Plasma Astrophysics Chapter 3: Kinetic Theory. Yosuke Mizuno Institute of Astronomy National Tsing-Hua University

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1 Plasma Asrophysics Chaper 3: Kineic Theory Yosuke Mizuno Insiue o Asronomy Naional Tsing-Hua Universiy

2 Kineic Theory Single paricle descripion: enuous plasma wih srong exernal ields, imporan or gaining insigh ino physical processes involved For a sysem wih a large number o paricles i is neiher possible nor desirable o deermine he moion o every single paricle => saisical approaches, average macroscopic properies Kineic heory averages ou microscopic inormaion o obain saisical, kineic equaions. No knowledge o individual paricle moion is required o describe observable phenomena.

3 Paricle Phase Space A paricle s dynamical sae can be speciied using is posiion and velociy: r =(x, y, z) and v =(v x,v y,v z ) Combining posiion and velociy inormaion gives paricle s posiion in phase space: (r, v) =(x, y, z, v x,v y,v z ) The sae space or posiion and momenum (or velociy) is a 6D phase space Volume o a small elemen o velociy space is dv x dv y dv z = d 3 v = dv Volume elemen in phase space is d 3 rd 3 v

4 Velociy disribuion uncions Single-paricle approach has limied applicaion where collecive moion no imporan. Non-zero elecric ields in a plasma generally arise sel-consisenly, so mus consider collecive moion o many plasma paricles. Sae o plasma described by he velociy disribuion uncion : (x, y, z, v x,v y,v z,) Gives he number o paricles per uni volume as a posiion r and a ime wih velociy, v x, v y, v z. Has 7 independen variables deining a 6D phase space. Number o paricles in a phase space volume d 3 rd 3 v is: dn = (r, v, )dxdydzdv x dv y dv z = (r, v, )d 3 rd 3 v The oal number o paricles is hereore n = (r, v, )d 3 rd 3 v

5 Momens Le (x) be any uncion ha is deined and posiive on an inerval [a, b]. The momens o his uncion is deined as Zeroh momen M 0 = Firs momen M 1 = Second momen M 2 = a b a b a b (x)dx x(x)dx x 2 (x)dx n h momen M n = b a x n (x)dx

6 Momens (con.) In paricular case ha disribuion is a probabiliy densiy, p(x), hen M 0 =1 M 1 = M 2 = a b a b xp(x)dx = x = mean(x) x 2 p(x)dx = variance(x) Higher order momens correspond o skewness and kurosis. Skewness: a measure o symmery or lack o symmery Kurosis: a measure o wheher he daa are peaked or la relaive o normal disribuion

7 Momens o disribuion uncion Velociy disribuion uncion gives microscopic descripion o saisical inormaion on paricles. However, mos imporan use is in deermining macroscopic (i.e., averages) values such as densiy, curren, ec. Zeroh order momen o (r, v, ) is: n(r,)= (r, v,)d 3 v Firs order momen is bulk velociy: u = 1 n v(r, v,)d 3 v Charge and curren densiies o spices (s) can be expressed in using momens: e = q s n s j = q s n s u s s Second order momen relaes o kineic energy s 1 2 mv2 = 1 n 1 2 mv2 (r, v,)d 3 v

8 Derivaion o Bolzmann Equaion Evoluion o (r,v,) is described by he Bolzmann Equaion. Consider paricles enering and leaving a small volume o space. Since r and v is independen, can rea separaely. Posiion: Number o paricles leaving d 3 r per second hrough is surace ds is (r, v,)ṙ ds = (r, v,)v ds Velociy: Number o paricles leaving d 3 v per second hrough is surace ds v is (r, v,) v ds v = (r, v,)a ds v So he ne number o paricles leaving he phase space volume d 3 rd 3 v is (r, v,)v dsd 3 v + (r, v,)a ds v d 3 r

9 Derivaion o Bolzmann Equaion (con.) The rae o change o paricle number in d3rd3v is: d3 rd3 v = v dsd3 v + a ds v d3 r As oal number o paricles in d3rd3v is conserved: d3 rd3 v + v dsd3 v + Recall Gauss Divergence Theorem: a ds v d3 r = 0 ( V F )dv = S (F n)ds Can change inegral over ds o d3r: or d3 rd3 v + d3 rd3 v + r r ( v)d3 rd3 v + ( v)d3 rd3 v + v v ( a)d3 rd3 v = 0 ( a)d3 rd3 v = 0

10 Derivaion o Bolzmann Equaion (con.) The phase space volume can be arbirarily small, such ha inegrals are consan wihin he volume. Thereore we have + r (v)+ v Bu since r and v are independen variables, we can ake v ouside d/dr and similarly or a. Then we can wrie + v r + a Replacing a=f/m, we have (a) =0 v =0 + v r + F m v =0 (3.1) This is he collisionless Bolzmann equaion. Can be used in ho plasma where collisions can be negleced

11 Vlasov equaion Previous equaion wrien in erms o generalized orce. For plasmas, Lorenz orce is o ineres, so + v r + q m [E +(v B)] v =0 This is called he Vlasov equaion. Can also be wrien as + v + q m [E +(v B)] v =0 (3.2) This is one o he mos imporan and widely used equaions in kineic heory o plasmas. Maxwell s equaions or E and B and he Vlasov equaion represen a complee se o sel-consisen equaions.

12 Convecive derivaive in phase space Disribuion uncion (r, v, ) depends on 7 independen variables. Toal ime derivaive o is: d = d + x x + y y + z z This can be wrien as + v x v x + v y v y + v z v z d d = + v r + a To appreciae meaning o his equaion, consider =(r,): d d = + v D r D Called he convecive derivaive or Lagrangian derivaive. Second erm gives change in measured by an observed moving in he luid rame. v

13 Phase space evoluion A plasma paricle s sae (r, v) evolves in phase space. In absence o collisions, poins move along coninuous curves and obeys he coninuiy equaion: + r,v [(ṙ, v)] =0 Called Liouville equaion The Liouville equaion describes he ime evoluion o he phase space disribuion uncion. Liouvilles heorem saes ha lows in phase space are incompressible. In Caresian coordinaes, equaion reduces o + r (ṙ)+ ( v) =0 v + v r + a v =0 Which is in orm o he collisionless Bolzmann equaion. The Bolzmann and Vlasov equaions ollow rom Liouville equaion.

14 Collisional Bolzmann and Vlasov equaions In he presence o collisions, he Bolzmann equaion can be wrien + v + F m v = where he erm on he righ is he ime rae o change o due o collisions. This is he collisional Bolzmann equaion. Similarly, he Vlasov equaion can be wrien + v + q m [E +(v B)] v = This is he collisional Vlasov equaion. Describes change in paricle disribuion due o shor-range ineracions. When here are collisions wih neural aoms: n where n is he neural aom disribuion coll uncion, and τ is he collision ime. Called Krook collision model coll coll

15 Kineic descripion o plasma Kineic descripion o plasma is highly applicable reamen or collisionless plasma (wave-paricle ineracion, collisionless shock, paricle acceleraion) Bu evaluaion o a 6D disribuion uncion is diicul: analyical soluions o a kineic equaion are rare and numerical are expensive. Asrophysical applicaion: Dark maer evoluion in cosmological simulaion Neurino ranspor in core-collapse supernova simulaion Sellar inerior (equaion o sae) Collisionless shock (supernova blas wave) Paricle acceleraion (asrophysical shock)

16 Momens o Bolzmann -Vlasov equaion Under cerain assumpions no necessary o obain acual disribuion uncion i only ineresed in he macroscopic values. Insead o solving Bolzmann or Vlasov equaion or disribuion uncion and inegraing, can ake inegrals over collisional Bolzmann-Vlasov equaion and solve or he quaniies o ineres. + v + q m [E +(v B)] v = Called aking he momens o Bolzmann-Vlasov equaion coll (3.3) Resuling equaions known as he macroscopic ranspor equaions, and orm he oundaion o plasma luid heory. Resuls in derivaion o he equaions o magneohydrodynamics (MHD).

17 Zeroh-order momen: coninuiy equaion Lowes order momen obained by inegraing eq. (3.3): dv + v dv + q m [E +(v B)] v dv = The irs-erm gives: dv = Since v and r are independen, v is no eeced by gradien operaor: v dv = vdv From previous one, he irs order momen o disribuion uncion is dv = n (3.4) coll dv u = 1 n v(r, v,)dv Thereore, v dv = (nu) (3.5)

18 Zeroh-order momen: coninuiy equaion (con.) For he hird erm, consider E and B separaely. E erm vanishes as E where using Gauss divergence heorem in velociy space. The surace area o velociy space goes as v 2. As v =>, => 0 more quickly han S => (i.e., ypically goes as 1/v 2. A Maxwellian goes as e -v^2 ). Inegral o v = ininiy goes o zero. Using vecor ideniy, (v B) v dv = v (E)dv = E ds =0 (aa) =A a + a A v dv = v (v B)dv v (3.6a). The v x B erm is (v B)dv = (v B) ds v (v B)dv =0 (3.6b) The irs erm on righ again vanishes as => 0 more quickly han S =>. The second erm vanishes as v x B is perpendicular o d/dv

19 Zeroh-order momen: coninuiy equaion (con.2) Las-erm is on righ-hand side o eq. (3.3) : dv = dv =0 This assumes ha he oal number o paricles remains consan as collisions proceed. Combing eq. (3.4)-(3.7) yields he equaion o coninuiy n coll + (nu) =0 (3.8) (3.7) Firs-erm represens rae o change o paricle concenraion wihin a volume, second-erm represens he divergence o paricles o he low o paricles ou o he volume. Eq (3.8) is he irs o he equaions o magneohydrodynamics (MHD). Eq (3.8) is a coninuiy equaion or mass or charge ranspor i we muliply m or q.

20 Firs-order o momen: momenum ranspor Re-wrie eq.(3.3) : Nex momen o he Bolzmann equaion is obained by muliplying Eq (3.3) by mv and inegraing over dv. (3.9) The righ-hand side is he change o he momenum due o collisions and will be given he erm P ij. The irs-erm gives + v + q m [E +(v B)] v = m v dv+m v(v )dv+q v[e+(v B)] v dv = m m v dv = m vdv coll coll dv = m (nu) (3.10)

21 Firs-order o momen: momenum ranspor (con.) Nex consider hird-erm: v[e +(v B)] v dv = v [v(e + v B)]dv v v (E + v B)dv (E + v B) v vdv The irs and second o inegrals on he righ vanishes or same reasons as beore. Thereore have, q v[e +(v B)] dv = q (E + v B)dv v = qn(e + u B) To evaluae second-erm o eq.(3.9), use ha v does no depend on gradien operaor: v(v )dv = (vv)dv = vvdv (3.11)

22 Firs-order o momen: momenum ranspor (con.) Since he average o a quaniy is 1/n imes is weighed inegral over v, we have vvdv = (n vv ) Now separae v ino average luid velociy u and a hermal velociy w: v = u + w Since u is already averaged, so we have (n vv )= (nuu)+ (n ww )+2 (nu w ) (3.12) The average hermal velociy is zero => < w > = 0 and is he sress ensor. P = mn ww (3.13) P is a measure o he hermal moion in a luid. I all paricles moved wih same seady velociy v, hen w = 0 and hus P = 0 (i.e., a cold plasma).

23 Firs-order o momen: momenum ranspor (con.3) Remaining erm in Eq (3.9) can be wrien (nuu) =u (nu)+n(u )u (3.14) Correcing eq. (3.10), (3.11), (3. 13), and (3.14), we have m (nu)+mu (nu)+mn(u )u + P qn(e + u B) =P ij Combing irs wo erms (using con. eq.), we obain he luid equaion o moion: mn u +(u )u = qn(e + u B) P + P ij (3.15) This describes low o momenum also called momenum ranspor equaion. Eq (3.15) is a saemen o conservaion o momenum and represens orce balance on componens o plasma. On righ are he Lorenz orce, pressure, and collisions

24 Summary o momens o Vlasov equaion Equaions o MHD and muli-luid heory are obained by aking he momens o he Vlasov equaion, corresponding o mass, momenum and energy. (Vlasov equaion)dv (Vlasov equaion)vdv conservaion o mass conservaion o momenum (Vlasov equaion)v 2 /2dv conservaion o energy Zeroh momen o he Vlasov equaion resuls in he MHD mass coninuiy equaion (eq. 3.8). Firs momen o he Vlasov equaion gives he MHD momenum equaion (eq. 3.15) Second momen o he Vlasov equaion give he MHD energy ranspor equaion