8.6: Linearized Boltzmann Equation
( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand\Dalpha
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[1], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\Dbeta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[2], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\Dgamma
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[3], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\Ddelta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[4], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\Depsilon
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[5], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\Dvarepsilon
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[6], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\Dzeta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[7], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\Deta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[8], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\Dtheta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[9], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\Dvartheta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[10], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\Diota
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[11], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\Dkappa
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[12], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\Dlambda
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[13], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\Dvarpi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[14], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\DGamma
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[15], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\DDelta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[16], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\DTheta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[17], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\Vmu
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[18], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\Vnu
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[19], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\Vxi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[20], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\Vom
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[21], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\Vpi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[22], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\Vvarpi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[23], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\Vrho
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[24], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\Vvarrho
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[25], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\Vsigma
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[26], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\Vvarsigma
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[27], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\Vtau
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[28], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\Vupsilon
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[29], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\Vphi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[30], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\Vvarphi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[31], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\Vchi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[32], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\Vpsi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[33], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\Vomega
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[34], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\VGamma
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[35], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\VDelta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[36], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\newcommand\BI{\mib I}}
\)
\newcommand { M}
\newcommand { m}
}
\( \newcommand\tcb{\textcolor{blue}\)
\( \newcommand\tcr{\textcolor{red}\)
1$#1_$
\newcommand\SZ{\textsf Z}} \( \newcommand\kFd{k\ns_{\RF\dar}\)
\newcommand\mutB{\tilde\mu}\ns_\ssr{B}
\( \newcommand\xhihOZ
Callstack: at (Template:MathJaxArovas), /content/body/div/span[1], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
\( \newcommand\labar
Callstack: at (Template:MathJaxArovas), /content/body/div/span[2], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.06:_Linearized_Boltzmann_Equation), /content/body/p/span, line 1, column 23
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
(∂f∂t)†coll=∫d3p∗1∫d3p′∫d3p′1 w(p′,p′1|p,p∗1){f(p′)f(p′1)−f(p)f(p∗1)} .
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,
(∂f∂t)†coll=f0(p)ˆLψ+O(ψ2) ,
where the action of the linearized collision operator is given by
ˆLψ=∫d3p∗1∫d3p′∫d3p′1 w(p′,p′1|p,p∗1)f0(p∗1){ψ(p′)+ψ(p′1)−ψ(p)−ψ(p∗1)}=∫d3p∗1∫dΩ|v−v∗1|∂σ∂Ωf0(p∗1){ψ(p′)+ψ(p′1)−ψ(p)−ψ(p∗1)} ,
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(p∗1)=f0(p′)f0(p′1) . 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 (ˆL−∂∂t)ψ=Y, where, for point particles, Y=1kBT{ε(p)−c∗pTTv⋅∇T+mv∗αv∗βQ∗αβ−k∗Bε(p)c∗V∇⋅V−F⋅v} . Equation [LBE] is an inhomogeneous linear equation, which can be solved by inverting the operator ˆL−∂∂t.
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, ⟨ψ∗1|ψ∗2⟩≡∫d3pf0(p)ψ∗1(p)ψ∗2(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: ⟨ψ∗1|ˆLψ∗2⟩=⟨ˆLψ∗1|ψ∗2⟩ .
We can now define the spectrum of normalized eigenfunctions of ˆL, which we write as ϕ∗n(p). The eigenfunctions satisfy the eigenvalue equation, ˆLϕ∗n=−λ∗nϕ∗n , and may be chosen to be orthonormal, ⟨ϕ∗m|ϕ∗n⟩=δ∗mn . Of course, in order to obtain the eigenfunctions ϕ∗n we must have detailed knowledge of the function w(p′,p′1|p,p∗1).
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 ϕ∗n with eigenvalue λ∗n=0. One can check that these normalized eigenfunctions are ϕ∗n(p)=1√nϕ∗p∗α(p)=p∗α√nmkBTϕ∗ε(p)=√23n(ε(p)kBT−32) .
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 ∂ψ∂t=ˆLψ . We can therefore write the most general solution in the form ψ(p,t)=∑n′C∗nϕ∗n(p)e−λ∗nt , where the prime on the sum reminds us that collisional invariants are to be excluded. All the eigenvalues λ∗n, 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 C∗n=0 for the five collisional invariants.
Recall also the particle, energy, and thermal (heat) currents, j=∫d3pvf(p)=∫d3pf0(p)vψ(p)=⟨v|ψ⟩j∗ε=∫d3pvεf(p)=∫d3pf0(p)vεψ(p)=⟨vε|ψ⟩j∗q=∫d3pv(ε−μ)f(p)=∫d3pf0(p)v(ε−μ)ψ(p)=⟨v(ε−μ)|ψ⟩ . Note j∗q=j∗ε−μj.
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 ϕ∗n and write ψ=∑nC∗nϕ∗n. Applying ˆL and taking the inner product with ϕ∗j, we have C∗j=−1λ∗j⟨ϕ∗j|Y⟩ . Thus, the formal solution to the linearized Boltzmann equation is ψ(p)=−∑n1λ∗n⟨ϕ∗n|Y⟩ϕ∗n(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=1kBT2∂T∂x⋅X∗κ , where X∗κ≡(ε−c∗pT)v∗x. Under the conditions of no particle flow (j=0), we have j∗q=−κ∂∗xTˆx. Then we have ⟨X∗κ|ψ⟩=−κ∂T∂x .
Viscosity
For the viscosity, we take Y=mkBT∂V∗x∂y⋅X∗η , with X∗η=v∗xv∗y. We then Π∗xy=⟨mv∗xv∗y|ψ⟩=−η∂V∗x∂y . Thus, ⟨X∗η|ψ⟩=−ηm∂V∗x∂y .
Variational approach
Following the treatment in chapter 1 of Smith and Jensen, define ˆH≡−ˆL. 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|ψ⟩⋅⟨ϕ|ˆH|ϕ⟩≥⟨ϕ|ˆH|ψ⟩2 , for any two Hilbert space vectors |ψ⟩ and |ϕ⟩. Consider now the above calculation of the thermal conductivity. We have ˆHψ=−1kBT2∂T∂xX∗κ and therefore κ=kBT2(∂T/∂x)2⟨ψ|ˆH|ψ⟩≥1kBT2⟨ϕ|X∗κ⟩2⟨ϕ|ˆH|ϕ⟩ . Similarly, for the viscosity, we have ˆHψ=−mkBT∂V∗x∂yX∗η , from which we derive η=kBT(∂V∗x/∂y)2⟨ψ|ˆH|ψ⟩≥m2kBT⟨ϕ|X∗η⟩2⟨ϕ|ˆH|ϕ⟩ .
In order to get a good lower bound, we want ϕ in each case to have a good overlap with X∗κ,η. One approach then is to take ϕ=X∗κ,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 η≥m2kBT⟨v∗xv∗y|v∗xv∗y⟩2⟨v∗xv∗y|ˆH|v∗xv∗y⟩ . Now the linearized collision operator ˆL acts as ⟨ϕ|ˆL|ψ⟩=∫d3pg0(p)ϕ(p)∫d3p∗1∫dΩ∂σ∂Ω|v−v∗1|f0(p∗1){ψ(p)+ψ(p∗1)−ψ(p′)−ψ(p′1)} . Here the kinematics of the collision guarantee total energy and momentum conservation, so p′ and p′1 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 ∂σ∂Ω=|d(b2/2)dsinχ| , where b is the impact parameter. The scattering angle is χ(b,u)=π−2∞∫rpdrb√r4−b2r2−2U(r)r4˜mu2 , where ˜m=12m is the reduced mass, and r∗p is the relative coordinate separation at periapsis, the distance of closest approach, which occurs when ˙r=0, 12˜mu2=ℓ22˜mr2p+U(r∗p) , where ℓ=˜mub is the relative coordinate angular momentum.
We work in center-of-mass coordinates, so the velocities are v=V+12uv′=V+12u′v∗1=V−12uv′1=V−12u′ , with |u|=|u′| and ˆu⋅ˆu′=cosχ. Then if ψ(p)=v∗xv∗y, we have Δ(ψ)≡ψ(p)+ψ(p∗1)−ψ(p′)−ψ(p′1)=12(u∗xu∗y−u′xu′y) . We may write u′=u(sinχcosφˆe∗1+sinχsinφˆe∗2+cosχˆe∗3) , where ˆe∗3=ˆu. With this parameterization, we have 2π∫0dφ12(u∗αu∗β−u′αu′β)=−πsin2χ(u2δ∗αβ−3u∗αu∗β) . Note that we have used here the relation e∗1αe∗1β+e∗2αe∗2β+e∗3αe∗3β=δ∗αβ , which holds since the LHS is a projector ∑3i=1|ˆe∗i⟩⟨ˆe∗i|.
It is convenient to define the following integral: R(u)≡∞∫0dbbsin2χ(b,u) . Since the Jacobian |det(∂v,∂v∗1)(∂V,∂u)|=1 , we have ⟨v∗xv∗y|ˆL|v∗xv∗y⟩=n2(m2πkBT)3∫d3V∫d3ue−mV2/k∗BTe−mu2/4k∗BT⋅u⋅3π2u∗xu∗y⋅R(u)⋅v∗xv∗y . This yields ⟨v∗xv∗y|ˆL|v∗xv∗y⟩=π40n2⟨u5R(u)⟩ , where ⟨F(u)⟩≡∞∫0duu2e−mu2/4k∗BTF(u)/∞∫0duu2e−mu2/4k∗BT .
It is easy to compute the term in the numerator of Equation [varvisc]: ⟨v∗xv∗y|v∗xv∗y⟩=n(m2πkBT)3/2∫d3ve−mv2/2k∗BTv2xv2y=n(kBTm)2 . Putting it all together, we find η≥40(kBT)3πm2/⟨u5R(u)⟩ .
The computation for κ is a bit more tedious. One has ψ(p)=(ε−c∗pT)v∗x, in which case Δ(ψ)=12m[(V⋅u)u∗x−(V⋅u′)u′x] . Ultimately, one obtains the lower bound κ≥150kB(kBT)3πm3/⟨u5R(u)⟩ . Thus, independent of the potential, this variational calculation yields a Prandtl number of Pr=νa=ηc∗pmκ=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=dsinϕ∗0=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)⟩=13d3⟨u5⟩=128√π(kBTm)5/2d2 and one finds η≥5(mkBT)1/216√πd2,κ≥75kB64√πd2(kBTm)1/2 .