Cercignani Shen Kuščer

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1 Cercignani Shen Kuščer Maxwell Knudsen Grad Nocilla Epstein Cercignani Lampis Lord Knudsen Kuščer

2 2

3 1 Gas Surface Interaction Galilei Darrozes Guiraud 2 elastic collision model specular reflection model f ref ξ[ f ref ξ n, ξ t1, ξ t2 ] f in ξ n, ξ t1, ξ t2, ξ n >. 1 f in f ref ξ n ξ ξ t1 ξ t2 3

4 T w T w 1 f ref ξ σ d 2π exp ξ 2 : σ 3/2 d MT w, ξ, ξ n >. 2 2 σ d ξ n< ξ nf in dξ + ξ n > ξ nf ref dξ σ d 2π ξ n< ξ n f in ξdξ diffuse reflection model perfect/complete accommodation 2 Maxwell Maxwell Maxwell ξ diffuse reflection with incomplete accommodation Lord 7.5 Cercignani Lampis

5 perfect/complete accommodation accommodation coefficient 5 [1] Knudsen T 1 T 2 T 2 T 2 a T 2 T 1 T 2 T 1 [2, 3] ξ φξ ξ n φξf in ξdξ ξ n φξf ref ξdξ αφ; f in ξ n < ξ n >, 4 ξ n φξf in ξdξ ξ n φξσ d MT w ; ξdξ ξ n < ξ n > α φ accommodation coefficient for φ φξ ξ 5

6 φ αφ; f in 1 f ref ξ σ d MT w ; ξ 4 φ 1 2 ξ 2 ξ t1 ξ n 4 3 ξ n ξ 2 f in ξdξ ξ n ξ 2 f ref ξdξ α e f in : ξ n< ξ n>, 5 ξ n ξ 2 f in ξdξ ξ n ξ 2 σ d MT w ; ξdξ ξ n< ξ n> ξ n ξ t1 f in ξdξ ξ n ξ t1 f ref ξdξ α t f in : ξ n< ξ n> ξ n ξ t1 f in ξdξ ξ n ξ t1 σ d MT w ; ξdξ ξ n < ξ n > ξ n ξ t1 f ref ξdξ ξ 1 n >, 6 ξ n ξ t1 f in ξdξ ξ n < ξ n 2 f in ξdξ ξ n 2 f ref ξdξ α p f in : ξ n < ξ n >. 7 ξ n 2 f in ξdξ ξ n 2 σ d MT w ; ξdξ ξ n < ξ n > α e α t α p energy accommodation coefficient[4], accommodation coefficient for energy thermal accommodation coefficient[5] tangential momentum accommodation coefficient[4] accommodation coefficient for tangential momemtum[3] normal momentum accommodation coefficient accommodation coefficient for normal momemtum[3] 3 4 6

7 e tf s tf p tf 1 1 e tf f in ξ n ξ n < 2 ξ 2 f in ξdξ + ξ n ξ n > 2 ξ 2 f ref ξdξ, 8 s tf f in ξ n ξ t1 f in ξdξ + ξ n ξ t1 f ref ξdξ, 9 ξ n < ξ n > p tf f in ξnf 2 in ξdξ + ξnf 2 ref ξdξ, 1 ξ n < ξ n > α e α t α p e tf α ef in ξ n ξ 2 σ d MT w ; ξdξ ξ n ξ 2 f in ξdξ, 11 2 ξ n > ξ n < s tf α t f in ξ n ξ t1 σ d MT w ; ξdξ ξ n ξ t1 f in ξdξ, 12 ξ n > ξ n < p tf [2 α p f in ] ξ n 2 f in ξdξ + α p f in ξ n 2 σ d MT w ; ξdξ, ξ n < ξ n > 13 f in ξ α e α t α p α e α t α p f in Handbuch der Physik Schaaf [6] α e α t α p 6 4 f ref f in 5 α e 6 Schaaf α e α t α p α p Schaaf α t 7

8 ξ ξ n < ξ ξ + dξ ξ n > Rξ ξdξ ξ ξ +dξ ξ n f in ξ dξ Rξ ξdξ ξ n f ref ξdξ ξ n f ref ξ Rξ ξ ξ n f in ξ dξ, ξ n >, 14 ξ n < Rξ ξ R 14 7 R i ii Rξ, ξ, ξ n <, ξ n >, 15 Rξ, ξdξ 1, ξ n <, 16 ξ n > iii reciprocity detailed balance ξ n MT w ; ξ Rξ, ξ ξ n MT w ; ξr ξ, ξ, ξ n <, ξ n >, 17 8 iii 9 iii i iii 17 ξ n < ξ 16 ξ n MT w; ξ Rξ ξ ξ n MT w; ξ dξ, ξ n >, ξ n < 7 Kuščer [7] 8 R Kuščer 1969 [3] 9 iii 8

9 14 R Rξ ξ δξ + ξ n n ξ t, 18 Rξ ξ 2π ξ n MT w ; ξ. 19 ξ t, ξ t1, ξ t2 R 3 δ g ξ n MT w ; ξ Rξ, ξgξ dξ ξ n < ξ n MT w ; ξ δξ + ξ n n ξ t gξ dξ ξ n < ξ n MT w ; ξg ξ n n + ξ t ξ n MT w ; ξ δξ ξ n < n + ξ n δξ t ξ t gξ dξ ξ n MT w ; ξ δ ξ nn + ξ n n + ξ ξ n < t ξ t gξ dξ ξ n MT w ; ξ δ ξ ξ nn + ξ tgξ dξ ξ n < ξ n MT w ; ξ R ξ ξ gξ dξ. ξ n < π F, G : F ξgξ ξ n MT w ; ξdξ, 2 ξ n> 2π 1 1 1,

10 f in ξ σ d MT w ; ξ[1 + ψ in ξ], ξ n <, 22 f ref ξ σ d MT w ; ξ[1 + ψ ref ξ], ξ n >, 23 f in f ref ψ in ψ ref f in f ref ξ n f in ξdξ σ d ξ n MT w ; ξdξ, ξ n< ξ n> ξ n f in ξdξ ξ n f ref ξdξ, ξ n< ξ n> ψ in ψ ref 1 1, ψ in ψ in, 1, 1, ψ ref ψ ref, αφ; ψ in σ d P φ, 1 + ψ in σ d φ, 1 + ψ ref σ d P φ, 1 + ψ in σ d φ, 1 φ, ψ ref 1 P φ φ, 1 + P φ, ψ in. P P F ξ F ξ R 3 F ξ P t F ξ n, ξ t F ξ n, ξ t P t P t P t F, G F, P t G. P P t F ξ n F even P F even P t F even P F even, G F even, P t G P t F even, G φ ξ n φ even φ even ξ n, ξ t φ even ξ n, ξ t. P φ even φ even φ even ξ n, ξ t φ even ξ n, ξ t ξ t P φ even φ even, 1 αφ even ; ψ in αφ even ; ψ in 1 φ even, ψ ref P φ even, ψ in. ψ ref ψ in f ref f in R ξ n f ref ξ ξ n f in ξ Rξ ξdξ, ξ n >, ξ n < 1

11 R ξ n MT w ; ξ[1 + ψ ref ξ] ξ n MT w ; ξ [1 + ψ in ξ ]Rξ ξdξ ξ n < ξ n MT w ; ξ [1 + ψ in ξ ]R ξ ξ dξ ξ n < ξ n MT w ; ξ 1 + ψ in ζr ξ ζdζ, ξ n >, ζ n> ψ ref ξ R ξ ζψ in ζdζ, ξ n >, ζ n > Cercignani R + R 2 R + R 2 A AF ξ : R ξ ζf ζdζ ζ n > ζ n > ζ n MT w ; ζ ξ n MT w ; ξ R ζ ξf ζdζ, ξ n >, [3] ψ ref ψ in Aψ ref ψ in A 2π AF, G ξ n > 2π ξ n> 2π ξ n > 2π F, AG. ζ n > ζ n > ζ n> ζ n > ξ n > R ξ ζf ζdζ ξ n MT w ; ξgξdξ ζ n MT w ; ζ ξ n MT w ; ξ R ζ ξf ζdζ ξ n MT w ; ξgξdξ ζ n MT w ; ζr ζ ξf ζdζ Gξdξ R ζ ξgξdξ ζ n MT w ; ζf ζdζ αφ even ; ψ in αφ even ; ψ in 1 Aφ even, ψ in P t φ even, ψ in. 25 P F even P t F even Cercignani Kuščer [3, 7] 11

12 5.2 Cercignani Cercignani αφ even ; ψ in 25 3 i iii AF even λp t F even F even 1 1 φ even λ φ λ even αφλ even; ψ in αφ λ even; ψ in 1 Aφλ even, ψ in P t φ λ 1 λ, 26 even, ψ in Cercignani Cercignani αφ λ even φ λ even AF even λp t F even R Cercignani φ λ even R Cercignani Lampis AF even λp t F even Shen Kuščer Cercignani Shen Kuščer 25 ψ in φ even + const 12 1 λ 1 F even 1 α 11 Cercignani [3] Cercignani Boltzmann BGK 12 const ψ in 1 12

13 α ij : αφ i even; ψ j in 1 Aφi P t φ i even, ψ j in even, ψ j in, A P t α ij φ 1 even 1 2 ξ 2 ψ 1 in φ1 φ 2 even ξ t1 ψ 2 in φ2 even φ 3 even ξ n ψ 3 in φ3 even 3 3 a E : α 11 1 A 1 2 ξ 2 2, 1 2 ξ ξ 2 2 2, a : α Aξ t 1, ξ t1 ξ t1 2, a : α 33 1 A ξ n π ξ n π 2 2, ξ n, π 2 2 even 2 π 2 a E accommodation coefficient for energy a accommodation coefficient for parallel momentum a accommodation coefficient for normal momentum ψ in α e α t α p ψ in p/σ d T w +ξ v/ ξ 2 / 5 T/T w a E f in T T w + T p σ d T w T T/T w 1 a f in T T w p σ d T w v n v t2 v t1 v t1 v t1 / 1 a f in T T w v t1 v t2 v n v n p σ d T w 1 v n π 2 v n / 1 α e α t a E a Kuščer a E [7] 6 α e α t 13

14 1: [8] Gas Reservoir Source Stage1 2 sample UHV chamber skimmer collimator molecular beam 13 [8, 1] ξ 2 exp ξ u 2 /2RT dξ monoenergetic collimated beam, monochromatic beam thermal beam effusive beam supersonic nozzle beam 14 14

15 ξ 2 exp ξ u 2 /2RT dξ [9] f beam ξdξdω beam R ω beam f in f in ξdξ f beam ξdξdω beam f beam ξδω ω beam dξdω, f beam f in ξ ξ 2 f beam ξ δω ω beam, f ref ξ n f ref ξ, ω Rξ ξ ξ n f in ξ dξ ξ n< Rξ, ω ξ, ωξ ω n f beam ξ δω ω beam dξ dω ω n< Rξ, ω beam ξ, ωξ f beam ξ dξ cosπ θ beam, θ, ϕ ω beam θ beam, ϕ beam D ref D ref ω : ξ 2 ξ n f ref ξ, ωdξ Rξ, ω beam ξ, ωξ ξ 2 f beam ξ dξ dξ cosπ θ beam, 27 u u 15

16 ω n cos θ > ω beam n cos θ beam < D ref ω ξ 3 cos θσ d MT w ; ξdξ 1 RTw π 2π σ d cos θ 1 ξ n f in dξ cos θ π ξ n< 1 ξf beam ξdξ cosπ θ beam cos θ, π i ϕ ii θ cos θ θ D ref ξ n f ref ξ f ref θ D ref cos θ Lambert a 2b a b 2: 15 Lambert n n beam beam

17 7 7.1 Maxwell Maxwell 1 α α f ref ξ 1 αf in ξ n n + ξ t + ασ d MT w ; ξ, ξ n >, 28 2π σ d ξ n f in ξdξ. 29 ξ n < Maxwell α α 1 α α 1 Maxwell 2π Rξ ξ 1 αδξ + ξ n n ξ t + α ξ n MT w ; ξ. 3 i iii Maxwell 2π Aφ even 1 αδ ξ + ζ n n ζ t + α ζ n MT w ; ζ φ even ζdζ ζ n> 1 αp t φ even ξ + α 1, φ even, Aφ even, ψ in 1 α P t φ even, ψ in + α 1, φ even 1, ψ in 1 α P t φ even, ψ in, 25 φ even ψ in αφ even ; ψ in α, 31 17

18 D ref ω 1 α + α Rξ, ω beam ξ, ωξ ξ 2 f beam ξ dξ dξ cosπ θ beam 1 α + α π 1 δξ + ξ n n ξ t ξ ξ 2 f beam ξ dξ dξ cosπ θ beam ξ ξ 2 f beam ξ 2π ξ n MT w ; ξdξ dξ cosπ θ beam ξ 3 f beam ξ dξ cosπ θ beam δω + P t ω beam ξf beam ξdξ cosπ θ beam cos θ. Maxwell cos θ δ 2b 7.2 Knudsen Grad Nocilla Nocilla b f ref ξ σ N 2πRT N exp ξ U N 2, ξ 3/2 n >. 32 2RT N σ N ξ n > ξ n f ref ξdξ ξ n < ξ n f in ξdξ σ N Cu N, T N ξ n f in ξdξ, 33 ξ n < RT N Cu N, T N 2 2π e u N + u N 2 [1 + erfu N] U N / 2RT N u N, v N, w N T N U N U N Knudsen 1911 T N T w Grad 1958 [1] Knudsen Grad Nocilla Rξ ξ Cu N, T N ξ n MT N ; ξ U N, 35 18

19 D ref ω Rξ, ω beam ξ, ωξ ξ 2 f beam ξ dξ dξ cosπ θ beam Cu N, T N ξ n MT N ; ξ U N ξ ξ 2 f beam ξ dξ dξ cosπ θ beam Cu N, T N ξ f 2πRT N 3/2 beam ξ dξ cosπ θ beam ξ 3 exp ξ U Nω ω N 2 + UN 2 1 ω ω N 2 dξ cos θ 2RT N Cu N, T N 2RT N 1/2 ξ f beam ξ dξ cosπ θ beam π 3/2 ζ + u N ω ω N 3 e ζ 2 dζ e u2 N 1 ω ω N 2 cos θ u N ω ω N Cu N, T N RTN ξ f beam ξ dξ cosπ θ beam e u2 N π 2π 1 + u 2 N ω ω N 2 + πu N ω ω N u 2 N ω ω N [1 + erfu Nω ω N ]e u2 N ω ω N 2 cos θ. ω N U N / U N ω ω N cos θ cos θ N +sin θ sin θ N cosϕ ϕ N u N Nicolla [11] α e α t α p 1 Knudsen Grad Nocilla iii iii Knudsen Grad Nocilla 7.3 Epstein Epstein 1967 Maxwell Rξ ξ 2π ξ n MT w ; ξ αξαξ + [1 αξ]δξ + ξ 1, α nn ξ t. αξ αξ 1 ξ i iii Maxwell α ξ α ξ 19

20 δ Cercignani i iii Epstein [3] 2π Rξ ξ ξ n MT w ; ξ[hξh ξ + Kξ, ξ ], ξ n <, ξ n >, 1 1, K, ξ Hξ. 1 1, K, ξ, 1 Kζ, ξ Epstein ζ Bζ K Kζ, ξ Bζδζ P t ξ Epstein K δ 7.4 Cercignani Lampis Cercignani Lampis 1971 i iii i AF even λp t F even Rξ ξ Maxwell λ 1 λ 1 λ 2 1 α ii ξ 2 n ξ t 1 ξ t2 Laguerre Hermite Cercignani Lampis CL [12] 2 p 1 1 q 1 Rξ 1 1 p ξ 2π 2 1 p1 q 2 ξ ξ n ξ n n I 1 p exp ξ2 n + p ξ n p ξ t q ξ t q 2, ξ n >, ξ n <. 36 I I x : 1 2π 2π expx cos ϕdϕ, 2

21 1 Bessel R 2π Rξ ξ ξ n MT w ; ξ Ψ lmn ξ l,m,n p l q m+n Ψ lmn ξ Ψ lmn ξ, 1 ξ 2 l L n H m m! n! 2 m+n 2 ξt1 ξt2 H n 2RTw 2RTw, R L l l Laguerre H m m Hermite Ψ lmn AF even λp t F even p l q m+n Ψ lmn AΨ ijk p i q j+k Ψ ijk AΨ ijk, Ψ lmn p i q j+k δ il δ jm δ kn R 2 i iii ii 2 2 iii Hermite H n x 1 n H n x 2 i Ψ lmn l, m, n p q 1 p q Cercignani Lampis f in Cercignani 1 26 Laguerre Hermite A R A1 1 1, ψ in AΨ 1, ψ in P t Ψ 1, ψ in A 1 2 ξ2 n, ψ in 1 P t 2 ξ2 n, ψ in, AΨ 1, ψ in P t Ψ 1, ψ in Aξ t 1, ψ in P t ξ t1, ψ in, AΨ 2, ψ in P t Ψ 2, ψ in A 1 2 ξ2 t 1, ψ in 1 P t 2 ξ2 t 1, ψ in, 17 L x 1 L 1 x 1 x H x 1 H 1 x 2x H 2 x 2x φ even Ψ 2 Ψ 11 m+n i AF even λp t F even Cercignani Lampis 21

22 φ even Ψ 1 1 ξ2 n 1 p 2 ξ t1 Ψ 1 1 q RTw 1 ξ 2 Ψ 2 2 t1 1 1 q 2 2 ξ t1 ξ t2 Ψ 11 1 q 2 1: Cercignani α Cercignani Lampis p q α n 1 p α t 1 q α e α e ψ in 1 p ξ2 n, ψ in + q 2 ξ t 2, ψ in ξ 2. 37, ψ in α p Cercignani Lampis D ref ω exp Rξ, ω beam ξ, ωξ ξ 2 f beam ξ dξ dξ cosπ θ beam 1 ξ cos θ p 2π 2 1 p1 q 2 I ξ ξ cos θ beam cos θ 1 p ξ2 cos 2 θ + p ξ 2 cos 2 θ beam q2 ξ 2 sin 2 θ beam sin 2 ϕ ϕ beam 2 1 p 2 1 q 2 ξ sin θ qξ sin θ beam cosϕ ϕ beam q 2 ξ ξ 2 f beam ξ dξ dξ 1 cosπ θ beam p ξ ξ 2π 2 1 p1 q 2 I cos θ beam cos θ 1 p exp ξ2 cos 2 θ + p ξ 2 cos 2 θ beam q2 ξ 2 sin 2 θ beam sin 2 ϕ ϕ beam 2 1 p 2 1 q 2 ξ sin θ qξ sin θ beam cosϕ ϕ beam q 2 ξ ξ 3 f beam ξ dξ dξ 2 Cercignani α t.1 α n K ξ ξ2 n 1 2 ξ2 t ξ2 t ξ2 n 1 2 ξ2 t ξ2 t 2 t 1 t 2 cosπ θ beam cos θ. 22

23 191 K [13] [3] Cercignani Cercignani Lampis [3] 36 p q Rξ ξ ξ n 2π 2 w exp p ξ n ξ n wp, qi 1 p 2 ξ2 n + p ξ n 2 1 p ξ t q ξ t q 2 dqdp, ξ n >, ξ n <. wp, q, wp, q1 p1 q 2 dqdp 1, wp, q 1 p 1 1 q 2 1 δp p δq q., Ψ lmn α n α t. Cercignani, α n α t 2 i Ψ lmn 2 Hilbert AF even λp t F even φ i even l,m,n ai lmn Ψ lmn R 2π Rξ ξ ξ n MT w ; ξ λ i φ i RT evenξ φ i evenξ w i 2π ξ n MT w ; ξ λ i a i I RT Ψ Iξ a i J Ψ Jξ w i,i,j 2π ξ n MT w ; ξ b IJ Ψ I ξ Ψ J ξ I,J I J l, m, n b IJ i λi a i I ai J ii p pξ t, ξ t p ξ t, ξ t q qξ n, ξ n q ξ n, ξ n ξ t ξ t ξ n ξ n i iii α n α t 23

24 7.5 Lord Cercignani Lampis CL Lord CL [14] 2 CL p q CL ξ ξ Rξ ξ 2 2RTw π 1 p ξ n p p ξ ξ I ξ ξ 1 exp ξ 2 + p ξ 2. 1 p 2 1 p I Bessel I 1 x x 6π 2π sin 2 θ expx cos θdθ. i iii R ξ n cos θ p p 1 α t 1 p p 1 diffuse elastic reflection 7.6 Knudsen Kuščer 3 Knudsen Knudsen Knudsen T 1 T 2 T 2 Kuščer Knudsen i iii 2 Lord Cercignani Lampis Lord CLL CLL Cercignani Lampis 2 Petek & Kuščer 1979 Lord CL DSMC CL [14] 24

25 iv T in T ref T ref [15] Kuščer Knudsen Knudsen Kuščer Knudsen Fokker Planck CL iv T in T ref ξ n MT ref ; ξ Tref T in ξ n < ξ n MT in ; ξ Rξ ξdξ, 38 i iii Kuščer Fokker Planck t P ξ [ ap ξbp ]. a ξ b 2 P ξ, t δξ ξ P ξ, t 1 [2π 1 e 2t ] exp ξ ξ e t 2 3/2 2 1 e 2t, t ξ t ξ P ξ ξ; t P P ξ n 1 ξ n ; t [2π 1 e 2t ] exp ξ n ξ ne t 2 1/2 2 1 e 2t, 39 P ξ t 1 ξ t ; t 2π 1 e 2t exp ξ t ξ te t e 2t, 4 P ξ ξ; t P ξ n ξ n ; tp ξ t ξ t ; t P 25

26 ξ n exp ξ n 2 P n ξ n ξ n 2 ξ n exp ξ n 2 P n ξ n ξ 2RT n, ξ n >, ξ n <, w P n ξ n ξ n ; tdξ n 1, P n P n ξ n ξ n ; tp ξ t ξ t ; t i iii 5.1 A n A n F ξ n : P n ξ n ζ n ; tf ζ n dζ n, ξ n >, M 1 T w ; ξ n 2π 1/2 exp ξ 2 n/2 2π A n F, G n 2π 2π F, G n : F, A n G n, ξ n M 1 T w ; ξ n F ξ n Gξ n dξ n, ξ n M 1 T w ; ξ n P n ξ n ζ n ; tf ζ n dζ n Gξ n dξ n ξ n M 1 T w ; ξ n ζ n M 1 T w ; ζ n ξ n M 1 T w ; ξ n P n ζ n ξ n ; tf ζ n dζ n Gξ n dξ n ζ n M 1 T w ; ζ n P n ζ n ξ n ; tgξ n dξ n F ζ n dζ n P n A n Fokker Planck t P n ξn [ a n P n ξ n bp n ] : DP n, ξ n >, 41 P n Kuščer t t t t ζ n P n ζ n ξ n ; t e td δξ n ζ n, 42 26

27 A n F, G n 2π A n F, G n dζ n Gζ n ζ n M 1 T w ; ζ n dξ n P n ζ n ξ n ; tf ξ n 2π dζ n Gζ n ζ n M 1 T w ; ζ n dξ n F ξ n [e td δξ n ζ n ] 2π dξ n F ξ n e td dζ n Gζ n ζ n M 1 T w ; ζ n δξ n ζ n 2π dξ n F ξ n e td [Gξ n ξ n M 1 T w ; ξ n ], 2π F, A n G n e td n 1 n! tn D n dξ n Gξ n D[F ξ n ξ n M 1 T w ; ξ n ] dξ n Gξ n e td [F ξ n ξ n M 1 T w ; ξ n ], dξ n F ξ n D[Gξ n ξ n M 1 T w ; ξ n ], D P n A n dξ n GDF ξ n M 1 dξ n F [ a n Gξ n M b Gξ nm 1 ] dξ n F [ a n ξ n M 1 G a n ξ n M 1 G b Gξ nm 1 ] dξ n F [ a n ξ n M 1 G a n ξ n M 1 G ] F b Gξ nm dξ n F b Gξ n M 1 [a n ξ n M 1 F G bξ nm 1 F G ]dξ n 1 2 bm 1 F G [ 1 2 bξ nm 1 F G + a n ξ n M 1 F G ]dξ n, F G a n b a n ξ n M bξ nm 1 a n 1 2 [ db 1 + b ξ ] n, 43 dξ n ξ n Knudsen 38 Rξ ξ P n ξ n ξ n ; tp ξ t ξ t ; t ξ n M 1 T ref ; ξ n 2πRT ref 1 exp ξ t 2 2RT ref, 27

28 Tref dξ T n ξ n M 1 T in ; ξ np n ξ in ξ n < n ξ n 2πRT in 1 dξ t2πrt w 1 1 e 2t 1 exp ξ t e t ξ t e 2t ξ t 2 2RT in Tref dξ T n ξ n M 1 T in ; ξ np n ξ in ξ n < n ξ n 2πRT in 1 2π 1 1 e 2t 1 dξ t ξ t 2 exp 2 1 e 2t T w + T in T w e 2t e t T in 2 T in 1 e 2t [ξ t T w + T in T w e 2t ξ t] 2 Tref ξ t 2 e 2t T ] in e 2t T w + T in T w e 2t dξ T n ξ n M 1 T in ; ξ np n ξ in ξ n < n ξ n 1 2πR[T w + T in T w e 2t ] exp ξ t 2 2R[T w + T in T w e 2t, ] ξ t ξ n M 1 T ref ; ξ n T ref T w + T in T w e 2t, 44 Tref T in ξ n < dξ n ξ n M 1 T in ; ξ np n ξ n ξ n, 42 ξ n M 1 T ref ; ξ n Tref T in Tref ξ n < dξ n ξ n M 1 T in ; ξ ne td δξ n + ξ n e td ξ n M 1 T in ; ξ n, T in T in T ref ξ n M 1 T ref ; ξ n P n ξ n, t ξ n M 1 T in ; ξ n 41 T w T in t ξ 2 n 1 dt ref 1 2RT ref T ref dt Tin T ref ξ n M 1 T ref ; ξ n, T in T ref ξ n M 1 T ref ; ξ n 41 28

29 Tin a n ξ n M 1 T ref ; ξ n + 1 [ Tin ] b ξ n M 1 T ref ; ξ n T ref 2 T ref Tin [ 1 T ref 2 b a n ξ n M 1 T ref ; ξ n + 1 ] ξ 2 b n M 1 T ref ; ξ n Tin T ref 1 2 b a n 1 2 b ξ n RT ref 1 1 2RT ref Tref T w 1 1 2RT ref Tref T w 1 ξ n M 1 T ref ; ξ n ξ n T in bξ 2 T nm 1 T ref ; ξ n ref T [ in ξ b 2 ] ξ n 2b n 1 ξ n M 1 T ref ; ξ n, T ref 2RT ref ξ n 43 b T ref dt ref dt b R Tref 1, T w T ref 44 b 2 43 a n ξn 1 ξn 2 Fokker Planck t P n ξn ξ n ξ n P n + ξn P n : DP n, ξ n >, i iii Knudsen P n ξ n ξ n ; tp ξ t ξ t ; t P 4 P n ξ n x ξ 2 n/2 x F G D[F xgx] dx 1 dx 2 x F G + F G dξ n 2RTw x dξ n x 2 x 1 F G + 2 xf G x 2xF G + 2xF G x F G 2xF G + F G + 2xF G + 2F G + F G x F G 2xF + 1 xf G + 2[2x 1G + 2xG ]F + 1 x x[2x 1G + 2xG ] F, 29

30 2x 1G+2xG G G const xe x D[F x xe x ] 2xF + 1 xf xe x, Laguerre Laguerre L k x Laguerre xy + 1 xy + ky, Laguerre e x F Gdx Hilbert F x k f ktl k x D[F x, t xe x ] 2 t [F x, t xe x ] kf k tl k x xe x, k k df k dt L kx xe x, df k dt 2kf kt f k t const. e 2kt, F xe x 45 ξ n exp 2RTw ξ2 n 2 k ξ 2 n e 2kt L k C k, C k, 2 45 P n δξ n + ξ n t ξ n exp 2RTw ξ2 n 2 k ξ 2 n L k C k ξ 2RT n δξ n +ξ n, ξ n >, ξ n <, w C k ξ n Laguerre L k xl k y e x δx y, k C k ξ n 2 L k ξ n, 2RTw 2 2 3

31 Laguerre k s k L k xl k y 1 1 s exp s 1 s x + y P n ξ n ξ n ; t ξ n exp ξ2 n 2 ξ n 1 1 e 2t exp k ξ 2 n 2 sxy I 1 s, e 2kt L k L k ξ n 2 2 ξ2 n + e 2t ξ n 2 I 2 1 e 2t 2 e t ξ n ξ n 1 e 2t P Rξ ξ n ξ; t 2π 2 1 e 2t 2 I exp e t ξ n ξ n 1 e 2t ξ2 n + e 2t ξ n e 2t ξ t ξ te t e 2t CL p q 2 e 2t α n α t α e α e α n 37 Kuščer CL 1 1 Knudsen 2 T in T ref T ref ξ n M 1 T ref ; ξ n M 1 T ref ; ξ t1 M 1 T ref ; ξ t2 Tref ξ T n MT in ; ξ Rξ ξdξ. in ξ n < Knudsen CL 21., [1] Petek & Kuščer

32 [2] C. Cercignani, Rarefied Gas Dynamics, Cambridge, 2. [3] C. Cercignani, The Boltzmann Equation and Its Applications, Springer, New York, [4] M. N. Kogan, Rarefied Gas Dynamics, Plenum, New York, [5] G. A. Bird, Molecular Gas Dynamics and The Direct Simulation of Gas Flows, Clarendon, Oxford, [6] S. A. Schaaf, Mechanics of rarefied gases, in Handbuch der Physik, Band VIII/2, Springer, Berlin, 1963, pp [7] F. R. W. McCourt, J. J. M. Beenakker, W. E. Köhler, I. Kuščer, Nonequilibrium Phenomena in Polyatomic Gases, Vol. 2, Clarendon, Oxford, [8] G. Attard, C. Barnes, Surfaces, Oxford, Oxford, [9] I. Kinefuchi, H. Yamaguchi, S. Shiozaki, Y. Sakiyama, Y. Matsumoto, Out-of-plane scattering distribution of Nitrogen molecular beam on Graphite 1 surface, in Rarefied Gas Dynamics, AIP, New York, 25, pp [1] H. Grad, Principles of the kinetic theory of gases, in Handbuch der Physik, Band XII, Springer, Berlin, 1958, pp [11] S. Nocilla, The surface re-emission law in free molecule flow, in Rarefied Gas Dynamics, Academic, New York, 1963, pp [12] C. Cercignani, Models for gas surface interactions: Comparison between theory and experiment, in Rarefied Gas Dynamics, Editorice Tecnico Scientifica, Pisa, 1971, pp [13] J. J. Hinchen, W. M. Foley, Scattering of molecular beams by metallic surfaces, in Rarefied Gas Dynamics, Vol. II, Academic, New York, 1966, p [14] R. G. Lord, Some extensions to the Cercignani Lampis gas-surface scattering kernel, Phys. Fluids A, 3, [15] I. Kuščer, J. Mozins, F. Krizanic, The Knudsen model of thermal accomodation, in Rarefied Gas Dynamics, Editorice Tecnico Scientifica, Pisa, 1971, pp

() n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (5) (6 ) n C + nc + 3 nc n nc n (7 ) n C + nc + 3 nc n nc n (

3 n nc k+ k + 3 () n C r n C n r nc r C r + C r ( r n ) () n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (4) n C n n C + n C + n C + + n C n (5) k k n C k n C k (6) n C + nc