C. Franck E. Kovacevic A. v. Keudell Managing Editors D. Naujoks

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K.-H. Spatschek, Düsseldorf Associate Editors:

C. Franck E. Kovacevic A. v. Keudell Managing Editors D. Naujoks

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ContributionstoPlasmaPhysics CPP

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Dynamic Collision Frequency in Kelbg-Pseudopotential-Modelled Plasmas and the Method of Moments with Local Constraints

Yu. V. Arkhipov¹, A. Askaruly¹, D.Yu. Dubovtsev¹, L.T. Erimbetova¹,andI.M. Tkachenko²^∗

1IETP, Department of Physics and Technology, Al-Farabi Kazakh National University, al-Farabi av. 71, 050040 Almaty, Kazakhstan

2Instituto de Matem´atica Pura y Aplicada, Universidad Polit´ecnica de Valencia, Camino de Vera s/n, 46022 Valencia, Spain

Received 01 November 2014, revised 14 January 2015, accepted 26 January 2015 Published online 25 March 2015

Key words Strongly-coupled plasmas, dynamic collision frequency, method of moments

The dense plasma dynamic collision frequency is modeled by the ﬁrst two terms of its asymptotic expansion at high frequencies and its values at a few interpolation points on the real axis. This makes the dynamic collision frequency a non-rational function whose extension onto the upper half-plane of the complex frequency is holomorphic with a non-negative imaginary part and with a continuous extension to the real axis. The validity of the suggested analytic form of the latter is tested against the simulation data, where the Kelbg effective potential was used.

c 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 The problem set-up

1.1 Dynamic collision frequency

One of the problems of modern plasma physics is to obtain an analytic expression for the dielectric function determining the screening effects, the dispersion relations and other dynamic characteristics, as the conductivity, collision frequency, etc. In the present we discuss the long-wavelength generalized Drude-Lorentz model intro- duced in [1], see also corresponding references therein and the discussion in [2]. Precisely, the following model expression for the two-component plasma IDF was suggested:

⁻¹_gDL(z) = 1 + ω²_p

z²−ω_p²+izν(z), Imz≥0, (1)

whereν(z)is the dynamic collision frequency (DCF) deﬁned in a way that the static conductivity σ₀= lim

ω→0

ω 4πi

1

⁻¹_gDL(ω)−1

= ω_p²

4πν(0). (2)

It is known that the classical method of moments provides an important insight for the reconstruction of some physical dynamic characteristics such as the dynamic structure factor [3, 4], which must possess some mathematical properties [2, 5] obeying independently derived sum rules and possessing some speciﬁc asymptotic behavior.

In the case we consider here, the problem of reconstruction of the DCF real part or of the (generalized Drude- Lorentz) loss function (LF),

L(ω) =−Im⁻¹_gDL(ω)

ω , (3)

∗ Corresponding author. E-mail:[email protected]

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374 Yu.V. Arkhipov et al.: Dynamic collision frequency

which includes both real and imaginary parts of the DCF, L(ω) = ω_p²Reν(ω)

ω²−ω²_p−ωImν(ω)₂

−ω²Reν²(ω) (4)

which was studied within the method of moments before [2]. This choice is due to the fact that the moment problem possesses solutions only if the characteristic under construction is positive-deﬁnite [6] and the DCF is the complex function ofωwith a real part which is a positive deﬁnite function as well as the LF.

Given a positive-deﬁnite and even functionf(t), t∈R, one can introduce its Cauchy transform F(z) =

∞

−∞

f(t)dt

t−z , Imz >0,

whose asymptotic expansion along any ray in the upper half-plane, F(z→ ∞) =

∞

−∞

f(t)dt t−z =−1

z ∞

−∞

f(t)dt 1−^t_z −1

z ∞

−∞

1 + t

z+ t² z² +t³

z³ +· · ·

f(t)dt=

= −c₀ z +c₂

z³ +· · · ,

where the coefﬁcients are the power moments of the functionf(t):

c_k = _∞

−∞

t^kf(t)dt, k= 0,1,2, . . . . (5)

Note that the odd power moments of an even function vanish and that the above expansion does not necessarily converge. Even more, the functionf(t), due to its slow asymptotic decay might not possess all of its moments convergent. For example, if

f(t→+∞)At⁻^α, α∈R,

only the moments {c_k}^k_k⁼₌₀^K withK < 1 +αwill converge. Besides, the numerical precision of a physical problem will limit the number of convergent moments as well.

The (truncated Hamburger) moment problem actually consists in the determination of all positive-deﬁnite (and for simplicity even) functionsf(t)given the (positive-deﬁnite) set of moments{c_k}^k_k⁼²₌₀ⁿ. The positivity of the latter set means that the determinants of the Hankel matrices

c₀, c₀ 0

0 c₂ ,

c₀ 0 c₂ 0 c₂ 0 c₂ 0 c₄

,· · ·

are all positive. The number of the solutions of the truncated Hamburger problem is, certainly, inﬁnite. In what follows we are interested in the so called non-canonical, i.e., continuous, solutions [5] parametrized by the Nevanlinna parameter functionQ_n(z)via the Nevanlinna formula [5, 6]:

∞

−∞

f(t)dt

z−t = E_n₊₁(z) +Q_n(z)E_n(z)

D_n₊₁(z) +Q_n(z)D_n(z) , Imz >0. (6) Notice that the polynomialsD_n(z)can be easily found by the Gram-Schmidt procedure from the canonical set 1, z, z², z³, . . .

using the weight functionf(t): D₀(z) = 1, D₁(z) =z, D₂(z) =z−

c₂/c₀, . . .

c2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.cpp-journal.org

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and the polynomialsE_n(z)are their conjugate:

E_n(z) = ∞

−∞

D_n(z)−D_n(t) t−z f(t)dt .

What matters is that in physical problems the power moments {c_k}^kk⁼²=0ⁿ are actually sum rules and they can be found independently from the functionf(t), for example, within the Kubo linear reaction theory [5]. The Nevanlinna parameter function Q_n(z)is a Nevanlinna function (it must be analytic in the upper half-plane Imz >0and must have there a non-negative imaginary part) with an additional property:

zlim→∞

Q_n(z) z = 0.

The latter requirement guarantees that the moment conditions (5) are satisﬁed for any adequateQ_n(z).Generally speaking, the Nevanlinna parameter function can be found from some additional information on the physical properties of the function under construction,f(t), e.g., its asymptotic behaviour [2, 5].

In the present work we supplement the above method by imposing local constraints as it was done in [6–8].

Doing so we employ the molecular-dynamics data in which the Kelbg effective potential was used [1, 9]. We use this DCF numerical data to calculate numerically only two moments,c₀andc,and three local constraintsw_sat pointst₁,t₂t₃arbitrarily chosen on the real axis:

w_s=F(t₁) = lim

ε↓0

∞

−∞

f(t)dt

t−t_s−iε, s= 1,2,3.

Thus we extend the efforts which have proven their fruitfulness in [8] and in [11] for the DCF and also for the loss function generated from the DCF data. First we do so withf(t)being the DCF real part and then with f(t)chosen as the loss function (lfd) corresponding to the generalized Drude-Lorentz model. The mathematical background of the work [8, 10, 11] is provided in the Appendix. Notice that Eq. (1) stems directly from (6) and the Kramers-Kronig relations forn= 1andf(t) = Im⁻¹(t)withQ(z) =iν(z).

Of further interest is a strongly coupled hydrogen-like plasma which exists in stellar like interiors [12] and can be encountered in various nuclear fusion devices [13].

Fig. 1 The real part of the dynamical collision frequency, Reν(ω), in comparison with the simulation data of [9] for Γ = 1andT = 80kK.

Fig. 2 The loss function, (3), in comparison with the simulation data of [9] forΓ = 1andT = 80kK.

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376 Yu.V. Arkhipov et al.: Dynamic collision frequency

2 Numerical procedure

Since we deal here with non-negative functions, the real part of the dynamic collision frequency and the corresponding loss function, the solvability of the moment problem is not an issue. In each case the absolutely continuous non-negative measure with the chosen density is just one of the solutions to the moment problem.

To apply the Schur-like algorithm described above, one has to know not only the values of some power moments of the distribution densityf(t)under investigation,

c_k = _∞

−∞

t^kf(t)dt , k= 0,1, . . . ,2n , n= 1,2, . . . ,

but also the values of the Nevanlinna function at the set of points{t₁, ..., t_p}:

w_s=ϕ(t_s) =P _∞

−∞

f(t)dt

t−t_s +iπf(t_s).

In all cases considered here we use only two moments,n = 1, and three interpolation nodes,p= 3; the latter principal value integrals have been computed numerically and the orthogonal polynomials are directly calculated as:

D₀(z) = 1, D₁(z) =z, D₂(z) =z²−ω₁², E₀(z)≡0, E₁(z) =c₀, E₂(z) =c₀z,

whereω²₁=c₂/c₀. The distribution densityf(t)employed was taken from the numerical data provided in [1, 9]

for two-component hydrogen-like plasmas modelled using the Kelbg effective potential.

To ﬁnd the free parameterα∈(0,1)of the auxiliary function

u_s(z) = exp

⎧⎨

⎩ α πi

t_s+1 t_s−1

1 +tz

t−z ln|t−t_s| dt t²+ 1

⎫⎬

⎭ , s= 1,2,3,

we make use of the Shannon entropy S(α) =−

_+∞

−∞

F(α, t) ln (F(α, t))dt,

maximization procedure [14] where the densityF(α, t)is reconstructed in the adopted algorithm described in the Appendix and represents the imaginary part (divided byπ) of the model function obtained in the Schur-algorithm procedure. The densityF(α, t)has no real poles and is positive over the whole real axis, hence it was quite easy to solve the maximization procedure equation:dS(α)/dα= 0.

3 Results and conclusions

The numerical results were compared to the simulation data of [9] for Γ = 1, T = 80000K; Γ = 1, T = 100000K;Γ = 1, T = 350000K, and are presented in ﬁgures 1-6. In all ﬁgures the thick lines correspond to our results, the squares correspond to the data of [9] withω_pbeing the plasma frequency.

We can conclude that the suggested algorithm leads to a quantitative agreement between the simulation data on the plasma dynamic characteristics and their non-rational counterparts reconstructed by a few integral characteristics, i.e., the power moments and the local constraints.

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4 Appendix. Mathematical background

Consider the mixed L¨owner-Nevanlinna problem [10, 15, 16], see also [17] for the matrix version of the problem.

Problem 1. Given a set of real numbers(c₀, ..., c₂_n), a ﬁnite set of points(t₁, ..., t_p)on the real axis, and a set of complex numbers(w₁, ..., w_p)with non-negative imaginary parts, ﬁnd a positive functionf(t), t∈ Rsuch that

_∞

−∞

t^kf(t)dt=c_k, k= 0,1, ...,2n (7)

and

w_s= lim

ε↓0

∞

−∞

f(t)dt

t−t_s−iε, s= 1, .., p . (8)

Problem 1 is a mixture of the truncated Hamburger moment problem [6,10] with the L¨owner-type interpolation problem in the class of Nevanlinna functions [15].

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378 Yu.V. Arkhipov et al.: Dynamic collision frequency 4.1 The mixed problem solution

4.1.1 Solvability and contractive functions

Assume that the set of moments is positive-deﬁnite so that the truncated Hamburger moment problem is solvable [6], and that there exists an inﬁnite set of non-negative measuresgon the real axis such that (dg(t) =f(t)dt)

_∞

−∞

t^kdg(t) =c_k, k= 0,1, ...,2n. (9)

Then, the formula _∞

−∞

dg(t)

t−z =−E_n₊₁(z) +ζ(z)E_n(z)

D_n₊₁(z) +ζ(z)D_n(z), Imz >0, n= 0,1,2, . . . (10) according to Nevanlinna’s theorem [6], establishes a one-to-one correspondence between the set of all measures g(t)satisfying (9) and the Nevanlinna functionsζ(z) ∈ R, i.e., functions which are analytic in the half-plane Imz > 0, continuous on its closure Imz = 0, having a positive imaginary part atImz ≥ 0 and such that limz→∞ζ(z)/z= 0, Imz >0.

The polynomials{D_k}ⁿ₀⁺¹ form the orthogonal system with respect to each g-measure satisfying (9) and can be found by the Gram-Schmidt procedure applied to the basis

1, t, t², . . . , tⁿ⁺¹

, while {E_k}ⁿ₀⁺¹ is the corresponding set of conjugate polynomials [6]. Notice that the zeros of each orthogonal polynomialD_k(z)are real and, by virtue of the Schwarz-Christoffel identity [6], the zeros ofD_k₋₁(z)alternate with the zeros ofD_k(z) as well as with the zeros ofE_k₋₁(z).

To meet constraints (8) it is enough now to substitute into the right hand side of (10) any functionζ(z)which satisﬁes the following conditions:

ξ_s=ζ(t_s) =−w_sD_n₊₁(t_s) +E_n₊₁(t_s)

w_sD_n(t_s) +E_n(t_s) , s= 1, .., p. (11)

Note thatImξ_s>0.Thus, Problem 1 reduces to

Problem 2 Given a ﬁnite number of distinct points t₁, ..., t_p of the real axis and a set of complex numbers w₁, ..., w_pwith positive imaginary parts, ﬁnd a set of functionsζ(z)∈Rthat are continuous in the closed upper half-plane and satisfy conditions (11).

Each Nevanlinna functionζ(z)in the upper half-plane admits the following Caley representation ζ(z) =i1 +θ(z)

1−θ(z), (12)

where

θ(z) =ζ(z)−i

ζ(z) +i (13)

is a holomorphic function in the upper half-plane withcontractivevalues, i.e.|θ(z)| ≤1, Imz >0. Therefore, Problem 2 is equivalent to the following problem for contractive functions.

LetBbe the set of all contractive functions which are holomorphic in the upper half-plane and continuous on its closure.

Problem 3 Given a ﬁnite number of distinct pointst₁, ..., t_pof the real axis and a set of pointsλ₁, ..., λ_p, λ_s= ξ_s−i

ξ_s+i, |λ_s| ≤1, s= 1, ..., p, (14)

ﬁnd a set of functionsθ∈Bsuch that

θ(t_s) =λ_s, s= 1, ..., p. (15)

Problem 3 is a limiting case of the Nevanlinna-Pick problem [6, 16] with interpolation nodes on the real axis.

Its solvability for any interpolation dataλ₁, ..., λ_p inside the unit circle was actually proven in [18]. The point

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is that the associated Pick matrix is automatically positive-deﬁnite for any given contractive interpolation values as soon as the interpolation nodes are close enough to the axis; this guarantees that the approximate Nevanlinna- Pick problem is solvable if the interpolation nodes are close enough to the real axis. Then, the Vitali-Montel theorem is applied to take the limit of the interpolation nodes approaching the real axis which also implies that the Nevanlinna-Pick problem is solvable even if some or all|λ_s|= 1.

We describe below an algorithm of solving Problem 3 when all|λ_s| <1, which is a simple modiﬁcation of the Schur algorithm. An alternative algorithm, similar to the Lagrange method of the interpolation theory, can be applied if some or even all|λ_s|= 1[10].

4.1.2 Schur algorithm

Note that a functionθ ∈Bsatisﬁes the conditionθ(t₁) = λ₁, |λ₁|<1,if and only if it admits the following representation

θ(z) = φ(z) +λ₁

λ₁φ(z) + 1, (16)

whereφ ∈ Bandφ(t₁) = 0. In case of the Nevanlinna-Pick problem, i.e., when t₁ belongs to the upper half-plane, the functionφ(z)is of the form

φ(z) = z−t₁ z−t₁χ(z) ,

whereχ(z)is an arbitrary contractive function in the upper half-plane. There is no such simple relation for the contractive functionφ(z)whent₁∈R.

Here we carry out the reconstruction procedure using the non-rational functions, as suggested in [7]:

φ(z) =θ₁(z) exp

⎧⎨

⎩ α πi

t₁+1 t₁−1

1 +tz

t−z ln|t−t₁| dt t²+ 1

⎫⎬

⎭:=θ₁(z)u₁(z), (17) with a unique free parameterα∈(0,1). Hereθ₁is any function fromBsuch that

θ₁(t_s) =λ_s= 1 u₁(t_s)

λ_s−λ₁

1−λ₁λ_s, s= 2, ..., p. (18)

Such a choice ofθ₁(z)guarantees that conditions (15) are all veriﬁed. Hence, Problem 3, initially formulated for pnodes of interpolation on the real axis and strictly contractive values of the functions to be found at these nodes, reduces to the same problem but withp−1nodes of interpolation and modiﬁed values at these nodes given by (18). Repeating the above procedurep−1times with a suitable choice of the parameterαand modifying the values of emerging contractive functions at the remaining pointst_s₊₁, ..., t_paccording to (18), permits to obtain a solution to Problem 3. Note that contrary to the Nevanlinna-Pick problem with nodes in the open upper half- plane, Problem 3 is always solvable if the values of the function to be reconstructed are contractive at the nodes of interpolation.

Letθ_s₋₁ ∈Bbe a contractive function emerging after thes−1steps in the course of solving Problem 3 by the above method, and letλ⁽s^s⁻¹⁾=θ_s₋₁(t_s), λ⁽⁰⁾₁ =λ₁. It follows from the above arguments that should the initial parametersλ₁, ..., λ_p be strictly contractive, there exists a set of solutions to Problem 3 described by the formula

θ(z) =a(z)μ(z) +b(z)

c(z)μ(z) +d(z), (19)

where the matrix elements of linear fractional transformation (19) are non-rational functions constructed as described above andμ(z)runs the subset of all functions fromBsatisfying the conditionμ(t_p) = λ⁽_p^p⁻¹⁾. This matrix can be calculated as

a(z) b(z) c(z) d(z)

=

p−1 s=1

u_s(z) λ⁽_s^s⁻¹⁾ λ⁽_s^s⁻¹⁾u_s(z) 1

, (20)

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380 Yu.V. Arkhipov et al.: Dynamic collision frequency where the indexson the right hand side increases from left to right.

Observe that the simplest choice for the functionμ(z)in (19) is just to assumeμ(z)≡λ⁽_p^p⁻¹⁾. Hence, if the initial parametersλ₁, ..., λ_pin Problem 3 are strictly contractive, then, among the solutions to this problem there are non-rational functions of the type we consider.

A numerical testing of this representation is given in Section 2.

Acknowledgements The authors acknowledge the ﬁnancial support of the Science Committee of the Ministry of Education and Sciences of the Republic of Kazakhstan (Grants numbers 3119/GF4, 3831/GF4), Yu.V. Arkhipov expresses gratitude for the ﬁnancial support provided by the Ministry by a grant “The Best Professor” and I.M.T. is grateful to the KazNU for its hospitality. We are also thankful to I.V. Morozov for providing the numerical data.

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