Transport coe cients in plasmas spanning weak to strong

Transport coefficients in plasmas spanning weak tostrong correlation

Scott D. Baalrud1,2 and Jerome Daligault1

1Theoretical Division, Los Alamos National Laboratory2Department of Physics and Astronomy, University of Iowa

APS Division of Plasma Physics Meeting, October 29, 2012

APS-DPP — October 29, 2012, p 1

Outline

• Calculate transport coefficients directly from the Boltzmann collision

operator using an effective potential

1) Show that this approach can capture weak, moderate and into strongly coupledregimes

2) Test the theory using classical molecular dynamics simulations for e+− i+ temper-ature relaxation

• Avoid the conventional weak-coupling approximations by:

1) Exploiting symmetries of moments of the full Boltzmann collision integral (no smallangle scattering, or hierarchy, expansions are used)

2) Obtain a convergent theory using an effective potential that includes screeningat long range (one example is the Debye shielded (Yukawa) potential, but we alsoconsider others)

• For temperature relaxation, we find a generalized Coulomb logarithm

• For resistivity, diffusion, etc (processes where deviation from Maxwellian

distributions are important), we calculate the collision integrals of the

Chapman-Enskog matrix (Ωi,j integrals)


Boltzmann collision operator

• The Boltzmann collision operator

CB(fs, fs′) =

∫~v′

∫Ω

d3v′dΩσ u [fs(~v)f ′s′(~v′)− fs(~v)f ′s′(~v

′)]

• Derived assuming binary collisions and the following three properties of

the collision process:

(i) d3vd3v′ = d3vd3v′ (property of Jacobian)

(ii) u = u (from conservation of momentum, or energy)

(iii) σ(~v, ~v′ → ~v, ~v′) = σ(~v, ~v′ → ~v, ~v′) (from time reversal invariance)

• Here v denotes velocities after binary scattering, σ is the differential

scattering cross section, σd3vd3v the scattering probability and dΩ the

solid angle

• For calculating transport properties (moments of CB) the difficult part

to deal with is ~v: what is f(~v)?


Weak coupling: assume small scattering angle

• Need to do something with fs(~v) and fs′(~v′)

• Small scattering angle expansion: ~v = ~v + ∆~v, ~v ∆~v

CB(fs) =∑s′

[∂fs

∂~v· −

ms

ms′fs(~v)

∂

∂~v·]〈∆~v〉s/s′

∆t

+

[1

2

∂2fs(~v)

∂~v∂~v: +

1

2

m2s

m2s′fs(~v)

∂2

∂~v∂~v: −

ms

ms′

∂fs(~v)

∂~v·∂

∂~v·]〈∆~v∆~v〉s/s′

∆t

where

〈∆~v〉s−s′

∆t=mss′

ms

∫d3v′fs′(~v

′)∆~u

〈∆~v∆~v〉s−s′

∆t=m2ss′

m2s

∫d3v′fs′(~v

′)∆~u∆~u

and in which

∆~u =

∫dΩσ u∆~u,

∆~u∆~u =

∫dΩσ u∆~u∆~u

where

∆~u = u[sin θ cosφ x+ sin θ sinφ y − 2 sin2(θ/2)u]


Logarithmic divergence for bare Coulomb potential

• For bare Coulomb potential, get Rutherford scattering cross section

σ =q2sq

2s′

4m2ss′u

4 sin4(θ/2)

•Writing in terms of impact parameter bdbdφ = σdΩ:

∆~u = −4πu~ub2min

∫ bmax

0

dbb

b2 + b2min

= −4πq2

sq2s′~u

m2ss′u

3ln(b2

max/b2min + 1)

where bmin = qsqs′/mss′u2 is the distance of closest approach

• Nominally, bmax =∞, but then this diverges logarithmically!

• Typically impose screening in an ad-hoc way by taking bmax = λD, so

ln(b2max/b

2min + 1) ≈ 2 ln(λD/bmin) = 2 ln Λ

• This only makes sense when λD bmin (weakly coupled)


Transport coefficients come from moments of C(fs, fs′)

• Friction force density: ~Rs−s′ =∫d3vms~vC(fs, fs′)

• Energy exchange density: Qs−s′ =∫d3v 1

2ms(~v − ~Vs)

2C(fs, fs′)

• Resistivity comes from ~Re−i, thermal conductivity from Qe−i, etc.

• Need to input the distribution functions fs(~v) and fs′(~v′)

• For Maxwellian distributions, one finds (for weak coupling)

~Rs−s′ = −nsmsνs−s′(~Vs − ~Vs′)

and

Qs−s′ = −3mss′

ms′nsνs−s′(Ts − Ts′)

where the collision frequency is

νs−s′ =16√πq2

sq2s′ns′

3msmss′v3T

ln Λ

to first order in |~Vs − ~Vs′|/v2T 1, and v2

T = v2Ts + v2

Ts′

• For some processes, deviations from Maxwellian are essential – there

are methods for dealing with this (Chapman-Enskog, Grad’s moment

method)


Two effects must be dealt with when ln Λ . 10

• (1) Can’t use a small angle scattering expansion

– Ends up ordering terms according to Λ 1

– First two terms in hierarchy are ∝ ln Λ, but all others O(1)

• (2) Can’t use the bmax = λD cutoff

– By definition bmin λD in strongly coupled plasmas

– Particle interactions are all long range (outside the Debye sphere)

• To avoid (1): We work straight from the Boltzmann collision operator,

and exploit symmetries to calculate transport coefficients

• To avoid (2): We use an effective potential that accounts for screening

(as well as close interaction physics that arises at strong coupling)


Can exploit symmetries of the Boltzmann equation• Transport coefficients come from velocity moments of the form

〈χ〉s−s′ =

∫d3v χs(~v)CB(fs, fs′)

• Apply same properties used to derive the Boltzmann collision operator:

(i) d3vd3v′ = d3vd3v′, (ii) u = u, (iii) σ(~v, ~v′ → ~v, ~v′) = σ(~v, ~v′ → ~v, ~v′)

• Applying these properties and the Boltzmann collision operator yields

〈χ〉s−s′ =

∫d3vd3v′ ∆χsfs(~v)fs′(~v

′)

in which ∆χs =∫dΩσ u∆χs, and ∆χs = χs(~v)− χs(~v).

• Only depends on the distribution functions before scattering: no f(v)!

• Can relate χs(~v) to χs(~v) using conservation laws

• Evaluating for momentum, χs = ms~v, gives the friction force density

~Rs−s′ = mss′

∫d3u ∆~u

∫d3v′ fs(~u+ ~v′)fs′(~v

′)

• Evaluating for χs = 12ms(~v − ~Vs)

2, gives the energy exchange density

Qs−s′ = mss′

∫d3u∆~u ·

∫d3v′(~v′ − ~Vs +

mss′

ms

~u)fs(~u+ ~v′)fs′(~v′)


For Maxwellians: A generalized Coulomb log

• Evaluating ~Rs−s′ and Qs−s′ for flowing Maxwellian fs and fs′, one finds the

same coefficients as weakly coupled theory, except that ln Λ is replaced

by a generalized Coulomb logarithm:

Ξ =1

2

∫ ∞0

dξ e−ξ2ξ5σs(ξ,Λ)

σo

where ξ ≡ u/vT , and σo = πλ2D/Λ

2cl

• Here σs is the momentum-scattering cross section

σs = 4π

∫ π

0

dθ sin θ sin2(θ/2)σ

• Maxwellians are just an example, the theory can be applied to any f

• Only other input is the electric potential surrounding individual particles

(which determines σ)

• Next, we will consider e-i temperature relaxation: dTe/dt = 2Qe−i/(3ne)


Collision integrals of Chapman-Enskog (or Grad)• Many processes (such as diffusion, resistivity, thermal conduction, etc.)

can be described by Chapman-Enskog equations, or Grad’s moment equa-

tions, which account for deviations from Maxwellian

• In these theories, collisions arise through the integrals

Ω(l,r)ss′ =

3

16

ms

mss′

νss′

ns′

Ξ(l,r)ss′

Ξss′in which νss′ ≡

16√πq2

sq2s′ns′

3msmss′v3ss′

Ξss′

is a collision frequency corresponding to the (l, r)th matrix element, Ξss′ =

Ξ(1,1)ss′ is the generalized Coulomb logarithm associated with the lowest

order (1, 1) term, and

Ξ(l,r)ss′ =

1

2

∫ ∞0

dξ ξ2r+3e−ξ2 σ

(l)ss′(ξ,Λ)

σo

is the generalized Coulomb logarithm for the (l, r)th matrix element,

σ(l)ss′ ≡ 2π

∫ π

0

dθ sin θ(1− cosl θ)σss′

is the lth momentum-transfer cross section and

σo ≡πq2

sq2s′

m2ss′v

4ss′

=πλ2

Λ2

is a reference value for the differential scattering cross section.


The effective potential is externally determined

• A common potential in plasmas is the Debye-screened (Yukawa) potential

φt =qt

re−r/λD

• This can be justified if: (1) the transport timescale is long compared to

the plasma frequency for a given process (νs−s′/ωps 1),

– Then, the polarization response is adiabatic and satisfies the Boltzmann relation

ns = no exp(−qsφ/Ts)

• (2) The interaction potential is weak: |qsφ/Ts| 1

– Then, Poisson’s equation ∇2φ = −4πρq gives the Debye-screened potential

• Adiabaticity (νs−s′/ωps 1) for e-i collisions: νe−i/ωpe ' Ξ/Λ

– Weak coupling (Λ 1), Ξ = ln Λ, νe−i/ωpe ∼ ln Λ/Λ 1

– Strong coupling (Λ 1), Ξ = Λ2 ln2 Λ, νe−i/ωpe ∼ Λ ln2 Λ 1

• If Γ is large |qφ/T | 1 may not hold at close distances

• The effective potential (and hence correlation effects) are related to the

pair correlation function g(r)


Kinetic equations implicitly approximate g(r)– The pair correlation function in the BBGKY hierarchy evolves according to:

∂g2(1, 2; t)

∂t=[L0

1 + L02

]g2(1, 2; t) + L12f(1; t)f(2; t) (1a)

+ L12g2(1, 2; t) (1b)

+

∫d3 [L13f(1; t)g2(2, 3; t) + L13f(3; t)g2(1, 2; t) + (1↔ 2)] (1c)

+

∫d3(L13 + L23)g3(1, 2, 3; t) (1d)

where L0i = −~vi · ∇i, Li,j = ∇Vi,j · ∂i,j and Vi,j = φ(|~ri − ~rj|)

• Equilibrium limit: f(1; t) = nfM(~p1), g2(1, 2; t) = n2fM(~p1)fM(~p2)h(|~r1−~r2|) where h(r) = g(r)− 1 and g(r) is the pair distribution function

• Landau collision operator: Assumes terms (1b)-(1d) = 0, gives

g(r) = 1− eφ(r)/kBT where φ = q/r

• Lenard-Balescu: Assumes terms (1b) and (1d) = 0, gives

g(r) = 1− eφsc/kBT where φsc = qe−r/λD/r

• Boltzmann Equation: Assumes terms (1c) and (1d) = 0, gives

g(r) = e−eφ(r)/kBT where φ is whatever model you want to use

• Closures give g(r)↔ φ(r): correlations can be imposed through g(r)


Hypernetted chain (HNC) theory can be used

• HNC is accurate for Γ . 100 (checked with classical MD)

• Yukawa gives a good approximation for Γ . 10

• Here Γ = e2/(aekBTe) where ae = [3/(4πne)]1/3

10−3 10−2 10−1 100 1010

0.5

1

1.5

r/ae

gie(r

)

Yukawa

Bare Coulomb

HNC unlike charge

Γ = 0.001

0.01

0.1

1

10

like charge


Scattering cross section in the Yukawa potential

• Recall that the lth momentum-transfer cross section is

σ(l)ss′ ≡ 2π

∫ π

0

dθ sin θ(1− cosl θ)σss′

• For the Deybe screened (Yukawa) potential in a classical regime θ =

π − 2Θ where

Θ = b

∫ ∞ro

dr

r2√

1− Ueff(r, b)(2)

and

Ueff = ±2

Λclξ2

λD

re−r/λD +

b2

r2(3)

in which + refers to the repulsive (qsqs′ > 0), and − to the attractive

(qsqs′ < 0) Yukawa potentials.

• ro is the distance of closest approach from Ueff(ro) = 1.

• Exact analytic solutions of these are not known, but we will look at

numerical solutions and asymptotic limits

• One interesting feature is that repulsive and attractive cases give different

solutions


Ueff has a barrier for attractive potential• If Λclξ

2 < 1/13.2, an effective “potential barrier” can form

• Barrier extends the distance of closest approach

• Multiple roots of Ueff = 1, only want largest

0 5 10−2

−1

0

1

2

3Attractive

r/λD

Ueff

0 5 10−2

−1

0

1

2

3Repulsive

r/λD

Ueff

3.04.04.74.9

b/λD

Λc lξ2 = 1/30 Λc lξ

2 = 1/30


Need to take maxrmin for attractive potential

• Only one root in the repulsive case

• Need to be careful to take the largest root in attractive case (beyond the

barrier Λclξ2 . 1/13.2)

10−1

100

101

102

10−4

10−3

10−2

10−1

100

101

102

b/λD

rmin

Repu lsive

Λc lξ2 = 1/100

Attractive root 2

Attractive root 1


Asymptotic limits return known results

• For no barrier: Λclξ2 & 1/13.2

σcls ' 4σoξ

−4 ln(1 + Λclξ2)

• For this the generalized Coulomb logarithm is

Ξcl,a = exp(Λ−1cl )E1(Λ

−1cl )

(this covers weak, moderate, and part of strong correlation)

• For Λ 1 this gives

Ξcl,a ' ln(Λ)− γ = ln(e−γΛ) = ln(0.56Λ) = ln(0.79/g)

where g ≡ Zie2/(TeλDe) =√

2/Λ for Te ' Ti• Previous theories get the number 0.79 using complicated renormalization

techniques (see e.g., Landau & Lifshitz, Aono, BPS)

• Beyond the barrier: Λclξ2 . 1/13.2

σs ' 0.81σoΛ2cl[ln

2(Λclξ2)− 2 ln(Λclξ

2) +O(1)]

• For this, the generalized Coulomb logarithm is

Ξcl,b = 0.4Λ2cl[ln

2(Λcl) + ln(Λcl)(1− γ) +O(1)]


Momentum scattering cross section

• Attractive and repulsive cases differ for 10−2 . Λclξ2 . 10

• Figure shows: numerical solutions (black), weakly coupled solution (red),

below barrier asymptote (blue), above barrier asymptote (green)

10−4

10−2

100

102

104

106

10−10

10−5

100

105

1/Λc lξ2

σs/λ2 D

Attractive (classical)

Repul sive (classical)

σs,wc

σs,a

σs,b


Theory gives excellent agreement with MD

• Classical MD simulations for like-charge e+ − i+ thermal relaxation:

dTe/dt ≈ 2Qe−i/3ne

• Screening length is the electron Debye length (λDe)

10−2 10−1 100 1010

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

g

Ξ

Ξ (repulsive)

lnΛ

ln(0.765Λ)

MD data best fit

Simulations: Dimonte and Daligault, PRL 101, 135001 (2008).


Solutions for the generalized Coulomb logarithm

• Numerical solutions (solid lines) and asymptotic solutions (dashed)

• Attractive and repulsive cases can be distinguished for 10−2 . Λclξ2 . 10

• Asymptote to a common value in weak and strong correlation limits


Collision integrals for Chapman-Enskog matrix

• Smoother curves are for repulsive (other for attractive). Red dashed line

shows the weakly coupled result keeping up to O(1) terms

10010−4 10−2 1020

1

2

1/Λ

Ξ(1,1

) /Ξ

10010−4 10−2 1021

2

3

1/Λ

Ξ(1,2

) /Ξ

10010−4 10−2 1020

5

10

1/Λ

Ξ(1,3

) /Ξ

10010−4 10−2 1021020.5

1

1.5

2

1/Λ

Ξ(2,1

) /Ξ

10010−4 10−2 1021

2

3

1/Λ

Ξ(2,2

) /Ξ

10010−4 10−2 1024

6

8

10

1/Λ

Ξ(2,3

) /Ξ

10010−4 10−2 102

1

2

3

1/Λ

Ξ(3,1

) /Ξ

10010−4 10−2 1022

3

4

1/Λ

Ξ(3,2

) /Ξ

10010−4 10−2 1026

8

10

12

1/ΛΞ

(3,3

) /Ξ


Conclusions

• Demonstrated a way to calculate transport coefficients without assuming

small angle scattering

– Based on the Boltzmann equation (which has a binary collision assumption built in)

– Collisions are binary, but in an effective potential that accounts for screening andcorrelation effects

– Theory agrees with MD simulations of temperature relaxation

• For temperature relaxation a generalized Coulomb logarithm is required

– The generalized Coulomb logarithm is more complicated for moderate-strong cou-pling, e.g., attractive and repulsive collisions can be distinguished

– The Chapman-Enskog collision integrals can be written in terms of generalizedCoulomb logarithms

• Future work will address going beyond the Yukawa potential using HNC,

and testing the theory for diffusion, resistivity, etc., with MD


Documents

Transport coe cients in plasmas spanning weak to strong