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)3/2∫d3uexp(−mu22kBT)×{1+mkBTu⋅v+23(mu22kBT−32)(mv22kBT−32)}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)=∑rℓmarℓm(t)Srℓ+12(x)xℓ/2Yℓm(ˆn), with x=mv2/2kBT, and thus express the action of the linearized Boltzmann operator algebraically on the expansion coefficients arℓm(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)=n∑m=0Γ(α+n+1)(−x)mΓ(α+m+1)(n−m)!m! , and satisfy the generalized orthogonality relation ∞∫0dxe−xxαSnα(x)Sn′α(x)=Γ(α+n+1)n!δ†nn′ .
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=−γ∑rℓmarℓm(t)Srℓ+12(x)xℓ/2Yℓm(ˆn) + γ2π3/2∑rℓmarℓm(t)∞∫0dx†1x1/21e−x†1×∫dˆn∗1[1+2x1/2x1/21ˆn⋅ˆn∗1+23(x−32)(x†1−32)]Srℓ+12(x†1)xℓ/21Yℓm(ˆn1) , where we’ve used u=√2kBTmx1/21,du=√kBT2mx−1/21dx†1 . Now recall Y00(ˆn)=1√4π and Y11(ˆn)=−√38πsinθeiφY10(ˆn)=√34πcosθY1−1(ˆn)=+√38πsinθe−iφN∑NS01/2(x)=1S03/2(x)=1S11/2(x)=32−x , which allows us to write N∑N1=4πY00(ˆn)Y00∗(ˆn1)ˆn⋅ˆn1=4π3[Y10(ˆn)Y10∗(ˆn1)+Y11(ˆn)Y11∗(ˆn1)+Y1−1(ˆn)Y1−1∗(ˆn1)] . We can do the integrals by appealing to the orthogonality relations for the spherical harmonics and Sonine polynomials: N∑N∫dˆnYℓm(ˆn)Yl′m′∗(ˆn)=δll′δmm′∞∫0dxe−xxαSnα(x)Sn′α(x)=Γ(n+α+1)Γ(n+1)δnn′ . Integrating first over the direction vector ˆn1, Lψ=−γ∑rℓmarℓm(t)Srℓ+12(x)xℓ/2Yℓm(ˆn)+2γ√π∑rℓmarℓm(t)∞∫0dx†1x1/21e−x†1∫dˆn∗1[Y00(ˆn)Y00∗(ˆn1)S01/2(x)S01/2(x†1)+23x1/2x1/211∑m′=−1Y1m′(ˆn)Y1m′∗(ˆn1)S03/2(x)S03/2(x†1)+23Y00(ˆn)Y00∗(ˆn1)S11/2(x)S11/2(x†1)]Srℓ+12(x†1)xℓ/21Yℓm(ˆn1) , we obtain the intermediate result Lψ=−γ∑rℓmarℓm(t)Srℓ+12(x)xℓ/2Yℓm(ˆn)+2γ√π∑rℓmarℓm(t)∞∫0dx†1x1/21e−x†1[Y00(ˆn)δl0δm0S01/2(x)S01/2(x†1)+23x1/2x1/211∑m′=−1Y1m′(ˆn)δl1δmm′S03/2(x)S03/2(x†1)+23Y00(ˆn)δl0δm0S11/2(x)S11/2(x†1)]Srℓ+12(x†1)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 x†1. For the first term in brackets, we invoke the orthogonality relation with n=0 and α=12, giving Γ(32)=12√π. 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=−γ∑rℓm′arℓm(t)Srℓ+12(x)xℓ/2Yℓm(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 ψrℓm(v)=NrℓmSrℓ+12(x)xℓ/2Yℓm(ˆn), is an eigenfunction of L with eigenvalue −γ if (r,ℓ,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 arℓm is (La)rℓm={−γarℓmif (r,ℓ,m)∉CI=0if (r,ℓ,m)∈CI The quantity τ=γ−1 is the relaxation time.
It is pretty obvious that L is self-adjoint, since ⟨ϕ|Lψ⟩≡∫d3vf0(v)ϕ(v)L[ψ(v)]=−γn(m2πkBT)3/2∫d3vexp(−mv22kBT)ϕ(v)ψ(v)+γn(m2πkBT)3∫d3v∫d3uexp(−mu22kBT)exp(−mv22kBT)×ϕ(v)[1+mkBTu⋅v+23(mu22kBT−32)(mv22kBT−32)]ψ(u)=⟨Lϕ|ψ⟩ , where n is the bulk number density and f0(v) is the Maxwellian velocity distribution.