evolution & actions · 2008. 3. 30. · the darwin approximation “neglects retardation” so...

Approaches to the Darwin ApproximationEquivalence of Darwin Formulations

Darwin Action PrinciplesSummary

Evolution & ActionsUnifying Approaches to the Darwin Approximation

Todd B. Krause, [email protected]

Institute for Fusion StudiesThe University of Texas at Austin

April 3, 2008Tech-X Corporation

Todd B. Krause Unifying Darwin



Collaborators

Amit S. ApteCentre for AppliedMathematics, Tata Instituteof Fundamental Research,Bangalore, [email protected]

Philip J. MorrisonInstitute for Fusion Studies,The University of Texas atAustin, Austin, [email protected]


mailto:[email protected]

[email protected]

[email protected]

mailto:[email protected]

[email protected]

[email protected]



Outline

1 Approaches to the Darwin ApproximationQuasistatic ApproachOperator Approximation Approach

2 Equivalence of Darwin FormulationsDarwin IntegralEquivalent Potentials

3 Darwin Action PrinciplesDarwin Field ActionDarwin Particle-Field & Particle ActionsVlasov-Darwin Action




Quasistatic ApproachOperator Approximation Approach

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 lawsThou shalt not suffer a witch to live (Ex. 22:18)If [a man seizes a woman] in her house, it is the woman’ssin, and she is to die (Hitt. Laws)F = −GMmr/r2 (Newton)

C. G. Darwin (1920)

The order-(v/c)2 approximation to relativistic interaction ofclassical charged particles





Basics of the Darwin Approximation

The order-(v/c)2 approximation to the relativisticinteraction of classical charged particles:

L =n∑

a=1

(maq2

a2

+maq4

a8c2

)− 1

2

∑a 6=b

eaeb

rab

+12

∑a 6=b

eaeb

2c2rab[qa · qb + (qa · rab)(qb · rab)]

The interaction stems from the coupling with the fields ofother particles via Aµ = (φ, A), where

φ(x, t) =∑

b

eb

‖x− qb‖, A(x, t) =

∑b

eb[qb + (qb · rb)rb]

2c‖x− qb‖





So What Good Is It?

Grandson of the CharlesDarwinEvidently a ferventeugenist!

But that’s another talk. . .

Darwin’s originalmotivation

Bohr-Sommerfeld atom,hydrogen spectrum(Darwin 1920)





Uses in Plasma Physics

Mirrortron simulationsPlasma column withstationary ions,counterstreamingelectronsShear Alfven waves





Outline








Coulomb Gauge

We start with Maxwell’s equations for φ and A

∇2φ +1c

∂

∂t(∇ · A) = −4πρ,

∇2A− 1c2

∂2A∂t2 −∇

(∇ · A +

1c

∂φ

∂t

)= −4π

cJ

Employ Coulomb condition ∇ · A = 0, yielding

∇2φ = −4πρ,

∇2A− 1c2

∂2A∂t2 = −4π

cJ +

1c

∂∇φ

∂t





Coulomb Solution

The solution for the scalar potential is the friendly

φC(x, t) =

∫d3x ′

ρ(x′, t)r

The solution for the vector potential is more monstrous

AC(x, t) =1c

∫d3x ′

1r[J(x′, t ′)− r

(r · J(x′, t ′)

)]ret

+ c∫

d3x ′1r

∫ r/c

0dττ

[3r(r · J(x′, t − τ)

)−J(x′, t − τ)

]where r := ‖x− x′‖ and r = (x− x′)/r , “ret” denotes thatthe quantities in brackets are evaluated at the retardedtime t ′ = t − r/c





Quasistatic Coulomb Solution

The instantaneous, or ‘quasistatic’, form of the vectorpotential

Substitute J(x′, t − τ) → J(x′, t) in the second integralRemove “ret” from the first integral, i.e. evaluate the currentat the time t

AqsC (x, t) =

12c

∫d3x ′

1r[J(x′, t) + r

(r · J(x′, t)

)]Clearly the continuum analogue of

A(x, t) =∑

b

eb[qb + (qb · rb)rb]

2c‖x− qb‖

in Darwin’s original particle action





Turning the TablesWhat Equations Do φC & Aqs

C Satisfy?

Clearly φC satisfiesPoisson’s equation

∇2φC = −4πρ

What about AqsC ? Compute:

∇2AqsC = −4π

cJ +

1c

∫d3x ′

1r3 [J′ − 3(J′ · r)r]

What’s that mess on the right?





Turning the TablesVector Decomposition

We can decompose any vector into longitudinal andtransverse components

In particular, decompose the current

JT (x, t) =1

4π∇×∇×

∫d3x ′ J(x′, t)

r,

JL(x, t) = − 14π∇∫

d3x ′∇′ · J(x′, t)r

Re-express the longitudinal component

JL(x, t) =1

4π

∫d3x ′ 1

r3 [J′ − 3(J′ · r)r]





Turning the TablesEquation for Aqs

C

Just What We Need

∇2AqsC (x, t) = −4π

cJT (x, t)

This is Poisson’s equation in vector formThe source is not the full current, but only the transversecurrent JT





Outline








Do Over

Rather than approximate the solution, approximate thegoverning equations directlyArgue the form of the differential operator

The term (1/c2)∂2A/∂t2 leads to field retardationThe Darwin approximation “neglects retardation”So drop the time derivative of vector potential

Maxwell’s equations take the approximate form

∇2φ +1c

∂

∂t(∇ · A) = −4πρ,

∇2A−∇(∇ · A +

1c

∂φ

∂t

)= −4π

cJ





You Don’t Appreciate What You Have until. . .Continuity Equation

Now you’ve done it! You just lostconservation!

You can’t drop terms fromMaxwell’s equations withoutlosing the continuity equation

We now have the constraint

∂ρ

∂t+∇ · J = − 1

4πc∂2

∂t2∇ · A

For sources that conserve charge, we have a naturalansatz: ∇ · A = 0





AnSatz is Good Satz

We make the ansatz∇ · A = 0, yielding

∇2φ = −4πρ,

∇2A− 1c

∂∇φ

∂t= −4π

cJ

and see if the solution AOAsatisfies ∇ · AOA = 0





Problem Solved!Almost. . .

The first equation gives the Coulomb scalar potential, andthe second equation gives

AOA(x, t) =1c

∫d3x ′

J(x′, t)r

− 14πc

∂

∂t

∫d3x ′

∇′φC(x′, t)r

Check the ansatz

∇ · AOA =1c

∫d3x ′

1r

[∇′ · J(x′, t) +

∂ρ(x′, t)∂t

]

AOA is divergence-free when (ρ, J) satisfy chargeconservationConsistent with the ansatz!





Making It Work for YouEmploying Charge Conservation

Given that Coulomb gauge and charge conservation areequivalent, we can now use that to our advantage

Use the continuity equation in JL

JL(x, t) = − 14π∇∫

d3x ′∇′ · J(x′, t)r

=1

4π

∂∇φC(x, t)∂t

Then AOA satisfies

∇2AOA = −4π

c(J− JL) = −4π

cJT





Speak My Language!In Terms of Fields

Let E = ET + EL, with ∇ · ET = 0 and ∇× EL = 0. Thenthe preceding is equivalent to

1c

∂EL

∂t−∇× B = −4π

cJ, ∇ · EL = 4πρ,

1c

∂B∂t

+∇× ET = 0, ∇ · B = 0

for EL = −∇φ, ET = −(1/c)∂A/∂t , and B = ∇× AChanging gauge by function χ

4π

cJ = ∇× B′ +

1c

∂∇φ′

∂t= ∇× B +

1c

∂∇φ

∂t+

1c2

∂2χ

∂t2

so gauge invariant only when the function χ is of the sameorder in 1/c as A





Pros & Cons

ProsEliminates high-frequency, transverse electromagneticeffects

1c2

∂2A∂t2 ∼ 0 ⇐⇒ field time-variation

speed of light∼ 0

Retains electrostatic and low-frequency inductive effects

ConsNumerical instabilitiesDifficulties with boundaryconditions





Loch LommondThe High Road vs. the Low Road

L[A] = J

AC = L−1[J] L[A] = J

AqsC = L−1[J] AOA = L−1[J]

��

��

��

��

solve

��??

????

????

????

??

approx.

��

approx.

��

solve

oo //?=





Loch LommondToy Model

dxdt

=1

1− x

t =

∫ x1

(1− x)dx t =

∫ x2 dx1 + x

x1(t) = 1−√

1− 2t ≈ t x2(t) = et − 1

zztttttttttttsolve

$$JJJJJJJJJJJapprox.

��

approx.

��

solve

oo //?=




Darwin IntegralEquivalent Potentials

Outline








The Heart of the Matter

We have the Darwin field equations

∇2φOA(x, t) = −4πρ(x, t),

∇2AOA(x, t) = −4π

cJT (x, t)

A direct solution for the vector potential is

AOA(x, t) =1

4πc

∫d3x ′

r

[∇′ ×∇′ ×

∫d3x ′′

J(x′′, t)rx ′x ′′

]But Darwin’s original potential is

AqsC (x, t) =

12c

∫d3x ′

1r[J(x′, t) + r

(r · J(x′, t)

)]Are they equal?

If not, then AOA isn’t really Darwin





Setting Up

Using vector identities, we bring AOA to the form

AOA =1c

∫d3x ′

J(x′, t)‖x− x′‖

+1

4πc

∫d3x ′

(J′ · ∇

)∇∫

d3x ′′

‖x− x′′‖‖x′ − x′′‖

The first term has the form we need for AqsC

Everything hinges on the second termWe need to perform the integral

ID :=

∫d3x ′′

‖x′′ − x‖‖x′′ − x′‖=

∫d3y

‖y‖‖y− z‖

letting y := x′′ − x′ and z := x− x′





The Darwin IntegralWhat’s a Little Infinity among Friends?

The integral is simple if we use Legendre polynomials

1‖y− z‖

=∞∑

l=0

r l<

r l+1>

Pl(cos γ),

∫ 1

−1Pk (ξ)Pl(ξ)dξ =

2δkl

2l + 1

Then we just calculate

ID =

∫d3y

‖y‖‖y− z‖= 2π

∞∑l=0

∫ ∞

0

r l<

r l+1>

ydy∫ 1

−1P0(ξ)Pl(ξ)dξ

= 4π

∫ ∞

0

yr>

dy = 4π

∫ z

0

yz

dy + 4π

∫ ∞

z

yy

dy

= 4π · ∞ − 2πz





Outline








Regularize

The infinity is actually hidden behind a derivativeWe are free to subtract a term canceling the x′′ divergence

AOA =1c

∫d3x ′

J(x′, t)‖x− x′‖

+1

4πc

∫d3x ′

(J′ · ∇

)∇[∫

d3x ′′

‖x− x′′‖‖x′ − x′′‖

−∫

d3x ′′

‖x′′‖2

]=

1c

∫d3x ′

J(x′, t)‖x− x′‖

− 12c

∫d3x ′

(J′ · ∇

)∇‖x− x′‖





There Can Be Only One!

This gives us exactly what we need

AOA =12c

∫d3x ′

J(x′, t) + r′ (r′ · J(x′, t))‖x− x′‖

= AqsC

The two potentials are in fact equal!




Darwin Field ActionDarwin Particle-Field & Particle ActionsVlasov-Darwin Action

Outline








Got Milk?

Got Action?





Wanna Li’l Action. . .

Joe Monaghan (2004), talk entitled “SPH and Simulation”

Conservation of general properties may be more importantthan high order integration

Continuing. . .Lagrangians, when they can be used,are good because the physics can beadded consistently and invariants canbe satisfied more easily





Putting Words into ActionMaxwell Field Action

The standard Maxwell field action is

SMf [φ, A] =

∫dtd3x

{−ρφ +

1c

J · A +1

8π

[E2 − B2

]}where E = −∇φ− (1/c)∂A/∂t and B = ∇× A are justshorthand notationVariation gives Maxwell’s equations

∇2φ +1c

∂

∂t(∇ · A) = −4πρ,

∇2A− 1c2

∂2A∂t2 −∇

(∇ · A +

1c

∂φ

∂t

)= −4π

cJ





Putting Words into ActionDarwin Field Action

Similarly, we write a Darwin field action

SDf [φ, A] =

∫dtd3x

{−ρφ +

1c

J · A +1

8π

[E2 − E2

T − B2]}

where EL = −∇φ and ET = −(1/c)∂A/∂tVariation gives the equations

∇2φ +1c

∂

∂t(∇ · A) = −4πρ,

∇2A−∇(∇ · A +

1c

∂φ

∂t

)= −4π

cJ





Putting Words into ActionDarwin Field Action with Constraint

Setting ∇ · A = 0 is equivalent to imposing chargeconservation, and so

∇2φ = −4πρ,

∇2A− 1c

∂∇φ

∂t= −4π

cJ

=⇒

∇2φ = −4πρ,

∇2A = −4π

cJT

So we have an action for theDarwin field equations!





Outline








Brick by BrickDarwin Particle-Field Action

We may add the 2nd-order relativistic particle action,together with a coupling term

SD[q;φ, A]

=

∫dt

{∑a

[maq2

a2

+maq4

a8c2

]+∑

a

ea

∫d3xδ(3)(x− qa)

[−φ(x, t) +

qa

c· A(x, t)

]+

18π

∫d3x

[E2(x, t)− E2

T (x, t)− B2(x, t)]}

Variation over φ and A gives Darwin field equationsHow do we get particle equations?





Particle Equations

Vary φ and A to obtain the Darwin field equations

∇2φ = −4πρ, ∇2A = −4π

cJT

Express sources in terms ofparticles themselves

ρ(x, t) =∑

b

ebδ(3)(x− qb(t)),

J(x, t) =∑

b

ebqb(t)δ(3)(x− qb(t))





ParticlizationIt’s All the Rage!

Solve for potentials in terms of sources

φ(x, t) =

∫d3x ′K (x|x′)ρ(x′, t) =

∑b

ebK (x|qb),

Ai(x, t) =1c

∫d3x ′Kij(x|x′)Jj(x′, t) =

1c

∑b

ebKij(x|qb)qbj

where

K (x|x′) :=1

‖x− x′‖,

Kij(x|x′) :=1

2‖x− x′‖

[δij +

(xi − x ′i )(xj − x ′j )

‖x− x′‖2

]





On the MoveParticle Equations of Motion

Insert these solutions back into the action

SD[q] =

∫dt

{∑a

(maq2

a2

+maq4

a8c2

)

+12

∑a 6=b

eaeb

[K (qa|qb) +

qai qbj

c2 Kij(qa|qb)

]Vary over particle coordinates q

mqi = eEi +ec

(q× B)i − eq2

c2

(δij

2+

qi qj

q2

)ELj

E and B are shorthand for combinations of φ and Aφ and A are shorthand for combinations of q and q





Outline








Changing Viewpoints

We can Legendre (convex) transform the DarwinLagrangian to obtain the Darwin Hamiltonian

HD(q, p, t) =1

2m

(p− e

cA)2− 1

8m3c2

(p− e

cA)4

+eφ(q, t)

Letting P := p− (e/c)A(q, t), Hamilton’s equationsbecome

q =Pm

(1− P2

2m2c2

),

P = e[E +

(1− P2

2m2c2

)P

mc× B

]





Heretical RevelationsNoncanonical Hamiltonian Systems

Given a Hamiltonian H, and Z := (q, P), we may look atphase space actions of the form

S[Z ] =

∫ t1

t0

[θµ(Z , t)Zµ − H(Z , t)

]dt

Variation givesωµν Z ν = H,µ + ∂tθµ

with ωµν := θν,µ − θµ,ν

If Jνσωσµ = δνµ, then

Z ν = Jνµ (H,µ + ∂tθµ)

give Hamilton’s equations of motion





Darwin Phase Space Action

We form the noncanonical Darwin phase space action

SD[q, P] =

∫dt[(

P +ec

A)· q− P2

2m+

P4

8m3c2 − eφ

]The equations Z ν = Jνµ (H,µ + ∂tθµ) become(

qP

)=

(03×3 13×3−13×3

ec Bij

)( e ∂φ∂q + e

c∂A∂t(

1− P2

2m2c2

)Pm

)

where Bij := εijkBk





What’s in a Label?Passing to the Continuum

Envision a continuum of particles in phase spaceLabel each particle by the initial conditions Z0 := (q0, P0) ofits trajectory: q(q0, P0, t) and P(q0, P0, t), succinctly Z (Z0, t)This is an invertible map of phase space: Z0(Z , t)

Associated with each trajectory is a phase space numberdensity (distribution function) f0(Z0)

Like the initial conditions, f0(Z0) constant on the trajectoryTo find f at an observation point Ξ := (x,Π) at time t , weneed the f associated with the trajectory Z that hits Ξ at t

So start at Ξ and map back to initial conditions Z0:Ξ = Z (Z0, t). This gives the Euler-Lagrange map

f (Ξ, t)d6Ξ = f0(Z0(Ξ, t))d6Z0





The Whole ShebangVlasov-Darwin Action

Phase Space, Vlasov-Darwin Action

SVD[q, P;φ, A]

=

∫dt{∫

d6Z0 fR0(Z0)

[P · q− P2

2m+

P4

8m3c2

]+

∫d6Z0 fR0(Z0)

[−eφ(q, t) +

ec

A(q, t) · q]

+1

8π

∫d3x

[E2(x, t)− E2

T (x, t)− B2(x, t)]}





The Whole ShebangVlasov-Darwin Equations

Variation of SVD with respect to q and P yields the Darwinequations of motion

q =Pm

(1− P2

2m2c2

),

P = e[E +

(1− P2

2m2c2

)P

mc× B

]These together with the Euler-Lagrange map imply theVlasov equation in the Darwin approximation

∂fD∂t

+

(1− Π2

2m2c2

)Π

m· ∂fD

∂x

+ e[E +

(1− Π2

2m2c2

)Π

mc× B

]· ∂fD∂Π

= 0




Summary

Darwin ActionsOne action encompasses both particle and field formulations ofthe Darwin approximation

Double troubleTwo common formulations of the Darwin approximation:particle and field

There Can Be Only One!The Darwin particle-field action encapsulates bothformulations from one consistent viewpointProvides a consistent check on orders of approximation

Vlasov-DarwinNoncanonical Hamiltonian systemsUnified action principle for Vlasov equation with Darwinparticle-field dynamics




Select Bibliography

C. G. DarwinThe 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, ControlledFusion


evolution & actions · 2008. 3. 30. · the darwin approximation “neglects retardation” so...

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