6.6. Aspectos metodológicos
6.6.7. Resultados esperados
with the correction terms
∆F(irrev)(H(1) S ) = 1 βD ρ(0) ⊗ωβ(HB)kωβ(HS(1)+V+HB) , (11.11) ∆F(res)(H( f ) S ) = 1 βD ωβ(H( f )S +V+HB)kωβ(H(0)S) ⊗ωβ(HB) . (11.12) Since both terms are expressed as relative entropies, it is clear that they are positive. The optimal work that can be extracted from the initial state is then given by optimizing the HamiltoniansHS(1)andHS( f ). Further below, we will derive the exact conditions that char- acterize the solutions of this minimization problem. We will also provide the exact solu- tions to second order in the coupling strength. Before we come to that, however, let us first discuss the two penalty terms heuristically.
The first penalty term, given by ∆F(irrev), describes the dissipation that occurs when the system is coupled to to the bath and compares the state after equilibration with the state before equilibration. Due to the interaction, there is an unavoidable build-up of correlations, which are not present in the initial state. Hence, in general this term cannot be made zero for any finite interaction. In the weak coupling limit, on the other hand, the term can be made vanishingly small by choosingHS(1)as the modular HamiltonianHρ(β0)(cf. section 5). The second penalty term, given by∆F(res), can be interpreted as the residual free energy with respect to the initial, uncoupled Hamiltonian that is left at the end of the isothermal protocol. This free energy could in principle be extracted, if the Hamiltonian could be quenched globally instead of only on the system. Again, in the weak-coupling limit this term becomes arbitrarily small by takingHS( f )=H(0)S.
Observe that the above discussion already shows that, if we treat the interaction as a perturbation, all corrections to the weak coupling bound vanish to first order in the interac- tion strength. Since the relative entropy vanishes if and only if the two arguments coincide, we can also expect already that corrections to first order in the coupling strength vanish as well. We will later verify these statements explicitly. Let us now discuss general properties of the correction terms.
11.1.1
General properties of the correction terms
The correction terms formally depend on the whole state and Hamiltonian of the system and bath together. In principle, they could therefore scale extensively with the size of the bath, as is typically the case for non-equilibrium free energies. The first result that we will discuss shows that this is not the case.
Lemma 11.1 (Scaling of corrections). For any ρ(0) >0, H(0)S, V and HBthe correction
terms fulfill ∆F(res)(H( f ) S ) ≤2 kVk + H(0)S−H ( f ) S , (11.13) ∆F(irrev)(H(1) S ) ≤2 kVk + H (1) S −H β ρ(0) . (11.14)
Proof. For any two HamiltoniansA, B, we have 1 βD(ωβ(A)kωβ(B)) =Tr (A−B)(ωβ(B) −ωβ(A)) −1 βD(ωβ(B)kωβ(A)) ≤2kA−Bk, (11.15)
where the last inequality follows from the positivity of the relative entropy, the triangle inequality and the definition of the operator norm. The claim then follows by inserting the corresponding Hamiltonians and interpreting ρ(0)as thermal state of the modular Hamil- tonianHρ(β0).
In particular, the lemma shows that the optimal corrections fulfill
∆F(res) min :=min HS(f) ∆F(res)(H( f ) S ) ≤2kVk, (11.16) ∆F(irrev) min :=min H(S1) ∆F(irrev)(H(1) S ) ≤2kVk. (11.17)
These relations show that the strong coupling corrections are indeed negligible in the macroscopic limit if the interaction between two macroscopic bodies is given by a local interaction, since in such a case the norm of the interactionkVkonly scales like the con- tact area between the two bodies. The weak coupling workW(weak), on the other hand, generally scales like the volume of the systemS and hence dominates.
All of the above could be deduced without knowing the exact Hamiltonians that min- imize the correction terms. We will now derive conditions on the optimal Hamiltonians, which will be instrumental for determining the perturbative expansion of the correction terms.
Lemma 11.2 (Minimizing dissipation). Let ρ(0)have full support. Then any Hamiltonian XSthat minimizes∆F(irrev)has to fulfill
ρ(0) =TrB ωβ(XS+V+HB) . (11.18)
Proof. Since we consider arbitrary large, but finite systems,∆F(irrev)(HS(1))is a smooth, positive function ofHS(1). Let one minimum be attained byXS and consider the Hamilto-
nians
XS(t):=XS+tYS, X(t) =XS(t) +V+HB, (11.19)
whereYSis an arbitrary perturbation. Then we have
d∆F(irrev)(X(t)) dt t=0 =0. (11.20)
Calculating the derivative yields
d∆F(irrev)(X(t)) dt t=0 =Tr ρ(0) ⊗ωβ(HB) −ωβ(X(0)) YS⊗1 =Tr ρ(0) −TrB(ωβ(XS+V+HB)) YS =0. (11.21) Since this relation has to fold for arbitraryYS, the claim follows.
The lemma can easily be interpreted. It tells us that to optimize the dissipation in the protocol, we have to do the first quench in such a way that the initial state can be interpreted as the marginal of the corresponding interacting Hamiltonian. Even though this minimizes the dissipation, the dissipation does not vanish, but is given by
∆F(irrev) min =
1
βD TrB(ωβ(X(0))) ⊗ωβ(HB)kωβ(X(0)) .
(11.22)
This quantity vanishes only if ωβ(X(0)) is a product-state and hence non-interacting. It can thus be seen as a measure of the correlations which are induced by interaction between S and B. It is, however, not a standard-measure of correlations, even though it might easily be confused with the mutual information
I(S : B)ω
A Q UA N T U M O F T H E R M O DY N A M I C S 139
There are two important differences: First, the order of the arguments in the relative entropy is reversed, somewhat similar in how the "vacancy" in chapter 3 behaves in comparison to the non-equilibrium free energy. Second, the first argument in (11.22) is not given by the product of the marginals of ωβ(X(0))in contrast to the mutual information.
Similar to the condition on the optimal Hamiltonian to minimize dissipation, we can also derive a condition on the Hamiltonian H( f )S that minimizes the residual free energy. This condition is stated in the following Lemma. Its proof is technically more demanding and therefore given in section 14.8.2. To state the result, we define the following function on arbitrary Hermitian operatorsY, X:
YX := Z 1
0 e
−βτXYeβτXdτ. (11.24)
YX can be interpreted as the operatorY averaged over the imaginary-time evolution under
the operatorX.
Lemma 11.3 (Minimizing residual free energy). LetRSbe a Hamiltonian that minimizes
the residual free energy∆F(res)
HS( f )and defineR = RS+V+HB. ThenR has to
fulfill TrB ωβ(R) = TrB ωβ(R) (RS+V−H(0)S)−R Tr ωβ(R)(RS+V−H(0)S) . (11.25)
In particular, we find that forV=0 we can choose RS=H(0)Sas expected. Unfortu-
nately, in general and for finiteV, the optimal choice RSis much more difficult to interpret
than the correspondingXS minimizing the dissipation. Let us therefore now consider the
perturbative expansion of the correction terms in powers of the coupling strength, which will yield a clearer interpretation.