A recent paper by Davies et al. [Phys. Plasmas 28, 012305 (2021)] presents rational interpolants for the resistive (α) and thermoelectric (β) transport coefficients as functions of the Hall parameter χ and the ion charge state Z. Here, this discussion is augmented by showing that, at least in the Lorentz limit Z → ∞⁠, the proposed rational interpolants for α ⊥ and β ⊥ can be made even more accurate over a larger range of χ by making slight modifications to ensure the exact asymptotic limits are preserved. Although the authors do not discuss the conductivity (κ) coefficients, this exercise can also be repeated for the rational interpolants of κ ⊥ and κ ∧ published in a similar work [Phys. Rev. Lett. 126, 075001 (2021)].

In a recent paper,1 Davies et al. provided rational interpolants to the resistive (α) and thermoelectric (β) transport coefficients as functions of χ and Z. Here, it is shown that through minor modifications (namely, reducing the number of fit parameters), the rational interpolants for α ⊥ and β ⊥ provided by Ref. 1 can be brought into stronger agreement with the Lorentz transport coefficients when Z → ∞⁠. In particular, the correct asymptotic scaling of β ⊥ is restored by removing a term in the interpolant entirely, enabling it to be used over the entire range of possible Hall values. Hence, with regard to the chosen functional form, the rational interpolants of Ref. 1 can be considered even more accurate than implied by the original paper.

To begin, as originally shown in Ref. 2, applying the Chapman–Enskog procedure to the electron kinetic equation subjected to the Lorentz collision operator yields the following expressions for the resistive (α) and thermoelectric (β) transport coefficients:
(1a)
(1b)
(1c)
(2a)
(2b)
(2c)
Here, M is the magnetization parameter of Ref. 2 defined as M ≐ 4 χ / 3 π⁠, and also
(3)
Although Ref. 1 do not discuss heat conduction, one can similarly show that the conductivity (κ) coefficients for the Lorentz operator take the following form:
(4a)
(4b)
(4c)
Having obtained analytical forms for the Lorentz transport coefficients, their limiting behavior as M → 0 and M → ∞ can be readily obtained. First, using a second-order Taylor expansion of I m ( M )⁠, one can show that
(5)
(5a)
(5b)
(5c)
(5d)
(5e)
(5f)
Similarly, using the asymptotic forms for I m ( M ) derived in the appendix of Ref. 3, one can show that
(6a)
(6b)
(6c)
(6d)
(6e)
(6f)
One can readily verify that, after taking Z → ∞⁠, the interpolants for α ∧ and β ∧ of Ref. 1 do satisfy the limits (5) and (6), but those for α ⊥ and β ⊥ do not. This can be remedied while maintaining the same functional form of the rational interpolants by imposing the asymptotic relationship between certain coefficients and then re-optimizing the remaining free numerical constants. Specifically, the interpolant for α ⊥ given by Eq. (A27) in Ref. 1 is modified as
(7)
while the interpolant for β ⊥⁠, given nominally by Eq. (A27) as well, is instead replaced by a modified form of the alternate interpolant given by Eq. (A14) of Ref. 1,
(8)
Note that the modification is not simply to constrain free constants, but also to remove entirely the χ 5 / 3 term in the denominator of Eq. (A14) to ensure Eq. (5) is satisfied.

Now the interpolants for α ⊥ and β ⊥ depend only on two free constants c1 and c2, making the subsequent optimization problem considerably simpler. A suitable choice of constants c1 and c2 for Eqs. (7) and (8) obtained via basic optimization procedures in Mathematica is given in Table I, and a comparison of the resulting interpolants is given in Fig. 1. From the figure, it is clear that the modified interpolants have better agreement with the exact result for large values of χ. They also maintain strong agreement in the opposite limit of small χ as did the original interpolants (not shown).

TABLE I.

Coefficients for the interpolated transport coefficients provided in Eqs. (7)–(10).

c1 c2 c3
α ⊥  0.02  0.348   
β ⊥  36.8  −0.290   
κ ⊥  15.5     
κ ∧  1107  18.4  447 
c1 c2 c3
α ⊥  0.02  0.348   
β ⊥  36.8  −0.290   
κ ⊥  15.5     
κ ∧  1107  18.4  447 
View Large
FIG. 1.
Comparison of the exact expression, the old rational interpolant obtained by Ref. 1, and the newly obtained interpolant for the α ⊥ and β ⊥ transport coefficients.
View largeDownload slide

Comparison of the exact expression, the old rational interpolant obtained by Ref. 1, and the newly obtained interpolant for the α ⊥ and β ⊥ transport coefficients.

FIG. 1.
Comparison of the exact expression, the old rational interpolant obtained by Ref. 1, and the newly obtained interpolant for the α ⊥ and β ⊥ transport coefficients.
View largeDownload slide

Comparison of the exact expression, the old rational interpolant obtained by Ref. 1, and the newly obtained interpolant for the α ⊥ and β ⊥ transport coefficients.

Close modal
Even though Ref. 1 do not consider the κ coefficients, this exercise can be repeated with the rational interpolants for κ ⊥ and κ ∧ provided in Ref. 4. Again, Eq. (5) of Ref. 4 supplementary material is modified as
(9)
and Eq. (9) of Ref. 4 supplementary material is modified as
(10)
The interpolants with coefficients specified in Table I are compared with the previous interpolants and the exact expression in Fig. 2. Once again, the new interpolants demonstrate visible improvement over the previous interpolants.
FIG. 2.
Same as Fig. 1 but for the κ ⊥ and κ ∧ interpolants provided by Ref. 4.
View largeDownload slide

Same as Fig. 1 but for the κ ⊥ and κ ∧ interpolants provided by Ref. 4.

FIG. 2.
Same as Fig. 1 but for the κ ⊥ and κ ∧ interpolants provided by Ref. 4.
View largeDownload slide

Same as Fig. 1 but for the κ ⊥ and κ ∧ interpolants provided by Ref. 4.

Close modal

In summary, Ref. 1 provides an accurate rational interpolant to the classical resistive (α) and thermoelectric (β) transport coefficients. By modifying their expressions to ensure the exact asymptotic behavior is preserved, it is possible (at least in the Lorentz limit) to make their interpolants even more accurate over the entire range of χ, i.e., χ ∈ ( 0 , ∞ )⁠. This now enables one to use with confidence the interpolants of Ref. 1 for highly inhomogeneous systems that exhibit large variation of χ, with applications beyond magnetic-field transport.

The authors have no conflicts to disclose.

1.
J. R.
Davies
,
H.
Wen
,
J.-Y.
Ji
, and
E. D.
Held
,
Phys. Plasmas
28
,
012305
(
2021
).
https://doi.org/10.1063/5.0023445
Google Scholar
Crossref
Search ADS
 
2.
E. M.
Epperlein
,
J. Phys. D
17
,
1823
(
1984
).
https://doi.org/10.1088/0022-3727/17/9/007
Google Scholar
Crossref
Search ADS
 
3.
E. M.
Epperlein
 and
M. G.
Haines
,
Phys. Fluids
29
,
1029
(
1986
).
https://doi.org/10.1063/1.865901
Google Scholar
Crossref
Search ADS
 
4.
J. D.
Sadler
,
C. A.
Walsh
, and
H.
Li
,
Phys. Rev. Lett.
126
,
075001
(
2021
).
https://doi.org/10.1103/PhysRevLett.126.075001
Google Scholar
Crossref
Search ADS
PubMed
 
© 2024 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).