122 Linear response theory VI.2.2 Dissipation Consider a system subject to a harmonic perturbation Ŵ cos ωt with constant Ŵ. A straightforward application of time-dependent perturbation theory in quantum mechanics yields for the transition rate from an arbitrary initial state |φi i to the set of corresponding final states |φf i 2 π X hφf | Ŵ |φi i δ(ωfi − ω) + δ(ωif − ω) . Γi→f = 2 (VI.30) 2~ f If the system absorbs energy from the perturbation, then Ef > Ei , so that only the first δ-term contributes. On the other hand, energy release from the system, corresponding to “induced emission”, requires Ef < Ei , i.e. involves the second δ-term. We apply this result to the system of Sec. VI.1.1, initially at thermodynamic equilibrium. Let f (t) = fω cos ωt, with constant real fω , be a “force” coupling to an operator Â acting on the system, leading to perturbation (VI.7) in the Hamiltonian. The probability per unit time that the system absorb a quantum ~ω of energy starting from an initial state |φn i is given by Eq. (VI.30) with |φi i = |φn i and keeping only the first δ-term. Multiplying this probability by the initial state population πn and by the absorbed energy ~ω, and summing over all possible initial states, one finds the total energy received per unit time by the system 2 πf 2 X Pgain = ω2 πn ~ω Ann0 δ(ωn0 n − ω). 2~ 0 n,n An analogous reasoning gives for the total energy emitted by the system per unit time −Ploss = 2 X 2 πfω2 X Ann0 2 δ(ωnn0 − ω) = πfω π ~ω πn0 ~ω Ann0 δ(ωn0 n − ω), n 2 2 2~ 2~ 0 0 n,n n,n where in the second identity we have exchanged the roles of the two dummy indices. Note that since the system is in thermodynamic equilibrium, the lower energy levels are more populated, so that the absorption term is larger than the emission term. Adding Pgain and Ploss yields the net total energy exchanged—and actually absorbed by the system—per unit time, dEtot /dt, when it is submitted to a perturbation −fω Â cos ωt, namely 2 dEtot πf 2 ω X = ω (πn − πn0 )Ann0 δ(ωn0 n − ω). dt 2~ 0 n,n Under consideration of the definition (VI.20) of the spectral density, this also reads dEtot f2 = ω ω ξ˜A†A (ω). dt 2 (VI.31) The spectral function ξ˜A†A (ω) thus characterizes the dissipation of energy in the system when it is submitted to a small perturbation proportional to Â cos ωt. VI.3 Properties and interrelations of the correlation functions 127 VI.3.2 Properties and relations of the time-correlation functions VI.3.2 b Interrelations between time-correlation functions :::::::::::::::::::::::::::::::::::::::::::::::::::::::::: The explicit expression (VI.26) of the generalized susceptibility shows that it is simply related to the inverse Fourier transform (VI.19) of the spectral density according to χBA (τ ) = 2iΘ(τ ) ξBA (τ ). (VI.48) Since χBA (τ ), which was defined for Hermitian operators only, is real-valued (see last remark of Sec. VI.1.2 a), one recovers property VI.47b. Let us define an operator Ȧˆ by the relation 1 Ȧˆ ≡ Â, Ĥ0 , i~ (VI.49) i.e. such that its matrix elements are given by (Ȧ)nn0 = (En0 − En )Ann0 /i~ = iωnn0 Ann0 . If Â is an (81) The properties involving CBA , SBA or ξBA can be read at once from their definitions (VI.12) resp. (VI.16), or invoking their respective spectral representations. Those pertaining to KBA can be shown with the help of the decomposition (VI.23), in particular using the invariance of the ratio under the exchange n ↔ n0 . 128 Linear response theory observable, then Ȧˆ coincides with the value taken at t = 0 by the derivative dÂI (t)/dt for a system evolving with Ĥ0 only, i.e. in the absence of external perturbation. Replacing Â by Ȧˆ in the spectral form (VI.23) of Kubo’s correlation function, one finds X πn − πn0 KB Ȧ (τ ) = i Bnn0 An0 n e−iωn0 n τ , β~ 0 n,n i.e. KB Ȧ (τ ) = 2i ξBA (τ ). β (VI.50) In turn, relation (VI.48), becomes χBA (τ ) = βΘ(t)KB Ȧ (τ ). (VI.51) This relation is sometimes referred to as Kubo formula, since in his original article  Kubo expressed the linear response to an perturbation with the help of βKB Ȧ (τ ) instead of the retarded propagator χBA (τ ) used in Sec. VI.1.2. Identifying the right-hand sides of Eqs. (VI.38) and (VI.37) and differentiating the resulting relation with respect to time, one finds dKBA (t) 2i = − ξBA (t). dt β Equation (VI.50) then yields dKBA (t) = −KB Ȧ (t). (VI.52) dt 138 Linear response theory VI.4 Examples and applications The presentation of this Chapter is missing. VI.4.1 Green–Kubo relation An important application of the formalism of linear response is the calculation of the transport coefficients which appear in the phenomenological constitutive relations of non-equilibrium thermodynamics. VI.4.1 a Linear response revisited :::::::::::::::::::::::::::::::::: Under consideration of relation (VI.48) or (VI.51) and dropping the nonlinear terms, the Kubo formula (VI.8) can be recast as either Z ∞ ξBA (τ ) f (t − τ ) dτ (VI.83) B̂I (t) n.eq. = B̂ eq. + 2i 0 or equivalently B̂I (t) n.eq. = B̂ eq. + ∞ Z 0 βKB Ȧ (τ ) f (t − τ ) dτ, (VI.84) where the latter is in fact the form originally given by Kubo. Let us assume that the expectation value of B̂ at equilibrium vanishes: for instance, B̂ is a flux as introduced in Chapter I. To emphasize this interpretation, we denote the responding observable by Ĵ b , instead of B̂. Accordingly, to increase the similarity with Sec. I.2, we call the generalized force Fa (t) instead of f (t), and we rename Âa the observable coupling to Fa (t). Fourier transforming relations (VI.83) or (VI.84) leads to ˜ Ĵ b (ω) n.eq. = Lba (ω) F˜a (ω), (VI.85a) where we also adopted a new notation for the susceptibility: Z ∞ Z ∞ iωτ ξJb Aa(τ ) eiωτ dτ. Lba (ω) ≡ β KJb Ȧa(τ ) e dτ = 2i (VI.85b) 0 0 Summing over different generalized forces, whose effects simply add up at the linear approximation, the Fourier-transformed Kubo formula (VI.85a) is a straightforward generalization of Eq. (I.31) accounting for frequency dependent fluxes and affinities. The real novelty is that the “generalized kinetic coefficient” Lba (ω) is no longer a phenomenological factor as in Sec. I.2: instead, there is now an explicit formula to compute it using time-correlation functions at equilibrium of the system, Eq. (VI.85b). In Chapter I, the considered affinities and fluxes were implicitly quasi-static—the gradients of intensive thermodynamic quantities were time-independent—, which is the regime for which transport coefficients are defined. For instance, the electrical conductivity σel. is defined as the proportionality factor between a constant electric field and the ensuing direct current. Accordingly, the kinetic coefficients Lba in the constitutive relation I.31 are actually the zero-frequency limits of the corresponding susceptibilities: Lba 1 = lim ω→0 kB T Z 0 ∞ iωτ KJb Ȧa(τ ) e 1 dτ = kB T Z 0 ∞ KJb Ȧa(τ ) dτ. (VI.86) This constitutes the general form of the Green–Kubo-relation [57, 58, 50]. VI.4.1 b Example: electrical conductivity ::::::::::::::::::::::::::::::::::::::::: As example of application of the Green–Kubo-relation, consider a system made of electrically charged particles, with charges qi , submitted to a classical external electric field E~ (t), which plays 139 VI.4 Examples and applications the role of the generalized force. In the dipolar approximation, this field perturbs the Hamiltonian by coupling to the electric dipole moment of the system X ~ˆ where D ~ˆ ≡ Ŵ (t) = −E~ (t) · D qi~rˆi with ~rˆi the position operator of the i-th particle. i Quite obviously, we are interested in the electric current due to this perturbation, namely X d~rˆi (t) ˆ J~ el. (t) ≡ qi . dt i ~ˆ along the direction of E~ —let us In the notations of the present Chapter, Â is the component of D for simplicity denote this component by D̂z —, while the role of B̂ is played by any component Jˆel.,j ˆ of J~ el. . Additionally, one sees that B̂ = Ȧˆ in the case B̂ = Jˆel.,z . Focussing on the latter case, Kubo’s linear response formula reads in Fourier space Jˆel.,z (ω) = χ̃Jz Dz (ω)Ez (ω) ≡ σzz (ω)Ez (ω), where for obvious reasons we have introduced for the relevant generalized susceptibility the alternative notation σzz (ω). The zero-frequency limit of σzz is the electrical conductivity, which according to the Green–Kubo-relation (VI.86) is given by Z ∞ 1 KJz Jz (τ ) dτ, σel. = kB T 0 i.e. by the integral of the time-autocorrelation function of the component of the electric current along the direction of the electric field. Let us again emphasize that given a microscopic model of the system, the canonical correlation function can be calculated, which then allows one to compute the electrical conductivity, which in paragraph I.2.3 c was a mere phenomenological coefficient.