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CHAPTER 6: Vlasov Equation

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CHAPTER 6: Vlasov Equation CHAPTER 6: Vlasov Equation Part 1 6. 1 Introduction Distribution Function, Kinetic Equation, and Kinetic Theory : The distribution function ft(xv , , ) gives the particle density of a certitain speci es i n th th6die 6-dimensi onal ph ase space ofdtf xv and at time t. Thus, ftdxdv(xv , , )33 is the total number of particles in the differen- tial volume dxdv33 at point (xv , ) and time t. A kinetic equation describes the time evolution of f (xv , ,t ). The kinetic theories in Chs. 3-5 derive various forms of kinetic equations. In most cases,,,p however, the plasma behavior can be described by an approximate kinetic equation, called the Vlasov equation, which simply neglects the complications caused by collisions. By ignoring collisions , we may star t out without the knowledge of Chs. 3-5 and proceed directly to the derivation of the Vlasov equation (also called the collisionless Boltzmann equation). 1 6.1 Introduction (continued) The Vlasov Equation : As shown in Sec. 2.9, a collision can result in an abrupt change of two colliding pa rticles' velocities and their instant escape from a small element in the xv- space, which contains the colliding particles before the collision. However, if collisions are neglected(validonatimescaled (valid on a time scale the collision time, see Sec. 1.6), particles in an element at position A in the xv- space will wander in continuous curves to v position B ( see fig fiure) . Th us, th e t ot al numbithber in the B element is conserved, and ft(xv , , ) obeys an equation A of continuity, which takes the form (see next page): x t ft(xv , , )xv, [ ft ( xv , , )( x , v )] 0 (1) In (1), xv, [ ( , , , , , )] is a 6 - dimensional xyzvxyz v v divergence operator and (x , v ) [ (xyzv , , , x , v yz,)]v can be regarded as a 6-dimensional "velocity" vector in the xv- space. 2 6.1 Introduction (continued) To show that t ft(xv , , )xv, [ ft ( xv , , )( x , v )] 0 [(1)] implies conservation of particles, we integrate it over v ds6 an arbitrary volume VS66 enclosed by surface V6 S6 in the xv- space: x d 3 333 ftdx(xv , , ) dvxv, [(,,)(,)] fxv t x v dxdv 0 dt VV66 Nt() ft(,,)(,)xv x v d s6 s6 (by 6-dimensional divergence theorem) d Nt( ) f (x ,vx,td )( , vs )6 0, (2) dt s6 where Nt( ) is the total number of particles in V6 , f(xv , , t )( x , v ) is the 6-dimensional "particle flux" in the xv- space, ds6 is a 6-dimensional differential surface area of SS66, with a direction normal to pointing outward. Thus, (2) states that the rate of increase (decrease) of the total number of particles in V6 equals the partcle flux into (out of) V6. 3 6.1 Introduction (continued) Rewrite: t ft(xv , , )xv, [ ft ( xv , , )( x , v )] 0 (1) In (1), xv, [[(ft (xv,, )( x, v )] (fx ) ( fy ) ( fz ) ( fv xyz ) ( fv ) ( fv ) xyzvxyz v v ()ffxv () (6.1) xv xv ffffxxvvv+v v+v where, because v is an independent variable, we have x v=0 (3) q 1 In general, v vvEvB does not vanish. But for mc( ), v EE 0 [ does not depend on v] we have vvv()(vB B v )( v B )000 q 1 which gives v v= m v (EvBc ) 0 (4) q 1 Thus, xv, [(, [ffff (,xv , ,)(,)] t)( x , v )] v xff +mc ( E v B ) v q 1 and (1) becomes t ffv+EvBxvmc( ) f 0, (6.5) which is the Vlasov equation. 4 6.1 Introduction (continued) Physical Interpretation of the Vlasov Equation : q 1 The Valsov equation t ffv+EvBxvmc (()0 ) f 0 (6.5) can be written q 1 vEvB mc( ) dddft()(xv,,)0 fxv f f 0, (5) dtt dtxv dt df where is the total time derivative of f . The total time derivative dt (also called a convective derivative) fo llows the orbit* of a particle in the xv- space. It evaluates the variation of f due to the change of the particle position in the xv- space as well as the explicit time variation of f . Thus , (6 .5) or (5) can be in terpreted as: Along a particle'sorbit s orbit in the xv- space, the particle density ft( xv , , ) remains unchanged. *In the 6-dimensional xv- space, particles at a point ( xv , ) have the same velociy v. Hence, the orbit of a particle is also the orbitofant of an infinistesimal element containing the particle. In the 3-dimensional xx-space, by contrast, particles at a point have a range of velocities. 5 6.1 Introduction (continued) This interpretation is consistent with the fact that the sum of (3) and (4) gives xv, (xvx, vxv= ))0 x x v v= 0, (6) i.e. the 6-dimensional divergence of the 6-dimensional "velocity" (x , v ) vanishes. This implies that a collisionless v plliiasma is incompress iblHthlible. Hence, the volume of B an elements (thus the particle density f ) will be A unchanged as it moves from AB to . x A specific example : Consider the simple case of a group of particles initially located in a sqqp(,uare in the x-vAx space (area , lower figg)ure). If the p articles are force free, those on the upper edge will move at the fastest (equal and constant) speed, while those on the lower edge vx move at the slowest (equal and constant) speed. B Then, some time later, the square will become a A x parallelogram of the same area (area B, lower figure). 6 6.1 Introduction (continued) Effect of collisions : If there are collisions, they will cause a variation of f at the symbolic rate of ( f ) , which should be added to (6.5) to t coll give q 1 t ffvEvBx mc( ) v ff(),t coll while the spp()pecfic form of ()t f coll depends on interparticle forces. Throughout this course, the (t f )coll term will be neglected. 7 6.1 Introduction (continued) Complete Set of Equations : We now have the following set of self- consistent, coupled particle and field equations: q 1 t ffvEvBmc( ) v f 0 (6.5) B 0 (7) For simplicity, we shall E 4 (8) henthforth denote x by . 1 EB c t (9) BEJ 1 4 (10) cct ()(xxv,tftdt)( q ftd (,, )3v (11) where 3 Jx( ,tqftdv ) ( xv , , ) v (12) Each particle species, denoted by the subscript " ", is governed by a separate Valsov equation, and q carries the sign of the charge. 8 6.2 Equilibrium Solutions General Form of Equilibrium Solutions : As shown in (5), the Vlasov equation can be written as a total time derivative: dddft(xv , , ) fxv f f 0, (5) dtt dt dt v where d follows the orbit of a particle whose position and velocity dt at time t is xv and , respectively. Thus, any fuction of constants of the motion along the orbit of the particle, CCii (xv , , t ), is a solution of d f dCi the Vlasov equation, i.e. fCC(12 , ...) 0, dti Ci dt because,,y by the definition of constant of the motion, dC i CCddxv C0. dtt ii dt dt v i The equilibrium solution ((yp)denoted by subscript "0") of interest to us is a steady-state solution formed of constants of the motion that do not depend explicitly on tffCC, i.e. 0012( , ...) with CCii(xv , ). 9 6.2 Equilibrium Solutions (continued) Examples of Constants of the Motion : 1. If BE00 0, vvx , yz, and v are constants of the motion . 2. If EBe0000, Bconstvvz ., , z are constants of the motion. 3. The motion of a charged particle (mass mq and charge ) in EM fie lds (represen te d by b pot enti al s A and ) idbL'is governed by Lagrange's equation: [Goldstein, Poole, & Safko, "Classical Mechanics," 3rd ed., p. 21 ] q d LL, i 1, 2, 3 (13) mq d v EvB (13) dt qq dt c ii See Goldstein, Poole, & qi is a position coordinate. Safko, Sec. 1.5. where q Lmv1 2 vA qL [: Lagrangian; v: particle velocity] 2 c In cylindrical coordinates, we have qrzqrzii( , , ), ( , , ), 22222 v r r z, and vA rArz r A zA . If the fields A and are in depen dent o f , (13) gi ives d LL 0 . H ence, dt q L mrv rA const [canonical angular momentum] (14) c 10 6.2 Equilibrium Solutions (continued) Examples of Equilibrium Solutions: In contrast to the Boltzmann equation, which has only one equilibrium solution (the Maxwellian distribution), the Vlasov equation has an infinite number of possible equilibrium solutions. But they exist on a time scale short compared with the collision time. The choice of the equilibrium solution depends on how the plasma is formed. For example, if we inject two counter streaming electrons of velilocity v0 ez and v0 ez into a neutralizi ng bkbackgroun d of cold ions, we have the following equilibrium solutions for the electrons and ions (of equal density n0): f1 nv( ) ( v )[ ( vv ) ( vv )] e0 2 000xy z z (15) fnvvvi00 ()()()xyz which correctly represent the electron/ion distributions on a time scale short compared with the collision time. 11 6.2 Equilibrium Solutions (continued) Given sufficient time, collisions will first randomize electron velocities and eventually equalize elect ronandiontemperaturesron and ion temperatures. The fianl state will be an equilibrium solution in the form of the Maxwellian distribution for both the electrons and ions: 22 mmvv3/2 n0 fv00( ) n ( ) exp( )3/2 3 exp(2 ) (16) 22kT kT (2 ) vT 2vT where vkTmT / is the thermal speed, T is the same for both species, mmei fthltfor the electrons and mm f fthior the ions.
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