Correction: Abramov, R. The Random Gas of Hard Spheres. J 2019, 2, 162–205

Abramov, Rafail V.

doi:10.3390/j3030025

Open AccessCorrection

Correction: Abramov, R. The Random Gas of Hard Spheres. J 2019, 2, 162–205

by

Rafail V. Abramov

Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 S. Morgan St., Chicago, IL 60607, USA

J 2020, 3(3), 324-328; https://0-doi-org.brum.beds.ac.uk/10.3390/j3030025

Submission received: 9 September 2020 / Accepted: 11 September 2020 / Published: 16 September 2020

Download Versions Notes

In the published paper [1], we used the spatial correlation function

R (σ)

of two spheres, each of diameter

σ

, to construct a closure to the BBGKY hierarchy of hard spheres. In the subsequent derivation of the fluid dynamics equations in [1], we relied upon the assumption that

R (σ) = 1 + O (σ)

. We later discovered that

R (σ)

is the two-sphere cavity distribution function for hard spheres [2] in the constant sphere density limit, and thus is not

1 + O (σ)

. What follows below are the necessary corrections to the relevant equations in [1].

List of Changes

Equation (162) and the following sentence are changed as follows:

$F^{(2)} (t, x, y, v, w) \approx e^{- λ Θ_{α σ} (σ - ∥ x - y ∥)} R^{(2)} ((x + y) / 2, ∥ x - y ∥) f (t, x, v) f (t, y, w),$

(162)

where $R^{(2)}$ depends parametrically on $α$ and $λ$ : $R^{(2)} = R_{λ, α}^{(2)}$ , and, in the case of a spatially nonuniform distribution, is also generally a function of the midpoint between $x$ and $y$ .
The sentence following Equation (165) is changed as follows:
“We can apply the formula above to our set-up by setting $z_{1} = (x, v)$ , $z_{2} = (y, w)$ , $F = F^{(2)}$ , $ψ = e^{- λ Θ_{α σ} (σ - ∥ x - y ∥)} R^{(2)} ((x + y) / 2, ∥ x - y ∥)$ , and $p_{1} = p_{2} = {\bar{F}}^{(1)}$ from Equation (156).”
Equation (167) is changed as follows:

$\begin{matrix} \frac{\partial f}{\partial t} + v \cdot \frac{\partial f}{\partial x} = (K - 1) λ σ^{2} \int e^{- λ Θ_{α} (1 - r)} δ_{α} (r - 1) n \cdot (w - v) Θ (n \cdot (w - v)) \\ [R_{λ, α}^{(2)} (x + σ r n / 2, σ r) f (x, v^{'}) f (x + σ r n, w^{'}) - R_{λ, α}^{(2)} (x - σ r n / 2, σ r) f (x, v) f (x - σ r n, w)] r^{2} d r d n d w . \end{matrix}$

(167)
The first sentence in Section 6.2 is changed as follows:
“Above in Equation (167), we can formally assume that the contact zone is ‘thin’ that is, $α \to 0$ so that, for the values of r for which $δ_{α} (r - 1) > 0$ , we have $f (x \pm σ r n) \to f (x \pm σ n)$ , $R_{λ, α}^{(2)} (x \pm σ r n / 2, σ r) \to R_{λ, 0}^{(2)} (x \pm σ n / 2, σ)$ .”
Equation (169) is changed as follows:

$\begin{matrix} \frac{\partial f}{\partial t} + v \cdot \frac{\partial f}{\partial x} = (K - 1) σ^{2} (1 - e^{- λ}) \int n \cdot (w - v) Θ (n \cdot (w - v)) \\ [R_{λ, 0}^{(2)} (x + σ n / 2, σ) f (x, v^{'}) f (x + σ n, w^{'}) - R_{λ, 0}^{(2)} (x - σ n / 2, σ) f (x, v) f (x - σ n, w)] d n d w . \end{matrix}$

(169)
Equation (170) is changed as follows, with the new sentence appended immediately after it:

$\begin{matrix} \frac{\partial f}{\partial t} + v \cdot \frac{\partial f}{\partial x} = (K - 1) σ^{2} \int n \cdot (w - v) Θ (n \cdot (w - v)) \\ [R (x + σ n / 2) f (x, v^{'}) f (x + σ n, w^{'}) - R (x - σ n / 2) f (x, v) f (x - σ n, w)] d n d w, \end{matrix}$

(170)

where $R (x) = R_{\infty, 0}^{(2)} (x, σ)$ becomes the two-sphere cavity distribution function for hard spheres of diameter $σ$ .
The last sentence before Section 7 is changed as follows:
“This sets $R (x \pm σ n / 2) = 1$ , $f (x \pm σ n) = f (x, v)$ …”
Equation (172) is changed as follows:

$\begin{matrix} \frac{\partial g}{\partial t} + v \cdot \frac{\partial g}{\partial x} = \frac{K - 1}{K} \frac{6}{π ρ_{s p} σ} \int n \cdot (w - v) Θ (n \cdot (w - v)) \\ [R (x + σ n / 2) g (x, v^{'}) g (x + σ n, w^{'}) - R (x - σ n / 2) g (x, v) g (x - σ n, w)] d n d w . \end{matrix}$

(172)
Equation (173) is changed as follows:

$\begin{matrix} \frac{\partial g}{\partial t} + v \cdot \frac{\partial g}{\partial x} = \frac{6}{π ρ_{s p} σ} \int n \cdot (w - v) Θ (n \cdot (w - v)) \\ [R (x + σ n / 2) g (x, v^{'}) g (x + σ n, w^{'}) - R (x - σ n / 2) g (x, v) g (x - σ n, w)] d n d w . \end{matrix}$

(173)
Equation (175), together with the preceding sentence, are changed as follows:
We also note that

$R (x \pm σ n / 2) g (x, v) g (x \pm σ n, w) = R (x) g (x, v) g (x, w) \pm σ \sqrt{R (x)} g (x, v) n \cdot \frac{\partial}{\partial x} (\sqrt{R (x)} g (x, w)) + \dots$

(175)
Equation (180) is changed as follows:

$\begin{matrix} \frac{\partial g_{0}}{\partial t} + v \cdot \frac{\partial g_{0}}{\partial x} = \frac{6}{π ρ_{s p}} \int n \cdot (w - v) Θ (n \cdot (w - v)) [R (g_{0} (v^{'}) g_{1} (w^{'}) + g_{1} (v^{'}) g_{0} (w^{'}) - \\ - g_{0} (v) g_{1} (w) - g_{1} (v) g_{0} (w)) + n \cdot \sqrt{R} (g_{0} (v^{'}) \frac{\partial (\sqrt{R} g_{0} (w^{'}))}{\partial x} + g_{0} (v) \frac{\partial (\sqrt{R} g_{0} (w))}{\partial x})] d n d w . \end{matrix}$

(180)
Equations (185a) and (185b) are changed as follows:

$C [v] = \frac{6}{π ρ_{s p}} \int (v - v^{'}) n \cdot (w - v) Θ (n \cdot (w - v)) n \cdot \frac{\partial (\sqrt{R} g_{0} (w))}{\partial x} \sqrt{R} g_{0} (v) d n d w d v,$

(185a)

${C [∥ v ∥}^{2}] = \frac{6}{π ρ_{s p}} \int ({∥ v ∥}^{2} - {∥ v^{'} ∥}^{2}) n \cdot (w - v) Θ (n \cdot (w - v)) n \cdot \frac{\partial (\sqrt{R} g_{0} (w))}{\partial x} \sqrt{R} g_{0} (v) d n d w d v .$

(185b)
Equations (187a) and (187b) are changed as follows:

$C [v] = - \frac{6}{π ρ_{s p}} \int {(n \cdot (w - v))}^{2} Θ (n \cdot (w - v)) n \cdot \frac{\partial (\sqrt{R} g_{0} (w))}{\partial x} \sqrt{R} g_{0} (v) n d n d w d v,$

(187a)

${C [∥ v ∥}^{2}] = - \frac{6}{π ρ_{s p}} \int (n \cdot (w + v)) {(n \cdot (w - v))}^{2} Θ (n \cdot (w - v)) n \cdot \frac{\partial (\sqrt{R} g_{0} (w))}{\partial x} \sqrt{R} g_{0} (v) d n d w d v .$

(187b)
Equation (188) is changed as follows:

$C [v] = - \frac{4}{ρ_{s p}} \frac{\partial}{\partial x} (R ρ^{2} θ) {, C [∥ v ∥}^{2}] = - \frac{8}{ρ_{s p}} \frac{\partial}{\partial x} \cdot (R ρ^{2} θ u) .$

(188)
Equations (189a) and (189b) are changed as follows:

$\frac{\partial ρ}{\partial t} + \frac{\partial}{\partial x} \cdot (ρ u) = 0, \frac{\partial (ρ u)}{\partial t} + \frac{\partial}{\partial x} \cdot (ρ (u u^{T} + (1 + \frac{4 ρ}{ρ_{s p}} R) θ I)) = 0,$

(189a)

$\frac{\partial (ρ ϵ)}{\partial t} + \frac{\partial}{\partial x} \cdot (ρ (ϵ + (1 + \frac{4 ρ}{ρ_{s p}} R) θ) u) = 0 .$

(189b)
Equations (197a) and (197b) are changed as follows:

$\begin{matrix} \frac{6}{π ρ_{s p}} \int {(n \cdot (w - v))}^{2} ((n \cdot (w - v)) n n^{T} + n {(v - u)}^{T} + (v - u) n^{T}) Θ (n \cdot (w - v)) \\ (R (g_{0} (v) g_{1} (w) + g_{1} (v) g_{0} (w)) - n \cdot \sqrt{R} g_{0} (v) \frac{\partial (\sqrt{R} g_{0} (w))}{\partial x}) d n d w d v = \\ = ρ θ (\frac{\partial u}{\partial x} + {\frac{\partial u}{\partial x}}^{T} - \frac{2}{3} (\frac{\partial}{\partial x} \cdot u) I) - \frac{8}{3 ρ_{s p}} R ρ^{2} θ (\frac{\partial}{\partial x} \cdot u) I, \end{matrix}$

(197a)

$\begin{matrix} \frac{6}{π ρ_{s p}} \int {(n \cdot (w - v))}^{2} [{(n \cdot (w - v))}^{2} n + (n \cdot (w - v)) (I + 2 n n^{T}) (v - u) + \\ + ({∥ v - u ∥}^{2} I + 2 (v - u) {(v - u)}^{T}) n] Θ (n \cdot (w - v)) \\ (R (g_{0} (v) g_{1} (w) + g_{1} (v) g_{0} (w)) - n \cdot \sqrt{R} g_{0} (v) \frac{\partial (\sqrt{R} g_{0} (w))}{\partial x}) d n d w d v = 5 ρ θ \frac{\partial θ}{\partial x} - \frac{20}{ρ_{s p}} θ \frac{\partial (R ρ^{2} θ)}{\partial x} . \end{matrix}$

(197b)
Equations (198a)–(198d) are changed as follows:

$\begin{matrix} \int {(n \cdot (w - v))}^{2} ((n \cdot (w - v)) n n^{T} + n {(v - u)}^{T} + (v - u) n^{T}) \\ Θ (n \cdot (w - v)) R (g_{0} (v) g_{1} (w) + g_{1} (v) g_{0} (w)) d n d w d v = - \frac{16 \sqrt{π}}{5} R ρ^{2} \sqrt{θ} S, \end{matrix}$

(198a)

$\begin{matrix} \int {(n \cdot (w - v))}^{2} ((n \cdot (w - v)) n n^{T} + n {(v - u)}^{T} + (v - u) n^{T}) \\ Θ (n \cdot (w - v)) n \cdot \sqrt{R} g_{0} (v) \frac{\partial (\sqrt{R} g_{0} (w))}{\partial x} d n d w d v = \frac{4 π}{15} R ρ^{2} θ (\frac{\partial u}{\partial x} + {\frac{\partial u}{\partial x}}^{T} + (\frac{\partial}{\partial x} \cdot u) I), \end{matrix}$

(198b)

$\begin{matrix} \int {(n \cdot (w - v))}^{2} [{(n \cdot (w - v))}^{2} n + (n \cdot (w - v)) (I + 2 n n^{T}) (v - u) + \\ + ({∥ v - u ∥}^{2} I + 2 (v - u) {(v - u)}^{T}) n] Θ (n \cdot (w - v)) \\ R (g_{0} (v) g_{1} (w) + g_{1} (v) g_{0} (w)) d n d w d v = - \frac{64 \sqrt{π}}{15} R ρ^{2} \sqrt{θ} q, \end{matrix}$

(198c)

$\begin{matrix} \int {(n \cdot (w - v))}^{2} [{(n \cdot (w - v))}^{2} n + (n \cdot (w - v)) (I + 2 n n^{T}) (v - u) + \\ + ({∥ v - u ∥}^{2} I + 2 (v - u) {(v - u)}^{T}) n] Θ (n \cdot (w - v)) \\ n \cdot \sqrt{R} g_{0} (v) \frac{\partial (\sqrt{R} g_{0} (w))}{\partial x} d n d w d v = \frac{10 π}{3} θ \frac{\partial (R ρ^{2} θ)}{\partial x} + 2 π R ρ^{2} θ \frac{\partial θ}{\partial x} . \end{matrix}$

(198d)
Equations (199a) and (199b) are changed as follows:

$ρ S = (\frac{1}{R} + \frac{8 ρ}{5 ρ_{s p}}) {[ρ S]}_{B} = - (\frac{1}{R} + \frac{8 ρ}{5 ρ_{s p}}) μ (\frac{\partial u}{\partial x} + {\frac{\partial u}{\partial x}}^{T} - \frac{2}{3} (\frac{\partial}{\partial x} \cdot u) I),$

(199a)

$ρ q = (\frac{1}{R} + \frac{12 ρ}{5 ρ_{s p}}) {[ρ q]}_{B} = - (\frac{1}{R} + \frac{12 ρ}{5 ρ_{s p}}) \frac{15}{4} μ \frac{\partial θ}{\partial x}, μ = \frac{5 \sqrt{π} ρ_{s p} σ}{96} \sqrt{θ},$

(199b)
Equations (201a) and (201b) are changed as follows:

$\begin{matrix} \frac{\partial (ρ u)}{\partial t} + \frac{\partial}{\partial x} \cdot (ρ (u u^{T} + θ I + σ S)) = - \frac{4}{ρ_{s p}} \frac{\partial}{\partial x} \cdot (R ρ^{2} θ) + \\ + \frac{6 σ}{π ρ_{s p}} \int {(n \cdot (w - v))}^{2} n Θ (n \cdot (w - v)) \\ \sqrt{R} (- g_{1} (v) n \cdot \frac{\partial (\sqrt{R} g_{0} (w))}{\partial x} - g_{0} (v) n \cdot \frac{\partial (\sqrt{R} g_{1} (w))}{\partial x} + \frac{1}{2} g_{0} (v) n^{T} \frac{\partial^{2} (\sqrt{R} g_{0} (w))}{\partial x^{2}} n) d n d w d v, \end{matrix}$

(201a)

$\begin{matrix} \frac{\partial (ρ ϵ)}{\partial t} + \frac{\partial}{\partial x} \cdot (ρ ((ϵ + θ) u + σ S u + σ q)) = - \frac{4}{ρ_{s p}} \frac{\partial}{\partial x} \cdot (R ρ^{2} θ u) + \\ + \frac{3 σ}{π ρ_{s p}} \int (n \cdot (w + v)) {(n \cdot (w - v))}^{2} Θ (n \cdot (w - v)) \\ \sqrt{R} (- g_{1} (v) n \cdot \frac{\partial (\sqrt{R} g_{0} (w))}{\partial x} - g_{0} (v) n \cdot \frac{\partial (\sqrt{R} g_{1} (w))}{\partial x} + \frac{1}{2} g_{0} (v) n^{T} \frac{\partial^{2} (\sqrt{R} g_{0} (w))}{\partial x^{2}} n) d n d w d v . \end{matrix}$

(201b)
Equation (202), together with the preceding sentence, are changed as follows:
Above, we take advantage of the fact that, for $ψ (v) = v$ or $ψ (v) = {∥ v ∥}^{2}$ ,

$\int (ψ (v^{'}) - ψ (v)) n \cdot (w - v) Θ (n \cdot (w - v)) (n n^{T}) g_{0} (v) g_{0} (w) d n d w d v = 0 .$

(202)
Equations (203a)–(203d) are changed as follows:

$\begin{matrix} \int {(n \cdot (w - v))}^{2} n Θ (n \cdot (w - v)) \\ \sqrt{R} (g_{1} (v) n \cdot \frac{\partial (\sqrt{R} g_{0} (w))}{\partial x} + g_{0} (v) n \cdot \frac{\partial (\sqrt{R} g_{1} (w))}{\partial x}) d n d w d v = \frac{4 π}{15} \frac{\partial}{\partial x} \cdot (R ρ^{2} S), \end{matrix}$

(203a)

$\begin{matrix} \int {(n \cdot (w - v))}^{2} n Θ (n \cdot (w - v)) n^{T} \frac{\partial^{2} (\sqrt{R} g_{0} (w))}{\partial x^{2}} n \sqrt{R} g_{0} (v) d n d w d v = \\ = \frac{8 \sqrt{π}}{15} \frac{\partial}{\partial x} \cdot (R ρ^{2} \sqrt{θ} (\frac{\partial u}{\partial x} + {\frac{\partial u}{\partial x}}^{T} + (\frac{\partial}{\partial x} \cdot u) I)), \end{matrix}$

(203b)

$\begin{matrix} \int (n \cdot (w + v)) {(n \cdot (w - v))}^{2} Θ (n \cdot (w - v)) \\ \sqrt{R} (g_{1} (v) n \cdot \frac{\partial (\sqrt{R} g_{0} (w))}{\partial x} + g_{0} (v) n \cdot \frac{\partial (\sqrt{R} g_{1} (w))}{\partial x}) d n d w d v = \frac{8 π}{15} \frac{\partial}{\partial x} \cdot (R ρ^{2} (S u + \frac{3}{2} q)), \end{matrix}$

(203c)

$\begin{matrix} \int (n \cdot (w + v)) {(n \cdot (w - v))}^{2} Θ (n \cdot (w - v)) n^{T} \frac{\partial^{2} (\sqrt{R} g_{0} (w))}{\partial x^{2}} n \sqrt{R} g_{0} (v) d n d w d v = \\ = \frac{16 \sqrt{π}}{15} \frac{\partial}{\partial x} \cdot (R ρ^{2} \sqrt{θ} [(\frac{\partial u}{\partial x} + {\frac{\partial u}{\partial x}}^{T} + (\frac{\partial}{\partial x} \cdot u) I) u + \frac{5}{2} \frac{\partial θ}{\partial x}]) . \end{matrix}$

(203d)
Equations (204a) and (204b) are changed as follows:

$\begin{matrix} \frac{\partial (ρ u)}{\partial t} + \frac{\partial}{\partial x} \cdot (ρ (u u^{T} + θ I)) = - \frac{4}{ρ_{s p}} \frac{\partial}{\partial x} \cdot (R ρ^{2} θ) - \\ - σ \frac{\partial}{\partial x} \cdot ((1 + \frac{8 ρ}{5 ρ_{s p}} R) ρ S) + \frac{8 σ}{5 \sqrt{π} ρ_{s p}} \frac{\partial}{\partial x} \cdot (R ρ^{2} \sqrt{θ} (\frac{\partial u}{\partial x} + {\frac{\partial u}{\partial x}}^{T} + (\frac{\partial}{\partial x} \cdot u) I)), \end{matrix}$

(204a)

$\begin{matrix} \frac{\partial (ρ ϵ)}{\partial t} + \frac{\partial}{\partial x} \cdot (ρ (ϵ + θ) u) = - \frac{4}{ρ_{s p}} \frac{\partial}{\partial x} \cdot (R ρ^{2} θ u) - \\ - σ \frac{\partial}{\partial x} \cdot ((1 + \frac{8 ρ}{5 ρ_{s p}} R) ρ S u) - σ \frac{\partial}{\partial x} \cdot ((1 + \frac{12 ρ}{5 ρ_{s p}} R) ρ q) + \\ + \frac{8 σ}{5 \sqrt{π} ρ_{s p}} \frac{\partial}{\partial x} \cdot (R ρ^{2} \sqrt{θ} [(\frac{\partial u}{\partial x} + {\frac{\partial u}{\partial x}}^{T} + (\frac{\partial}{\partial x} \cdot u) I) u + \frac{5}{2} \frac{\partial θ}{\partial x}]) . \end{matrix}$

(204b)
Equations (205a) and (205b) are changed as follows:

$\begin{matrix} \frac{\partial (ρ u)}{\partial t} + \frac{\partial}{\partial x} \cdot (ρ (u u^{T} + (1 + \frac{4 ρ}{ρ_{s p}} R) θ I)) = \\ = \frac{\partial}{\partial x} \cdot (μ ((\frac{1}{R} + a_{1}) (\frac{\partial u}{\partial x} + {\frac{\partial u}{\partial x}}^{T}) - \frac{2}{3} (\frac{1}{R} + a_{2}) (\frac{\partial}{\partial x} \cdot u) I)), \end{matrix}$

(205a)

$\begin{matrix} \frac{\partial (ρ ϵ)}{\partial t} + \frac{\partial}{\partial x} \cdot (ρ (ϵ + (1 + \frac{4 ρ}{ρ_{s p}} R) θ) u) = \frac{15}{4} \frac{\partial}{\partial x} \cdot (μ (\frac{1}{R} + a_{3}) \frac{\partial θ}{\partial x}) + \\ + \frac{\partial}{\partial x} \cdot (μ ((\frac{1}{R} + a_{1}) (\frac{\partial u}{\partial x} + {\frac{\partial u}{\partial x}}^{T}) - \frac{2}{3} (\frac{1}{R} + a_{2}) (\frac{\partial}{\partial x} \cdot u) I) u) . \end{matrix}$

(205b)
Equations (206a)–(206c) are changed as follows:

$a_{1} (\frac{ρ}{ρ_{s p}}) = \frac{16 ρ}{5 ρ_{s p}} (1 + \frac{4 ρ}{5 ρ_{s p}} R (1 + \frac{12}{π})),$

(206a)

$a_{2} (\frac{ρ}{ρ_{s p}}) = \frac{16 ρ}{5 ρ_{s p}} (1 + \frac{4 ρ}{5 ρ_{s p}} R (1 - \frac{18}{π})),$

(206b)

$a_{3} (\frac{ρ}{ρ_{s p}}) = \frac{24 ρ}{5 ρ_{s p}} (1 + \frac{2 ρ}{15 ρ_{s p}} R (9 + \frac{32}{π})) .$

(206c)
The paragraph preceding Equations (207a)–(207c), as well as Equations (207a)–(207c), are removed. In the next paragraph, the reference [2] (which is [46] in the updated manuscript) is provided for cavity distribution functions for hard spheres.
The main results of [1] are unchanged, and the summary remains the same.

Conflicts of Interest

The author declares no conflict of interest.

References

Abramov, R. The Random Gas of Hard Spheres. J 2019, 2, 162–205. [Google Scholar] [CrossRef] [Green Version]
Boublík, T. Hard-Sphere Radial Distribution Function from the Residual Chemical Potential. Molec. Phys. 2006, 104, 3425–3433. [Google Scholar] [CrossRef]

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Abramov, R.V. Correction: Abramov, R. The Random Gas of Hard Spheres. J 2019, 2, 162–205. J 2020, 3, 324-328. https://0-doi-org.brum.beds.ac.uk/10.3390/j3030025

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Abramov RV. Correction: Abramov, R. The Random Gas of Hard Spheres. J 2019, 2, 162–205. J. 2020; 3(3):324-328. https://0-doi-org.brum.beds.ac.uk/10.3390/j3030025

Chicago/Turabian Style

Abramov, Rafail V. 2020. "Correction: Abramov, R. The Random Gas of Hard Spheres. J 2019, 2, 162–205" J 3, no. 3: 324-328. https://0-doi-org.brum.beds.ac.uk/10.3390/j3030025

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