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By Harrison D.M.

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Following the strategy of [JP4], one can show that the Planck law ρ(r) = (1 + eβ(r−µ) )−1 can be deduced from the stability requirement Ep(ωλ+ ) = 0 for a more general class of interactions Vj . For reasons of space we will not discuss this subject in detail in these lecture notes (the interested reader may consult [AJPP]). We will see below that the entropy production of the SEBB model is nonvanishing whenever the density operators of the reservoirs are not identical. 3 The Heat and Charge Fluxes Recall that the observables describing heat and charge currents out of the j-th reservoir are Φj = λ(a∗ (irfj )a(1) + a∗ (1)a(irfj )), Jj = λ(a∗ (ifj )a(1) + a∗ (1)a(ifj )).

Let ωT be the gauge-invariant quasi-free state on CAR(h) associated to T . Let Tn = pn T pn . Then lim ωTn (a∗ (pn f )a(pn g)) = ωT (a∗ (f )a(g)). n→∞ Assume that µ and η are two faithful ωT -normal states and let Ent(µ|η) be their Araki relative entropy. Let µn and ηn be the restrictions of µ and η to CAR+ (Λn ). Then the function n → Ent(µn |ηn ) = TrΛn (µn (log µn − log ηn )), is monotone increasing and lim Ent(µn |ηn ) = Ent(µ|η). 33. 4. The tight binding approximation for an electron in a single Bloch band of a d-dimensional (cubic) crystal is defined by h ≡ 2 (Zd ) with the translation invariant Hamiltonian 1 ψ(y), (37) (hψ)(x) ≡ 2d |x−y|=1 where |x| ≡ i |xi |.

Let Tn = pn T pn . Then lim ωTn (a∗ (pn f )a(pn g)) = ωT (a∗ (f )a(g)). n→∞ Assume that µ and η are two faithful ωT -normal states and let Ent(µ|η) be their Araki relative entropy. Let µn and ηn be the restrictions of µ and η to CAR+ (Λn ). Then the function n → Ent(µn |ηn ) = TrΛn (µn (log µn − log ηn )), is monotone increasing and lim Ent(µn |ηn ) = Ent(µ|η). 33. 4. The tight binding approximation for an electron in a single Bloch band of a d-dimensional (cubic) crystal is defined by h ≡ 2 (Zd ) with the translation invariant Hamiltonian 1 ψ(y), (37) (hψ)(x) ≡ 2d |x−y|=1 where |x| ≡ i |xi |.

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