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This relation comes directly from the definition of work, (Equation presented), which implies (Equation presented). This is equivalent to Eq. (3), which, together with Eq. (2) gives (Equation presented)
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This relation comes directly from the definition of work, (Equation presented), which implies (Equation presented). This is equivalent to Eq. (3), which, together with Eq. (2) gives (Equation presented).
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The non-negativity of the nonadiabatic work is the content of the minimal work principle, which is valid well beyond the cases we address here, as discussed in Ref. [27].
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We notice that Eq. (11) is equivalent to two explicit relationships among relative entropies, which read TBD (ρτ ρB)-TAD (ρτ ρA)=TBD (ρA ρB) =-TAD (ρB ρA)+ (SA-SB) (TA-TB). These two equalities hold for every process (Equation presented) bringing (Equation presented) into (Equation presented); that is, they hold for every final state (Equation presented). Proof: each of the three sides entering Eqs. (18) and (19) is equal to (Equation presented) [as one can show by using Eq. (11) for the first term and by direct evaluation for the others]
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We notice that Eq. (11) is equivalent to two explicit relationships among relative entropies, which read TBD (ρτ ρB)-TAD (ρτ ρA)=TBD (ρA ρB) =-TAD (ρB ρA)+ (SA-SB) (TA-TB). These two equalities hold for every process (Equation presented) bringing (Equation presented) into (Equation presented); that is, they hold for every final state (Equation presented). Proof: each of the three sides entering Eqs. (18) and (19) is equal to (Equation presented) [as one can show by using Eq. (11) for the first term and by direct evaluation for the others].
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A possible derivation of Eq. (16) goes as follows: Let (Equation presented) the generating function for the cumulants, with (Equation presented). Then, (Equation presented). Taking (Equation presented), one obtains (Equation presented), which gives Eq. (16).
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