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By Joseph Maciejko

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thermodynamics of nonequilibrium processes

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66) has renormalized the velocity in the driving term, vp → vp + ∇p Re Σ = ∇p ( p + Re Σ), Σ reminiscent of the wavefunction renorand introduced the factor 1 − ∂ Re ∂ω −1 malization factor Z in Fermi liquid theory. The other term [Σ < , Re G] gives some additional terms proportional to the E and B fields. The term Σ > G< − G> Σ < on the left-hand side is a collision term. 3 Applications We now consider a few applications of the nonequilibrium formalism. 1 Nonequilibrium Transport through a Quantum Dot In this section we study nonequilibrium transport through a quantum dot connected to two external metallic leads (two-probe system).

68, 2512 (1992). 58 References 26. Y. Meir, N. S. Wingreen, and P. A. Lee, Phys. Rev. Lett. 66, 3048 (1991). 27. Y. Meir, N. S. Wingreen, and P. A. Lee, Phys. Rev. Lett. 70, 2601 (1993).

As a result, the action becomes ¯ d] = Sleads [¯ ¯ d] S[¯ c , c , d, c , c ] + SQD [d, where Sleads [¯ c ,c ] ≡ dτ c¯kα (i∂τ − kα )ckα C kα is the action of the isolated leads, and ¯ d] ≡ SQD [d, dτ L0 ({d¯n }, {dn })− C kα,mn dτ1 d¯m (τ )t∗kα,m gkα (τ, τ1 )tkα,n dn (τ1 ) dτ C C is an effective action for the quantum dot, which contains the effect of the external leads (bath). The source term η¯d is unaffected by the change of variables, but the source term c¯J becomes c¯J = c¯kα (τ )Jkα (τ ) = kα dτ1 d¯m (τ1 )t∗kα,m gkα (τ1 , τ ) Jkα (τ ) c¯kα (τ ) + kα m C The generating functional thus becomes Z[¯ η , J] = ¯ i ¯ d]eiSleads [¯c ,c ] eiSQD [d,d] e D[¯ c , c , d, C ¯ ∗ g)J ) dτ (η¯d+(¯ c +dt where the integration measure D[¯ c, c] = D[¯ c , c ] is invariant under the change of variables since it is only a shift.

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