10 Thermodynamics in closed quantum sys tems using Generalized Gibbs Ensembles
10.4 The minimum work principle
10.4.6 Example: Work extraction violating the minimum work principle
For the second example, instead of taking a Gibbs state as the initial state of the bath-part of the system, we choose a GGE initial state ωGGE(B) , which we want to choose to find an example that violates the minimum work principle. From the previous discussions, we
A Q UA N T U M O F T H E R M O DY N A M I C S 133 5 10 15 20 25 30 0.00 0.05 0.10 0.15 0.20 N Work 5 10 15 20 25 30 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5Entropy production N
Figure 10.3: Left: Extracted work with quenches only on a single site of the chain of Fermions in the second example. Green points correspond to the workW computed from the exact unitary evolution, yellow points to the work WGGEcomputed from the effective description in terms GGE states. As an initial state we take the one specified byK=32, Tr(f†
1f1ρ(0)S) =0.1, n=150. For the initial Hamiltonian, e0=0.1, ei =1
∀i6=1, g=0.5. The protocol consists of a first quench to e1 =1.6, followed by N−1 equidistant quenches back to the original Hamiltonian. The exact evolution is obtained by letting system and bath interact for a random time between20/g and 100/g much larger than the equilibration time. Right: The entropy production in the same protocol. (Figures adapted from Ref. [3].)
learn that we should try to engineer it such that the final state of the reversible limit is not passive. We therefore choose the GGE in the following way:
TrωGGE(B) n(B)k = ( 1 if k≤K, 0 if k>K. (10.62)
Here,n(B)k denote the number operators of the normal modes of the bath HamiltonianHB
and we assume that the corresponding single-particle energies µ(B)k are ordered in decreas- ing fashion. Therefore only theK most energetic normal modes are populated in the initial state ωGGE(N). This ensures that the initial correlation matrix of the system is not passive. Note that such behaviour would be impossible to have in a Gibbs state with positive tem- perature. Nevertheless, if we would use an effective description in terms of Gibbs states, then the effective temperature of this state would be positive for large enough system sizes and fixedK since the energy-density in the state ω(B)GGEis well below the critical energy den- sity. This implies that from the point of view of the Gibbs equilibration model, we would predict that the minimum work principle would be fulfilled in a work-extraction protocol.
Fig. 10.3 shows the results of a work-extraction protocol from these initial condition. As can clearly be seen, the minimum work principle is violated: the extracted work mono- tonically decreases with the number of steps in the second part of the protocol and thus in the reversible limit. Indeed, we find that the final state in the reversible limit is highly non-passive, explaining the breakdown of the minimum work principle. This finishes our discussion of entropy production and the minimum work principle in Generalized Gibbs ensembles.
10.5
Summary
It is generally assumed that sufficiently interacting, non-integrable systems equilibrate and even thermalize after a quench of the Hamiltonian. This means that local observables can be described by global Gibbs states with the same energy-density as the initial state. Nevertheless, there are class of many-body systems for which this is not true, such as integrable systems [192–202] or many-body localizing systems, which equilibrate but do not thermalize [224–226]. The equilibrium states of such systems have to be described by Generalized Gibbs ensembles.
In this chapter, we built up a framework for thermal machines and studied its predic- tions for different kind of equilibration behaviour of many-body systems on the basis of an effective description in terms of Generalized Gibbs ensembles. In this framework, we assume that Generalized Gibbs ensembles provide a good description not only for single
quenches but, importantly, also for many consecutive quenches. This allowed us to derive general results about entropy production and the minimum work principle in a unifying language. Importantly, the statement that the entropy can only increase along a thermody- namic protocol of several quenches in an isolated system follows straightforwardly without any additional assumptions.
As expected, systems that thermalize allow us to derive standard statements of phe- nomenological thermodynamics, such as the minimum work principle. This is even true for finite baths, where standard thermodynamic arguments fail since the bath changes its temperature over the course of a thermodynamic protocol. Importantly, these results also hold true if we cannot clearly separate systems into "working systems" and "heat baths". What we call "system" and what we call "bath" is simply determined by our control capa- bilities, as in chapter 9.
For the case of arbitrary GGEs we found that the usual thermodynamic statements are not automatically true. In particular, we discussed an explicit example where the minimum work principle fails. This could be attributed to the fact that GGE states need not be passive states, even though the description on the basis of the constants of motion seems to be passive. Similarly, in the case of the time-average equilibration model and for Gibbs-states we also found that the minimum work principle holds, in a suitable formulation, when the relevant effective descriptions are given by passive states. This suggests that "equilibrium states", in the sense of the second law, are not simply maximum entropy ensembles, given the expectation values of the relevant conserved quantities, but also have to be passive. It is an intriguing open problem to study the role of passivity for Generalized Gibbs ensembles for more general systems then just free fermionic or free bosonic systems.
As discussed above, the framework introduced in this chapter assumes that we can use the effective description after a first quench also to predict the new effective description after a second quench. This goes beyond of what is usually considered in theory and experiments and deserves further studies both from a theoretical point of view, but also from the experimental point of view. Indeed, using optical lattices or trapped ion platforms, it seems entirely plausible that this assumption can be tested experimentally with similar set-ups as for single-quench experiments [169–174].
I already mentioned above that the results and the framework in this chapter are not restricted to any weak coupling limit. In the next chapter, we will now use this framework to study thermal machines in the strong coupling regime and derive general corrections to work-extraction bounds and efficiencies of heat engines.