BGK 2009 2 6
i X AGN Eckart [] Landau [2] [3] Israel Stewart Grad moment [4] BGK BGK
iii 2 7 2.............................. 7 2................................. 8 2..2......................... 9 2..3 Boltzmann...................... 0 2..4 H................................ 2 2..5................................ 4 2..6............................ 5 2..7 Chapman-Enskog....................... 8 2..8............................ 20 2..9.............................. 22 2..0 moment............................ 24 2.2............................... 32 2.2.......... 32 2.2.2 Boltzmann................. 33 2.2.3 H................................ 36 2.2.4................... 38 2.2.5 Eckart Landau-Lifshitz................ 43 2.2.6............................ 47 2.2.7 Chapman-Enskog....................... 5 2.2.8 moment............................ 54 2.2.9......................... 59 3 6 3. Boltzmann....................... 62
iv 3.. BGK............................. 62 3..2..................... 63 3..3 Result................................ 66............................... 66............................... 70 3..4 Discussion.............................. 72............. 72.................. 73 3..5......................... 74.............................. 74............................ 75........................... 76 3..6 continuous spectrum........................ 77 3.2 Boltzmann Marle........... 79 3.2. Marle BGK model........................ 79 3.2.2 Marle BGK................... 80 3.2.3.............. 82 3.2.4 Result................................ 86............................... 86 3.2.5 Discussion.............................. 93 3.3 Boltzmann Anderson-Witting.... 95 3.3. Anderson-Witting BGK model................. 95 3.3.2.............. 96 3.3.3 Result................................ 00 3.3.4 Discussion.............................. 07 3.4 Israel-Stewart............ 0 4 3 A 5 B moment 23 C 25 D Fredholm 27
v E 29 E. BGK......................... 29 E.2 Marle BGK...................... 32 E.3 Anderson-Witting BGK.................... 35 F 39 F. BGK................... 39 F.2 Anderson-Witting BGK.............. 4 F.3 Marle BGK................ 43
20 ev 20 X X AGN X 60 80
2 MRI saturation rate reconnection Navier-Stokes 2.2.9 Navie-Stokes [2] [22] [23] Navier-Stokes Boltzmann
3 Chapman Enskog 0 Knudsen [24] 0 Euler Navier-Stokes Burnett [25] [26] Knudsen Chapman-Enskog Knudsen Knudsen Burnett Chen [27] [28] BGK [8] Navier-Stokes Knudsen Knudsen Grad [8] [9] moment method Knudsen Hilbert moment moment 3 moment expansion leading order Navier-Stokes 5 moment moment 5 Knudsen moment [29] [30] [3] [32] Knudsen Grad moment Levermore [33] [34] Eckart [] Landau [2]
4 [3] Chapman-Enskog Chapman-Enskog Israel Stewart Grad moment [4] moment B moment BGK BGK BGK [2] [8] BGK moment [9] BGK BGK review BGK BGK BGK Israel-Stewart
5
7 2 review Boltzmann review r N /3 N : d Nd 3 (d/ r) 3 k B = 2. review review - [5] Mihalas [6]
8 2 2.. f(t, q, p) q q p dµ = dqdp fdµ. f(t, q, p) q q p fdµ = fdqdp f q r p p 2. f(t, r, p)d 3 p = n(t, r) (2.) ndv dv Boltzmann dv 3. dv d 3 dv L 3 d L 2
2. 9 d r L r L n r L n n n 2..2 p dp dp dp dp p p, p p, p p, p p, p w(p, p ; p, p ) fdp f dp dp dp (2.2) w(p, p ; p, p ) p, p p, p fdp w dp f f(t, r, p) f f(t, r, p ) w 4. dσ dσ = w(p, p ; p, p ) dp dp (2.3) v v w w(p, p ; p, p )dp dp = w(p, p ; p, p )dp dp (2.4) Ŝ
0 2 S in S nk = δ ik (2.5) n i = k S ni 2 = (2.6) n S ni 2 i n S in S nk = δ ik (2.7) n i = k S in 2 = (2.8) n 2 n = i S ni 2 = S in 2 (2.9) n i n i w 2..3 Boltzmann f f f t + f t + q (f q) + (fṗ) = 0 (2.0) p ( q f q + ṗ f p ) + f ( q q + ṗ ) = 0 (2.) p
2. Hamilton q = H p, ṗ = H q (2.2) q q = 2 H q p = ṗ p (2.3) (2.) 0 D Dt f = f ( t + q f ) q + ṗ f = 0 (2.4) p D/Dt f t ( ) f f + v f + ṗ p = t coll (2.5) ( f/ t) coll f f (x, v) 2 p p p, p p, p p p p dv dv dp w(p, p ; p, p )ff d 3 p d 3 p d 3 p (2.6) p, p p, p dv dp w(p, p ; p, p )f f d 3 p d 3 p d 3 p (2.7) dv dp (w f f wff )d 3 p d 3 p d 3 p (2.8)
2 2 w w(p, p ; p, p ), w w(p, p ; p, p ) (2.9) ( ) f = (w f f wff )d 3 p d 3 p d 3 p (2.20) t coll 2 (2.4) ( ) f = w (f f ff )d 3 p d 3 p d 3 p (2.2) t coll 0 (ṗ = 0) f t + v f = w (f f ff )d 3 p d 3 p d 3 p (2.22) (2.3) ( ) f = v rel (f f ff )dσd 3 p (2.23) t coll d 2..4 H f S = f ln e f dv d3 p (2.24) ( ds dt = f ln e ) dv d 3 p = ln f f t f t dv d3 p (2.25)
2. 3 (2.25) f t ( ) f f = v f F p + t coll (2.26) (2.25) ln f [ v f F f ] [ dv d 3 p = v + F p ] ( f ln f ) dv d 3 p (2.27) p e f = 0 0 ( ) ds f dt = ln f dv d 3 p (2.28) t 2.. q p ( ) f φ(p) dp (2.29) t φ p (2.20) ( ) f φ(p) dp = φ w(p, p ; p, p t )f f d 4 p φ w(p, p ; p, p )ff d 4 p (2.30) coll d 4 p = dpdp dp dp 4 p 2 p p p p ( ) f φ(p) dp = t coll coll coll (φ φ )w f f d 4 p (2.3) 2 p p p p /2 w ( ) f φ(p) dp = (φ + φ φ φ t coll 2 )w f f d 4 p (2.32) φ m p (v u) 2 /2 0 φ = ( ) f dp = t w (f f ff )d 4 p = 0 (2.33) coll
4 2 (2.32) (2.28) ds dt = 2 w f f ln f f d 4 pdv = ff 2 w ff x ln x d 4 pdv (2.34) d 4 p x = f f /ff 0 (2.33) /2 ds dt = w ff (x ln x x + ) d 4 pdv (2.35) 2 x > 0 x = 0 w f f H ds dt 0 (2.36) (2.35) d 4 p (2.2) molecular chaos ff v rel dσd 3 p d 3 pdv dt [5] 2..5 (2.22) p f(p) f(p) f(p )f(p ) f(p)f(p ) = 0 (2.37) log ln f(p ) + ln f(p ) = ln f(p) + ln f(p ) (2.38)
2. 5 ln f(p) 2 ln f(p) ln f(p) = α + βɛ (2.39) α β ɛ f (2.) β ρ f 0 = [ m(2πrt ) exp ɛ ] 3/2 T (2.40) R = m (2.4) m ρ = mn T Maxwell-Boltzmann f Maxwell-Boltzmann Boltzmann (2.35) 0 review - [7] 2..6 Boltzmann Boltzmann 5. l l nσ (2.42)
6 2 d σ d 2 n r 3 l r d l r ( r ) 2 l r (2.43) d L l L l 2. 6. d 3 vvf(t, r, v) = u : (2.44) d 3 v 2 (v u)2 f(t, r, v) = 3 ρrt ɛ : (2.45) 2 n u ɛ Boltzmann ( ) f f t + (vf) = t coll (2.46) m p (v u) 2 /2 d 3 v (2.32) φ m p (v u) 2 /2 0 Boltzmann m p (v u) 2 /2 p ρ + (ρu) = 0 t (2.47) t ρu α + Π αβ x β = 0 (2.48) n ɛ + q = 0 t (2.49) Π αβ q Π αβ = mv α v β fd 3 p (2.50) q = 2 (v u)2 vfd 3 p (2.5)
2. 7 f Π αβ q l L l L L/ v 0 f 0 (t, x) [ ] ρ(t, x) f 0 (t, x, v) = exp (v u(t, x))2 m(2πrt (t, x)) 3/2 2RT (t, x) (2.52) ρ u T f Boltzmann 0 0 Boltzmann Π αβ q u K v v = v + u Π αβ Π αβ = ρ v α v β = ρ (u α + v α)(u β + v β) = ρ(u α u β + v αv β ) (2.53) K v αv β = 3 v 2 δ αβ = T m δ αβ (2.54) v 2 = 3T/m Π αβ nt = P Π αβ Π αβ = ρu α u β + δ αβ P (2.55) P Boltzmann q K v 2 mv2 = 2 mv 2 + mu v + 2 mu2 (2.56) q [ mu 2 q = nu + m ] ( ) ρu 2 2 3 v 2 + ɛ = u 2 + h (2.57)
8 2 h = P + n ɛ Navier-Stokes review [2] 2..7 Chapman-Enskog Navier-Stokes f f 0 f l/l f f 0 ( ) 2 l l f = f 0 + f L + f 2 + (2.58) L (2.52) ρ u T 0 L f f 0 f 0 matching condition f 0 m(f f 0 )d 3 p = 0, v α (f f 0 )d 3 p = 0, 2 mv2 (f f 0 )d 3 p = 0 (2.59) T Maxwell-Boltzmann matching condition f Boltzmann l/l Boltzmann l l/l (2.23) f v rel dσ /l l/l f
2. 9 l D Dt f n ( ) f0 0 = t l D ( ) Dt f f 0 = t ( ) n ( ) l fn = L t D/Dt coll coll coll (2.60) l (2.6) L ( ) n l (2.62) L matching condition f 0 l/l f n f 0 f n f 0 ρ u T Boltzmann Chapman-Enskog f n Chapman-Enskog f f p f = f 0 + f, f = f 0 ɛ χ(p) = T f 0χ (2.63) χ matching condition mχd 3 p = 0, v α χd 3 p = 0, 2 mv2 χd 3 p = 0 (2.64) collision f 0 matching condition f 0 f 0 ρ u T f 0 Chapman-Enskog f 0 ( ) f0 0 = (2.65) t (2.52) f 0 u = 0 { f 0 ɛ cp T v T + T T [ vα v β R coll δ αβ ] } ɛ V αβ c v f 0 I(χ) (2.66) T
20 2 ɛ v2 2R, V αβ 2 ( uα + u ) β x β x α (2.67) I(χ) ( ) f = f 0 I(χ) (2.68) t coll T I(χ) = w f 0 (χ + χ χ χ )dp dp dp (2.69) T V ρ + (ρu) = 0 t (2.70) u + u u = P t ρ (2.7) s + u s = 0 t (2.72) c v dt P dρ = T ds ρ2 (2.73) P = ρrt (2.74) (2.66) l/l f f 2 f Navier-Stokes 2..8 (2.66) ɛ c p T T v T = I(χ) (2.75) χ = g T (2.76) g χ (2.75) T ɛ c p T T v = I(g) (2.77)
2. 2 matching condition(2.64) f 0 gdp f 0 ɛgdp 0 matching condition f 0 vgdp = 0 (2.78) χ (2.5) f = f 0 + δf q = T 2 (v u)2 vf 0 χdp = T 2 (v u)2 vf 0 (g T )dp (2.79) T q α = κ αβ (2.80) x β κ αβ = T 2 (v u)2 f 0 v α g β dp (2.8) κ αβ δ αβ κ αβ = κδ αβ, κ = κ αα 3 (2.82) q = κ T (2.83) κ = 3T 2 (v u)2 f 0 v gdp (2.84) H f g κ Chapman-Enskog l ( ) f v t l (f f 0) = vf 0 l T χ (2.85) coll (2.66) (2.68) I v l χ (2.86)
22 2 (2.75) g l (2.87) (2.66) f f 0 l/l Chapman-Enskog g κ κ cnl v (2.88) c l l 2..9 (2.66) Π αβ = P δ αβ + ρu α u β σ αβ (2.89) ( σ αβ = 2η V αβ ) 3 δ αβ u + ζδ αβ u (2.90) (2.90) shear viscosity bulk viscosity (2.66) mv α v β (V αβ ) ( mv 2 3 δ αβ u + 3 ɛ ) u = I(χ) (2.9) c v shear viscosity (2.9) u = 0 m (v α v β 3 ) δ αβv 2 V αβ = I(χ) (2.92) trace 0 χ = g αβ V αβ (2.93) g αβ V αβ trace 0 δ αβ χ g αβ trace 0 χ
2. 23 (2.92) V αβ m (v α v β 3 ) δ αβv 2 = I(g αβ ) (2.94) g αβ matching condition g αβ (2.50) σ αβ = m v α v β f 0 χdp = η αβγδ V γδ (2.95) T η αβγδ = m v α v β f 0 g γδ dp (2.96) T η αβγδ {α, β} {γ, δ} {γ, δ} 0 δ αβ η αβγδ = η [δ αγ δ βδ + δ αδ δ βγ 23 ] δ αβδ γδ (2.97) η = m v α v β f 0 g αβ dp (2.98) 0T σ αβ = 2ηV αβ η η η m vnl (2.99) κ nc η mn vl (2.00) bulk viscosity (2.9) 0 ( mv 2 χ 3 ɛ ) u = I(χ) (2.0) c v χ = g u (2.02)
24 2 shear viscosity (2.0) u mv 2 3 ɛ c v = I(g) (2.03) σ αβ shear viscosity ζδ αβ u ζ = m v 2 f 0 gdp (2.04) 3T ɛ = mv 2 /2 c v = 3/2 (2.0) 0 (2.03) I(g) = 0 g = 0 (2.04) ζ = 0 bulk viscosity 0 review [5] Mihalas [6] 2..0 moment Chapman-Enskog f l L l/l l/l l/l l/l f f Hermite polynomials Grad [8] [9] (2.52) [ ρ(t, x) c 2 ] f 0 (t, x, v) = exp m(2πrt (t, x)) 3/2 2RT (t, x) (2.05) c v u (2.06) matching condition f f(t, x, v) f 0 (t, x, v) ρ(t, x) u(t, x) T (t, x) a (n) (t, x) ρ u T a (n) moment ρ
2. 25 u T a (n) moment a (m) a (n) = 0 moment moment ξ = c RT (2.07) g = (RT )3/2 f (2.08) n N u T g gdξ = (2.09) ξgdξ = 0 (2.0) ξ 2 gdξ = 3 (2.) ] g 0 = exp [ ξ2 (2π) 3/2 2 (2.2) g Hermite polynomials g(ξ, x, t) = g 0 (ξ) = g 0 (ξ) f n! a(n) i n=0 (x, t)h (n) i (ξ) (2.3) ( a (0) i H (0) + a () i H () i + 2! a(2) ij H(2) ij + f(ξ, x, t) = f 0 (ξ) n! a(n) i n=0 (x, t)h (n) i (ξ) (2.4) Hermite polynomials H (n) i ξ n a (n) i n i = (i i n ) Cartesian Hermite polynomials a (n) i = gh (n) i d 3 ξ = fh (n) i d 3 v (2.5) n )
26 2 H (n) i H (n) i H (0) = (2.6) H () i H (2) ij = ξ i = ξ i ξ j δ ij H (3) ijk = ξ iξ j ξ k (ξ i δ jk + ξ j δ ik + ξ k δ ij ) H (4) ijkl = ξ iξ j ξ k ξ l (ξ i ξ j δ kl + ξ i ξ k δ jl + ξ i ξ l δ jk + ξ j ξ k δ il + ξ j ξ l δ ik + ξ k ξ l δ ij ) +(δ ij δ kl + δ ik δ jl + δ il δ jk ) H (m) j n = m j i ξ i ± a (n) i a (0) = (2.7) a () i = 0 a (2) ij a (3) ijk = = p ij p S ijk p RT a (4) ijkl = Q ijkl prt p (p ijδ kl + p ik δ jl + p il δ jk + p jk δ il + p jl δ ik + p kl δ ij ) +(δ ij δ kl + δ ik δ jl + δ il δ jk ) a (0) f = f 0 0 a (3) a (3) i = a (3) irr = 2q i p RT (2.8) Hermite moment p ij q i a (n) i (2.3) a (n) i
2. 27 a (2) ij a (2) ij t a (2) ij + u r + a (2) ir x r u j + a (2) u i rj + (a (2) ij + δ ij ) x r x r RT DRT Dt (2.9) + RT a(3) ijr x r + RT n a(3) ijr N + 3 x r 2RT a(3) ijr RT x r + u i + u j = J (2) ij x j x i D/Dt J (n) ij J (n) i = N 2 r,s=0 β (nrs) ijk a (r) j a (s) k (2.20) i = (i i n ), j = (j j r ), k = (k k s ) β (nrs) ijk = ( r!s! m B θ, ξ ) ξ g 0 (ξ)g 0 (ξ )H (r) j (ξ)h (s) k (ξ )[H (n) i (ξ)]dθ dɛ dξ dξ RT (2.2) [φ] = φ + φ φ φ (2.22) B (2.20) J (n) a (n) i (2.9) a (2) i a (n) i a (2) i a (n) i U r 4 (2.20) Maxwell f a (n) i moment 3 moment 3 moment expansion bulk viscosity 4 moment f { f = f 0 + p ij 2pRT v iv j q )} i prt v i ( v2 5RT (2.23)
28 2 p ij q i p ij t + (u r p ij ) + 2 ( qi + q j 2 ) x r 5 x j x i 3 δ q r ij x r (2.24) u j u i + p ir + p jr 2 x r x r 3 δ u r ijp rs x ( s ui + p + u j 2 ) x j x i 3 δ u r ij + βnp ij = 0 x r q i t + (u r q i ) + 7 x r 5 q u r r + 2 x j 5 q u r r + 2 x i 5 q u r i (2.25) x r + RT p ir x r + 7 2 p ir RT x r + 5 2 p RT x i + 2 3 βnq i = 0 p ir N P rs x s P ij β β(rt ) = 2 2π x 6 e x2 /2 5 0 { } m B(θ, x 2RT ) sin 2 θ cos 2 θ dθ dx (2.26) β (2.24) (2.25) p ij + βnp ij = 0 (2.27) t q i t + 2 3 βnq i = 0 (2.28) p ij q i /βn β (2.20) β v rel σ /βn (2.24) (2.25) Chapman-Enskog 2..7 f ( ui p + u j 2 ) x j x i 3 δ u r ij + βnp () ij = 0 (2.29) x r 5 2 p RT x i + 2 3 βnq() i = 0 (2.30) f p () ij q() i Chapman-Enskog
2. 29 λ = p βn = T nl v β (2.3) κ = 5 pr 4 βn = 5 RT 4 β cnl v (2.32) β λ η m (2.24) (2.25) /βn (2.24) (2.25) p ij + A ij + βnp ij = 0 (2.33) t q i t + B i + 2 3 βnq i = 0 (2.34) A ij B i /βn A ij B i source term (2.27) (2.28) p ij = A ij βn q i = 3 B i 2 βn (2.35) (2.36) p ij q i /βn A ij B i Chapman-Enskog /βn Chapman-Enskog (2.24) q i u i Landau- Lifshitz moment 2..7 Chapman-Enskog Navier-Stokes moment (2.9)
30 2 (2.24) (2.25) moment B moment moment moment Chapman-Enskog l/l moment moment 3-moment moment moment Hermite (2.3) moment moment free streaming moment free streaming moment moment moment 3 δt ρr + δq = 0 2 t (2.37) τ δ q + δq = κ δt t (2.38) τ κ exp[ nt + ikx] δt δq n 2 τ n + Kk2 = 0 (2.39) K = κ/τc v n n = 2τ [ ± 4τKk 2 ] (2.40)
2. 3.0 n 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8.0 k 2. K = τ = k = 0 n = 0 n = /τ couple couple BGK moment B
32 2 2.2 metric (+ ) c review de Groot [0] C. Cercignani [] 2.2. f(x, p) f f fd 3 xd 3 p f d 3 xd 3 p p d 4 p δ D (p µ p µ m 2 ) d 4 p δ D (p µ p µ m 2 ) (2.4) d 4 p δ D (p µ p µ m 2 ) = d 4 p 2E δ D(p 0 E) (2.42) p 0 >0, E= p 2 +m 2 d 3 p = 2p 0 p0 = p 2 +m 2 δ D (φ(x)) = i φ (a i ) δ D(x a i ) (2.43) (2.42) d 3 p p 0 = dp xdp y dp z p 0 (2.44) p 0 = p 2 +m 2
2.2 33 K K V K ( d d 3 xd 3 p = γd 3 x p 0 3 p ) = d 3 x d 3 p (2.45) d 3 xd 3 p f f N µ 7. N µ = (n, j) p 0 n j 8. T µν d 3 p p 0 pµ f : (2.46) d 3 p p 0 pµ p ν f : (2.47) 9. d S µ 3 p p 0 pµ f(ln f ) : (2.48) 2.2.2 Boltzmann f dµ = d 3 xd 3 p N(t) N = f(t + t, x + v t, p + F t)dµ(t + t) f(t, x, p)dµ(t) = 0 (2.49) v = p/p 0 F t dµ
34 2 dµ(t + t) = J dµ(t) (2.50) J = ( x (t + t), x 2 (t + t),, p 3 (t + t) ) (x (t), x 2 (t),, p 3 (2.5) (t)) t J J = + F i p i t + O [ ( t) 2] (2.52) f (2.49) [ N f t = f + vi t x i + F i f p i + f F i p i = [ f t f + vi x i + (ff i ) p i ] dµ(t) = 0 ] dµ(t) (2.53) t s [ DN f Ds = γ t [ = p0 f m t [ p ν = m f + vi x i + (ff i ) p i f + vi x i + (ff i ) p i f x ν + p0 m (ff i ) p i ] dµ(t) (2.54) ] dµ(t) ] dµ(t) = 0 ( p F K µ = m, p 0 ) m F (2.55) p µ K µ = 0 (2.56) (2.54) / p x p 0 p p0 p + p = p p 0 p 0 + p (2.54) p 0 (ff i [ ( ) ) m p i = p 0 fk p p 0 p 0 p 0 + ] p 0 p (fk) = fk0 p 0 = fkµ p µ + p (fk) (2.57) (2.58)
2.2 35 [ ] DN p ν Ds = f m x ν + (fkν ) p ν dµ(t) = 0 (2.59) 0 p (2.23) v rel dσ n n 2 σ = dσ v rel dv dt dν dν = σv rel n n 2dV dt (2.60) dν dν dν = An n 2 dv dt (2.6) A σv rel dv dt dν An n 2 n n 0 ndv n = γn 0 = p0 m n 0 (2.62) An n 2 Ap 0 p 0 2 Ap 0 p 0 2 p0 σv rel A p0 p 0 2 p µ p µ 2 p 0 = A p 0 2 p 0 p0 2 p (2.63) p 2
36 2 p 0 2 = m 2 p 2 = 0 σv rel A = σv rel A A = σv rel p µ p µ 2 p 0 p0 2 (2.64) v rel p µ p µ 2 p µ p µ 2 = m v 2 rel m 2 (2.65) v rel = m2 m2 2 (p µ p µ 2 )2 (2.66) p p 2 dν n = d 3 p f ( ) f t coll dµ(t) = dµ(t) (f f ff ) p µp µ p 0 p 0 = m ( ) f p 0 dµ(t) s coll v rel σd 3 p (2.67) p ν f x ν + ) m (fkν p ν = p 0 (f f ff ) p µp µ p 0 p 0 v rel σd 3 p (2.68) p 0 f t + pi f p 0 x i + (ff i ) p i = (f f ff ) p µp µ p 0 p 0 v rel σd 3 p (2.69) 2.2.3 H (2.68) H
2.2 37 2.2. µ S µ d µ S µ 3 p = p 0 pµ µ [f(ln f )] (2.70) d 3 p = ln f pµ µ f (2.68) p 0 d 3 p m p 0 ln f (fkµ ) d 3 p p µ p 0 ln f(f f ff )p µ p µ v relσ d3 p p 0 (2.7) d 3 p m p 0 ln f (fkµ ) p µ = m d 4 p p µ δ D(p 2 m 2 )θ(p 0 )f(ln f )K µ (2.72) θ(p 0 ) 0 φ(f f ff )p µ p µ v relσ d3 p d 3 p p 0 p 0 = (φ + φ φ φ 4 )(f f ff )p µ p µ v relσ d3 p d 3 p p 0 p 0 (2.73) µ S µ = ln ff ( 4 f f ff ) f f f f p µ p µ v relσ d3 p d 3 p p 0 p 0 (2.74) ( x) ln x x > 0 x = 0 µ S µ 0 (2.75) (2.68)
38 2 2.2.4 (2.68) p 0 0 0 f 0 f 0f 0 f 0 f 0 = 0 (2.76) ln ln f 0 + ln f 0 ln f 0 ln f 0 = 0 (2.77) ln f 0 ln f 0 = α + β µ p µ (2.78) f 0 n f 0 = [ 4πm 2 T K 2 (ζ) exp u µp µ T ] (2.79) ζ = m T n T u µ u µ u µ u µ = T Maxwell-Jüttner (2.74) 0 ζ u µ
f f f 2.2 39 0 0.2 0.4 0.6 0.8 v / c 2.2 ζ = 00 0 0.2 0.4 0.6 0.8 v / c 2.3 ζ = 0.99 0.98 0.97 0.96 0.95 0.94 0 0.2 0.4 0.6 0.8 v / c 2.4 ζ = 0.0 c ζ = 00 Maxwell ζ =
40 2 ζ = 0.0 N µ 2.2. d N µ 3 p n d 3 p = p 0 pµ f 0 = 4πm 2 T K 2 (ζ) p 0 pµ exp [ u µp µ T u µ ] (2.80) N µ = n u µ (2.8) u µ p = p d 3 p = p 2 dpdω = pp 0 dp 0 dω (2.82) z = p0 T (2.83) (2.80) (2.8) u µ u µ n n d 3 p = 4πm 2 T K 2 (ζ) p 0 p0 exp [ u µp µ ] (2.84) T n = ζ 2 dzz z K 2 (ζ) 2 ζ 2 e z (2.85) K n (ζ) = 2n n! (2n)! ζ = 2n (n )! (2n 2)! ζ n dz(z 2 ζ 2 ) n /2 e z (2.86) ζ ζ n dzz(z 2 ζ 2 ) n 3/2 e z (2.87) ζ (2.85) n = n N µ = nu µ 2.2. T µν T µν = d 3 p p 0 pµ p ν f 0 = n d 3 p 4πm 2 T K 2 (ζ) p 0 pµ p ν exp [ u µp µ ] T (2.88) u µ u ν η µν T µν = neu µ u ν P γ µν (2.89)
2.2 4 γ µν = η µν u µ u ν 2.2.5 (2.89) e P f 0 d ne = T µν 3 p u µ u ν = p 0 (pµ u µ ) 2 f 0 (2.90) P = 3 T µν γ µν = d 3 p 3 p 0 pµ p ν γ µν f 0 (2.9) N µ P = nt (2.92) e = m K (ζ) + 3T K 2 (ζ) (2.93) (2.92) recurrence relation K n+ (ζ) = 2n K n(ζ) ζ + K n (ζ) (2.94) (2.93) e = m K 3(ζ) K 2 (ζ) T (2.95) (2.92) (2.95) h = e + P n = mk 3(ζ) K 2 (ζ) (2.96) c P = ( ) ( ) h e, c V = T P T V (2.97) c P = Γ Γ = d K 3 (ζ) dζ K 2 (ζ) (2.98) c V = c P (2.99) Γ recurrence relation d K n (ζ) dζ ζ n = K n+(ζ) ζ n (2.200)
42 2 (2.94) c P c P = Γ Γ = ζ2 + 5h T ( ) 2 h (2.20) T s d ns = S µ 3 p u µ = p 0 pµ u µ f(ln f ) (2.202) f 0 [ µ p µ ] u µ f 0 = exp (2.203) T (2.202) (2.85) (2.90) ns = n (e µ) + n (2.204) T µ = e + P n T s = h T s (2.205) µ T ζ ζ ζ massless K n (ζ) [ π 2ζ e ζ + 4n2 + (4n2 )(4n 2 ] 9) 8ζ 2!(8ζ 2 + ) (ζ ) (2.206) (2.95) (2.96) (2.20) e m + 3 2 T + 5 8 h m + 5 2 T + 5 Γ 5 3 5 3 T 2 m + (2.207) T 2 m + (2.208) 8 T m + (2.209)
2.2 43 lim ζ 0 ζn K n (ζ) = 2 n (n )! (2.20) e = 3T (2.2) h = 4T (2.22) Γ = 4 3 (2.23) P = nt (2.2) P = ne 3 (2.24) trt µν = 0 (2.25) 2.2.5 Eckart Landau-Lifshitz f time like u µ 0. u µ : (2.26) u µ u µ = (2.27). γ µν : (2.28) γ µν = g µν u µ u ν (2.29) g Minkowski η µν 2.2.
44 2 T µν = η µρ η νσ T ρσ (2.220) = (γ µρ + u µ u ρ )(γ νσ + u ν u σ )T ρσ = (T ρσ u ρ u σ )u µ u ν + (u ρ T ρσ γ σµ )u ν + (u ρ T ρσ γ σν )u µ + {(γ µρ γ νσ 3 ) γµν γ ρσ + 3 } γµν γ ρσ T ρσ f 0 T µν = neu µ u ν (2.22) + {(q µ + h E γ µν N ν )u ν + u µ (q ν + h E γ νµ N ν )} + p µν (P + Π)γ µν N µ N µ = nu µ + ν µ (2.222) ν µ u µ u µ ν µ = 0 (2.223) 2. n N µ u µ : (2.224) p µν (γ µρ γ νσ 3 ) γµν γ ρσ T ρσ : (2.225) P + Π 3 γµν γ ρσ T ρσ : + bulk viscosity (2.226) q µ u ρ T ρσ γ σµ h E γ µν N ν (2.227) = u ρ T ρσ γ σµ h E ν µ : e n T ρσu ρ u σ : (2.228) p µν traceless T 00 T µν u µ u ν
2.2 45 u µ u µ (2.22) (2.222) T µν T 0µ N µ Eckart [] 3. u µ E : Eckart velocity (2.229) N µ = nu µ E (2.230) T µν = neu µ E uν E (2.23) + (q µ u ν E + u µ E qν ) + p µν (P + Π)γ µν N µ u µ u µ E f u µ E N µ N µ = d 3 p N µ n p 0 pµ f (2.232) Landau-Lifshitz [2] 4. u µ L : Landau Lifshitz velocity (2.233) N µ = n L u µ L + νµ (2.234) T µν = n L e L u µ L uν L (2.235) + p (µν) L (P L + Π L )γ µν T µν u µ u µ L f u µ L T µν u Lν ulρ T ρσ T στ u τ L = u Lν ulρ T ρσ T στ u τ L d 3 p p 0 pµ p ν f (2.236) Eckart Landau-Lifshitz T µν u µ E uµ L u µ L = uµ E + U µ (2.237)
46 2 U µ u Lµ u µ L = = (uµ E + U µ )(u Eµ + U µ ) + 2U µ u Eµ (2.238) U µ u µ E u Lµ u µ E = (2.239) N µ = nu µ E = n Lu µ L + νµ (2.240) u µ L (2.239) n = n L (2.24) Eckart Landau- Lifshitz (2.237) (2.240) U µ = νµ n (2.242) T µν = neu µ E uν E + (q µ u ν E + u µ E qν ) + p µν (P + Π)γ µν (2.243) = n L e L u µ L uν L + p (µν) L (P L + Π L )γ µν (2.237) P L = P, e L = e, Π L = Π, p (µν) L = p µν, ν µ = h E q µ (2.244) h E = e + P/n Eckart Landau-Lifshitz u µ E = uµ L nh E q µ (2.245) q µ Eckart Landau-Lifshitz
2.2 47 j = ρv (2.246) (2.207) (2.22) (2.222) 2.2.6 (2.68) p µ µ f = ( ) f s coll (2.247) p ν d 3 p/p 0 (2.73) µ N µ = 0 (2.248) µ T µν = 0 (2.249) (2.22) (2.222) Eckart Landau-Lifshitz µ u µ µ = η µν ν = u µ u ν ν + γ µν ν u µ D + µ (2.250) D u µ µ = t µ γ µν ν (2.25) = x i (2.252)
48 2 = u µ D u µ Lagrange µ (2.248) Dn = n µ u µ µ ν µ + ν µ Du ν (2.253) Eckart ν µ = 0 Dn = n µ u µ (2.254) Landau-Lifshitz ν µ = q µ /h E Dn = n µ u µ + µ q µ h E q µ h E Du ν (2.255) (2.22) γ ρ ν W µ u ν T νρ γ ρµ = q µ + h E ν µ (2.256) T µν (2.22) T µν = neu µ u ν + {W µ u ν + u µ W ν } + p µν (P + Π)γ µν (2.257) (2.256) nh E Du µ = µ P + µ Π γ ν µ σ p νσ (2.258) + (p µν Du ν γ ν µ DW ν W µ ν u ν W ν ν u µ ) u µ Eckart W µ = q µ nh E Du µ = µ P + µ Π γ ν µ σ p νσ (2.259) + (p µν Du ν γ ν µ Dq ν q µ ν u ν q ν ν u µ ) Landau-Lifshitz W µ = 0 nh E Du µ = µ P + µ Π γ ν µ σ p νσ (2.260) +p µν Du ν (2.22) u ν D(ne) = nh E µ u µ + p µν ν u µ µ W µ + 2W µ Du µ (2.26)
2.2 49 (2.255) Dn nde = P µ u µ Π µ u µ + p µν ν u µ µ W µ + e µ ν µ (2.262) +(2W µ eν µ )Du µ (2.26) (2.262) Eckart D(ne) = nh E µ u µ + p µν ν u µ µ q µ + 2q µ Du µ (2.263) nde = P µ u µ Π µ u µ + p µν ν u µ µ q µ + 2q µ Du µ (2.264) Landu-Lifshitz D(ne) = nh E µ u µ + p µν ν u µ (2.265) nde = P µ u µ Π µ u µ + p µν q µ ν u µ e µ + e q µ Du µ h E h E (2.266) P dn De + P Dn = (2.262) (2.253) De Dn n(de+p Dn ) = Π µ u µ +p µν ν u µ µ q µ ν µ µ h E +(2q µ +h E ν µ )Du µ (2.267) 0 Eckart n(de + P Dn ) = Π µ u µ + p µν ν u µ µ q µ + 2q µ Du µ (2.268) Landau-Lifshitz n(de + P Dn ) = Π µ u µ + p µν ν u µ + qµ h E µ h E q µ Du µ (2.269) q µ p µν Π (2.253) (2.258) (2.262) Dn = n µ u µ (2.270) nh E Du µ = µ P (2.27) nde = P µ u µ (2.272)
50 2 u µ exp( ik µ x µ ) = exp( iωt + ikx) (2.273) δu i = δu x ω kn 0 0 δn 0 ωnh E 0 k δu x 0 kp ωnc V 0 δt T 0 n δp iωδn = iknδu x (2.274) iωnh E δu x = ikδp (2.275) iωnδe = ikp δu x (2.276) δe = c V δt (2.277) δp = nδt + T δn (2.278) = 0 0 0 0 (2.279) n 2 ω ( h E c V ω 2 c P T k 2) = 0 (2.280) ω = ±C s k (2.28) Γ C s = T (2.282) h E m h E 2.2.4 (2.208) (2.209) h E m Γ 5 3 5T/3m (2.22) (2.23) h E 4T Γ 4 3 c/ 3
2.2 5 2.2.7 Chapman-Enskog Chapman-Enskog Chapman-Enskog 2..7 2.2.5 Eckart Landau-Lifshitz Eckart f f(t, x, p) f 0 (t, x, p) + f (t, x, p) = f 0 (t, x, p)[ + φ(t, x, p)] (2.283) f l L l/l f 0 (t, x, p) n f 0 (t, x, p) = [ 4πm 2 T K 2 (ζ) exp u µp µ T ] (2.284) n T u i matching condition Eckart d 3 p p 0 pµ f 0 φ = 0 (2.285) d 3 p p 0 (pµ u µ ) 2 f 0 φ = 0 (2.286) (2.285) f 0 n u µ (2.286) e Landau-Lifshitz u µ d 3 p p 0 pµ f 0 φ = 0 (2.287) u µ d 3 p p 0 pµ p ν f 0 φ = 0 (2.288) 0 f 0 f
52 2 p µ µ f 0 = I[φ] (2.289) I[φ] (2.270) (2.27) (2.272) f 0 { T [ ( u c V 3 ζ2 c V (G 2 ζ 2 4Gζ ζ 2 µ ) p µ ) T ( u 3 (ζ2 + 5Gζ ζ 2 G 2 µ ) ] 2 p µ 4) ν u ν p µp ν T T µ u ν + p ( µ T 2 (p νu ν h E ) µ T T )} µ P = I[φ] nh E G (2.290) G = K 3(ζ) K 2 (ζ) = h E m (2.29) φ φ = [ a 0 + a u µ p µ + a 2 (u µ p µ ) 2] ν u ν (2.292) [ + p µ (a 3 + a 4 p ν u ν ) µ T T ] µ P + a 5 p µ p ν µ u ν nh E φ (2.290) matching condition(2.285) (2.286) a 0 a a 2 a 3 a 4 a 5 φ bulk viscosityπ q µ p µν 2.2.5 traceless (2.292) Π = µ µ u µ (2.293) [ q µ = κ µ T T ] µ P (2.294) nh E p µν = 2η µ u ν (2.295) bulk viscosity e = c V T m T
2.2 53 P ( µ ) ( ) he d = T nt 2 dt + dp nt (2.296) µ q µ = κ nt 2 ( µ µ ) h E T (2.297) Y (P, T ) = 0 dv j [ e (E j+p V )/T = exp µ ] T (2.298) a i [0] [] µ = P 2 T m 2 I (20G + 3ζ 3G 2 ζ 2Gζ 2 + 2G 3 ζ 2 ) 2 ( 5Gζ ζ 2 + G 2 ζ 2 ) 2 (2.299) κ = 3P 2 m 2 (ζ + 5G G 2 ζ) 2 I I 2 (2.300) η = 30P 2 T m 2 G 2 2I 6I 2 + 3I 3 (2.30) I I 2 I 3 αr n n µ = T (20G + 3ζ 3G 2 ζ 2Gζ 2 + 2G 3 ζ 2 ) 2 ζ 4 K 2 (ζ) 2 64π σ ( 5Gζ ζ 2 + G 2 ζ 2 ) 2 (2K 2 (2ζ) + ζk 3 (2ζ)) κ = 3 (ζ + 5G G 2 ζ) 2 ζ 4 K 2 (ζ) 2 64πσ (ζ 2 + 2)K 2 (2ζ) + 5ζK 3 (2ζ) η = 5 T ζ 4 K 3 (ζ) 2 64π σ (2 + 5ζ 2 )K 2 (2ζ) + (3ζ 3 + 49ζ)K 3 (2ζ) (2.302) (2.303) (2.304) bulk viscosity
54 2 (2.302) (2.303) (2.304) ζ 2.2.4 µ 25 mt 256σ π ζ 2 κ 75 mt 256mσ π η 5 mt 64σ π ( 83 6 ( + 3 6 ) ζ + ) ζ + ) ( + 25 6 ζ + (2.305) (2.306) (2.307) bulk viscosity ζ µ [ ( T 49 288π σ ζ4 + 2 + 76 ln ζ ) 2 + 6γ κ ( ) 2πσ 4 ζ2 + η 3 ( T + ) 0π σ 20 ζ2 + ] + bulk viscosity (2.308) (2.309) (2.30) 2.2.8 moment Chapman-Enskog moment moment bulk viscosity 3 moment Grad 3-moment expansion bulk viscosity 4 moment Eckart Cercignani [] moment ψ =, p µ, p µ p ν d 3 p/p 0 µ N µ = 0 (2.3) µ T µν = 0 (2.32) µ T µνρ = P νρ (2.33)
2.2 55 (2.33) d T µνρ 3 p p 0 pµ p ν p ρ f (2.34) P νρ d 3 p d 3 p 2 p 0 p 0 (p ν p ρ + p ν p ρ pν p ρ p ν p ρ )ff p µ p µ v relσ (2.35) n, u µ, T, Π, p µν, q µ 4 moment T µνρ 4-moment moment moment f 4-moment f T µνρ u µ s 2.2. f s uµ n 4-moment 4 constraint d 3 p p 0 p µf(ln f ) (2.36) N µ u µ = u µ d 3 p p 0 pµ f (2.37) T µν u µ = u µ d 3 p p 0 pµ p ν f (2.38) T µν ρ u ρ = u ρ d 3 p p 0 p µ p ν p ρ f (2.39) N µ T µν moment constraint constraint F = uµ d 3 p n p 0 p µf(ln f ) + λ ( d + λ ν T µν 3 p u µ u µ p 0 pµ p ν f ( N µ u µ u µ d 3 p ) p 0 pµ f ) + λ µν ( T µν ρ u ρ u ρ d 3 p p 0 p µ p ν p ρ f (2.320) ) f 4-moment moment f Euler-Lagrange f [ ] f = exp n(λ + λ µ p µ + λ µν p µ p ν ) (2.32) 4 λ λ µ λ µν 4 moment λ = λ E + λ NE, λ µ = λ E µ + λ NE µ, λ µν = λ NE µν (2.322)
56 2 f f = exp [ n(λ E + λ E µ p µ ) ] exp [ ] n(λ NE + λ NE µ p µ + λ NE µν p µ p ν ) (2.323) f 0 moment f { } f = f 0 n(λ NE + λ NE µ p µ + λ NE µν p µ p ν ) (2.324) λ NE µ λne µν uµ λ NE µ = λu µ + λ ν γµ ν (2.325) λ NE µν = Λu µu ν + ( 2 Λ ρ(γµu ρ ν + γνu ρ µ ) + Λ ρσ γµγ ρ ν σ ) 3 γρσ γ µν (2.326) f f = f 0 { nλ NE n( λu µ + λ ν γµ)p ν µ (2.327) n [Λu µ u ν + 2 Λ ρ(γ ρµu ν + γ ρνu µ ) + Λ ρσ (γ ρµγ σν 3 )] } γρσ γ µν p µ p ν ) f Πq µ p µν f d N µ 3 p d 3 p = p 0 pµ f = p 0 pµ f 0 (2.328) N µ u µ γ ν µ ( λ NE + λm G ) ( + Λm 2 + 3 G ) = 0 (2.329) ζ ζ λ µ γ µν + mgλ µ γ µν = 0 (2.327) T µν T µν 2.2.5 p µν = 2n 2 m 2 T G ζ Λ µν (2.330) [ ( Π = n 2 T λ NE + Gm λ + + 5 G ) ] m 2 Λ ζ [ ( q µ = n 2 T Gmγ µν λν + + 5 G ) ] m 2 γ µν Λ ν ζ ( λ NE G ) + (3 ζ λm Gζ ) + + Λm (5 2 Gζ 2 + 2ζ ) + G = 0
2.2 57 4 (2.329) (2.330) 4 f { Π f = f 0 + α 0 (α + α 2 u µ p µ + ζm ) P 2 u µu ν p µ p ν (2.33) ( q µ G + α 3 P m pµ ) m 2 u νp µ p ν + p } µν ζ P 2Gm 2 pµ p ν α 0 = 5Gζ ζ 2 + G 2 ζ 2 20G + 3ζ 3G 2 ζ 2Gζ 2 + 2G 3 ζ 2 (2.332) α = 5G + 2ζ 6G2 ζ + 5Gζ 2 + ζ 3 G 2 ζ 3 5Gζ ζ 2 + G 2 ζ 2 α 2 = 3ζ 6G + ζ G 2 ζ m 5Gζ ζ 2 + G 2 ζ 2 ζ α 3 = ζ + 5G G 2 ζ f T µνρ P µν T µνρ = (nc + C 2 Π)u µ u ν u ρ + 6 (nm2 nc C 2 Π)(η µν u ρ + η µρ u ν + η νρ u µ ) +C 3 (η µν q ρ + η µρ q ν + η νρ q µ ) 6C 3 (u µ u ν q ρ + u µ u ρ q ν + u ν u ρ q µ ) +C 4 (p µν u ρ + p µρ u ν + p νρ u µ ) (2.333) P µν = B (η µν 4u µ u ν ) Π + B 2 (u µ q ν + u ν q µ ) + B 3 p µν (2.334) C i C = m2 (ζ + 6G) (2.335) ζ C 2 = 6m 2ζ 3 5ζ + (9ζ 2 30)G (2ζ 3 45ζ)G 2 9ζ 2 G 3 ζ 20G + 3ζ 3G 2 ζ 2ζ 2 G + 2ζ 2 G 3 C 3 = m ζ + 6G G 2 ζ ζ ζ + 5G G 2 ζ C 4 = m (ζ + 6G) Gζ
58 2 B i 2.2.7 I i B = ζ ( ζ 2 5ζG + ζ 2 G 2 )I 3m 2 P 20G + 3ζ 3G 2 ζ 2ζ 2 G + 2ζ 2 G 3 (2.336) B 2 = ζ I I 2 3m 2 P ζ + 5G ζg 2 ζ 2I 6I 2 + 3I 3 B 3 = 30m 2 P G (2.33) C 2 2 DΠ + 2 (m2 + C )Dn ζ 2T nc DT 5C 3 µ q µ (2.337) + 6 (nm2 + 5nC ) µ u µ = 3B Π 5C 3 Dq µ [ (m 2 C ) µ n + ζ ] 6 T nc µ T C 2 µ Π C 4 ν p µν (2.338) 6 (nm2 + 5nC )Du µ = B 2 q µ C 4 Dp µν + 2C 3 µ q ν + 3 (nm2 nc ) µ u ν = B 3 p µν (2.339) ζ (2.337) (2.338) Dn Du µ DT Dn + n µ u µ = 0 (2.340) nh E Du µ = µ (P + Π) ν p µν Dq µ nc V DT = P µ u µ µ q µ (2.337) (2.338) C 2 2 DΠ n 2 (m2 + C ) µ u µ + ζ 2T C ζ 2 + 5ζG ζ 2 G 2 ( µ q µ + P µ u µ ) (2.34) 5C 3 µ q µ + 6 (nm2 + 5nC ) µ u µ = 3B Π 5C 3 Dq µ [ (m 2 C ) µ n + ζ ] 6 T nc µ T C 2 µ Π C 4 ν p µν (2.342) [ (nm 2 + 5nC ) µ (P + Π) ν p µν Dq µ] = B 2 q µ 6nh E (2.339) (2.340) (2.34) (2.342) 4-moment (2.339) (2.34) (2.342) p µν Π q µ (2.339) (2.34) (2.342) p µν Π q µ
2.2 59 0 Chapman-Enskog 2.2.9 Chapman-Enskog Grad moment Chapman-Enskog Navier-Stokes Chapman-Enskog (2.83) Chapman-Enskog Navier-Stokes c P T t + ( κ T ) (2.343) c P κ T t = χ T (2.344) χ κ/c P T = 0 T (t = 0, x) = const δ D (x) T (t, x) = const ) ( 2 πχt exp x2 4χt (2.345) t x = 0 Chapman-Enskog Chapman-Enskog
60 2 [3] causal cone causal cone Lorentz causal cone Chapman-Enskog 2.5 causal cone Lorentz [2] Grad moment 2..0 moment Chapman-Enskog Chapman-Enskog moment
6 3 2.2.9 moment Chapman-Enskog Chapman-Enskog BGK BGK Navier-Stokes BGK BGK BGK Eckart Marle [3] BGK
62 3 Landau-Lifshitz Anderson-Witting [4] BGK 3. Boltzmann 3.. BGK (2.22) A k Bhatnagar Gross Krook [8] BGK ( ) f = f(t, x, v) f 0(t, x, v) (3.) t coll τ f 0 (t, x, v) = ρ (v u) 2 e 2RT (3.2) m(2πrt ) d/2 f 0 (t, x, v) f local matching condition (f 0 f) ψ µ d 3 v = 0 (3.3) ψ µ = m (, v, 2 ) (v u)2 BGK f t = f(t, x, v) f 0(t, x, v) τ (3.4) f f 0 e t/τ τ f 0 τ mean flight time f f 0 v [9] A τ 2 0 2 0 BGK
3. Boltzmann 63 f0 f ψ µ d 3 v = 0 (3.5) τ ψ µ = m (, v, 2 ) (v u)2 matching condition(3.3) (3.5) τ v matching condition mean flight time τ v v C s 3..2 BGK E ( ) ( ) Df(t, x, v) f = Dt t + v f(t, x, v) = t coll (3.6) BGK ( ) f t coll = f(t, x, v) f 0(t, x, v) τ (3.7) f 0 (t, x, v) = ρ(t, x) (v u(t,x)) 2 e 2RT (3.8) m(2πrt (t, x)) 3/2 f 0 source term f matching condition f 0 f (f 0 f) ψ µ d 3 v = 0 (3.9) ψ µ = m (, v, 2 ) (v u)2 (3.0) f 0 (v) δf = f f 0, δf 0 = f 0 f 0, δτ = τ τ ( τ f 0 ) (3.)
64 3 f 0 ( ) t + v δf = δf δf 0 τ (3.2) τ τ τ δf = δ fe iω(t t 0)+ik x (3.3) (3.2) ( ) iω + ik v δf = τ τ δf 0 (3.4) δf 0 [ δf 0 = f δρ 0 + v δu + ρ 0 RT 0 ρ 0 f 0 = e v m(2πrt 0 ) 3/2 f 0 ( v 2 2RT 2 0 3 ) ] δt 2T 0 (3.5) 2 /2RT 0 (3.6) δρ δu δt matching condition δf ( ) iω + ik v δf(v) (3.7) τ = d 3 v f [ 0 (v) m + v ( v v 2 + m τ ρ 0 RT 0 2RT0 2 3 ) ( v 2 3 )] δf(v ) 2T 0 2RT 0 2 ωτ ω, 2RT0 τk k, v 2RT0 v (3.8) δf(v) = d 3 v K(v, v )δf(v ) (3.9) [ f K(v, v 0 (v) m ) + 2v v + m ( v 2 3 ) ( v 2 3 )] (3.20) iω + ik v ρ 0 T 0 2 2 iω + ikv 0 iω + ikv = 0 ω Fredholm 2 K(v, v )
3. Boltzmann 65 D v v 2 3/2 ω k E iω ( b δρ+ δρ + 2ib ) { k k δu b } πe b 2 Erfc(b) { ( b b 2 + } πe b 2) 2 Erfc(b) δt = 0 b b = iω k (3.2) (3.22) k v { ( k δu + ib b b 2 + } πe b 2) 2 Erfc(b) δt (3.23) ( i δρ + 2ib ) ( k k δu b πe Erfc(b)) b2 = 0 k v kv y { k } πe b2 Erfc(b) δu y = 0 (3.24) v 2 /2 3/2 (3.2) [ { 3k 2b( b 2 ) + (2b 4 b 2 + ) }] πe b2 Erfc(b) δt 2 + [ { 3k 2 b (b 2 ) }] πe b2 Erfc(b) δρ 2 [ { 2b b (b 2 ) } ] k δu πe b2 Erfc(b) 3 = 0, (3.25) k 3.2 δu δρ δt { ( b 2 b b 2 + } πe b 2) 2 Erfc(b) { ( 3 k 2b + 2 ) ( + 2 k { ( ( kb) + + 2b 2 2b = 0, { 3k + 2b 2 2b k ) ( b πe b2 Erfc(b)) k ( 2b( b 2 ) + (2b 4 b 2 + ) πe b2 Erfc(b) (3.26) ) ( b + ( b 2 ) πe Erfc(b)) } b2 } )}
66 3 k πe b2 Erfc(b) = 0 (3.27) ω δρ δu δt δf δf(v) = n C n f0 (v) iω n + ik v C n [ ( δρωn + 2v δu ωn + v 2 3 ) ] δtωn e i(ω nt+k x) ρ 0 2 T 0 (3.28) Erfc(z) arg(z) < π/4 ω (3.28) ψ µ (3.28) matching condition matching condition (3.2) b (3.23) i ωδρ + k δu = 0 (3.29) iω + ikv = 0 kv continuous mode (3.7) 0 = d 3 v f 0 (v) τ = δf 0 [ m ρ 0 + v ( v v 2 + m RT 0 2RT0 2 3 ) ( v 2 3 )] δf(v ) (3.30) 2T 0 2RT 0 2 ρ u T f moment (3.30) 3..6 3..3 Result
3. Boltzmann 67 00 0 (ω τ) 0. 0.0 0.00 0. 0 k τ C s 3. 00 0 (ω τ) 0. 0.0 0.00 0. 0 k τ C s 3.2 00 0 (ω τ) 0. 0.0 0.00 0. 0 k τ C s 3.3 shear flow
68 3 ω ω k 2 Navier-Stokes Navier-Stokes shear flow 00 0 (ω τ) 0. 0.0 0.00 0. 0 k τ C s 3.4 shear flow 00 0 (ω τ) 0. 0.0 0.00 0. 0 k τ C s 3.5
3. Boltzmann 69 00 0 (ω τ) 0. 0.0 0.00 0. 0 k τ C s 3.6 2 Im ω 0.8 Re (k/ ω) 0.6 0.4 0.2 0 0.0 0. 0 / τ ω 3.7
70 3 0.5 0.4 Im (k/ ω) 0.3 0.2 0. 0 0.0 0. 0 / τ ω 3.8 Bhatnagar [9] Sirovich [35] (3.2) (3.23) (3.25) δρ 0 δ ρ δ u δ T 0. 0.0 0. 0 k τ C s 3.9
3. Boltzmann 7 0 δ ρ δ u δ T 0. 0. 0 k τ C s 3.0 00 δ ρ δ u δ T 0 0. 0. 0 k τ C s 3. 00 δ ρ δ u δ T 0 0. 0. 0 k τ C s 3.2 2
72 3 δt δu 0 δu δρ δu δt 2 ω 3..4 Discussion (3.26) (3.27) shear (3.27) ω = 0 ( iω)/k b Erfc(x) Erfc(x) = 2 π e x2 2 π x e t2 dt (3.3) ( 2x 2 + 3 ) (2n )!! 2 2 ( )n x4 2 n x 2n (3.32) (3.27) k b + ( ) 2b 3 + O b 4 = 0 (3.33) b 4 b = ( iω)/k k 2( + iω) 3 {k2 2( + iω)iω} = 0 (3.34) iω k 3 iω = k2 2 + O(k4 ) (3.35)
3. Boltzmann 73 shear flow k = 0 iω = (3.26) b b (3.26) ω 4 ( ) ( 3k 8 b + 5 kb 2 + 3 b 3 2 2kb 4 2b 2 + 3 2b ( 4 kb + 2 kb 2 b 2 + 3 2kb 3 + 9 2b 4 ) ) ( 3k 3 b + 5 3b 3 ) = 0 (3.36) b = ( iω)/k ω 4 k k 2 (72 + 268k 2 )(iω) 4 (2 + 22k 2 )(iω) 3 + 78k 2 (iω) 2 0k 2 (iω) + 5k 4 = 0 (3.37) iω k 2 k = 0 ω = 0 3 iω = k2 2 + O(k3 ), k 2 5 2 ± i 6 k + O(k3 ) (3.38) A (A.45) (A.46) A BGK λ = /τ (A.45) (A.46) (A.45) (A.46) k / 2 k 2RT 0 A RT 0 k Im ω Re ω k b (3.34) b 3..3 k = 0 ω = 0 shear flow k = 0 ω = i
74 3 A shear flow k = 0 ω = i A BGK ω = i (3.26) (3.2) b 0 /b (3.26) b = 0 k ω = i ω = iτ Im ω < F. [35] 3..5 collision n collision time n δρ δu k n collision BGK τ v k collision collisionless k collision δe Chapman-Enskog k
3. Boltzmann 75 collision n momentum density pressure viscosity Eigenvalue 0. 0. 0 k τ C s 3.3 00 momentum density pressure viscosity Eigenvalue 0 0. 0. 0 k τ C s 3.4
76 3 00 momentum density pressure viscosity Eigenvalue 0 0. 0. 0 k τ C s 00 momentum density pressure viscosity Eigenvalue 0 0. 0. 0 k τ C s 3.5 2
3. Boltzmann 77 3..6 continuous spectrum 3..2 iω + ikv = 0 continuous spectrum v y v z (3.30) C. Cercignani [40] v 0 δf δf = δf + Dδ D (v v 0 ) (3.39) δ D Dirac D δf (3.7) δf ( ) ( iω + ik v δf + τ τ iω + ik v 0 = d 3 v f [ 0 (v) m + v ( v v 2 + m τ ρ 0 RT 0 2RT0 2 3 ) ( v 2 3 2T 0 2RT 0 2 + f [ 0 (v) m + v ( v 2 v 0 + m 3 ) ( v 2 0 3 )] D τ ρ 0 RT 0 2T 0 2RT 0 2 2RT 2 0 continuous spectrum ) D (3.40) )] δf (v ) ω = i τ + k v 0 (3.4) ω d 3 v [ f0 (v) m δf = + v ( v v 2 + m iτk (v v 0 ) ρ 0 RT 0 2RT0 2 3 ) ( v 2 3 )] δf (v ) 2T 0 2RT 0 2 + [ f 0 (v) m + iτk (v v 0 ) ρ 0 v ( v 2 v 0 + m RT 0 2RT0 2 3 ) ( v 2 0 3 )] D 2T 0 2RT 0 2 (3.42) Fredholm 2 3..2 shear flow v y { k } πe b2 Erfc(b) δu y = πe b2 Erfc(b) D v 0y (3.43) b = ik v 0 k = iv 0x (3.44)
78 3 continuous spectrum δu y k [ ] k δu y = πe v0xerfc( iv 2 0x ) D v 0y (3.45) δρ δu δt continuous spectrum v (3.30) continuous spectrum δf continuous spectrum A Maxwell BGK [7] r n n > 2 continuous mode f L 2
3.2 Boltzmann Marle 79 3.2 Boltzmann Marle BGK BGK Eckart Marle BGK 3.2. Marle BGK model Marle [3] BGK ( ) f = m t coll τ (f(t, x, p) f 0(t, x, p)) (3.46) τ p µ f 0 [ n(t, x) f 0 (t, x, p) = 4πm 2 T (t, x)k 2 (ζ(t, x)) exp p µu µ ] (t, x) T (t, x) ζ = m T (3.47) (3.48) p µ Maxwell-Jüttner n u µ T x µ u µ u 2 = ( ) p µ µ f = p 0 t + v f = m τ (f(t, x, v) f 0(t, x, v)) (3.49) p µ µ N µ = m τ µ T µν = m τ d 3 p p 0 (f f 0) (3.50) d 3 p p 0 pν (f f 0 ) (3.5) 0 n u µ T 0
80 3 d 3 p p 0 (f f 0) = p 0 nk (ζ) mk 2 (ζ) = 0 (3.52) d 3 p p 0 pµ (f f 0 ) = N µ N µ 0 = 0 (3.53) matching condition f 0 matching condition n u e Marle BGK model n matching condition /e matching condition e 0 f 0 e e 0 Marle BGK e e 0 0 bulk viscosity (3.53) Eckart 3.2.2 Marle BGK BGK [] ζ = m/t Marle BGK /ζ τ (3.46) BGK τ τ ζ ζ τ BGK τ BGK ( ) t + v f = τ (f(t, x, v) f 0(t, x, v)) (3.54) f (3.54) [ f(t) = f(0) + t ] e t/τ f 0 (t )dt e t/τ (3.55) τ 0
3.2 Boltzmann Marle 8 τ τ Marle BGK (3.49) t ] f(t) = [f(0) + τ e t /τ f 0 (t )dt e t/τ (3.56) 0 τ = p0 m τ (3.57) τ τ τ f(t, x, p) Marle BGK (3.49) p µ µ f(t, x, p) = p 0 ( t + v ) f(t, x, p) = m τ (f(t, x, p) f 0(t, x, p)) (3.58) p = 0 t f(t, x, 0) = m τ (f(t, x, 0) f 0(t, x, 0)) (3.59) BGK τ Marle BGK τ τ τ BGK τ τ τ Marle BGK τ τ τ M τ M m τ M p 0 τ (3.60) τ M 3.2. (3.52) τ M = m n d 3 p p 0 f 0 τ = K (ζ) τ (3.6) K 2 (ζ)
82 3 K n n K (ζ)/k 2 (ζ) ζ ζ 0 ζ/2 τ (3.6) Marle BGK 3.2.3 Marle BGK ( ) D Ds f = pµ µ f = p 0 t + v f = m (f f 0 ) (3.62) τ M f 0 n f 0 = [ 4πm 2 T K 2 (ζ) exp p µu µ T ζ = m T ] (3.63) (3.64) f 0 source term f f 0 f (f 0 f) ψ µ d3 p p 0 = 0 (3.65) ψ µ = (, p µ ) (3.66) (3.65) f 0 f matching condition ( 3.2. f 0 (p µ ) δf = f f 0, δf 0 = f 0 f 0, δτ M = τ M τ M ( τ M f 0 ) (3.67) f 0 ( ) t + v δf = δf δf 0 (3.68) τ M τ M = p0 m τ M (3.69)
3.2 Boltzmann Marle 83 τ M τ M τ M δf = δ fe ik µx µ = δ fe iω(t t 0)+ik x (3.70) (3.68) ( ) iω + ik v δf = δf 0 (3.7) τ M τ M u µ δf 0 [ δf 0 = f δn 0 + ( + p0 + K 2 n 0 T 0 f 0 = n 0 4πm 2 T 0 K 2 (ζ 0 ) exp [ p0 ζ 0 K 2 ] T 0 ) δt p δu ] T 0 T 0 (3.72) (3.73) f 0 δu 3.2. Eckart K n K n ζ δn δu δt matching condition δf ( ) iω + ik v δf(p) (3.74) τ M d 3 p [ ) f0 (p) p 0 = p 0 + ( + p0 + K 2 A(ζ) p 0 ζ 0 p ] p δf(p ) τ M n 0 T 0 K 2 T 0 T 0 δt A(ζ) ω τ ω, τk k, p 0 T 0 z (3.75) d 3 p δf(p) = p 0 K(p, p )δf(p ) (3.76) [ ( ) f K(p, p 0 (p) p 0 ) ( ) + + z + K 2 A(ζ) p 0 ζ 0 p ] p iω + ik p K z n 0 K 2 T 0 T 0 p 0 K 2 ζ (3.77) ( iω + ik p ) K z p 0 K 2 ζ 0 (3.78)
84 3 (3.78) ω Fredholm 2 K(p, p ) p p 0 d 3 p/p 0 ω k E.2 [ K 2 k ζk 2 P (0) πk 2 e c] δn (3.79) [( + 3 K ) (kk 2 ζ K 2 ( πk 2 e c c 2 K + [ ik 2ζ k πk 2 e c i k 2 { ( ζ K2 k P (n) b ζk 2 P (0) ) + ζk 2 (kk ζp ()) )] δt ζ K 2 ) } P (0) iωp () K K { iω( + c) + K 2 ζ K }] k δu = 0 P (n) = dy e ζy y n arctan ζ y 2 b b = ζ ( ) K2 iωy k K (3.80) (3.8) c = K 2ζ K Im(ω) ζ c = iωk 2 ω 2 /k 2 k 2 + K ( K2 kk c F.3 ) 2 + ω2 k 2 if Im(ω) < K 2 K (3.82) if Im(ω) > K 2 K
3.2 Boltzmann Marle 85 k p [ ( ζ ik + i K ) 2ζ ωζ P (0) + K k k P () + πk { e c [ ( ) + iζ 3K K2 + i K 2ζ 2 + ωζ2 k + π k e c [ω ( 3P () K ζ [ { ζ 2 + k 2 K 2 ( iω) K 2ζ K k { + π k e c k 2 ω( + c) + i K 2 K ζ }] δn (3.83) ( ζ + ζk 2 3P (0) K ) ζ P (0) + ζp () K 2 K k K 2 ) P () + ζp (2) K 2 { } ( + c) K 2 ζ + c 2 + c + + i K ( )]] 2 ζ c + K 2 ζ δt K 2 K K 2 ( ) K2 P (0) iωp () + iωζ ( ) } K2 P () iωp (2) ζk K k K ω 2 (c 2 + 2c + 2) + 2iω K ( ) }] 2 2 K2 ζ ζ( + c) k δu = 0 K k p kp y [ [ 2kK dy e ζy bζ y 2 + { b 2 + ζ 2 (y 2 ) } ] arctan ζ y 2 b (3.84) π ) {(c ζ e c 2 + 2c + 2) ( ω2 k 2 2iK ( 2 K 2 k 2 ζω( + c) + 2 ζ 2 )}] K k 2 K 2 ζ 2 δu y = 0 p 0 [ ζp () K k + π ] ζ e c ( + c) δn (3.85) [ ( + ζ 3P () K ) ζp () + ζp (2) + πζ { }] K e c ( + c) K 2 ζ + c 2 + c + δt 2 K 2 [ + iζ { K 2 ζ ( )} K2 P () iωp (2) k k K + π i { ζ e c iω(c 2 + 2c + 2) + K }] 2 ζ( + c) k δu = 0 k K F.3 matching condition Marle BGK matching condition δn δu 3 (3.79) iζ/k k (3.85) K
86 3 ωζ/k (3.83) iωδn + ik δu = 0 δρ δu δt ω δρ δu δt δf δf(v) = n C n f0 (v) ( iω + ik p [ δnωn + n 0 p 0 ) K z K 2 ζ ( + p0 T 0 + K 2 K 2 ζ 0 ) δtωn T 0 p δu ω n T 0 ] e i(ω nt+k x) (3.86) C n (3.78) ω p x continuous mode Cercignani [40] 3..6 δf = δf + Dδ D ( ( iω + ik p p 0 ) ) K z K 2 ζ (3.74) δf (3.87) 3.2.4 Result ζ = 00 ζ = ζ = 0.0
3.2 Boltzmann Marle 87 00 0 damping rate n τ 0. 0.0 0.00 0.000 e-05 e-06 e-07 0.0 0. 0 00 wave number k τ c 3.6 ζ = 00 00 0 damping rate n τ 0. 0.0 0.00 0.000 e-05 0.0 0. 0 00 wave number k τ c 3.7 ζ = 00 0 damping rate n τ 0. 0.0 0.00 0.000 e-05 0.0 0. 0 wave number k τ c 3.8 ζ = 0.0
88 3 Chapman-Enskog k 2 k 2 Chapman-Enskog ζ = 00 3..3 Marle BGK BGK k ζ = 00 k ζ ζ = 0.0 ζ = 00 0.0
3.2 Boltzmann Marle 89 00 0 Re (ω τ) 0. 0.0 0.00 0.0 0. 0 00 wave number k τ c 3.9 ζ = 00 0 damping rate n τ 0. 0.0 0.00 0.0 0. 0 wave number k τ c 3.20 ζ = 0 Re (ω τ) 0. 0.0 0.00 0.0 0. 0 wave number k τ c 3.2 ζ = 0.0
90 3 00 0 damping rate n τ 0. 0.0 0.00 0.000 e-05 e-06 e-07 0.0 0. 0 00 wave number k τ c 3.22 ζ = 00 0 0. Re (ω τ) 0.0 0.00 0.000 e-05 0.0 0. 0 wave number k τ c 3.23 ζ = 0 damping rate n τ 0. 0.0 0.00 0.000 e-05 0.0 0. 0 wave number k τ c 3.24 ζ = 0.0
3.2 Boltzmann Marle 9 ω ζ ζ = 00 3..3 ζ = ζ = 0.0 k = 0 C BGK shear flow ζ = 00 0.0
92 3 00 0 damping rate n τ 0. 0.0 0.00 0.000 e-05 e-06 e-07 0.0 0. 0 00 wave number k τ c 3.25 ζ = 00 shear flow 000 00 damping rate n τ 0 0. 0.0 0.00 0.000 e-05 0.0 0. 0 00 wave number k τ c 3.26 ζ = shear flow 000 00 damping rate n τ 0 0. 0.0 0.00 0.000 e-05 0.0 0. 0 00 wave number k τ c 3.27 ζ = 0.0 shear flow
3.2 Boltzmann Marle 93 shear flow shear flow ζ ζ = 00 3..3 BGK 3.2.5 Discussion Marle BGK Eckart 3.2.3 3.2.4 Chapman-Enskog k 2 BGK Anderson-Witting Marle BGK ζ BGK 3.2.4 ζ = 00 BGK Marle BGK ζ BGK ζ moment shear flow moment k = 0 n 0 Anderson-Witting BGK shear flow Israel-Stewart [4] moment Israel-Stewart BGK k couple n /τ
94 3 couple n /τ k n Israel-Stewart couple F.3 3.2.3 (3.70) /τ Cercignani [4] Anderson-Witting BGK Marle BGK n BGK Sirovich [35] BGK
3.3 Boltzmann Anderson-Witting 95 3.3 Boltzmann Anderson-Witting Anderson-Witting BGK [4] Marle BGK Eckart Landau-Lifshitz 3.3. Anderson-Witting BGK model Anderson Witting [4] BGK ( ) f = p ν f(t, x, p) f 0 (t, x, p) u ν t τ coll (3.88) τ p µ u µ f 0 f 0 (t, x, p) = ζ = m T [ n(t, x) 4πm 2 T (t, x)k 2 (ζ(t, x)) exp p µu µ ] (t, x) T (t, x) (3.89) (3.90) p µ Maxwell-Jüttner n u µ T x µ u µ u 2 = p µ µ f = p 0 ( t + v ) f = p ν u ν f(t, x, p) f 0 (t, x, p) τ (3.9) p µ µ N µ = u ν τ µ T µρ = u ν τ d 3 p p 0 pν (f f 0 ) (3.92) d 3 p p 0 pν p ρ (f f 0 ) (3.93) 0 n u µ T 0
96 3 u ν d 3 p p 0 pν (f f 0 ) = u ν N ν u ν N ν 0 = 0 (3.94) u ν d 3 p p 0 pν p ρ (f f 0 ) = u ν T νρ u ν T νρ 0 = 0 (3.95) matching condition Anderson-Witting BGK (3.94) f 0 n f (3.95) 0 f 0 e f BGK matching condition (3.95) matching condition (3.95) u µ T 0µ f 0 T 0µ 0 T 0µ 0 matching condition u µ Landau-Lifshitz 3.3.2 Anderson-Witting [4] BGK ( ) D Ds f = pµ µ f = p 0 t + v f = u ν p ν f f 0 τ (3.96) f 0 n f 0 = [ 4πm 2 T K 2 (ζ) exp p µu µ T ζ = m T ] (3.97) (3.98) K 2 BGK Landau-Lifshitz matching condition f 0 n e f ( 3.3. ) f 0 source term f f 0 f u ν d 3 p p 0 pν ψ µ (f 0 f) = 0 (3.99)
3.3 Boltzmann Anderson-Witting 97 ψ µ = (, p µ ) (3.00) (3.99) f 0 f matching condition f 0 (p µ ) δf = f f 0, δf 0 = f 0 f 0, δτ = τ τ ( τ f 0 ) (3.0) f 0 ( ) t + v δf = δf δf 0 τ (3.02) τ τ τ δf = δ fe ik µx µ = δ fe iω(t t 0)+ik x (3.03) (3.02) ( ) iω + ik v δf = τ τ δf 0 (3.04) u µ δf 0 [ δf 0 = f δn 0 + ( + p0 + K 2 n 0 T 0 f 0 = n 0 4πm 2 T 0 K 2 (ζ 0 ) exp [ p0 ζ 0 K 2 ] T 0 ) δt p δu ] T 0 T 0 (3.05) (3.06) f 0 δu 3.3. Landau-Lifshitz K n K n ζ δn δu δt matching condition δf ( ) iω + ik v δf(p) (3.07) τ = d 3 p f [ ) 0 (p) + ( + p0 + K 2 A(ζ) p 0 ζ 0 p ] p δf(p ) τ n 0 T 0 K 2 T 0 T 0
98 3 δt A(ζ) ω τ ω, τk k, p 0 T 0 z (3.08) δf(p) = d 3 p K(p, p )δf(p ) (3.09) [ ( ) f K(p, p 0 (p) ) iω + ik p + + z + K 2 A(ζ) p 0 ζ 0 p ] p p n 0 0 K 2 T 0 T 0 (3.0) iω + ik p p 0 0 (3.) (3.) ω Fredholm 2 K(p, p ) p 0 p 0 p (p 0 ) 2 d 3 p/p 0 ω k E.3 p 0 [( (ζq(2) K 2 k) δn + 3 K ) ] ζ ζq(2) + ζ 2 Q(3) δt (3.2) K 2 i ( 3K2 + ζk bζ 2 Q(3) ) k δu = 0 k Q(n) b Q(n) = b = iω k dy e ζy y n arctan y2 b y (3.3) (3.4) p 0 k p ( 3K2 + ζk bζ 2 Q(3) ) δn (3.5) [ + K ζ i [ ζk 3 b k ( 3 + K ζ K 2 ) + (3 + ζ 2 )K 2 bζ 2 ( 3Q(3) K { (2 + ζ 2 )K 2 + 3ζK bζ 3 Q(4) }] k δu = 0 )] ζq(3) + ζq(4) K 2 δt
3.3 Boltzmann Anderson-Witting 99 p 0 k p kp 0 p y [ 2kζK3 ζ 3 { (b 2 + )Q(4) Q(2) } + b { (2 + ζ 2 )K 2 + 3ζK 2 }] δuy = 0 (3.6) (p 0 ) 2 [ k(3k2 + ζk ) ζ 2 Q(3) ] δn i [ bζ 3 Q(4) (ζ 2 ] + 2)K 2 3ζK k δu (3.7) [ ( k + k {(3 + ζ 2 )K 2 ζk 3 + K )} ( ζ ζ 2 3Q(3) K )] Q(3)ζ + ζq(4) δt = 0 K 2 K 2 matching condition Anderson- Witting BGK Landau-Lifshitz iωδn + ik δu = 0 (3.95) µ T µν = δ(ne) + (neδu) (3.8) t = iω n [ ( 0T 0 (3K 2 + ζk )δn + {(3 + ζ 2 )K 2 ζk 3 + K )} ] ζ δt K 2 K 2 + i ζk 3 K 2 n 0 T 0 k δu =0 matching condition δu matching condition (3.5) δt matching condition (3.7) (3.5) (3.7) b (3.8) δρ δu δt ω δρ δu δt δf δf(v) = n C n C n f0 (v) iω + ik p (3.9) p [ 0 ) δnωn + ( + p0 + K 2 δtωn ζ 0 p δu ] ω n e i(ω nt+k x) n 0 T 0 K 2 T 0 T 0 (3.) ω p x continuous mode Cercignani [40]
00 3 3..6 3.2.3 ( δf = δf + Dδ D iω + ik p ) p 0 (3.20) (3.07) δf 3.3.3 Result Marle BGK 3.3.4 ζ = 00 ζ = ζ = 0.0
3.3 Boltzmann Anderson-Witting 0 00 0 damping rate n τ 0. 0.0 0.00 0.000 e-05 e-06 e-07 0.0 0. 0 00 wave number k τ c 3.28 ζ = 00 0 damping rate n τ 0. 0.0 0.00 0.000 e-05 0.0 0. 0 wave number k τ c 3.29 ζ = 0 damping rate n τ 0. 0.0 0.00 0.000 e-05 0.0 0. 0 wave number k τ c 3.30 ζ = 0.0
02 3 Chapman-Enskog k 2 k 2 Chapman-Enskog ζ = 0.0 ζ = 00 3..3 Anderson-Witting BGK BGK k ζ = 00 k ζ ζ = 00 0.0
3.3 Boltzmann Anderson-Witting 03 00 0 Re (ω τ) 0. 0.0 0.00 0.0 0. 0 00 wave number k τ c 3.3 ζ = 00 00 0 Re (ω τ) 0. 0.0 0.00 0.0 0. 0 00 wave number k τ c 3.32 ζ = 00 0 Re (ω τ) 0. 0.0 0.00 0.0 0. 0 00 wave number k τ c 3.33 ζ = 0.0
04 3 00 0 damping rate n τ 0. 0.0 0.00 0.000 e-05 e-06 e-07 0.0 0. 0 00 wave number k τ c 3.34 ζ = 00 0 damping rate n τ 0. 0.0 0.00 0.000 e-05 0.0 0. 0 00 wave number k τ c 3.35 ζ = 0 damping rate n τ 0. 0.0 0.00 0.000 e-05 0.0 0. 0 00 wave number k τ c 3.36 ζ = 0.0
3.3 Boltzmann Anderson-Witting 05 ω ζ ζ = 00 3..3 ζ = ζ = 0.0 k = 0 C BGK shear flow ζ = 00 0.0
06 3 00 0 damping rate n τ 0. 0.0 0.00 0.000 e-05 e-06 e-07 0.0 0. 0 00 wave number k τ c 3.37 ζ = 00 shear flow 000 00 damping rate n τ 0 0. 0.0 0.00 0.000 e-05 0.0 0. 0 00 wave number k τ c 3.38 ζ = shear flow 000 00 damping rate n τ 0 0. 0.0 0.00 0.000 e-05 0.0 0. 0 00 wave number k τ c 3.39 ζ = 0.0 shear flow
3.3 Boltzmann Anderson-Witting 07 shear flow shear flow ζ Marle BGK ζ = 00 3..3 BGK shear flow 00 damping rate n τ 0 0. 0.0 0.0 0. 0 00 wave number k τ c 3.40 ζ = 00 shear flow 00 0 Re (ω τ) 0. 0.0 0.00 0.0 0. 0 00 wave number k τ c 3.4 ζ = 0.0 shear flow moment 3.3.4 Discussion Anderson-Witting BGK Landau-Lifshitz 3.3.2
08 3 3.3.3 Chapman-Enskog k 2 BGK Marle Anderson-Witting BGK ζ BGK 3.3.3 ζ = 00 BGK Anderson-Witting BGK ζ BGK ζ moment shear flow moment k = 0 n 0 shear flow Israel-Stewart [4] moment Israel-Stewart Landau-Lifshitz moment couple n /τ Marle BGK couple n /τ k n Israel-Stewart Landau-Lifshitz couple
3.3 Boltzmann Anderson-Witting 09 Marle BGK 3.2.4 Marle BGK Anderson-Witting BGK Landau-Lifshitz Marle BGK Eckart ζ = 00 3..3 BGK C BGK
0 3 3.4 Israel-Stewart Discussion Israel-Stewart [4] Israel-Setewart 2.2.8 review moment (2.339) (2.340) (2.34) (2.342) (2.339) (2.34) (2.342) Π = µ( ν u ν + β 0 Π α0 ν q ν ) (3.2) ( ) q µ = κγ µν T νt + u ν + β q ν α 0 ν Π α ρ p ρ ν (3.22) p µν = 2η( µ u ν + β 2 ṗ µν α µ q ν ) (3.23) α 0 α β 0 β β 2 2.2.8 Israel-Stewart [5] ( ) n > 0 (3.24) Θ T ( ) ne > 0 (3.25) T Θ β 0, β, β 2 > 0 (3.26) β (ne + P ) > 0 (3.27) ( ) ne > 0 (3.28) T Θ = (ρ + P nst )/T n /T Isreal-Stewart 2.2.7 2.2.8 moment l L l/l l/l Israel-Stewart moment (2.339) (2.34) (2.342) Chapman-Enskog n
3.4 Israel-Stewart B moment Israel-Stewart 4 4 Longitudinal-mode [] 0 damping rate n τ 0. 0.0 0.00 0. 0 wave number k τ C s 3.42 ζ = 00 Israel-Stewart damping rate n τ 0. 0.0 0.00 0. 0 wave number k τ C s 3.43 ζ = 0.0 Israel-Stewart 3.42 3.43
2 3 BGK Israel-Stewart ζ cτk = Chapman-Enskog c C s C s τk = k 2 moment moment moment [36] Israel-Stewart Israel-Stewart Chapman-Enskog Israel-Stewart
3 4 BGK Cauchy BGK BGK Marle BGK Anderson-Witting BGK Cauchy Israel-Stewart Israel-Stewart Isreal-Stewart [5]
4 4
5 A P. Résibois [6] f 0 f = f 0 + δf, δf f 0 ( ) t + v δf = ncδf (A.) (A.2) C collision operator Cδf d 3 v dωσ(χ; v rel )v rel [δf ϕ eq + ϕ eq δf δfϕ eq ϕeq δf ] (A.3) ϕ eq = f 0 N ϕ eq ϕ eq = ϕ eq ϕ eq Cδf = d 3 v [ ( ) ( ) δf δf ϕ eq + ϕ eq dωσ(χ; v rel )v rel ϕ eq ϕ eq (A.4) ( ) ( ) ] δf δf ϕ eq ϕ eq (A.5) collision operator 2..4 k h d 3 vϕ eq (v) k(v) Ch(v) = d 3 vd 3 v dωσ(χ; v rel )v rel ϕ eq ϕ eq (A.6) 4 [ ( ) ( ) ( ) ( ) ] k k k k + ϕ eq [ ( h ϕ eq ) + ϕ eq ) ( h ϕ eq ϕ eq ( h ϕ eq ) ϕ eq ( h ϕ eq ) ]
6 A collision operator Hilbert k h = d 3 vϕ eq (v) k(v) h(v) (A.7) collision operator C (A.6) k C h = h C k (A.8) λ 0 j nc φ 0 j = λ 0 j φ 0 j (A.9) (A.6) h C h 0 (A.0) λ 0 j = φ0 j nc φ0 j φ 0 j φ0 j 0 (A.) (A.6) φ 0 j /ϕeq collision operator C 0 φ 0 j Hilbert φ 0 j(v) φ 0 j(v ) = δ(v v )ϕ eq (v) (A.2) j φ 0 j φ 0 j = (A.3) j t δf = ncδf (A.4) δf δf(t, v) = c 0 jφ 0 j(v)e λ0 j t j c 0 j = φ 0 j δf(t = 0) = d 3 vϕ eq (v) φ 0 j(v) δf(0, v) (A.5) (A.6)
7 t λ 0 j 0 0 5 lim δf(t, v) = t α= c 0 αφ 0 α(v) (A.7) δρ δu δt collision operator C φ 0 j(v) φ 0 rlm(v) = φ rl (v)y lm (θ v, Φ v ) (A.8) Y lm U r 4 Maxwell σ(χ, v rel )v rel F (χ) collision operator C Cδf = d 3 v dωf (χ)ϕ eq ϕ eq [ ( ) ( ) ( ) ( ) ] δf δf δf δf ϕ eq + ϕ eq ϕ eq ϕ eq (A.9) (A.20) v rel {v, v } {v, v } δf = ϕ eq h n (v) (A.2) Wang Chang Uhlenbeck [39] ( ) ( ) φ 0 rlm = A rl ϕ eq (v)s (r) v 2 l v l+/2 Y lm (θ v, Φ v ) 2RT 2RT (A.22) S r l Laguerre π [ ( λ 0 rl = 2πn dχ sin χf (χ) cos 2r+ χ ) Pl 0 (cos χ 0 2 2 ) ( + sin 2r+l χ ) ( χ ) ] Pl 0 δ r,0 δ l,0 2 2 (A.23)
8 A δ r,l L. Sirovich [36] ( ) t + ξ x g = L(g) = x m [g] = g + g g g f 0 = ρ 0 exp (2πRT 0 ) 3/2 ( ξ2 ) = 2RT 0 f 0 [g] B(θ) dɛ dθ dξ ρ 0 (RT 0 ) 3/2 ω (A.24) (A.25) (A.26) ξ ξ g f = f 0 ( + g) (A.27) f ν = /τ νt = t, xν (RT 0 ) /2 = x, ξ (RT 0 ) /2 = ξ, ρ 0 Bν m = B (A.28) ( ) t + ξ x g = L(g) = x ω [g] B(θ) dɛ dθ dξ (A.29) L shear ψ rl = Sr l+/2 (ξ2 /2)ξ l P l (cos θ) 2 l+ Γ(r+l+3/2) π /2 r!(2l+) (A.30) Sl r P l Laguerre Legendre ωψ rl ψ r l dξ = δ rr δ ll (A.3) ψ } 2(r + l + 3/2) ξ x ψ rl = (l + ) {ψ r,l+ (2l + )(2l ) ψ 2r r,l+ (2l + 3)(2l + ) } 2(r + l + /2) + l {ψ r,l (2l + )(2l ) ψ 2(r + ) r+,l (2l + )(2l ) (A.32)
9 L { λ rl = 2π dθb(θ) cos 2r+l θ ( 2 P l cos θ ) + sin 2r+l θ 2 2 P l ( sin θ ) } ( + δ r0 δ l0 ) 2 (A.33) λ r0 = λ r, (A.34) λ rl [36] [39] λ 00 = λ 0 = λ 0 = 0 (A.35) λ [36] ψ rl g = b rl ψ rl, b rl = ω g ψ rl dξ (A.36) r,l b µν 3 b 00 = ρ, b 0 = u, b 0 = 2 T, b 02 = 3 2 2 p, b = 5 S (A.37) ( ) t λ µν b µν + x ( 2(µ + ν + 3/2) ν {b µ,ν (2ν + )(2ν ) b µ+,ν 2µ + 2 (2ν )(2ν + ) }) 2(µ + ν + 3/2) +(ν + ) {b µ,ν+ (2ν + 3)(2ν + ) b 2µ µ,ν+ = 0 (2ν + 3)(2ν + ) (A.38) 3 (A.38) } ρ t + u x = 0 u t + p x + x (ρ + T ) = 0 T t + 2 S 3 x + 2 u 3 x = 0 (A.39) (A.40) (A.4)
20 A b rl = ˆb rl e σt ikx (A.42) ˆbrl ( } 2(µ + ν + 3/2) (σ λ µν )ˆb µν ik ν {ˆbµ,ν (2ν + )(2ν ) ˆb 2µ + 2 µ+,ν (2ν )(2ν + ) (A.43) +(ν + ) {ˆbµ,ν+ 2(µ + ν + 3/2) (2ν + 3)(2ν + ) ˆb µ,ν+ 2µ (2ν + 3)(2ν + ) 0 D(σ, k) = 0 6 6 [36] k = 0 (A.43) }) = 0 σ = λ rl (A.44) λ rl σ k = 0 k = 0 λ rl k k = 0 σ = 0 3 [36] σ = k2 + O(k 4 ) λ 5 σ = ±i 3 [k + O(k3 )] + k 2 (A.45) ( + 2 ) + O(k 4 ) (A.46) 3λ 3λ 02 Maxwell Maxwell n > 2 U r n Pao [7] BGK g (A.36) ψ rl L ( ) t + ξ x f 0 g = L(g) = f 0 λ rl b rl ψ rl x r,l (A.47)
2 λ 00 = λ 0 = λ 0 = 0 λ rl λ ( ) t + ξ x f 0 g = f 0 λ b rl ψ rl (A.48) x r,l )} 32 = λ f 0 {g (ψ 00 ρ + ψ 0 u + ψ 0 T (A.49) λ = /τ BGK (3.2) (A.46) λ 02 BGK λ 02 λ BGK
23 B moment moment moment moment [0] A Boltzmann ˆLb = [ iωi + iak Λ] b = 0 (B.) I b A Λ b j = φ 0 j δf, A ij = φ 0 i v φ 0 j, Λ ij = φ 0 i C φ 0 j (B.2) C collision operator A Maxwell (A.43) [36] (B.) Λ source term ˆP ˆP = iωi + iak (B.3) Q = det ˆP = det( iωi + iak) = 0 (B.4) (B.4) k ω [20] A weakly hyperbolic
24 B moment moment (B.) b (b, b) iω(b, b) + ik(b, Ab) (b, Λb) = 0 (B.5) A (b, Λb) A (b, Ab) ω = ω R in (B.5) n(b, b) (b, Λb) = 0 iω R (b, b) + ik(b, Ab) = 0 (B.6) (B.7) (B.6) n n = (b, Λb) (b, b) (B.8) (b, Λb) moment λ (B.8) n 0 n λ (B.9) f moment moment n k λ collisionless moment moment (B.9) moment λ moment moment collisionless moment consistent
25 C Cercignani [42] p µ µ f = Q(f, f) (C.) Q p µ f f = f 0 ( + h) (C.2) f 0 Maxwell-Jüttner h p µ µ h = ˆLh (C.3) ˆL collision operator H = L 2 (f 0 dω) (g, h) = ḡhf 0 dω (C.4) (h, ˆLh) [43] h = g(p µ ) exp( ik ν x ν ) (C.5) k µ = (ω, k)
26 C g (C.3) ik µ p µ g = ˆLg (C.6) Cauchy k ω = k(a + ib) a (C.6) ikp 0 (a + ib)g + ikp x g = ˆLg (C.7) g H (C.7) g ik(a + ib)(g, p 0 g) + ik(g, p x g) = (g, ˆLg) (C.8) (C.8) a(g, p 0 g) + (g, p x g) = 0 (C.9) a = (g, px g) (g, p 0 g) < (C.0) p x < p 0 g H [42]