Transport coefficients in plasmas spanning weak to strong correlation

Transcription

1 Transport coefficients in plasmas spanning weak to strong correlation Scott D. Baalrud 1,2 and Jerome Daligault 1 1 Theoretical Division, Los Alamos National Laboratory 2 Department of Physics and Astronomy, University of Iowa APS Division of Plasma Physics Meeting, October 29, 212 APS-DPP October 29, 212, p 1

2 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 coupled regimes 2) Test the theory using classical molecular dynamics simulations for e + i + temperature relaxation Avoid the conventional weak-coupling approximations by: 1) Exploiting symmetries of moments of the full Boltzmann collision integral (no small angle scattering, or hierarchy, expansions are used) 2) Obtain a convergent theory using an effective potential that includes screening at long range (one example is the Debye shielded (Yukawa) potential, but we also consider 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 (Ω i,j integrals) APS-DPP October 29, 212, p 2

3 Boltzmann collision operator The Boltzmann collision operator C B (f s, f s ) = d 3 v dω σ u [ ˆf s (ˆ v) ˆf s (ˆ v ) f s ( v)f s ( v )] v Ω Derived assuming binary collisions and the following three properties of the collision process: (i) d 3ˆvd 3ˆv = d 3 vd 3 v (property of Jacobian) (ii) û = u (from conservation of momentum, or energy) (iii) ˆσ( v, v ˆ v, ˆ v ) = ˆσ(ˆ v, ˆ v v, v ) (from time reversal invariance) Here ˆv denotes velocities after binary scattering, σ is the differential scattering cross section, ˆσd 3ˆvd 3 v the scattering probability and dω the solid angle For calculating transport properties (moments of C B ) to deal with is ˆ v: what is ˆf(ˆ v)? the difficult part APS-DPP October 29, 212, p 3

4 Weak coupling: assume small scattering angle Need to do something with ˆf s (ˆ v) and ˆf s (ˆ v ) Small scattering angle expansion: ˆ v = v + v, v v C B (f s ) = {[ fs v m s f s ( v) ] v s/s m s s v t [ 1 2 f s ( v) + 2 v v : + 1 m 2 s f 2 m 2 s ( v) 2 s v v : m s f s ( v) m s v where and in which v s s t = m ss m s v v s s = m2 ss t m 2 s { u} = { u u} = d 3 v f s ( v ){ u} d 3 v f s ( v ){ u u} dω σ u u, dω σ u u u v ] v v s/s t } where u = u[sin θ cos φ ˆx + sin θ sin φ ŷ 2 sin 2 (θ/2)û] APS-DPP October 29, 212, p 4

5 Logarithmic divergence for bare Coulomb potential For bare Coulomb potential, get Rutherford scattering cross section σ = q 2 s q2 s 4m 2 ss u 4 sin 4 (θ/2) Writing in terms of impact parameter bdbdφ = σdω: { u} = 4πu ub 2 min bmax b db b 2 + b 2 min = 4πq2 s q2 s u m 2 ss u 3 ln(b 2 max /b2 min + 1) where b min = q s q s /m ss u 2 is the distance of closest approach Nominally, b max =, but then this diverges logarithmically! Typically impose screening in an ad-hoc way by taking b max = λ D, so ln(b 2 max /b2 min + 1) 2 ln(λ D/b min ) = 2 ln Λ This only makes sense when λ D b min (weakly coupled) APS-DPP October 29, 212, p 5

6 Transport coefficients come from moments of C(f s, f s ) Friction force density: R s s = d 3 v m s vc(f s, f s ) Energy exchange density: Q s s = d 3 v 1 2 m s( v V s ) 2 C(f s, f s ) Resistivity comes from R e i, thermal conductivity from Q e i, etc. Need to input the distribution functions f s ( v) and f s ( v ) For Maxwellian distributions, one finds (for weak coupling) R s s = n s m s ν s s ( V s V s ) and where the collision frequency is Q s s = 3 m ss n s ν s s (T s T s ) m s ν s s = 16 πq 2 s q2 s n s 3m s m ss v 3 T ln Λ to first order in V s V s / v 2 T 1, and v2 T = v2 T s + v2 T s For some processes, deviations from Maxwellian are essential there are methods for dealing with this (Chapman-Enskog, Grad s moment method) APS-DPP October 29, 212, p 6

7 Two effects must be dealt with when ln Λ 1 (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 b max = λ D cutoff By definition b min λ D 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) APS-DPP October 29, 212, p 7

8 Can exploit symmetries of the Boltzmann equation Transport coefficients come from velocity moments of the form χ s s = d 3 v χ s ( v)c B (f s, f s ) Apply same properties used to derive the Boltzmann collision operator: (i) d 3ˆvd 3ˆv = d 3 vd 3 v, (ii) û = u, (iii) ˆσ( v, v ˆ v, ˆ v ) = ˆσ(ˆ v, ˆ v v, v ) Applying these properties and the Boltzmann collision operator yields χ s s = d 3 vd 3 v { χ s }f s ( v)f s ( v ) in which { χ s } = dω σ u χ s, and χ s = χ s (ˆ v) χ s ( v). Only depends on the distribution functions before scattering: no f(ˆv)! Can relate χ s (ˆ v) to χ s ( v) using conservation laws Evaluating for momentum, χ s = m s v, gives the friction force density R s s = m ss d 3 u { u} d 3 v f s ( u + v )f s ( v ) Evaluating for χ s = 1 2 m s( v V s ) 2, gives the energy exchange density Q s s = m ss d 3 u{ u} d 3 v ( v V s + m ss u)f s ( u + v )f s ( v ) m s APS-DPP October 29, 212, p 8

9 For Maxwellians: A generalized Coulomb log Evaluating R s s and Q s s for flowing Maxwellian f s and f s, one finds the same coefficients as weakly coupled theory, except that ln Λ is replaced by a generalized Coulomb logarithm: Ξ = 1 2 dξ e ξ2 ξ 5σ s(ξ, Λ) σ o where ξ u/ v T, and σ o = πλ 2 D /Λ2 cl Here σ s is the momentum-scattering cross section σ s = 4π π dθ sin θ sin 2 (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: dt e /dt = 2Q e i /(3n e ) APS-DPP October 29, 212, p 9

10 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 equations, which account for deviations from Maxwellian In these theories, collisions arise through the integrals Ω (l,r) ss = 3 16 m s m ss ν ss n s Ξ (l,r) ss Ξ ss in which ν ss 16 πq 2 s q2 s n s 3m s m ss v 3 ss Ξ ss is a collision frequency corresponding to the (l, r) th matrix element, Ξ ss = Ξ (1,1) ss is the generalized Coulomb logarithm associated with the lowest order (1, 1) term, and Ξ (l,r) ss = 1 2 dξ ξ 2r+3 e ξ2 σ (l) ss (ξ, Λ) σ o is the generalized Coulomb logarithm for the (l, r) th matrix element, σ (l) ss 2π π dθ sin θ(1 cos l θ)σ ss is the l th momentum-transfer cross section and σ o πq2 s q2 s m 2 ss v 4 ss = πλ2 Λ 2 is a reference value for the differential scattering cross section. APS-DPP October 29, 212, p 1

11 The effective potential is externally determined A common potential in plasmas is the Debye-screened (Yukawa) potential φ t = q t r e r/λ D This can be justified if: (1) the transport timescale is long compared to the plasma frequency for a given process (ν s s /ω ps 1), Then, the polarization response is adiabatic and satisfies the Boltzmann relation n s = n o exp( q s φ/t s ) (2) The interaction potential is weak: q s φ/t s 1 Then, Poisson s equation 2 φ = 4πρ q gives the Debye-screened potential Adiabaticity (ν s s /ω ps 1) for e-i collisions: ν e i /ω pe Ξ/Λ Weak coupling (Λ 1), Ξ = ln Λ, ν e i /ω pe ln Λ/Λ 1 Strong coupling (Λ 1), Ξ = Λ 2 ln 2 Λ, ν e i /ω pe Λ ln 2 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) APS-DPP October 29, 212, p 11

12 Kinetic equations implicitly approximate g(r) The pair correlation function in the BBGKY hierarchy evolves according to: g 2 (1, 2; t) t = [ L 1 + L 2] g2 (1, 2; t) + L 12 f(1; t)f(2; t) (1a) + L 12 g 2 (1, 2; t) (1b) + d3 [L 13 f(1; t)g 2 (2, 3; t) + L 13 f(3; t)g 2 (1, 2; t) + (1 2)] (1c) + d3(l 13 + L 23 )g 3 (1, 2, 3; t) (1d) where L i = v i i, L i,j = V i,j i,j and V i,j = φ( r i r j ) Equilibrium limit: f(1; t) = nf M ( p 1 ), g 2 (1, 2; t) = n 2 f M ( p 1 )f M ( p 2 )h( r 1 r 2 ) where h(r) = g(r) 1 and g(r) is the pair distribution function Landau collision operator: Assumes terms (1b)-(1d) =, gives g(r) = 1 eφ(r)/k B T where φ = q/r Lenard-Balescu: Assumes terms (1b) and (1d) =, gives g(r) = 1 eφ sc /k B T where φ sc = qe r/λ D /r Boltzmann Equation: Assumes terms (1c) and (1d) =, gives g(r) = e eφ(r)/k BT where φ is whatever model you want to use Closures give g(r) φ(r): correlations can be imposed through g(r) APS-DPP October 29, 212, p 12

13 Hypernetted chain (HNC) theory can be used HNC is accurate for Γ 1 (checked with classical MD) Yukawa gives a good approximation for Γ 1 Here Γ = e 2 /(a e k B T e ) where a e = [3/(4πn e )] 1/3 1.5 HNC u nlike charge Yukawa Bare Coulomb gie (r ) 1.5 Γ = like charge r/a e APS-DPP October 29, 212, p 13

14 Scattering cross section in the Yukawa potential Recall that the l th momentum-transfer cross section is σ (l) ss 2π π dθ sin θ(1 cos l θ)σ ss For the Deybe screened (Yukawa) potential in a classical regime θ = π 2Θ where dr Θ = b r 2 (2) 1 U eff (r, b) and in which + refers to the repulsive (q s q s (q s q s < ) Yukawa potentials. r o U eff = ± 2 Λ cl ξ 2 λ D r e r/λ D + b2 r 2 (3) r o is the distance of closest approach from U eff (r o ) = 1. > ), and to the attractive 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 APS-DPP October 29, 212, p 14

15 U eff has a barrier for attractive potential If Λ cl ξ 2 < 1/13.2, an effective potential barrier can form Barrier extends the distance of closest approach Multiple roots of U eff = 1, only want largest 3 2 Attractive 3 2 Repulsive Λ c l ξ 2 = 1/3 Λ c l ξ 2 = 1/3 1 1 Ue ff 1 Ue ff 1 b/λ D r/λ D r/λ D APS-DPP October 29, 212, p 15

16 Need to take max{r min } 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 Λ cl ξ 2 1/13.2) 1 2 Λ c l ξ 2 = 1/1 r min Repu lsive Attractive ro ot 1 Attractive ro ot b/λ D APS-DPP October 29, 212, p 16

17 Asymptotic limits return known results For no barrier: Λ cl ξ 2 1/13.2 σ cl s 4σ oξ 4 ln(1 + Λ cl ξ 2 ) For this the generalized Coulomb logarithm is Ξ cl,a = exp(λ 1 cl )E 1 (Λ 1 cl ) (this covers weak, moderate, and part of strong correlation) For Λ 1 this gives Ξ cl,a ln(λ) γ = ln(e γ Λ) = ln(.56λ) = ln(.79/g) where g Z i e 2 /(T e λ De ) = 2/Λ for T e T i Previous theories get the number.79 using complicated renormalization techniques (see e.g., Landau & Lifshitz, Aono, BPS) Beyond the barrier: Λ cl ξ 2 1/13.2 σ s.81σ o Λ 2 cl [ln2 (Λ cl ξ 2 ) 2 ln(λ cl ξ 2 ) + O(1)] For this, the generalized Coulomb logarithm is Ξ cl,b =.4Λ 2 cl [ln2 (Λ cl ) + ln(λ cl )(1 γ) + O(1)] APS-DPP October 29, 212, p 17

18 Momentum scattering cross section Attractive and repulsive cases differ for 1 2 Λ cl ξ 2 1 Figure shows: numerical solutions (black), weakly coupled solution (red), below barrier asymptote (blue), above barrier asymptote (green) 1 5 Attractive (classical) σ s,a 1 Repul si ve (classical) σs/λ 2 D σ s,wc σ s,b /Λ c l ξ 2 APS-DPP October 29, 212, p 18

19 Theory gives excellent agreement with MD Classical MD simulations for like-charge e + i + dt e /dt 2Q e i /3n e thermal relaxation: Screening length is the electron Debye length (λ De ) Ξ (repulsive) ln Λ ln(.765λ) MD data b est fit Ξ g Simulations: Dimonte and Daligault, PRL 11, 1351 (28). APS-DPP October 29, 212, p 19

20 Solutions for the generalized Coulomb logarithm Numerical solutions (solid lines) and asymptotic solutions (dashed) Attractive and repulsive cases can be distinguished for 1 2 Λ cl ξ 2 1 Asymptote to a common value in weak and strong correlation limits APS-DPP October 29, 212, p 2

21 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 2 3 Ξ ( 1,1) /Ξ 1 Ξ ( 1,2) /Ξ 2 Ξ ( 1,3) /Ξ / Λ / Λ /Λ Ξ ( 2,1) /Ξ Ξ ( 2,2) /Ξ 2 Ξ ( 2,3) /Ξ / Λ / Λ /Λ Ξ ( 3,1) /Ξ / Λ Ξ ( 3,2 ) /Ξ /Λ Ξ ( 3,3) /Ξ /Λ APS-DPP October 29, 212, p 21

22 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 and correlation 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 coupling, e.g., attractive and repulsive collisions can be distinguished The Chapman-Enskog collision integrals can be written in terms of generalized Coulomb logarithms Future work will address going beyond the Yukawa potential using HNC, and testing the theory for diffusion, resistivity, etc., with MD APS-DPP October 29, 212, p 22