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8.10: Appendix I- Boltzmann Equation and Collisional Invariants

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Problem : The linearized Boltzmann operator Lψ is a complicated functional. Suppose we replace L by L, where Lψ=γψ(v,t)+γ(m2πkBT)3/2d3uexp(mu22kBT)×{1+mkBTuv+23(mu22kBT32)(mv22kBT32)}ψ(u,t) .

Show that L shares all the important properties of L. What is the meaning of γ? Expand ψ(v,t) in spherical harmonics and Sonine polynomials, ψ(v,t)=rmarm(t)Sr+12(x)x/2Ym(ˆn), with x=mv2/2kBT, and thus express the action of the linearized Boltzmann operator algebraically on the expansion coefficients arm(t).

The Sonine polynomials Snα(x) are a complete, orthogonal set which are convenient to use in the calculation of transport coefficients. They are defined as Snα(x)=nm=0Γ(α+n+1)(x)mΓ(α+m+1)(nm)!m! , and satisfy the generalized orthogonality relation 0dxexxαSnα(x)Snα(x)=Γ(α+n+1)n!δnn .

Solution : The ‘important properties’ of L are that it annihilate the five collisional invariants, 1, v, and v2, and that all other eigenvalues are negative. That this is true for L can be verified by an explicit calculation.

Plugging the conveniently parameterized form of ψ(v,t) into L, we have Lψ=γrmarm(t)Sr+12(x)x/2Ym(ˆn) + γ2π3/2rmarm(t)0dx1x1/21ex1×dˆn1[1+2x1/2x1/21ˆnˆn1+23(x32)(x132)]Sr+12(x1)x/21Ym(ˆn1) , where we’ve used u=2kBTmx1/21,du=kBT2mx1/21dx1 . Now recall Y00(ˆn)=14π and Y11(ˆn)=38πsinθeiφY10(ˆn)=34πcosθY11(ˆn)=+38πsinθeiφNNS01/2(x)=1S03/2(x)=1S11/2(x)=32x , which allows us to write NN1=4πY00(ˆn)Y00(ˆn1)ˆnˆn1=4π3[Y10(ˆn)Y10(ˆn1)+Y11(ˆn)Y11(ˆn1)+Y11(ˆn)Y11(ˆn1)] . We can do the integrals by appealing to the orthogonality relations for the spherical harmonics and Sonine polynomials: NNdˆnYm(ˆn)Ylm(ˆn)=δllδmm0dxexxαSnα(x)Snα(x)=Γ(n+α+1)Γ(n+1)δnn . Integrating first over the direction vector ˆn1, Lψ=γrmarm(t)Sr+12(x)x/2Ym(ˆn)+2γπrmarm(t)0dx1x1/21ex1dˆn1[Y00(ˆn)Y00(ˆn1)S01/2(x)S01/2(x1)+23x1/2x1/211m=1Y1m(ˆn)Y1m(ˆn1)S03/2(x)S03/2(x1)+23Y00(ˆn)Y00(ˆn1)S11/2(x)S11/2(x1)]Sr+12(x1)x/21Ym(ˆn1) , we obtain the intermediate result Lψ=γrmarm(t)Sr+12(x)x/2Ym(ˆn)+2γπrmarm(t)0dx1x1/21ex1[Y00(ˆn)δl0δm0S01/2(x)S01/2(x1)+23x1/2x1/211m=1Y1m(ˆn)δl1δmmS03/2(x)S03/2(x1)+23Y00(ˆn)δl0δm0S11/2(x)S11/2(x1)]Sr+12(x1)x1/21.

Appealing now to the orthogonality of the Sonine polynomials, and recalling that Γ(12)=π,Γ(1)=1,Γ(z+1)=zΓ(z) , we integrate over x1. For the first term in brackets, we invoke the orthogonality relation with n=0 and α=12, giving Γ(32)=12π. For the second bracketed term, we have n=0 but α=32, and we obtain Γ(52)=32Γ(32), while the third bracketed term involves leads to n=1 and α=12, also yielding Γ(52)=32Γ(32). Thus, we obtain the simple and pleasing result Lψ=γrmarm(t)Sr+12(x)x/2Ym(ˆn) where the prime on the sum indicates that the set CI={(0,0,0) ,(1,0,0) ,(0,1,1) ,(0,1,0) ,(0,1,1)} are to be excluded from the sum. But these are just the functions which correspond to the five collisional invariants! Thus, we learn that ψrm(v)=NrmSr+12(x)x/2Ym(ˆn), is an eigenfunction of L with eigenvalue γ if (r,,m) does not correspond to one of the five collisional invariants. In the latter case, the eigenvalue is zero. Thus, the algebraic action of L on the coefficients arm is (La)rm={γarmif (r,,m)CI=0if (r,,m)CI The quantity τ=γ1 is the relaxation time.

It is pretty obvious that L is self-adjoint, since ϕ|Lψd3vf0(v)ϕ(v)L[ψ(v)]=γn(m2πkBT)3/2d3vexp(mv22kBT)ϕ(v)ψ(v)+γn(m2πkBT)3d3vd3uexp(mu22kBT)exp(mv22kBT)×ϕ(v)[1+mkBTuv+23(mu22kBT32)(mv22kBT32)]ψ(u)=Lϕ|ψ , where n is the bulk number density and f0(v) is the Maxwellian velocity distribution.


This page titled 8.10: Appendix I- Boltzmann Equation and Collisional Invariants is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Daniel Arovas.

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