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8.6: Linearized Boltzmann Equation

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Linearizing the collision integral

We now return to the classical Boltzmann equation and consider a more formal treatment of the collision term in the linear approximation. We will assume time-reversal symmetry, in which case

(ft)coll=d3p1d3pd3p1 w(p,p1|p,p1){f(p)f(p1)f(p)f(p1)} .

The collision integral is nonlinear in the distribution f. We linearize by writing f(p)=f0(p)+f0(p)ψ(p) ,

where we assume ψ(p) is small. We then have, to first order in ψ,

(ft)coll=f0(p)ˆLψ+O(ψ2) ,

where the action of the linearized collision operator is given by

ˆLψ=d3p1d3pd3p1 w(p,p1|p,p1)f0(p1){ψ(p)+ψ(p1)ψ(p)ψ(p1)}=d3p1dΩ|vv1|σΩf0(p1){ψ(p)+ψ(p1)ψ(p)ψ(p1)} ,

where we have invoked Equation [BEsig] to write the RHS in terms of the differential scattering cross section. In deriving the above result, we have made use of the detailed balance relation, f0(p)f0(p1)=f0(p)f0(p1) . We have also suppressed the r dependence in writing f(p), f0(p), and ψ(p).

From Equation [bwig], we then have the linearized equation (ˆLt)ψ=Y, where, for point particles, Y=1kBT{ε(p)cpTTvT+mvαvβQαβkBε(p)cVVFv} . Equation [LBE] is an inhomogeneous linear equation, which can be solved by inverting the operator ˆLt.

Linear algebraic properties of ˆL

Although ˆL is an integral operator, it shares many properties with other linear operators with which you are familiar, such as matrices and differential operators. We can define an inner product9, ψ1|ψ2d3pf0(p)ψ1(p)ψ2(p) . Note that this is not the usual Hilbert space inner product from quantum mechanics, since the factor f0(p) is included in the metric. This is necessary in order that ˆL be self-adjoint: ψ1|ˆLψ2=ˆLψ1|ψ2 .

We can now define the spectrum of normalized eigenfunctions of ˆL, which we write as ϕn(p). The eigenfunctions satisfy the eigenvalue equation, ˆLϕn=λnϕn , and may be chosen to be orthonormal, ϕm|ϕn=δmn . Of course, in order to obtain the eigenfunctions ϕn we must have detailed knowledge of the function w(p,p1|p,p1).

Recall that there are five collisional invariants, which are the particle number, the three components of the total particle momentum, and the particle energy. To each collisional invariant, there is an associated eigenfunction ϕn with eigenvalue λn=0. One can check that these normalized eigenfunctions are ϕn(p)=1nϕpα(p)=pαnmkBTϕε(p)=23n(ε(p)kBT32) .

If there are no temperature, chemical potential, or bulk velocity gradients, and there are no external forces, then Y=0 and the only changes to the distribution are from collisions. The linearized Boltzmann equation becomes ψt=ˆLψ . We can therefore write the most general solution in the form ψ(p,t)=nCnϕn(p)eλnt , where the prime on the sum reminds us that collisional invariants are to be excluded. All the eigenvalues λn, aside from the five zero eigenvalues for the collisional invariants, must be positive. Any negative eigenvalue would cause ψ(p,t) to increase without bound, and an initial nonequilibrium distribution would not relax to the equilibrium f0(p), which we regard as unphysical. Henceforth we will drop the prime on the sum but remember that Cn=0 for the five collisional invariants.

Recall also the particle, energy, and thermal (heat) currents, j=d3pvf(p)=d3pf0(p)vψ(p)=v|ψjε=d3pvεf(p)=d3pf0(p)vεψ(p)=vε|ψjq=d3pv(εμ)f(p)=d3pf0(p)v(εμ)ψ(p)=v(εμ)|ψ . Note jq=jεμj.

Steady state solution to the linearized Boltzmann equation

Under steady state conditions, there is no time dependence, and the linearized Boltzmann equation takes the form ˆLψ=Y . We may expand ψ in the eigenfunctions ϕn and write ψ=nCnϕn. Applying ˆL and taking the inner product with ϕj, we have Cj=1λjϕj|Y . Thus, the formal solution to the linearized Boltzmann equation is ψ(p)=n1λnϕn|Yϕn(p) . This solution is applicable provided |Y is orthogonal to the five collisional invariants.

Thermal conductivity

For the thermal conductivity, we take T=zTˆx, and Y=1kBT2TxXκ , where Xκ(εcpT)vx. Under the conditions of no particle flow (j=0), we have jq=κxTˆx. Then we have Xκ|ψ=κTx .

Viscosity

For the viscosity, we take Y=mkBTVxyXη , with Xη=vxvy. We then Πxy=mvxvy|ψ=ηVxy . Thus, Xη|ψ=ηmVxy .

Variational approach

Following the treatment in chapter 1 of Smith and Jensen, define ˆHˆL. We have that ˆH is a positive semidefinite operator, whose only zero eigenvalues correspond to the collisional invariants. We then have the Schwarz inequality, ψ|ˆH|ψϕ|ˆH|ϕϕ|ˆH|ψ2 , for any two Hilbert space vectors |ψ and |ϕ. Consider now the above calculation of the thermal conductivity. We have ˆHψ=1kBT2TxXκ and therefore κ=kBT2(T/x)2ψ|ˆH|ψ1kBT2ϕ|Xκ2ϕ|ˆH|ϕ . Similarly, for the viscosity, we have ˆHψ=mkBTVxyXη , from which we derive η=kBT(Vx/y)2ψ|ˆH|ψm2kBTϕ|Xη2ϕ|ˆH|ϕ .

In order to get a good lower bound, we want ϕ in each case to have a good overlap with Xκ,η. One approach then is to take ϕ=Xκ,η, which guarantees that the overlap will be finite (and not zero due to symmetry, for example). We illustrate this method with the viscosity calculation. We have ηm2kBTvxvy|vxvy2vxvy|ˆH|vxvy . Now the linearized collision operator ˆL acts as ϕ|ˆL|ψ=d3pg0(p)ϕ(p)d3p1dΩσΩ|vv1|f0(p1){ψ(p)+ψ(p1)ψ(p)ψ(p1)} . Here the kinematics of the collision guarantee total energy and momentum conservation, so p and p1 are determined as in Equation [finalps].

Now we have dΩ=sinχdχdφ , where χ is the scattering angle depicted in Fig. [scat_impact] and φ is the azimuthal angle of the scattering. The differential scattering cross section is obtained by elementary mechanics and is known to be σΩ=|d(b2/2)dsinχ| , where b is the impact parameter. The scattering angle is χ(b,u)=π2rpdrbr4b2r22U(r)r4˜mu2 , where ˜m=12m is the reduced mass, and rp is the relative coordinate separation at periapsis, the distance of closest approach, which occurs when ˙r=0, 12˜mu2=22˜mr2p+U(rp) , where =˜mub is the relative coordinate angular momentum.

[scat_impact] Scattering in the CM frame. O is the force center and P is the point of periapsis. The impact parameter is b, and \chi is the scattering angle. \phi_0 is the angle through which the relative coordinate moves between periapsis and infinity.
[scat_impact] Scattering in the CM frame. O is the force center and P is the point of periapsis. The impact parameter is b, and χ is the scattering angle. ϕ0 is the angle through which the relative coordinate moves between periapsis and infinity.

We work in center-of-mass coordinates, so the velocities are v=V+12uv=V+12uv1=V12uv1=V12u , with |u|=|u| and ˆuˆu=cosχ. Then if ψ(p)=vxvy, we have Δ(ψ)ψ(p)+ψ(p1)ψ(p)ψ(p1)=12(uxuyuxuy) . We may write u=u(sinχcosφˆe1+sinχsinφˆe2+cosχˆe3) , where ˆe3=ˆu. With this parameterization, we have 2π0dφ12(uαuβuαuβ)=πsin2χ(u2δαβ3uαuβ) . Note that we have used here the relation e1αe1β+e2αe2β+e3αe3β=δαβ , which holds since the LHS is a projector 3i=1|ˆeiˆei|.

It is convenient to define the following integral: R(u)0dbbsin2χ(b,u) . Since the Jacobian |det(v,v1)(V,u)|=1 , we have vxvy|ˆL|vxvy=n2(m2πkBT)3d3Vd3uemV2/kBTemu2/4kBTu3π2uxuyR(u)vxvy . This yields vxvy|ˆL|vxvy=π40n2u5R(u) , where F(u)0duu2emu2/4kBTF(u)/0duu2emu2/4kBT .

It is easy to compute the term in the numerator of Equation [varvisc]: vxvy|vxvy=n(m2πkBT)3/2d3vemv2/2kBTv2xv2y=n(kBTm)2 . Putting it all together, we find η40(kBT)3πm2/u5R(u) .

The computation for κ is a bit more tedious. One has ψ(p)=(εcpT)vx, in which case Δ(ψ)=12m[(Vu)ux(Vu)ux] . Ultimately, one obtains the lower bound κ150kB(kBT)3πm3/u5R(u) . Thus, independent of the potential, this variational calculation yields a Prandtl number of Pr=νa=ηcpmκ=23 , which is very close to what is observed in dilute monatomic gases (see Tab. [Prandtl]).

While the variational expressions for η and κ are complicated functions of the potential, for hard sphere scattering the calculation is simple, because b=dsinϕ0=dcos(12χ), where d is the hard sphere diameter. Thus, the impact parameter b is independent of the relative speed u, and one finds R(u)=13d3. Then u5R(u)=13d3u5=128π(kBTm)5/2d2 and one finds η5(mkBT)1/216πd2,κ75kB64πd2(kBTm)1/2 .


This page titled 8.6: Linearized Boltzmann Equation is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Daniel Arovas.

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