1 φ = 4π c t c J = 2 A = 4π c J T So we have an action for the Darwin field equations!

Brick by Brick Darwin Particle-Field Action Darwin Field Action Darwin Particle-Field & Particle Actions Vlasov-Darwin Action We may add the 2 nd -order relativistic particle action, together with a coupling term S D [q; φ, A] { [ ma q a 2 = dt + m a q 4 ] a 2 8c 2 a + e a d 3 xδ (3) (x q a ) a + 1 d 3 x 8π [ φ(x, t) + q a c [ E 2 (x, t) E 2 T (x, t) B2 (x, t) ] A(x, t) ] } Variation over φ and A gives Darwin field equations How do we get particle equations?

Particle Equations Darwin Field Action Darwin Particle-Field & Particle Actions Vlasov-Darwin Action Vary φ and A to obtain the Darwin field equations 2 φ = 4πρ, 2 A = 4π c J T Express sources in terms of particles themselves ρ(x, t) = b J(x, t) = b e b δ (3) (x q b (t)), e b q b (t)δ (3) (x q b (t))

Particlization It s All the Rage! Approaches to the Darwin Approximation Darwin Field Action Darwin Particle-Field & Particle Actions Vlasov-Darwin Action Solve for potentials in terms of sources φ(x, t) = d 3 x K (x x )ρ(x, t) = e b K (x q b ), b A i (x, t) = 1 d 3 x K ij (x x )J j (x, t) = 1 e b K ij (x q b ) q bj c c where K (x x ) := K ij (x x ) := 1 x x, 1 2 x x [ δ ij + (x i x i )(x j x x x 2 b j ) ]

On the Move Particle Equations of Motion Darwin Field Action Darwin Particle-Field & Particle Actions Vlasov-Darwin Action Insert these solutions back into the action { ( ma q a 2 S D [q] = dt + m a q 4 ) a 2 8c 2 + 1 2 a b Vary over particle coordinates q a [ e a e b K (q a q b ) + q ] ai q bj c 2 K ij (q a q b ) m q i = ee i + e c ( q B) i e q 2 c 2 ( δij 2 + q ) i q j q 2 E Lj E and B are shorthand for combinations of φ and A φ and A are shorthand for combinations of q and q

Changing Viewpoints Darwin Field Action Darwin Particle-Field & Particle Actions Vlasov-Darwin Action We can Legendre (convex) transform the Darwin Lagrangian to obtain the Darwin Hamiltonian H D (q, p, t) = 1 ( p e ) 2 2m c A 1 ( 8m 3 c 2 p e ) 4+eφ(q, c A t) Letting P := p (e/c)a(q, t), Hamilton s equations become q = P ) (1 P2 m 2m 2 c 2, [ ) ] Ṗ = e E + (1 P2 P 2m 2 c 2 mc B

Heretical Revelations Noncanonical Hamiltonian Systems Darwin Field Action Darwin Particle-Field & Particle Actions Vlasov-Darwin Action Given a Hamiltonian H, and Z := (q, P), we may look at phase space actions of the form S[Z ] = t1 t 0 [ ] θ µ (Z, t)ż µ H(Z, t) dt Variation gives ω µν Ż ν = H,µ + t θ µ with ω µν := θ ν,µ θ µ,ν If J νσ ω σµ = δ ν µ, then Ż ν = J νµ (H,µ + t θ µ ) give Hamilton s equations of motion

Darwin Phase Space Action Darwin Field Action Darwin Particle-Field & Particle Actions Vlasov-Darwin Action We form the noncanonical Darwin phase space action S D [q, P] = dt [ ( P + e ) ] c A q P2 2m + P4 8m 3 c 2 eφ The equations Ż ν = J νµ (H,µ + t θ µ ) become ( ) q = Ṗ where B ij := ɛ ijk B k ( ) ( 03 3 1 3 3 e ( φ 1 3 3 e c B ij q + e c A t 1 P2 2m 2 c 2 ) P m )

What s in a Label? Passing to the Continuum Darwin Field Action Darwin Particle-Field & Particle Actions Vlasov-Darwin Action Envision a continuum of particles in phase space Label each particle by the initial conditions Z 0 := (q 0, P 0 ) of its trajectory: q(q 0, P 0, t) and P(q 0, P 0, t), succinctly Z (Z 0, t) This is an invertible map of phase space: Z 0 (Z, t) Associated with each trajectory is a phase space number density (distribution function) f 0 (Z 0 ) Like the initial conditions, f 0 (Z 0 ) constant on the trajectory To find f at an observation point Ξ := (x, Π) at time t, we need the f associated with the trajectory Z that hits Ξ at t So start at Ξ and map back to initial conditions Z 0 : Ξ = Z (Z 0, t). This gives the Euler-Lagrange map f (Ξ, t)d 6 Ξ = f 0 (Z 0 (Ξ, t))d 6 Z 0

The Whole Shebang Vlasov-Darwin Action Darwin Field Action Darwin Particle-Field & Particle Actions Vlasov-Darwin Action Phase Space, Vlasov-Darwin Action S VD [q, P; φ, A] { ] = dt d 6 Z 0 f R0 (Z 0 ) [P q P2 2m + P4 8m 3 c 2 [ + d 6 Z 0 f R0 (Z 0 ) eφ(q, t) + e ] c A(q, t) q + 1 [ ] } d 3 x E 2 (x, t) ET 2 8π (x, t) B2 (x, t)

The Whole Shebang Vlasov-Darwin Equations Darwin Field Action Darwin Particle-Field & Particle Actions Vlasov-Darwin Action Variation of S VD with respect to q and P yields the Darwin equations of motion q = P ) (1 P2 m 2m 2 c 2, [ ) ] Ṗ = e E + (1 P2 P 2m 2 c 2 mc B These together with the Euler-Lagrange map imply the Vlasov equation in the Darwin approximation ) f D + (1 Π2 Π t 2m 2 c 2 m f D x [ ) ] + e E + (1 Π2 Π 2m 2 c 2 mc B f D Π = 0

Darwin Actions One action encompasses both particle and field formulations of the Darwin approximation Double trouble Two common formulations of the Darwin approximation: particle and field There Can Be Only One! The Darwin particle-field action encapsulates both formulations from one consistent viewpoint Provides a consistent check on orders of approximation Vlasov-Darwin Noncanonical Hamiltonian systems Unified action principle for Vlasov equation with Darwin particle-field dynamics

Select Bibliography C. G. Darwin The Dynamical Motions of Charged Particles. Rev. Mod. Phys., 39(233):537 551, 1920. A. Kaufman and P. Rostler. The Darwin Model as a Tool for... Plasma Simulation. Phys. Fluids, 14:446 448, 1971. T. B. Krause, A. Apte, and P. J. Morrison. A unified approach to the Darwin approximation. Phys. Plasmas, 14:102112, 2007. C. W. Nielson and H. R. Lewis. Particle-Code Models in the Nonradiative Limit. In J. Killeen, editor, Meth. Comp. Phys. 16, Controlled Fusion