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By L. Rosenfeld

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Let βeq and µeq be given equilibrium values of the inverse temperature and the chemical potential. The affinities (thermodynamic forces) conjugated to the currents Φj and Jj are Yj = βj µj − βeq µeq . Xj = βeq − βj , Indeed, it follows from the conservations laws (12) and (39) that M Ep(ωλ+ ) = (Xj ωλ+ (Φj ) + Yj ωλ+ (Jj )) . j=1 Since ρβj µj (r) = 1 , 1 + eβeq (r−µeq )−(Xj r+Yj ) we have ∂Xk ρβj µj (r)|X=Y =0 = δkj ρ(r)(1 − ρ(r)) r, ∂Yk ρβj µj (r)|X=Y =0 = δkj ρ(r)(1 − ρ(r)), where ρ ≡ ρβeq µeq .

Assumption (SEBB2) implies that the function G(z) ≡ e+ e− |f (r)|2 dr = −i r−z ∞ g(t) e−itz dt, 0 which is obviously analytic in the lower half-plane C− ≡ {z | Im z < 0}, is con¯ − . We denote by G(r − io) the value of this tinuous and bounded on its closure C function at r ∈ R. Assumption (SEBB3) For j = 1, · · · , M , the generator TRj is the operator of multiplication by a continuous function ρj (r) such that 0 < ρj (r) < 1 for r ∈ (e− , e+ ). Moreover, if ρj (r) sj (r) ≡ log , 1 − ρj (r) we assume that sj (r)fj (r) ∈ L2 ((e− , e+ ), dr).

5 Fermi Golden Rule (FGR) Thermodynamics Let λ ∈ R be a control parameter. We consider an open quantum system with coupling λV and write τλ for τλV , ωλ+ for ω+ , etc. The NESS and thermodynamics of the system can be described, to second order of perturbation theory in λ, using the weak coupling (or van Hove) limit. This approach is much older than the ”microscopic” Hamiltonian approach discussed so far, and has played an important role in the development of the subject. The classical references are [Da1,Da2,Haa,VH1,VH2,VH3].

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