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8.S: Summary

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References

  • H. Smith and H. H. Jensen, Transport Phenomena (Oxford, 1989) An outstanding, thorough, and pellucid presentation of the theory of Boltzmann transport in classical and quantum systems.
  • P. L. Krapivsky, S. Redner, and E. Ben-Naim, A Kinetic View of Statistical Physics (Cambridge, 2010) Superb, modern discussion of a broad variety of issues and models in nonequilibrium statistical physics.
  • E. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics (Pergamon, 1981) Volume 10 in the famous Landau and Lifshitz Course of Theoretical Physics. Surprisingly readable, and with many applications (some advanced).
  • M. Kardar, Statistical Physics of Particles (Cambridge, 2007) A superb modern text, with many insightful presentations of key concepts. Includes a very instructive derivation of the Boltzmann equation starting from the BBGKY hierarchy.
  • J. A. McLennan, Introduction to Non-equilibrium Statistical Mechanics (Prentice-Hall, 1989) Though narrow in scope, this book is a good resource on the Boltzmann equation.
  • F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, 1987) This has been perhaps the most popular undergraduate text since it first appeared in 1967, and with good reason. The later chapters discuss transport phenomena at an undergraduate level.
  • N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (3rd edition, North-Holland, 2007) This is a very readable and useful text. A relaxed but meaty presentation.

Summary

Boltzmann equation: The full phase space distribution for a Hamiltonian system, ϱ(φ,t), where φ=({qσ},{pσ}), satisfies ˙ϱ+˙φϱ=0. This is not true, however, for the one-particle distribution f(q,p,t). Rather, ˙f is related to two-, three-, and higher order particle number distributions in a chain of integrodifferential equations known as the BBGKY hierarchy. We can lump our ignorance of these other terms into a collision integral and write \[{\pz f\over\pz t}=\stackrel

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{\overbrace{\coll}}\ .\] In the absence of collisions, the distribution evolves solely due to the streaming term with ˙r=p/m and ˙p=Uext . If ˙p=Fext is constant, we have the general solution f(r,p,t)=ϕ(rptm+Fextt22m,pFexttm) , valid for any initial condition f(r,p,t=0)=ϕ(r,p). We write the convective derivative as DDt=t+˙rr+˙pp. Then the Boltzmann equation may be written DfDt=(ft)coll.

Collisions: We are concerned with two types of collision processes: single-particle scattering, due to a local potential, and two-particle scattering, due to interparticle forces. Let Γ denote the set of single particle kinematic variables, Γ=(px,py,pz) for point particles and Γ=(p,L) for diatomic molecules. Then (ft)coll=dΓ{w(Γ|Γ)f(r,Γ;t)w(Γ|Γ)f(r,Γ;t)} for single particle scattering, and (ft)coll=dΓ1dΓdΓ1{w(ΓΓ1|ΓΓ1)f2(r,Γ;r,Γ1;t)w(ΓΓ1|ΓΓ1)f2(r,Γ;r,Γ1;t)}dΓ1dΓdΓ1{w(ΓΓ1|ΓΓ1)f(r,Γ;t)f(r,Γ1;t)w(ΓΓ1|ΓΓ1)f(r,Γ;t)f(r,Γ1;t)} . for two-body scattering, where f2 is the two-body distribution, and where the approximation f2(r,Γ;r,Γ;t)f(r,Γ;t)f(r,Γ;t) in the second line closes the equation. A quantity A(r,Γ) which is preserved by the dynamics between collisions then satisfies dAdtddtddrdΓA(r,Γ)f(r,Γ,t)=ddrdΓA(r,Γ)(ft)coll . Quantities which are conserved by collisions satisfy ˙A=0 and are called collisional invariants. Examples include A=1 (particle number), A=p (linear momentum, if translational invariance applies), and A=εp (energy).

Time reversal, parity, and detailed balance: With Γ=(p,L), we define the actions of time reversal and parity as \boldsymbol{\Gamma^\sss{T}=(-\Bp,-\BL) \qquad,\qquad \Gamma^\sss{P}=(-\Bp,\BL) \qquad,\qquad \Gamma^{\sss{C}}=(\Bp,-\BL)\ ,} where C=PT is the combined operation. Time reversal symmetry of the underlying equations of motion requires \boldsymbol{w\big(\Gamma'\Gamma'_1 \, | \, \Gamma\Gamma_1\big)= w\big(\Gamma^\sss{T}\Gamma^\sss{T}_1 \, | \,\Gamma'{}^\sss{T}\Gamma'_1{}^\sss{T}\big)}. Under conditions of detailed balance, this leads to \boldsymbol{f^0(\Gamma)\,f^0(\Gamma\ns_1)=f^0(\Gamma'{}^\sss{T})\,f^0(\Gamma'_1{}^\sss{T})}, where f0 is the equilibrium distribution. For systems with both P and T symmetries, \boldsymbol{w\big(\Gamma'\Gamma'_1 \, | \, \Gamma\Gamma\ns_1\big)=w\big(\Gamma^\sss{C}\Gamma_1^\sss{C} \, | \, \Gamma'{}^\sss{C}\Gamma'_1{}^\sss{C}\big)}, whence w(p,p1|p,p1)=w(p,p1|p,p1) for point particles.

Boltzmann’s H-theorem: Let h(r,t)=dΓf(r,Γ,t)lnf(r,Γ,t). Invoking the Boltzmann equation, it can be shown that ht0, which means dHdt0, where H(t)=ddrh(r,t) is Boltzmann’s H-function. h(r,t) is everywhere decreasing or constant, due to collisions.

Weakly inhomogeneous gas: Under equilibrium conditions, f0 can be a function only of collisional invariants, and takes the Gibbs form f0(r,p)=Ce(μ+VpεΓ)/kBT. Assume now that μ, V, and T are all weakly dependent on r and t. f0 then describes a local equilibrium and as such is annihilated by the collision term in the Boltzmann equation, but not by the streaming term. Accordingly, we seek a solution f=f0+δf. A lengthy derivation results in {εΓhTvT+mvαvβQαβεΓh+TcpcV/kBVFextv}f0kBT+δft=(ft)coll , where v=εp is the particle velocity, h is the enthalpy per particle, Qαβ=12(Vαxβ+Vβxα), and Fext is an external force. For an ideal gas, h=cpT. The RHS is to be evaluated to first order in δf. The simplest model for the collision integral is the relaxation time approximation, where (ft)coll=δfτ. Note that this form does not preserve any collisional invariants. The scattering time is obtained from the relation nˉvrelστ=1, where σ is the two particle total scattering cross section and ˉvrel is the average relative speed of a pair of particles. This says that there is on average one collision within a tube of cross sectional area σ and length ˉvrelτ. For the Maxwellian distribution, \({\bar v}\ns_{rel}=\sqrt{2}\,{\bar v}=\sqrt

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\), so τ(T)T1/2. The mean free path  is defined as =ˉvτ=12nσ.

Transport coefficients: Assuming Fextα=Qαβ=0 and steady state, Eq. [bwig] yields δf=τ(εcpT)kBT2(vT)f0 . The energy current is given by jαε=dΓεΓvαδf=thermal conductivity καβ|nτkBT2vαvβεΓ(εΓcpT)Txβ . For a monatomic gas, one finds καβ=κδαβ with κ(T)=π8nˉvcpT1/2. A similar result follows by considering any intensive quantity ϕ which is spatially dependent through the temperature T(r). The ϕ-current across the surface z=0 is jϕ=nˆzvz>0d3vP(v)vzϕ(zcosθ)+nˆzvz<0d3vP(v)vzϕ(z+cosθ)=13nˉvϕzˆz . Thus, jϕ=KT, with K=13nˉvϕT the associated transport coefficient. If ϕ=εΓ, then ϕT=cp, yielding κ=13nˉvcp. If ϕ=px, then jzpx=Πxz=13nmˉvVxzηVxz, where η is the shear viscosity. Using the Boltzmann equation in the relaxation time approximation, one obtains η=π8nmˉv. From κ and η, we can form a dimensionless quantity Pr=ηcp/mκ, known as the Prandtl number. Within the relaxation time approximation, Pr=1. Most monatomic gases have Pr23.

Linearized Boltzmann equation: To go beyond the phenomenological relaxation time approximation, one must grapple with the collision integral, (ft)coll=d3p1d3pd3p1 w(p,p1|p,p1){f(p)f(p1)f(p)f(p1)} , which is a nonlinear functional of the distribution f(p,t) (we suppress the t index here). Writing f(p)=f0(p)+f0(p)ψ(p), we have (ft)coll=f0(p)ˆLψ+O(ψ2), with ˆLψ(p)=d3p1dΩ|vv1|σΩf0(p1){ψ(p)+ψ(p1)ψ(p)ψ(p1)} . The linearized Boltzmann equation (LBE) then takes the form (ˆLt)ψ=Y, where Y=1kBT{ε(p)52kBTTvT+mvαvβQαβkBε(p)cVVFv} . for point particles. To solve the LBE, we must invert the operator ˆLt. Various useful properties follow from defining the inner product ψ1|ψ2d3pf0(p)ψ1(p)ψ2(p), such as the self-adjointness of ˆL: ψ1|ˆLψ2=ˆLψ1|ψ2. We then have ˆL|ϕn=λn|ϕn, with ϕm|ϕn=δmn and real eigenvalues λn. There are five zero eigenvalues corresponding to the collisional invariants: ϕ1(p)=1n,ϕ2,3,4(p)=pαnmkBT,ϕ5(p)=23n(ε(p)kBT32) . When Y=0, the formal solution to ψt=ˆLψ is ψ(p,t)=nCnϕn(p)eλnt. Aside from the collisional invariants, all the eigenvalues λn must be positive, corresponding to relaxation to the equilibrium state. One can check that the particle, energy, and heat currents are given by j=v|ψ, jε=vε|ψ, and jq=v(εμ)|ψ.

In steady state, the solution to ˆLψ=Y is ψ=ˆL1Y. This is valid provided Y is orthogonal to each of the collisional invariants, in which case ψ(p)=nCIλ1nϕn|Yϕn(p) . Once we have |ψ, we may obtain the various transport coefficients by computing the requisite currents. For example, to find the thermal conductivity κ and shear viscosity η, κ:Y=1kBT2TxXκ,Xκ(ε52kB)vxκ=Xκ|ψT/xNNη:Y=mkBTVxyXη,Xηvxvyη=mXη|ψVx/y .

Variational approach: The Schwarz inequality, ψ|ˆL|ψϕ|ˆH|ϕϕ|ˆH|ψ2, holds for the positive semidefinite operator ˆHˆL. One therefore has κ1kBT2ϕ|Xκ2ϕ|ˆH|ϕ,ηm2kBTϕ|Xη2ϕ|ˆH|ϕ . Using variational functions ϕκ=(ε52kBT)vx and ϕη=vxvy, one finds, after tedious calculations, κ75kB64πd2(kBTm)1/2,η5(mkBT)1/216πd2 . Taking the lower limit in each case, we obtain a Prandtl number Pr=ηcpmκ=23, which is close to what is observed for monatomic gases.

Quantum transport: For quantum systems, the local equilibrium distribution is of the Bose-Einstein or Fermi-Dirac form, f0(r,k,t)={exp(ε(k)μ(r,t)kBT(r,t))1}1 , with k=p/, and (ft)coll=d3k1(2π)3d3k(2π)3d3k1(2π)3 w{ff1(1±f)(1±f1)ff1(1±f)(1±f1)} where w=w(k,k1|k,k1), f=f(k), f1=f(k1), f=f(k), and f1=f(k1), and where we have assumed time-reversal and parity symmetry. The most important application is to electron transport in metals and semiconductors, in which case f0 is the Fermi distribution. With f=f0+δf, one has, within the relaxation time approximation, δftecv×Bδfkv[e\boldmath{E}+εμTT]f0ε=δfτ , where \boldmath{E}=(ϕμ/e)=Ee1μ is the gradient of the ‘electrochemical potential’ ϕe1μ. For steady state transport with B=0, one has j=2eˆΩd3k(2π)3vδfL11\boldmath{E}L12Tjq=2ˆΩd3k(2π)3(εμ)vδfL21\boldmath{E}L22T where Lαβ11=e2Jαβ0, Lαβ21=TLαβ12=eJαβ1, and Lαβ22=1TJαβ2, with Jαβn14π3dετ(ε)(εμ)n(f0ε)dSεvαvβ|v| . These results entail \boldmath{E}=ρj+QT,jq=\boldmath{}jκT , or, in terms of the Jn, ρ=1e2J10,Q=1eTJ10J1,\boldmath{}=1eJ1J10,κ=1T(J2J1J10J1) . These results describe the following physical phenomena:

(T=B=0): An electrical current j will generate an electric field \boldmath{E}=ρj, where ρ is the electrical resistivity.

(T=B=0): An electrical current j will generate an heat current jq=j, where is the Peltier coefficient.

(j=B=0): A temperature gradient T gives rise to a heat current jq=κT, where κ is the thermal conductivity.

(j=B=0): A temperature gradient T gives rise to an electric field \boldmath{E}=QT, where Q is the Seebeck coefficient.

For a parabolic band with effective electron mass m, one finds \boldsymbol{\rho={m^*\over ne^2\tau} \quad,\quad Q=-{\pi^2 k_\ssr{B}^2 T\over 2 e\,\ve\ns_\ssr{F}} \quad,\quad \kappa = {\pi^2 n\tau k_\ssr{B}^2 T\over 3m^*}} with =TQ, where \boldsymbol{\ve\ns_\ssr{F}} is the Fermi energy. The ratio κ/σT=π23(kB/e)2=2.45×108V2K2 is then predicted to be universal, a result known as the Wiedemann-Franz law. This also predicts all metals to have negative thermopower, which is not the case. In the presence of an external magnetic field B, additional transport effects arise:

(Tx=Ty=jy=0): An electrical current j=jxˆx and a field B=Bzˆz yield an electric field \boldmath{E}. The Hall coefficient is RH=Ey/jxBz.

(Tx=jy=jq,y=0): An electrical current j=jxˆx and a field B=Bzˆz yield a temperature gradient Ty. The Ettingshausen coefficient is P=Ty/jxBz.

(jx=jy=Ty=0): A temperature gradient T=Txˆx and a field B=Bzˆz yield an electric field \boldmath{E}. The Nernst coefficient is Λ=Ey/TxBz.

(jx=jy=Ey=0): A temperature gradient T=Txˆx and a field B=Bzˆz yield an orthogonal gradient Ty. The Righi-Leduc coefficient is L=Ty/TxBz.

Stochastic processes: Stochastic processes involve a random element, hence they are not wholly deterministic. The simplest example is the Langevin equation for Brownian motion, ˙p+γp=F+η(t), where p is a particle’s momentum, γ a damping rate due to friction, F an external force, and η(t) a stochastic random force. We can integrate this first order equation to obtain p(t)=p(0)eγt+Fγ(1eγt)+t0dsη(s)eγ(st) . We assume that the random force η(t) has zero mean, and furthermore that η(s)η(s)=ϕ(ss)Γδ(ss) , in which case one finds p2(t)=p(t)2+Γ2γ(1e2γt). If there is no external force, we expect the particle thermailzes at long times, p22m=12kBT. This fixes Γ=2γmkBT, where m is the particle’s mass. One can integrate again to find the position. At late times tγ1, one finds x(t)=const.+Ftγm , corresponding to a mean velocity p/m=F/γ. The RMS fluctuations in position, however, grow as x2(t)x(t)2=2kBTtγm2Dt , where D=kBT/γm is the diffusion constant. Thus, after the memory of the initial conditions is lost (tγ1), the mean position advances linearly in time due to the external force, and the RMS fluctuations in position also increase linearly.

Fokker-Planck equation: Suppose x(t) is a stochastic variable, and define δx(t)x(t+δt)x(t) . Furthermore, assume δx(t)=F1(x(t))δt and [δx(t)]2=F2(x(t))δt, but that [δx(t)]nO(δt2) for n>2. One can then show that the probability density P(x,t)=δ(xx(t)) satisfies the Fokker-Planck equation, Pt=x[F1(x)P(x,t)]+122x2[F2(x)P(x,t)] . For Brownian motion, F1(x)=F/γmu and F2(x)=2D. The resulting Fokker-Planck equation is then Pt=uPx+DPxx, where Pt=Pt , Pxx=2Px2 , The Galilean transformation xxut then results in Pt=DPxx, which is known as the diffusion equation, a general solution to which is given by P(x,t)=dxK(xx,tt)P(x,t), where K(Δx,Δt)=(4πDΔt)1/2e(Δx)2/4DΔt is the diffusion kernel. Thus, \boldsymbol{\RDelta x\ns_\ssr{RMS}=\sqrt{2D\RDelta t}}.

Endnotes

  1. Indeed, any arbitrary function of p alone would be a solution. Ultimately, we require some energy exchanging processes, such as collisions, in order for any initial nonequilibrium distribution to converge to the Boltzmann distribution.
  2. Recall from classical mechanics the definition of the Poisson bracket, {A,B}=ArBpBrAp. Then from Hamilton’s equations ˙r=Hp and ˙p=Hr, where H(p,r,t) is the Hamiltonian, we have dAdt={A,H}. Invariants have zero Poisson bracket with the Hamiltonian.
  3. See Lifshitz and Pitaevskii, Physical Kinetics, §2.
  4. The function g(x)=xlnxx+1 satisfies g(x)=lnx, hence g(x)<0 on the interval x[0,1) and g(x)>0 on x(1,]. Thus, g(x) monotonically decreases from g(0)=1 to g(1)=0, and then monotonically increases to g()=, never becoming negative.
  5. In the chapter on thermodynamics, we adopted a slightly different definition of cp as the heat capacity per mole. In this chapter cp is the heat capacity per particle.
  6. Here we abbreviate QDC for ‘quick and dirty calculation’ and BRT for ‘Boltzmann equation in the relaxation time approximation’.
  7. The difference is trivial, since p=mv.
  8. See the excellent discussion in the book by Krapivsky, Redner, and Ben-Naim, cited in §8.1.
  9. The requirements of an inner product f|g are symmetry, linearity, and non-negative definiteness.
  10. We neglect interband scattering here, which can be important in practical applications, but which is beyond the scope of these notes.
  11. The transition rate from |k to |k is proportional to the matrix element and to the product f(1f). The reverse process is proportional to f(1f). Subtracting these factors, one obtains ff, and therefore the nonlinear terms felicitously cancel in Equation [qobc].
  12. In this section we use j to denote electrical current, rather than particle number current as before.
  13. To create a refrigerator, stick the cold junction inside a thermally insulated box and the hot junction outside the box.
  14. Note that it is Ej and not \boldmath{E}j which is the source term in the energy continuity equation.
  15. Remember that physically the fixed quantities are temperature and total carrier number density (or charge density, in the case of electron and hole bands), and not temperature and chemical potential. An equation of state relating n, μ, and T is then inverted to obtain μ(n,T), so that all results ultimately may be expressed in terms of n and T.
  16. The cgs unit of viscosity is the Poise (P). 1P=1g/cms.
  17. We further demand βn=0=0 and P1(t)=0 at all times.
  18. A discussion of measure for functional integrals is found in R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals.
  19. In this section, we use the notation ˆχ(ω) for the susceptibility, rather than ˆG(ω)

This page titled 8.S: Summary is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Daniel Arovas.

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