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=({q∗σ},{p∗σ}), satisfies ˙ϱ+˙φ⋅∇ϱ=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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∙ 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=(p∗x,p∗y,p∗z) for point particles and Γ=(p,L) for diatomic molecules. Then (∂f∂t)†coll=∫dΓ′{w(Γ|Γ′)f(r,Γ′;t)−w(Γ′|Γ)f(r,Γ;t)} for single particle scattering, and (∂f∂t)†coll=∫dΓ∗1∫dΓ′∫dΓ′1{w(ΓΓ∗1|Γ′Γ′1)f∗2(r,Γ′;r,Γ′1;t)−w(Γ′Γ′1|ΓΓ∗1)f∗2(r,Γ;r,Γ∗1;t)}≈∫dΓ∗1∫dΓ′∫dΓ′1{w(ΓΓ∗1|Γ′Γ′1)f(r,Γ′;t)f(r,Γ′1;t)−w(Γ′Γ′1|ΓΓ∗1)f(r,Γ;t)f(r,Γ∗1;t)} . for two-body scattering, where f∗2 is the two-body distribution, and where the approximation f∗2(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 dAdt≡ddt∫ddrdΓA(r,Γ)f(r,Γ,t)=∫ddrdΓA(r,Γ)(∂f∂t)†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=ε∗p (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′,p′1|p,p∗1)=w(p,p∗1|p′,p′1) 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 ∂h∂t≤0, which means dHdt≤0, 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+V⋅p−ε∗Γ)/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 {ε∗Γ−hTv⋅∇T+mv∗αv∗βQ∗αβ−ε∗Γ−h+Tc∗pc∗V/k∗B∇⋅V−Fext⋅v}f0kBT+∂δf∂t=(∂f∂t)†coll , where v=∂ε∂p is the particle velocity, h is the enthalpy per particle, Q∗αβ=12(∂Vα∂xβ+∂Vβ∂xα), and Fext is an external force. For an ideal gas, h=c∗pT. The RHS is to be evaluated to first order in δf. The simplest model for the collision integral is the relaxation time approximation, where (∂f∂t)†coll=−δfτ. Note that this form does not preserve any collisional invariants. The scattering time is obtained from the relation nˉv∗relστ=1, where σ is the two particle total scattering cross section and ˉv∗rel 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 ˉv∗relτ. For the Maxwellian distribution, \({\bar v}\ns_{rel}=\sqrt{2}\,{\bar v}=\sqrt
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∙ Transport coefficients: Assuming Fextα=Q∗αβ=0 and steady state, Eq. [bwig] yields δf=−τ(ε−c∗pT)kBT2(v⋅∇T)f0 . The energy current is given by jαε=∫dΓε∗Γvαδf=−thermal conductivity καβ⏞|nτkBT2⟨vαvβε∗Γ(ε∗Γ−c∗pT)⟩∂T∂xβ . For a monatomic gas, one finds καβ=κδαβ with κ(T)=π8nℓˉvc∗p∝T1/2. 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 j∗ϕ=nˆz∫v∗z>0d3vP(v)v∗zϕ(z−ℓcosθ)+nˆz∫v∗z<0d3vP(v)v∗zϕ(z+ℓcosθ)=−13nˉvℓ∂ϕ∂zˆz . Thus, j∗ϕ=−K∇T, with K=13nℓˉv∂ϕ∂T the associated transport coefficient. If ϕ=⟨ε∗Γ⟩, then ∂ϕ∂T=c∗p, yielding κ=13nℓˉvc∗p. If ϕ=⟨p∗x⟩, then jzp∗x=Π∗xz=−13nmℓˉv∂V∗x∂z≡−η∂V∗x∂z, where η is the shear viscosity. Using the Boltzmann equation in the relaxation time approximation, one obtains η=π8nmℓˉv. From κ and η, we can form a dimensionless quantity Pr=ηc∗p/mκ, known as the Prandtl number. Within the relaxation time approximation, Pr=1. Most monatomic gases have Pr≈23.
∙ Linearized Boltzmann equation: To go beyond the phenomenological relaxation time approximation, one must grapple with the collision integral, (∂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)} , 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 (∂f∂t)†coll=f0(p)ˆLψ+O(ψ2), with ˆLψ(p)=∫d3p∗1∫dΩ|v−v∗1|∂σ∂Ωf0(p∗1){ψ(p′)+ψ(p′1)−ψ(p)−ψ(p∗1)} . The linearized Boltzmann equation (LBE) then takes the form (ˆL−∂∂t)ψ=Y, where Y=1kBT{ε(p)−52kBTTv⋅∇T+mv∗αv∗βQ∗αβ−k∗Bε(p)c∗V∇⋅V−F⋅v} . for point particles. To solve the LBE, we must invert the operator ˆL−∂∂t. Various useful properties follow from defining the inner product ⟨ψ∗1|ψ∗2⟩≡∫d3pf0(p)ψ∗1(p)ψ∗2(p), such as the self-adjointness of ˆL: ⟨ψ∗1|ˆLψ∗2⟩=⟨ˆLψ∗1|ψ∗2⟩. We then have ˆL|ϕ∗n⟩=−λ∗n|ϕ∗n⟩, with ⟨ϕ∗m|ϕ∗n⟩=δ∗mn and real eigenvalues λ∗n. There are five zero eigenvalues corresponding to the collisional invariants: ϕ∗1(p)=1√n,ϕ∗2,3,4(p)=p∗α√nmkBT,ϕ∗5(p)=√23n(ε(p)kBT−32) . When Y=0, the formal solution to ∂ψ∂t=ˆLψ is ψ(p,t)=∑nC∗nϕ∗n(p)e−λ∗nt. Aside from the collisional invariants, all the eigenvalues λ∗n 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⟩, j∗ε=⟨vε|ψ⟩, and j∗q=⟨v(ε−μ)|ψ⟩.
In steady state, the solution to ˆLψ=Y is ψ=ˆL−1Y. This is valid provided Y is orthogonal to each of the collisional invariants, in which case ψ(p)=∑n∉CIλ−1n⟨ϕ∗n|Y⟩ϕ∗n(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=1kBT2∂T∂xX∗κ,X∗κ≡(ε−52kB)v∗x⇒κ=−⟨X∗κ|ψ⟩∂T/∂xN∑Nη:Y=mkBT∂V∗x∂yX∗η,X∗η≡v∗xv∗y⇒η=−m⟨X∗η|ψ⟩∂V∗x/∂y .
∙ Variational approach: The Schwarz inequality, ⟨ψ|−ˆL|ψ⟩⋅⟨ϕ|ˆH|ϕ⟩≥⟨ϕ|ˆH|ψ⟩2, holds for the positive semidefinite operator ˆH≡−ˆL. One therefore has κ≥1kBT2⟨ϕ|X∗κ⟩2⟨ϕ|ˆH|ϕ⟩,η≥m2kBT⟨ϕ|X∗η⟩2⟨ϕ|ˆH|ϕ⟩ . Using variational functions ϕ∗κ=(ε−52kBT)v∗x and ϕ∗η=v∗xv∗y, one finds, after tedious calculations, κ≥75kB64√πd2(kBTm)1/2,η≥5(mkBT)1/216√πd2 . Taking the lower limit in each case, we obtain a Prandtl number Pr=ηc∗pmκ=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/ℏ, and (∂f∂t)†coll=∫d3k∗1(2π)3∫d3k′(2π)3∫d3k′1(2π)3 w{f′f′1(1±f)(1±f∗1)−ff∗1(1±f′)(1±f′1)} where w=w(k,k∗1|k′,k′1), f=f(k), f∗1=f(k∗1), f′=f(k′), and f′1=f(k′1), 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, ∂δf∂t−eℏcv×B⋅∂δf∂k−v⋅[e\boldmath{E}+ε−μT∇T]∂f0∂ε=−δfτ , where \boldmath{E}=−∇(ϕ−μ/e)=E−e−1∇μ is the gradient of the ‘electrochemical potential’ ϕ−e−1μ. For steady state transport with B=0, one has j=−2e∫ˆΩd3k(2π)3vδf≡L11\boldmath{E}−L12∇Tj†q=2∫ˆΩd3k(2π)3(ε−μ)vδf≡L21\boldmath{E}−L22∇T where Lαβ11=e2Jαβ0, Lαβ21=TLαβ12=−eJαβ1, and Lαβ22=1TJαβ2, with Jαβn≡14π3ℏ∫dετ(ε)(ε−μ)n(−∂f0∂ε)∫dSεvαvβ|v| . These results entail \boldmath{E}=ρj+Q∇T,j†q=\boldmath{⊓}j−κ∇T , or, in terms of the Jn, ρ=1e2J−10,Q=−1eTJ−10J†1,\boldmath{⊓}=−1eJ†1J−10,κ=1T(J2−J†1J−10J†1) . 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 j∗q=⊓j, where ⊓ is the Peltier coefficient.
(j=B=0): A temperature gradient ∇T gives rise to a heat current j∗q=−κ∇T, where κ is the thermal conductivity.
(j=B=0): A temperature gradient ∇T gives rise to an electric field \boldmath{E}=Q∇T, 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=2.45×10−8V2K−2 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:
(∂T∂x=∂T∂y=j†y=0): An electrical current j=j†xˆx and a field B=B†zˆz yield an electric field \boldmath{E}. The Hall coefficient is RH=E†y/j†xB†z.
(∂T∂x=j†y=j†q,y=0): An electrical current j=j†xˆx and a field B=B†zˆz yield a temperature gradient ∂T∂y. The Ettingshausen coefficient is P=∂T∂y/j†xB†z.
(j†x=j†y=∂T∂y=0): A temperature gradient ∇T=∂T∂xˆx and a field B=B†zˆz yield an electric field \boldmath{E}. The Nernst coefficient is Λ=E†y/∂T∂xB†z.
(j†x=j†y=E†y=0): A temperature gradient ∇T=∂T∂xˆx and a field B=B†zˆz yield an orthogonal gradient ∂T∂y. The Righi-Leduc coefficient is L=∂T∂y/∂T∂xB†z.
∙ 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)e−γt+Fγ(1−e−γt)+t∫0dsη(s)eγ(s−t) . We assume that the random force η(t) has zero mean, and furthermore that ⟨η(s)η(s′)⟩=ϕ(s−s′)≈Γδ(s−s′) , in which case one finds ⟨p2(t)⟩=⟨p(t)⟩2+Γ2γ(1−e−2γt). 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 t≫γ−1, 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=2kBTtγm≡2Dt , where D=kBT/γm is the diffusion constant. Thus, after the memory of the initial conditions is lost (t≫γ−1), 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)⟩=F∗1(x(t))δt and ⟨[δx(t)]2⟩=F∗2(x(t))δt, but that ⟨[δx(t)]n⟩−O(δt2) for n>2. One can then show that the probability density P(x,t)=⟨δ(x−x(t))⟩ satisfies the Fokker-Planck equation, ∂P∂t=−∂∂x[F∗1(x)P(x,t)]+12∂2∂x2[F∗2(x)P(x,t)] . For Brownian motion, F∗1(x)=F/γm≡u and F∗2(x)=2D. The resulting Fokker-Planck equation is then P∗t=−uP∗x+DP∗xx, where P∗t=∂P∂t , P∗xx=∂2P∂x2 , The Galilean transformation x→x−ut then results in P∗t=DP∗xx, which is known as the diffusion equation, a general solution to which is given by P(x,t)=∞∫−∞dx′K(x−x′,t−t′)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
- 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.↩
- Recall from classical mechanics the definition of the Poisson bracket, {A,B}=∂A∂r⋅∂B∂p−∂B∂r⋅∂A∂p. Then from Hamilton’s equations ˙r=∂H∂p and ˙p=−∂H∂r, where H(p,r,t) is the Hamiltonian, we have dAdt={A,H}. Invariants have zero Poisson bracket with the Hamiltonian.↩
- See Lifshitz and Pitaevskii, Physical Kinetics, §2.↩
- The function g(x)=xlnx−x+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.↩
- In the chapter on thermodynamics, we adopted a slightly different definition of c∗p as the heat capacity per mole. In this chapter c∗p is the heat capacity per particle.↩
- Here we abbreviate QDC for ‘quick and dirty calculation’ and BRT for ‘Boltzmann equation in the relaxation time approximation’.↩
- The difference is trivial, since p=mv.↩
- See the excellent discussion in the book by Krapivsky, Redner, and Ben-Naim, cited in §8.1.↩
- The requirements of an inner product ⟨f|g⟩ are symmetry, linearity, and non-negative definiteness.↩
- We neglect interband scattering here, which can be important in practical applications, but which is beyond the scope of these notes.↩
- The transition rate from |k′⟩ to |k⟩ is proportional to the matrix element and to the product f′(1−f). The reverse process is proportional to f(1−f′). Subtracting these factors, one obtains f′−f, and therefore the nonlinear terms felicitously cancel in Equation [qobc].↩
- In this section we use j to denote electrical current, rather than particle number current as before.↩
- To create a refrigerator, stick the cold junction inside a thermally insulated box and the hot junction outside the box.↩
- Note that it is E⋅j and not \boldmath{E}⋅j which is the source term in the energy continuity equation.↩
- 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.↩
- The cgs unit of viscosity is the Poise (P). 1P=1g/cm⋅s.↩
- We further demand β∗n=0=0 and P∗−1(t)=0 at all times.↩
- A discussion of measure for functional integrals is found in R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals.↩
- In this section, we use the notation ˆχ(ω) for the susceptibility, rather than ˆG(ω)↩