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Lecture 7: Moments of the Boltzmann Equation

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Lecture 7: Moments of the Boltzmann Equation !"#$%&'()%"*#%*+,-./-*+01.2(.*3+456789* !"#$%&"'()'*+,"-$.'+/'0+1$2,3--4513.+6'78%39+-' Dr. Peter T. Gallagher Astrophysics Research Group Trinity College Dublin :2;$-$(01*%<*=,-./-*=0;"%/;"-* !"#$%#&!'()*%()#& +,)-"(./#012"(& 4,$50,671*)& 9*"5:%#))& 3*1*)& 87)21*)& ;<7#1*)& 3*%()5$&*=&9*"5:%#))& ;<7#1*)& +,)-"(&>7,?& 37"1/"(&& 3@4& >7,?$& >%/;"#.*%<*?%,#@/-""AB,-.%C*DE'-)%"* o Under certain assumptions not necessary to obtain actual distribution function if only interested in the macroscopic values. o Instead of solving Boltzmann or Vlasov equation for distribution function and integrating, can take integrals over collisional Boltzmann-Vlasov equation and solve for the quantities of interest. "f q "f & "f ) +v #$f + [E +(v % B)]# = ( + "t m "v ' "t *coll (7.1) o Called “taking the moments of the Boltzmann-Vlasov equation” . ! o Resulting equations known as the macroscopic transport equations, and form the foundation of plasma fluid theory. o Results in derivation of the equations of magnetohydrodynamics (MHD). F;$%#0A%$&;$*>%/;"#G*H%")"'2#1*DE'-)%"* o Lowest order moment obtained by integrating Eqn. 7.1: "f q "f ' "f * # dv + # v $%fdv + # [E +(v & B)]$ dv = # ) , dv "t m "v ( "t +coll The first term gives "f " "n o # dv = # fdv = (7.2) "t "t "t ! o Since v and r are independent, v is not effected by gradient operator: ! $ v " #fdv = # " $ vfdv o From before, the first order moment of distribution function is ! 1 u = " vf (r,v,t)dv n therefore, $ v " #fdv = # "(nu) (7.3) ! ! F;$%#0A%$&;$*>%/;"#G*H%")"'2#1*DE'-)%"* o For the third term, consider E and B separately. E term vanishes as #f # $ E " dv = $ "( fE)dv = $ fE "dS = 0 (7.4a) #v #v using Gauss’ Divergence Theorem in velocity space. The surface area in velocity space goes as v2. As v -> !, f ->0 more quickly than S ->! (e.g., f typically goes as 2 1/v2. A Maxwellian! goes as e " v ). Integral to v = infinity therefore goes to zero. o Using vector identity " # ( a A ) = A # " a + a " # A the v ! B term is ! $f $ $ % (v " B)# dv = % #( fv " B)dv & % f #(v " B)dv ! $v $v $v $ = % f (v " B)#dS& % f #(v " B)dv = 0 (7.4b) $v o The first term on right again vanishes as f ->0 more quickly than S ->!. The second! vanishes as v ! B is perpendicular to " / " v . ! F;$%#0A%$&;$*>%/;"#G*H%")"'2#1*DE'-)%"* o Last term is on right hand side of Eqn. 7.1: # "f & " ) % ( dv = [ ) fdv] = 0 (7.5) $ "t 'coll "t o This is assuming that the total number of particles remains constant as collisions proceed. ! o Combining Eqns. 7.2 – 7.5 yields the equation of continuity: "n + # $(nu) = 0 (7.6) "t o First term represents rate of change of particle concentration within a volume, while the second term represents the divergence of particles of the flow of particles out of the volume. ! o Eqn. 7.6 is the first of the equations of magnetohydrodynamics (MHD). Eqn. 7.6 is a continuity equation for mass or charge transport if we multiply m or q. I2$.#A%$&;$*>%/;"#G*>%/;"#'/*J$-".=%$#* o Re-write Eqn. 7.1: "f q "f & "f ) +v #$f + [E +(v % B)]# = ( + "t m "v ' "t *coll o Next moment of the Boltzmann equation is obtained by multiplying Eqn. 7.1 by mv and integrating over dv. ! "f "f ' "f * m # v dv + m # v(v $%) fdv + q # v[E +(v & B)]$ dv = # mv) , dv (7.7) "t "v ( "t +coll o The right-hand side is the change of the momentum due to collisions and will be ! given the term Pij . "f " o The first term gives m # v dv = m # vfdv "t "t "(nu) (7.8) = m "t ! I2$.#A%$&;$*>%/;"#G*>%/;"#'/*J$-".=%$#* o Next consider third term: $f $ % v[E +(v " B)]# dv = % #[ fv(E + v " B)]dv $v $v $ & % fv #(E + v " B)dv $v $ & % f (E + v " B)# vdv $v o The first to integrals on the right vanish for same reasons as before. ! $f o Therefore have, q % v[E +(v " B)]# dv = &q % f (E + v " B)dv $v = &qn(E + u " B) (7.9) o To evaluate the second integral of Eqn. 7.7, use fact that v does not depend on gradient operator: ! $ v(v "#) fdv = $ # "( fvv)dv = " # $ fvvdv ! ! I2$.#A%$&;$*>%/;"#G*>%/;"#'/*J$-".=%$#* o Since the average of a quantity is 1/n times its weighted integral over v, we have " # $ fvvdv = " #(n < vv >) o Now separate v into average fluid velocity u and a thermal velocity w: ! v = u + w o Since u is already and average, we have " #(n < vv >) = " #(nuu)+ " #(n < ww >)+ 2" #(nu < w >) (7.10) o The average thermal velocity is zero => < w > = 0 and ! P = mn < ww > (7.11) is the stress tensor. Also called the pressure tensor or dyad. ! o P is a measure of the thermal motion in a fluid. If all particles moved with same steady velocity v, then w = 0 and thus P = 0 (i.e., a cold plasma). I2$.#A%$&;$*>%/;"#G*>%/;"#'/*J$-".=%$#* o Remaining term in Eqn. 7.7 can be written " #(nuu) = u" #(nu)+ n(u #")u (7.12) o Collecting Eqns. 7.8, 7.9, 7.11 and 7.12, we have ! " m (nu)+ mu# $(nu)+ mn(u$ #)u + # $P% qn(E + u & B) = Pij "t o Combining the first two terms, we obtain the fluid equation of motion: ! %" u ( mn' +(u #$)u* = qn(E + u + B), $ #P+ Pij (7.13) & "t ) o This describes flow of momentum – also called momentum transport equation. ! o Eqn. 7.13 is a statement of conservation of momentum and represents force balance on components of plasma. On right are the Lorentz force, pressure, and collisions. K'//-$1*%<*>%/;"#.*%<*B,-.%C*DE'-)%"* o Equations of MHD and multi-fluid theory are obtained by taking the moments of the Vlasov equation, corresponding to mass, momentum and energy. " (Vlasov equation) dv => conservation of mass " (Vlasov equation) v dv => conservation of momentum ! " (Vlasov equation) v2/2 dv => conservation of energy ! o Zeroth moment of the Vlasov equation results in the MHD mass continuity !equation (Eqn. 7.6). o First moment of Vlasov equation gives MHD momentum equation (Eqn. 7.13). o Second moment of Vlasov equation give MHD energy transport equation. :2;$-$(01*%<*=,-./-*=0;"%/;"-* !"#$%#&!'()*%()#& +,)-"(./#012"(& 4,$50,671*)& 9*"5:%#))& 3*1*)& 87)21*)& ;<7#1*)& 3*%()5$&*=&9*"5:%#))& ;<7#1*)& +,)-"(&>7,?& 37"1/"(&& 3@4& >7,?$&.
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