The radiative self-force and collective interaction David A Burton (with Anthony Carr, Jonathan Gratus & Adam Noble1 ) Department of Physics, Lancaster University and the Cockcroft Institute of Accelerator Science and Technology, UK

Space Charge 2015 Trinity College Oxford 25th March 2015

1

Department of Physics, University of Strathclyde, Glasgow, UK

q2 τ= 6π0mc 3 (τ = 2re /3c ∼ 10−23 s for an electron)

SINGLE-PARTICLE RADIATION REACTION

Lorentz-Abraham equation Abraham (1905), Lorentz (1909): ma = F + mτ a˙

with F = q(E + v × B)

where E and B are background (i.e. applied) electric and magnetic fields, respectively.

Lorentz-Abraham equation Abraham (1905), Lorentz (1909): ma = F + mτ a˙

with F = q(E + v × B)

where E and B are background (i.e. applied) electric and magnetic fields, respectively. Contentious!

Lorentz-Abraham equation Abraham (1905), Lorentz (1909): ma = F + mτ a˙

with F = q(E + v × B)

where E and B are background (i.e. applied) electric and magnetic fields, respectively. Contentious! I

a = τ a˙ when F = 0 so a = a0 exp(t/τ )

Lorentz-Abraham equation Abraham (1905), Lorentz (1909): ma = F + mτ a˙

with F = q(E + v × B)

where E and B are background (i.e. applied) electric and magnetic fields, respectively. Contentious! I

a = τ a˙ when F = 0 so a = a0 exp(t/τ ) . . . runaway solution!

Lorentz-Abraham equation Abraham (1905), Lorentz (1909): ma = F + mτ a˙

with F = q(E + v × B)

where E and B are background (i.e. applied) electric and magnetic fields, respectively. Contentious! I I

a = τ a˙ when F = 0 so a = a0 exp(t/τ ) . . . runaway solution! t/τ R t 0 Introduce integrating factor: a = − emτ t0 F(t 0 )e −t /τ dt 0

Lorentz-Abraham equation Abraham (1905), Lorentz (1909): ma = F + mτ a˙

with F = q(E + v × B)

where E and B are background (i.e. applied) electric and magnetic fields, respectively. Contentious! I I I

a = τ a˙ when F = 0 so a = a0 exp(t/τ ) . . . runaway solution! t/τ R t 0 Introduce integrating factor: a = − emτ t0 F(t 0 )e −t /τ dt 0 Try constant F.
0 so need t0 = ∞ to get ma = F. Then ma = F 1 − exp t−t τ

Lorentz-Abraham equation Abraham (1905), Lorentz (1909): ma = F + mτ a˙

with F = q(E + v × B)

where E and B are background (i.e. applied) electric and magnetic fields, respectively. Contentious! I I I

a = τ a˙ when F = 0 so a = a0 exp(t/τ ) . . . runaway solution! t/τ R t 0 Introduce integrating factor: a = − emτ t0 F(t 0 )e −t /τ dt 0 Try constant F.
0 so need t0 = ∞ to get ma = F. Then ma = F 1 − exp t−t τ But t < ∞ so the particle must be prescient!

Side-step Ignore the pathologies! Follow Landau and Lifshitz (1951) and carry on. ma = F + mτ a˙ Reduction of order: ma = F + τ F˙ + O(τ 2 )

with F = q(E + v × B)

Side-step Ignore the pathologies! Follow Landau and Lifshitz (1951) and carry on. ma = F + mτ a˙

with F = q(E + v × B)

Reduction of order: ma = F + τ F˙ + O(τ 2 ) 

 q ˙ ˙ = q(E + v × B) + τ q(E + v × B) + F × B + O(τ 2 ) m

Side-step Ignore the pathologies! Follow Landau and Lifshitz (1951) and carry on. ma = F + mτ a˙

with F = q(E + v × B)

Reduction of order: ma = F + τ F˙ + O(τ 2 ) 

 q ˙ ˙ = q(E + v × B) + τ q(E + v × B) + F × B + O(τ 2 ) m and then drop O(τ 2 ). e.g. DAB, A Noble, Contemp. Phys. 55(2) 110 (2014)

Notation I

x 0 is inertial time and x 1 , x 2 , x 3 are Cartesian coordinates.

I

f (x) is short-hand for f (x 0 , x 1 , x 2 , x 3 )

I

∂a f = ∂f /∂x a

I

Einstein summation convention: sum over each up-down pair of indices e.g. ∂a V a = ∂V 0 /∂x 0 + ∇ · V where V = (V 1 , V 2 , V 3 )

I

On the following few slides x˙ a means dx a /dλ where λ is the proper time of a particle with world-line x a (λ)

I

Indices are lowered (raised) using the spacetime metric tensor ηab (its inverse η ab ) respectively where [ηab ] = diag(−1, 1, 1, 1). For example x˙ a = ηab x˙ b , i.e. x˙ 0 = −x˙ 0 , x˙ 1 = x˙ 1 , x˙ 2 = x˙ 2 , x˙ 3 = x˙ 3

Notation I

Electromagnetic field tensor:  0 E1 E2 E3  −E 1 0 −B 3 B 2 [Fab ] =   −E 2 B 3 0 −B 1 3 2 1 −E −B B 0

   

where a labels the row and b labels the column. I

Maxwell’s equations: ∂a Fbc + ∂b Fca + ∂c Fab = 0, ∂a F

ab

b

=J ,

(Faraday and no-mag-monopoles)

(Maxwell-Amp`ere and Gauss)

where J 0 is the charge density and J µ is the electric current (µ, ν = 1, 2, 3).

Notation

I

Stress-energy-momentum tensor of the electromagnetic field: 1 T ab = F ac F b c − η ab Fcd F cd 4 where T 00 is the energy density, T 0µ is the momentum flux and T µν is the stress tensor of the electromagnetic field.

I

∂a T ab = 0 in charged matter-free regions (energy-momentum conservation).

Lorentz-Abraham-Dirac equation

∂a T ab = 0

outside the worldline

=⇒ q a b ... F b x˙ + τ (δba + x˙ a x˙ b ) x b , m ηab x˙ a x˙ b = −1

x¨a = −

with τ = q 2 /6πm in units with 0 = µ0 = c = 1 PAM Dirac, Proc. Roy. Soc. (1938)

Lorentz-Abraham-Dirac equation

∂a T ab = 0

outside the worldline

=⇒ q a b ... F b x˙ + τ (δba + x˙ a x˙ b ) x b , m ηab x˙ a x˙ b = −1

x¨a = −

with τ = q 2 /6πm in units with 0 = µ0 = c = 1 PAM Dirac, Proc. Roy. Soc. (1938)

I

Requires an infinite negative bare mass to compensate for the infinite self-energy of the point particle.

For recent derivations of the LAD equation see, for example: MR Ferris, J Gratus, J. Math. Phys. 52, 092902 (2011) DAB, J Gratus, RW Tucker, Ann. Phys. 322 3, 599 (2007)

Landau-Lifshitz equation

Project out “unphysical” solutions to LAD equation by iteration and truncation: q a b ... F b x˙ + τ ∆a b x b m  q  q q a =⇒ x¨ = − F a b x˙ b − τ ∂d F ab x˙ b − ∆ab F b c F cd x˙ d + O(τ 2 ) m m m x¨a = −

where ∆a b = δ a b + x˙ a x˙ b LD Landau and EM Lifshitz, The Classical Theory of Fields (1962) F Rohrlich, Classical Charged Particles (2007)

ELEMENTS OF KINETIC THEORY

Non-relativistic Vlasov equation Postulate that the 1-particle distribution f (t, x, v) satisfies ∂f ∂f ∂f +v· + q(E + v × B) · =0 ∂t ∂x ∂v Z Electric charge density:

ρ(t, x) = q

f (t, x, v) d 3 v

Z Electric current density: j(t, x) = q vf (t, x, v) d 3 v = ρ u(t, x) Z µν Pressure tensor: P (t, x) = m (v − u)µ (v − u)ν f (t, x, v) d 3 v

Non-relativistic Vlasov equation Postulate that the 1-particle distribution f (t, x, v) satisfies ∂f ∂f ∂f +v· + q(E + v × B) · =0 ∂t ∂x ∂v Z Electric charge density:

ρ(t, x) = q

f (t, x, v) d 3 v

Z Electric current density: j(t, x) = q vf (t, x, v) d 3 v = ρ u(t, x) Z µν Pressure tensor: P (t, x) = m (v − u)µ (v − u)ν f (t, x, v) d 3 v I

Reasonable if collisions are negligible.

Non-relativistic Vlasov equation Postulate that the 1-particle distribution f (t, x, v) satisfies ∂f ∂f ∂f +v· + q(E + v × B) · =0 ∂t ∂x ∂v Z Electric charge density:

ρ(t, x) = q

f (t, x, v) d 3 v

Z Electric current density: j(t, x) = q vf (t, x, v) d 3 v = ρ u(t, x) Z µν Pressure tensor: P (t, x) = m (v − u)µ (v − u)ν f (t, x, v) d 3 v I

Reasonable if collisions are negligible.

I

Liouville: d 3 xd 3 v is preserved along the particle orbits.

Non-relativistic Vlasov equation Postulate that the 1-particle distribution f (t, x, v) satisfies ∂f ∂f ∂f +v· + q(E + v × B) · =0 ∂t ∂x ∂v Z Electric charge density:

ρ(t, x) = q

f (t, x, v) d 3 v

Z Electric current density: j(t, x) = q vf (t, x, v) d 3 v = ρ u(t, x) Z µν Pressure tensor: P (t, x) = m (v − u)µ (v − u)ν f (t, x, v) d 3 v I

Reasonable if collisions are negligible.

I

Liouville: d 3 xd 3 v is preserved along the particle orbits.

I

Vlasov =⇒ ∂ρ/∂t + ∇ · j = 0

Non-relativistic Vlasov equation Postulate that the 1-particle distribution f (t, x, v) satisfies ∂f ∂f ∂f +v· + q(E + v × B) · =0 ∂t ∂x ∂v Z Electric charge density:

ρ(t, x) = q

f (t, x, v) d 3 v

Z Electric current density: j(t, x) = q vf (t, x, v) d 3 v = ρ u(t, x) Z µν Pressure tensor: P (t, x) = m (v − u)µ (v − u)ν f (t, x, v) d 3 v I

Reasonable if collisions are negligible.

I

Liouville: d 3 xd 3 v is preserved along the particle orbits.

I

Vlasov =⇒ ∂ρ/∂t + ∇ · j = 0

I

Vlasov-Maxwell: include Maxwell for E, B with sources ρ, j

A GEOMETRIC APPROACH TO KINETIC THEORY

A GEOMETRIC APPROACH TO KINETIC THEORY Caution: x˙ a is now a coordinate!

Strategy

LAD equation   y

−−−−−−−−−−→ reduction of order

kinetic model −−−−−−−−−−→ reduction of order  moments in acceleration and velocity y

LL equation   y kinetic model  moments in velocity y

LAD warm fluid theory −−−−−−−−−−→ LL warm fluid theory reduction of order

LL kinetic & fluid: e.g. M Tamburini, et al, NIMA (2011) VI Berezhiani, et al, Phys. Rev. E (2004) RD Hazeltine and SM Mahajan, Phys. Rev. E (2004) LAD kinetic : A Noble, DAB, J Gratus, DA Jaroszynski, J. Math. Phys. 54 043101 (2013) LAD fluid : DAB, AC Carr, J Gratus, A Noble, Proc. SPIE 8779 (2013)

Strategy

LAD equation   y

−−−−−−−−−−→ reduction of order

kinetic model −−−−−−−−−−→ reduction of order  moments in acceleration and velocity y

LL equation   y kinetic model  moments in velocity y

LAD warm fluid theory −−−−−−−−−−→ LL warm fluid theory reduction of order

LL kinetic & fluid: e.g. M Tamburini, et al, NIMA (2011) VI Berezhiani, et al, Phys. Rev. E (2004) RD Hazeltine and SM Mahajan, Phys. Rev. E (2004) LAD kinetic : A Noble, DAB, J Gratus, DA Jaroszynski, J. Math. Phys. 54 043101 (2013) LAD fluid : DAB, AC Carr, J Gratus, A Noble, Proc. SPIE 8779 (2013)

Geometrical Vlasov equation

Ingredients: “phase space” Q, vector field LQ on Q, top-degree differential form ω on Q, 1-particle distribution f on Q

Geometrical Vlasov equation

Ingredients: “phase space” Q, vector field LQ on Q, top-degree differential form ω on Q, 1-particle distribution f on Q Vlasov equation: LLQ (f ω) = 0 where L is the Lie derivative

Geometrical Vlasov equation

Ingredients: “phase space” Q, vector field LQ on Q, top-degree differential form ω on Q, 1-particle distribution f on Q Vlasov equation: LLQ (f ω) = 0 where L is the Lie derivative =⇒ particle number conservation, but LLQ ω 6= 0 in general.

Geometrical Vlasov equation (Lorentz force)

Event-velocity “phase space”: Q = {(x, x) ˙ ∈ T M| ϕ = 0, x˙ 0 > 0} where

1 ϕ = (ηab x˙ a x˙ b + 1) 2 Solutions to the Lorentz equation are integral curves of L = x˙ a

∂ ∂ q − F a b (x)x˙ b a a ∂x m ∂ x˙

Geometrical Vlasov equation (Lorentz force) “Natural” top form on T M: ω ˆ = dx 0123 ∧ d x˙ 0123 where dx 0123 = dx 0 ∧ dx 1 ∧ dx 2 ∧ dx 3 , etc. Induced top form ω on Q: ω = ι∗ ω ˜ where ω ˆ=ω ˜ ∧ dϕ and ι : Q ,→ T M Vlasov equation for 1-particle distribution f ∈ ΓΛ0 Q: LLQ (f ω) = 0 where L = ι∗ LQ

Geometrical Vlasov equation (Lorentz force) “Natural” top form on T M: ω ˆ = dx 0123 ∧ d x˙ 0123 where dx 0123 = dx 0 ∧ dx 1 ∧ dx 2 ∧ dx 3 , etc. Induced top form ω on Q: ω = ι∗ ω ˜ where ω ˆ=ω ˜ ∧ dϕ and ι : Q ,→ T M Vlasov equation for 1-particle distribution f ∈ ΓΛ0 Q:

e.g.

LLQ (f ω) = 0 where L = ι∗ LQ p ι : (x, v) 7→ (x, x˙ 0 = 1 + v2 , x˙ = v), p ω = dx 0123 ∧ dv 123 / 1 + v2 , (N.B. LLQ ω = 0)

√ with d 3 v/ 1 + v2 the measure in proper velocity space.

Macroscopic fields I

Electron “fluid” 4-current: Ja = q

Z

x˙ a f √

where ∂a J a = 0 follows identically.

d 3v 1 + v2

Macroscopic fields I

Electron “fluid” 4-current: Ja = q

Z

x˙ a f √

d 3v 1 + v2

where ∂a J a = 0 follows identically. I

Maxwell equations: ∂a Fbc + ∂b Fca + ∂c Fab = 0,

b ∂a F ab = Jtot

a = J a if no additional charged matter fields. and Jtot

Macroscopic fields I

Electron “fluid” 4-current: Ja = q

Z

x˙ a f √

d 3v 1 + v2

where ∂a J a = 0 follows identically. I

Maxwell equations: ∂a Fbc + ∂b Fca + ∂c Fab = 0,

b ∂a F ab = Jtot

a = J a if no additional charged matter fields. and Jtot I

Electron “fluid” stress-energy-momentum tensor: Z d 3v ab S = m x˙ a x˙ b f √ 1 + v2 =⇒ ∂a (S ab + T ab ) = 0 if no additional fields (energy-momentum conservation).

Geometrical Vlasov equation (LAD)

Event-velocity-acceleration “phase space”: Q = {(x, x, ˙ x¨) ∈ T M ⊕ T M| ϕ1 = 0, ϕ2 = 0, x˙ 0 > 0} where

1 ϕ1 = (ηab x˙ a x˙ b + 1), ϕ2 = ηab x˙ a x¨b 2 Solutions to the LAD equation are integral curves of L = x˙ a

 ∂  ∂ ∂ q + x¨a a + x¨b x¨b x˙ a + τ −1 (¨ x a + F a b x˙ b ) a ∂x ∂ x˙ m ∂¨ xa

Geometrical Vlasov equation (LAD) “Natural” top form on T M ⊕ T M: ω ˆ = dx 0123 ∧ d x˙ 0123 ∧ d x¨0123 Induced top form ω on Q: ω = ι∗ ω ˜ where ω ˆ=ω ˜ ∧ dϕ1 ∧ dϕ2 and ι : Q ,→ T M ⊕ T M Vlasov equation for 1-particle distribution f ∈ ΓΛ0 Q: LLQ (f ω) = 0 where L = ι∗ LQ

Geometrical Vlasov equation (LAD) “Natural” top form on T M ⊕ T M: ω ˆ = dx 0123 ∧ d x˙ 0123 ∧ d x¨0123 Induced top form ω on Q: ω = ι∗ ω ˜ where ω ˆ=ω ˜ ∧ dϕ1 ∧ dϕ2 and ι : Q ,→ T M ⊕ T M Vlasov equation for 1-particle distribution f ∈ ΓΛ0 Q: LLQ (f ω) = 0 where L = ι∗ LQ p p ι : (x, v, a) 7→ (x, x˙ 0 = 1 + v2 , x˙ = v, x¨0 = a · v/ 1 + v2 , ¨x = a), ω = dx 0123 ∧ da123 ∧ dv 123 /(1 + v2 ), with d 3 a d 3 v/(1 + v2 ) the measure on proper velocity-acceleration space.

(N.B. LLQ ω = 3ω/τ )

Macroscopic fields I

Electron “fluid” 4-current: a

J =q

Z

x˙ a f

d 3a d 3v 1 + v2

where ∂a J a = 0 follows identically. I

Maxwell equations: ∂a Fbc + ∂b Fca + ∂c Fab = 0,

b ∂a F ab = Jtot

a = J a if no additional charged matter fields. and Jtot

Macroscopic fields I

Electron “fluid” 4-current: a

J =q

Z

x˙ a f

d 3a d 3v 1 + v2

where ∂a J a = 0 follows identically. I

Maxwell equations: ∂a Fbc + ∂b Fca + ∂c Fab = 0,

b ∂a F ab = Jtot

a = J a if no additional charged matter fields. and Jtot I

Electron “fluid” stress-energy-momentum tensor: Z d 3a d 3v S ab = m x˙ a x˙ b f 1 + v2 and, due to radiation reaction, ∂a (S ab + T ab ) 6= 0 in general.

Physical submanifold of Q f (x, v, a) =

p
1 + v2 g (x, v) δ (3) a − A(x, v)

Physical submanifold of Q f (x, v, a) = I

p
1 + v2 g (x, v) δ (3) a − A(x, v)

g (x, v) is the 1-particle distribution on event-velocity “phase space”.

Physical submanifold of Q f (x, v, a) =

p
1 + v2 g (x, v) δ (3) a − A(x, v)

I

g (x, v) is the 1-particle distribution on event-velocity “phase space”.

I

a = A(x, v) determines the physical submanifold of Q. N.B. Nothing to do with the vector potential!

Physical submanifold of Q f (x, v, a) =

p
1 + v2 g (x, v) δ (3) a − A(x, v)

I

g (x, v) is the 1-particle distribution on event-velocity “phase space”.

I

a = A(x, v) determines the physical submanifold of Q. N.B. Nothing to do with the vector potential!

I

LLQ (f ω) = 0 =⇒ µ ∂Aµ 1 q ν ∂A + A = Aa Aa v µ + (Aµ + F µ a x˙ a ), a ν ∂x ∂v τ m  µ p ∂g ∂ g A x˙ a a + 1 + v2 µ √ =0 ∂x ∂v 1 + v2 √ where A0 = v µ Aµ / 1 + v2 and µ, ν = 1, 2, 3

x˙ a

Physical submanifold of Q f (x, v, a) =

p
1 + v2 g (x, v) δ (3) a − A(x, v)

I

g (x, v) is the 1-particle distribution on event-velocity “phase space”.

I

a = A(x, v) determines the physical submanifold of Q. N.B. Nothing to do with the vector potential!

I

LLQ (f ω) = 0 =⇒ µ ∂Aµ 1 q ν ∂A + A = Aa Aa v µ + (Aµ + F µ a x˙ a ), a ν ∂x ∂v τ m  µ p ∂g ∂ g A x˙ a a + 1 + v2 µ √ =0 ∂x ∂v 1 + v2 √ where A0 = v µ Aµ / 1 + v2 and µ, ν = 1, 2, 3

x˙ a

I

Aµ = Aµ(0) + τ Aµ(1) + O(τ 2 ), (similar for g ) |{z} | {z } No RR

LL Vlasov

RELATIVISTIC FLUID THEORY

RELATIVISTIC FLUID THEORY Return to LLQ (f ω) = 0 and drop the decomposition √ f (x, v, a) = 1 + v2 g (x, v)δ (3) [a − A(x, v)].

Strategy

LAD equation   y

−−−−−−−−−−→ reduction of order

kinetic model −−−−−−−−−−→ reduction of order  moments in acceleration and velocity y

LL equation   y kinetic model  moments in velocity y

LAD warm fluid theory −−−−−−−−−−→ LL warm fluid theory reduction of order

LL kinetic & fluid: e.g. M Tamburini, et al, NIMA (2011) VI Berezhiani, et al, Phys. Rev. E (2004) RD Hazeltine and SM Mahajan, Phys. Rev. E (2004) LAD kinetic : A Noble, DAB, J Gratus, DA Jaroszynski, J. Math. Phys. 54 043101 (2013) LAD fluid : DAB, AC Carr, J Gratus, A Noble, Proc. SPIE 8779 (2013)

Strategy

LAD equation   y

−−−−−−−−−−→ reduction of order

kinetic model −−−−−−−−−−→ reduction of order  moments in acceleration and velocity y

LL equation   y kinetic model  moments in velocity y

LAD warm fluid theory −−−−−−−−−−→ LL warm fluid theory reduction of order

LL kinetic & fluid: e.g. M Tamburini, et al, NIMA (2011) VI Berezhiani, et al, Phys. Rev. E (2004) RD Hazeltine and SM Mahajan, Phys. Rev. E (2004) LAD kinetic : A Noble, DAB, J Gratus, DA Jaroszynski, J. Math. Phys. 54 043101 (2013) LAD fluid : DAB, AC Carr, J Gratus, A Noble, Proc. SPIE 8779 (2013)

Natural moments & field equations

S a1 ...a` :b1 ...bn (x) =

Z

x˙ a1 . . . x˙ a` x¨b1 . . . x¨bn f (x, v, a)

1 d 3a d 3v 1 + v2

Natural moments & field equations

S a1 ...a` :b1 ...bn (x) =

Z

x˙ a1 . . . x˙ a` x¨b1 . . . x¨bn f (x, v, a)

1 d 3a d 3v 1 + v2

∂a S a:∅ = 0, ∂a S ab:∅ − S ∅:b = 0,   q ∂a S a:b − S b:c c − τ −1 S ∅:b + F b c S c:∅ = 0, m abc:∅ (b:c) ∂a S − 2S = 0,   q ∂a S ab:c − S ∅:bc − S bc:d d − τ −1 S b:c + F c d S bd:∅ = 0, m  q d:(b c)  a:bc (b:c)d −1 ∅:bc ∂a S − 2S 2 S + S F d = 0, d −τ m ...

Natural moments & constraints

S

a1 ...a` :b1 ...bn

Z (x) =

x˙ a1 . . . x˙ a` x¨b1 . . . x¨bn f (x, v, a) S a a :∅ = −S ∅ , S a: a = 0, S ab b :∅ = −S a:∅ , S a a :b = −S ∅:b , S ab: a = 0, S a: ab = 0, S ab: bc = 0, S a a :bc = −S ∅:bc , ...

1 d 3a d 3v 1 + v2

Centred moments

R

   Z  a1 a1 a` a` (x) = x˙ − U (x) . . . x˙ − U (x)    1 b1 bn bn d 3a d 3v − A (x) . . . x¨ − A (x) f (x, v, a) 1 + v2

a1 ...a` :b1 ...bn

 × x¨b1 where

U a = S a:∅ /S ∅ , Aa = S ∅:a /S ∅

Centred moments

R

   Z  a1 a1 a` a` (x) = x˙ − U (x) . . . x˙ − U (x)    1 b1 bn bn d 3a d 3v − A (x) . . . x¨ − A (x) f (x, v, a) 1 + v2

a1 ...a` :b1 ...bn

 × x¨b1 where

U a = S a:∅ /S ∅ , Aa = S ∅:a /S ∅ Generalization of Amendt-Weitzner2 warm fluid theory √ I = 1 + U a Ua and R a1 ...a` :b1 ...bn = O(`+n ) I

2

Delete O(3 ) from the field equations and solve subject to the constraints to O(3 )

e.g. Weitzner in Relativistic Fluid Dynamics, Lecture Notes in Mathematics 1385 Springer-Verlag (1987)

Electric waves in magnetized plasma

Electric waves in magnetized plasma Dispersion relation ω(k) for longitudinal electric waves along a homogeneous magnetic field:   2 3 3k iτ − ωp θ − [ωp2 − (2k 2 + ωp2 )θ] + O(τ 2 , θ2 ) ω = ωp + 2 ωp 4 2 {z } | {z } | frequency shift

damping

where ωp is the plasma frequency and θ is the electron fluid’s thermal energy along the magnetic field lines normalized with respect to m.

Electric waves in magnetized plasma Dispersion relation ω(k) for longitudinal electric waves along a homogeneous magnetic field:   2 3 3k iτ − ωp θ − [ωp2 − (2k 2 + ωp2 )θ] + O(τ 2 , θ2 ) ω = ωp + 2 ωp 4 2 {z } | {z } | frequency shift

damping

where ωp is the plasma frequency and θ is the electron fluid’s thermal energy along the magnetic field lines normalized with respect to m. I

Bohm-Gross: ω = ωp + 3k 2 θ/2ωp + O(θ2 )

Electric waves in magnetized plasma Dispersion relation ω(k) for longitudinal electric waves along a homogeneous magnetic field:   2 3 3k iτ − ωp θ − [ωp2 − (2k 2 + ωp2 )θ] + O(τ 2 , θ2 ) ω = ωp + 2 ωp 4 2 {z } | {z } | frequency shift

damping

where ωp is the plasma frequency and θ is the electron fluid’s thermal energy along the magnetic field lines normalized with respect to m. I

Bohm-Gross: ω = ωp + 3k 2 θ/2ωp + O(θ2 )

I

“Cold” radiating plasma : ω = ωp − iτ ωp2 /2 + O(τ 2 )

Electric waves in magnetized plasma Dispersion relation ω(k) for longitudinal electric waves along a homogeneous magnetic field:   2 3 3k iτ − ωp θ − [ωp2 − (2k 2 + ωp2 )θ] + O(τ 2 , θ2 ) ω = ωp + 2 ωp 4 2 {z } | {z } | frequency shift

damping

where ωp is the plasma frequency and θ is the electron fluid’s thermal energy along the magnetic field lines normalized with respect to m. I

Bohm-Gross: ω = ωp + 3k 2 θ/2ωp + O(θ2 )

I

“Cold” radiating plasma : ω = ωp − iτ ωp2 /2 + O(τ 2 )

I

Shorter wavelength =⇒ lower damping

EM waves in magnetized plasma Left-handed circularly polarized EM wave k k B : k 2 ωωc + ω 2 − ωp2 = ω2 ω(ω + ωc ) ωp2 (ω 3 + 3ωωc2 + 4ω 2 ωc − 2ωc ωp2 ) +θ 2ω(ω + ωc )4  ωωp2 + iτ 2 ω + 2ωωc + ωc2  ωc ωp2 (4ωωc2 + ω 3 + ωωp2 + 5ω 2 ωc − 3ωc ωp2 ) +θ (ω + ωc )5 + O(τ 2 , θ2 )

EM waves in magnetized plasma Terms in Im(ω/ωp ) as a function of ω0 /ωp

ω = ω0 + τ α + θβ + τ θγ + O(τ 2 , θ2 )

EM waves in magnetized plasma Right-handed circularly polarized EM wave k k B

ω = ω0 + τ α + θβ + τ θγ + O(τ 2 , θ2 )

Summary I

A kinetic theory and a fluid theory including radiation reaction have been developed. I

Only scratched the surface here: quantum radiation reaction, vacuum polarization,. . .

Summary I

A kinetic theory and a fluid theory including radiation reaction have been developed. I

Only scratched the surface here: quantum radiation reaction, vacuum polarization,. . .

References: I

J. Math. Phys. 54 043101 (2013)

I

Proc. SPIE 8779 (2013) {arXiv:1303.7385}

I

Contemp. Phys. 55(2) 110 (2014)

Summary I

A kinetic theory and a fluid theory including radiation reaction have been developed. I

Only scratched the surface here: quantum radiation reaction, vacuum polarization,. . .

References: I

J. Math. Phys. 54 043101 (2013)

I

Proc. SPIE 8779 (2013) {arXiv:1303.7385}

I

Contemp. Phys. 55(2) 110 (2014)

Summary I

A kinetic theory and a fluid theory including radiation reaction have been developed. I

Only scratched the surface here: quantum radiation reaction, vacuum polarization,. . .

References: I

J. Math. Phys. 54 043101 (2013)

I

Proc. SPIE 8779 (2013) {arXiv:1303.7385}

I

Contemp. Phys. 55(2) 110 (2014)

Thank you for your attention!