APPENDIX C SOME IDENTITIES SATISFIED BY THE COULOMB POTENTIAL

Description

APPENDIX C : SOME IDENTITIES SATISFIED BY THE COULOMB POTENTIAL In this appendix, we use the notations of chapter 2. We shall give several formulas which hold for Landau's collisional operator Q(f, f) with a Coulomb potential. We assume that f is smooth and rapidly decreasing, so as to allow all formal manipulations. In this case, the matrix aij is given by (1) aij(v) = ( ?ij ? vivj |v|2 ) 1 |v| . Therefore, c = ∑ ij ∂ijaij = ?8pi?, where ? is the Dirac mass at the origin, and hence c = c ? f = ?8pif . Thus, the Landau operator can be rewritten as (2) Q(f, f) = aij∂ijf + 8pif 2. As pointed out to us to by Desjardins, the matrix aij can be com- puted “eplicitly” by the use of the so-called Riesz transform. Let us define f?(?) = ∫ R3 e?iv·?f(v) dv, Rij = ?∂ij(?∆)?1, so that in Fourier space, Rij is given by R?ij(?) = ?i?j |?|2 , and Rij satisfies the identities ∑ i Rii = I, ∂ij = Rij∆ = ∆Rij, ∑ ij Rij∂ij = ∆.

• ?2 ∫


• another formal


• dirac function


• fourier transform


• r6 aij


• lan- dau operator


• r3 cf


• ij ∂ijaij


• physically rel- evant

Sujets

• Dirac delta function
• Fourier transform

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 7
• Langue: English

Extrait

APPENDIX C : SOME IDENTITIES SATISFIED BY THE COULOMB POTENTIAL

In this appendix, we use the notations of chapter 2.We shall give several formulas which hold for Landau’s collisional operatorQ(f, f) with a Coulomb potential.We assume thatfis smooth and rapidly decreasing, so as to allow all formal manipulations. In this case, the matrixa ij is given by   v i v j 1 (1)a ij (v) = ij . 2 |v| |v| P Therefore,c= ∂ ij a ij =8 , whereis the Dirac mass at the ij origin, and hencec=cf=8 f. Thus,the Landau operator can be rewritten as 2 (2)Q(f, f) =a ij ∂ ij f+ 8. f As pointed out to us to by Desjardins, the matrixa ij can be com- puted “eplicitly” by the use of the so-called Riesz transform.Let us dene Z iv
 b f() =e f(v)dv, 3 R 1 R ij =∂ ij (
), so that in Fourier space,R ij is given by  i  j b R ij () =, 2 || andR ij diehtsesitasestitien X X R ii =I, ∂ ij =R ij
=
R ij , R ij ∂ ij =
. i ij It turns out that  i  j ab ij () = 8 , 4 || so that 1 (3)a ij =a ij f=8 R ij
f. Hence the equation (2) can be rewritten as (4)    1 21 2 Q(f, f) = 8(R ij
f)∂ ij f+f= 8(R ij
f)(R ij
f) +f . 1