Resurrecting Clausius: Why Nanomachines are More Efficient than We Thought
RResurrecting Clausius: Why Nanomachines are More Efficient than We Thought
Philipp Strasberg, Mar´ıa Garc´ıa D´ıaz, and Andreu Riera-Campeny
F´ısica Te`orica: Informaci´o i Fen`omens Qu`antics, Departament de F´ısica,Universitat Aut`onoma de Barcelona, 08193 Bellaterra (Barcelona), Spain (Dated: December 8, 2020)We study entropy production in nanoscale devices, which are coupled to finite heat baths. Thissituation is of growing experimental relevance, but most theoretical approaches rely on a formulationof the second law valid only for infinite baths. We fix this problem by pointing out that alreadyClausius’ paper from 1865 contains an adequate formulation of the second law for finite heat baths,which can be also rigorously derived from a microscopic quantum description. This Clausius’ in-equality shows that nonequilibrium processes are less irreversible than previously thought. We useit to correctly extend Landauer’s principle to finite baths and we demonstrate that any heat enginein contact with finite baths has a higher efficiency than previously thought. Importantly, our resultsare easy to study, requiring only the knowledge of the average bath energy.
More than 150 years ago, Clausius wrote down thefollowing inequality, which now bears his name [1]:Σ ≡ ∆ S S ( τ ) − (cid:90) ¯ dQ ( t ) T ( t ) ≥ . (1)For that inequality to be valid, Clausius imagined aprocess where a system S undergoes a nonequilibriumtransformation for a time τ while being in contact witha heat bath with a time-dependent temperature T ( t ).The change in thermodynamic entropy of the system is∆ S S ( τ ) = S S ( τ ) − S S (0) and the infinitesimal heat fluxfrom the bath into the system at time t is ¯ dQ ( t ). Anideal heat bath is characterized by the relation dS B ( t ) = − ¯ dQ ( t ) /T ( t ), where dS B ( t ) is the infinitesimal change inbath entropy. Consequently, Eq. (1) coincides with thetraditional statement of the second law: the thermody-namic entropy of the universe (i.e., the system and thebath) can not decrease. Hence, Eq. (1) is often called the entropy production , which we denote by Σ.In contrast, the standard framework of quantum ther-modynamics considers the following inequality as a ver-sion of the second law and Clausius’ inequality [2, 3]:Σ (cid:48) ≡ ∆ S S ( τ ) − Q ( τ ) T (0) ≥ . (2)Here, Q ( τ ) = (cid:82) ¯ dQ ( t ) denotes the total flux of heat intothe system. Equation (2) follows rigorously from Eq. (1)for an infinite heat bath, whose temperature T ( t ) = T (0)stays constant in time. In the following, we are inter-ested in finite heat baths, whose macroscopic parameterschange in time such that Σ (cid:54) = Σ (cid:48) .Indeed, many interesting recent experiments operatefar from the thermodynamic limit and finite size effects ofthe bath become visible [4–12]. While there is a growingnumber of theoretical studies looking at finite baths [2, 3],most of them still refer to Eq. (2) as the second law orClausius’ inequality [13].The reason for the popularity of Eq. (2), even in con-text of finite baths, is related to the fact that it is known how to microscopically derive an inequality, whichressembles it [14–18]. For that purpose, consider a sys-tem coupled to a bath described by a joint quantum state ρ SB ( t ) evolving unitarily in time. Initially, the system isassumed decorrelated from a bath described by a canon-ical ensemble at temperature T (0), which we denote by π B [ T (0)] ≡ e − H B /T (0) / Z B [ T (0)]. Here, H B is the bathHamiltonian and Z B ( T ) ≡ tr B { e − H B /T } the partitionfunction ( k B ≡ ρ SB (0) = ρ S (0) ⊗ π B [ T (0)], one can derive Eq. (2) ifone makes the following two identifications [14–18]. First,the thermodynamic entropy of the system is identifiedwith the von Neumann entropy, S S ( t ) ≡ S vN [ ρ S ( t )], with S vN ( ρ ) ≡ − tr { ρ ln ρ } . Second, the heat flux into the sys-tem is identified with minus the change in bath energy, Q ( t ) ≡ − ∆ E B ( t ), with E B ( t ) ≡ tr B { H B ρ B ( t ) } .This derivation of Eq. (2) is remarkable because no assumption about the dynamics and the bath size entersit. However, for a finite bath it has not been possible tolink the term − Q ( τ ) /T (0) to an entropy change. Hence,while remaining a valid mathematical inequality, Eq. (2)is no longer identical to the second law.In this paper, we advocate the use of Eq. (1) insteadof Eq. (2) for finite baths. In fact, very recently it wasshown that—under the same conditions as above—alsoEq. (1) can be microscopically derived [19, 20]. Thetime-dependent bath temperature T ( t ) appearing in thismicroscopic derivation is then fixed by demanding thatthe bath energy E B ( t ) matches the one computed witha canonical ensemble at temperature T ( t ): E B ( t ) ! = tr { H B π B [ T ( t )] } . (3)Note that we are not asserting that ρ B ( t ) = π B [ T ( t )]. If this is the case, then T ( t ) equals the conventional equilib-rium temperature. In general, however, Eq. (1) remainsvalid even if ρ B ( t ) is far from equilibrium, but now T ( t )obtained from Eq. (3) plays the role of an effective non-equilibrium temperature.We remark that our approach based on a microscopi-cally emerging definition of temperature generalizes the a r X i v : . [ qu a n t - ph ] D ec few notable previous studies, which also considered atime-dependent bath temperature. Those studies eitherassumed that T ( t ) is externally controllable in time [21–23] or T ( t ) was dynamically determined in the linear re-sponse regime for baths that do not develop nonequilib-rium features [24–28].Crucially, we find that using the correct second law,Eq. (1) instead of Eq. (2), reveals surprising insights:finite-time information erasure and heat engines havehigher efficiencies than previously thought. These gen-eral results follow from the central relation:Σ (cid:48) − Σ = D (cid:0) π B [ T ( τ )] (cid:12)(cid:12) π B [ T (0)] (cid:1) ≥ . (4)Here, D ( ρ | σ ) ≡ tr { ρ (ln ρ − ln σ ) } is the always positivequantum relative entropy, measuring the statistical dis-tance between two states ρ and σ . The derivation ofEq. (4) is simple. Let S B ( T ) denote the von Neumannentropy of π B ( T ). By virtue of definition (3), we obtainthe familiar relation − ¯ dQ ( t ) /T ( t ) = dS B [ T ( t )] and hence − (cid:82) ¯ dQ ( t ) /T ( t ) = S B [ T ( τ )] − S B [ T (0)]. Standard manip-ulations then imply Eq. (4). The supplementary material(SM) lists all the details of the derivation.The central result (4) tells us that the entropy pro-duction Σ is smaller than what one would naively expectbased on Σ (cid:48) . Thus, the process is actually less irreversible in reality. Physically speaking, we can explain Eq. (4)by pointing out that the available information about theheat flow Q is taken fully into account in Σ but not inΣ (cid:48) . The inequality Σ (cid:48) ≥ ignores the fact that the bath is finite.However, if one knows the heat flow Q , one can use thatinformation to gain a more accurate description via thedefinition (3) of a time-dependent temperature. This isthe information that Σ takes advantage of and the loss ofinformation resulting from ignoring the finiteness of thebath is quantified by the relative entropy in Eq. (4).Another interpretation of Eq. (4) is the following. Sup-pose that we have an additional infinitely large superbath at our disposal with fixed temperature T (0). After the fi-nite bath has interacted with the system, it is out of equi-librium with respect to this superbath if T ( τ ) (cid:54) = T (0).This nonequilibrium situation can be used to extractwork. The maximum extractable work equals the changein free energy: W maxext = F B [ T ( τ )] − F B [ T (0)] [2, 3]. Here, F B ( T ) ≡ E B ( T ) − T (0) S B ( T ) denotes the nonequilib-rium free energy with respect to the reference tempera-ture T (0). Then, it is not too hard to reveal that W maxext = T (0)(Σ (cid:48) − Σ) ≥ . (5)Thus, if we demand that the bath in our description getsreset after each process to its initial temperature, Eq. (4)tells us that we can always use this reset stage to extractuseful work, which remains unaccounted for in Eq. (2).In the following, we explicitly demonstrate the use andbenefit of Eq. (1) for two relevant cases: information era-sure and heat engines. Erasing one bit of information has become a paradig-matic example for a nonequilibrium thermodynamic pro-cess since Landauer’s groundbreaking work [29], wherehe argued that the minimal heat dissipation is − Q ( τ ) ≥ T (0) ln 2 (recall that heat is defined positive if it in-creases the system energy). More generally, Eq. (2) im-plies − Q ( τ ) ≥ − T (0)∆ S S ( τ ), which coincides with Lan-dauer’s principle when applied to the erasure of one bitof information (see Fig. 1 for a sketch).In the future, the design of energy-efficient computerswill become important. Hence, recent effort has been alsoput into obtaining tighter bounds on the heat generationduring information erasure for finite baths [30–32]. Thephysical nature of these bounds is, however, less transpar-ent as they are a consequence of mathematical identitiesnot directly related to the second law.As explained above, for a finite bath the second law isrelated to Eq. (1). Remarkably, we find that − Q ( τ ) ≥ − T (0) (cid:90) τ ¯ dQ ( t ) T ( t ) ≥ − T (0)∆ S S ( τ ) , (6)where the first inequality follows from Eq. (4) and the sec-ond from Eq. (1). The right inequality in Eq. (6) can befaithfully called ‘Landauer’s principle for a finite bath’.It is a logical consequence of applying the second law tothe case of bit erasure. For illustration, Fig. 1 comparesthe bounds (6) as well as the bounds from Refs. [30–32].We now turn to heat engines and extend our analy-sis to a system in contact with a finite hot and a finitecold bath. The initial system-bath state is generalizedto ρ S (0) ⊗ π C [ T C (0))] ⊗ π H [ T H (0))], where the subscript C/H refers to the cold/hot bath.For simplicity, we assume that the engine has operatedfor a sufficient amount of time (or cycles) such that itschange in entropy ∆ S S and internal energy ∆ U S is negli-gible compared to other terms appearing in the first andsecond law. This is called the steady state regime and itis well justified if the system is small in comparison withthe baths. However, our conclusions do not depend onthis assumption as shown in the SM.Under the conditions spelled out above, the Clausiusinequality can be generalized to [19, 20]Σ = − (cid:90) ¯ dQ C ( t ) T C ( t ) − (cid:90) ¯ dQ H ( t ) T H ( t ) ≥ , (7)where ¯ dQ C/H ( t ) denotes the infinitesimal heat flow fromthe baths at time t . Moreover, also Eq. (2) can be gen-eralized to [16]Σ (cid:48) = − Q C ( τ ) T C (0) − Q H ( τ ) T H (0) ≥ . (8)The first law reads W + Q C + Q H = 0 and our goal is toachieve W <
0, i.e., we want to extract work.Note that, in presence of two baths, the difference Σ (cid:48) − Σ is the sum of the relative entropies (4) with respect to ⇝ (a) (b) (c) (d) Figure 1. Landauer erasure protocol. (a)
A two-level system in a maximally mixed state ρ S (0) = ( | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ) / T (0). (b) The energy of state 1 is lifted toincrease the probability for a jump 1 → − Q into the bath, thereby increasing its temperature. (c) After some time τ , the energy of state 1 is returned to its initial value. The final state has a population imbalance towardsthe state 0 (populations are indicated by the size of the black disks). (d) Plot showing different bounds on the total heatdissipation − Q ( t ) (thick black line on top) over dimensionless time Ω t , where (cid:126) Ω is the energy gap of a spin-system coupledto a bath of five spins (all details are given in the SM together with complementary plots). Note the logarithmic scale. Thebounds are labeled with the publication date: 1961 [29] (lower thick black line), 2014 [30] (dotted turquoise line), 2020 [32](dash-dotted purple line) and 1865 [1] (dashed magenta line). The latter is investigated here for the first time and the differenceto − Q ( t ) is given by Eq. (4) times T (0). The bound from 2015 [31] is always zero for a maximally mixed initial state (see SM). the cold and hot bath. Hence, Σ (cid:48) − Σ ≥ − W/Q H , which implies the Carnot bound,is adapted to the situation of an engine operating be-tween two heat baths with a fixed temperature. This isnot the case here. Instead, we argue that an efficiency inabsence of fixed temperatures can be properly defined asfollows. Consider a positive entropy production Σ splitinto two contributions labeled A and B : Σ = A + B ≥ A <
0, which has to be compensated by investing
B > ≥
0. Then, we define η ≡ − AB = 1 − Σ B ≤ . (9)Importantly, independent of A , B and Σ, this efficiencyis always bounded by the same number (= 1).We start by applying this logic to Σ (cid:48) . Using the firstlaw, we rewrite Eq. (8) asΣ (cid:48) = W ( τ ) T C (0) + Q H ( τ ) T C (0) − Q H ( τ ) T H (0) ≥ . (10)Clearly, if operated as a heat engine, the first term isnegative and plays the role of the ‘ A -term’ above. Theremaining ‘ B -term’ proportional to Q H is then necessar- ily positive. Thus, we define η (cid:48) ≡ − W ( τ ) /T C (0) Q H ( τ )[1 /T C (0) − /T H (0)] ≤ . (11)Next, we apply this logic to Σ. Without too mucheffort one confirms thatΣ = W ( τ ) T C (0) + Q H ( τ ) T C (0) − Q H ( τ ) T H (0) + Σ − Σ (cid:48) ≥ . (12)Again, the first term is identified as the A -term, which,importantly, is the same term as before and thereforemakes a comparison meaningful. However, the B -term isnow different because the resources invested in order toextract work are differently counted. We define η ≡ − W ( τ ) /T C (0) Q H ( τ )[1 /T C (0) − /T H (0)] + Σ − Σ (cid:48) ≤ η (cid:48) η = 1 + Σ − Σ (cid:48) Q H ( τ )[1 /T C (0) − /T H (0)] ≤ . (14)We recapitulate our logic used to arrive at the generalconclusion η ≥ η (cid:48) . We started from two different inequal-ities Σ ≥ (cid:48) ≥
0. In both inequalities, the same microsopically defined heat and work fluxes enter, butthey are bounded in different ways. Based on these twoinequalities, we constructed two efficiencies η and η (cid:48) . Im-portantly, (i) these efficiencies are both bounded by oneand (ii) both quantify the amount of resources neededto extract the same quantity A = W/T C (0). They are (a) (b) (c) Figure 2. (a)
Sketch of a system (a qubit with energy gap ∆ S ) coupled to an infinite cold bath at temperature T C and a finitehot bath made up of N qubits with energy gap ∆ H . The cycle of the engine consists of two steps. First, a thermalizationstep with the cold bath (no contact with the hot bath). Second, a short and sudden interaction with a qubit of the hot bathmodeled by a swap operation. Each qubit of the hot bath is initially at temperature T H (0), interacts once with the system,and is afterwards found at a colder temperature (qubit populations are indicated by the size of the black disks). The model isexactly solved in the SM and work extraction is possible if ∆ S / ∆ H > T C /T H (0). (b) We plot Σ (large magenta circles) andΣ (cid:48) (small black circles) and the changing temperature T H ( t ) of the finite hot bath (inset) as a function of the number of cycles n for N = 100 qubits in the hot bath. (c) Plot of η (large magenta circles) and η (cid:48) (small black circles) as a function of n . Allour plots confirm our conclusions in the main text. Numerical parameters: ∆ S = 1, ∆ H = 3 / T H (0) = 1, T C = 1 / therefore comparable and we have shown in full general-ity that η ≥ η (cid:48) arises as a consequence of Σ (cid:48) ≥ Σ.To illustrate the above points, we have numericallysimulated a heat engine in contact with an infinite coldbath and a finite hot bath. Our setup combines the ideaof a swap engine [33] with the framework of repeatedinteractions [34] and is explained in detail in Fig. 2 andthe SM. Our numerical observations in Fig. 2 confirm ourgeneral claims.Before concluding, we emphasize another subtle point.Initially, we have said that Clausius’ inequality is iden-tical to the change in thermodynamic entropy of theuniverse for an ideal heat bath described by a chang-ing equilibrium temperature T ( t ). However, via defini-tion (3), Eq. (1) remains valid even for a nonequilibriumstate of the bath, but in this case − ¯ dQ ( t ) /T ( t ) no longerstrictly relates to a change in bath entropy. Remarkably,it is possible to generalize the second law to also accountfor information about the (coarse-grained) distribution ofbath energies [19, 20]. For the same reasons as explainedbelow Eq. (4), the corresponding efficiency with respectto this refined second law is even higher than η , whichwe show explicitly in the SM. Thus, having access to thenonequilibrium distribution of bath energies offers ad-ditional interesting benefits in unison with other recentfindings [35].To conclude, 155 years ago, Clausius wrote down aremarkable inequality, which remained unappreciated inquantum thermodynamics. Here, we emphasized thatthe original Clausius inequality (1) correctly quantifiesthe second law for a much larger class of situations thanthe conventionally employed inequality (2). We showedthat Eq. (1) can be fruitfully used to study a variety of nanomachines in contact with finite baths and, impor-tantly, it is easy to apply as it only relies on the knowl-edge of the bath energy. Most remarkably, the ancientClausius inequality offers the insight that nanoscale en-gines are more efficient than previously anticipated. Acknowledgements.—
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We here detail in chronological order additional information concerning (A) the derivation of Eq. (4), (B) theLandauer erasure protocol used for the numerics, (C) the swap engine coupled to a repeated interactions bath, (D)the fact that our conclusions remain true even if the system has not yet reached a steady state, (E) improved efficienciesif additional information about the nonequilibrium probability distribution of the bath energies is available.Furthermore, we sometimes make use of the inverse temperature β ( t ) = 1 /T ( t ) whenever convenient. (A) Details concerning the derivation of Eq. (4) We formulate our derivation using the inverse temperature β ( t ) and assume for the moment that it changes in adifferentiable way. We start by considering the integral − (cid:90) ¯ dQ ( t ) T ( t ) = (cid:90) τ dtβ ( t ) dE B ( t ) dt , (15)where we used − ¯ dQ ( t ) = dE B ( t ) = tr B { H B [ ρ B ( t + dt ) − ρ B ( t )] } . Next, a straightforward calculation reveals that dE B ( t ) dt = − ˙ β ( t )Var[ H B , β ( t )] , ˙ β ( t ) ≡ dβ ( t ) dt , Var( H B , β ) ≡ tr B { H B π B ( β ) } − tr B { H B π B ( β ) } . (16)Now, let S B ( β ) ≡ − tr B { π B ( β ) ln π B ( β ) } denote the von Neumann entropy of a canonical equilibrium state at inversetemperature β . It follows from another straightforward calculation that dS B [ β ( t )] dt = − β ( t ) ˙ β ( t )Var[ H B , β ( t )] . (17)Taken together, we confirm that (cid:90) τ dtβ ( t ) dE B ( t ) dt = (cid:90) τ dt dS B [ β ( t )] dt = S B [ β ( τ )] − S B [ β (0)] . (18)Next, we note the general identity − Q ( τ ) T (0) = tr B { π B [ β (0)] ln π B [ β (0)] } − tr B { π B [ β ( τ )] ln π B [ β (0)] } . (19)Combining this with Eq. (18), we confirmΣ (cid:48) − Σ = − Q ( τ ) T + (cid:90) τ ¯ dQ ( t ) T ( t ) = D (cid:2) π B [ β ( τ )] (cid:12)(cid:12) π B [ β (0)] (cid:3) , (20)which proves Eq. (4) of the main text.Two comments are important. First, we preferred to work with β ( t ) instead of T ( t ). This choice is related tothe phenomenon that for a bath with a finite Hilbert space (note that there can be finite baths with an infinite Hilbert space, e.g., a collection of particles in a box) the temperature T ( t ) becomes negative for a population invertedstate. Importantly, when the state changes continuously from a state ρ B ( t − ) without population inversion to a state ρ B ( t + ) with population inversion, the associated temperature T ( t ) changes via definition (3) from T ( t − ) = ∞ to T ( t + ) = −∞ , i.e., there is a sudden and discontinuous jump in the temperature. However, this jump can be avoidedwhen working with the inverse temperature, which changes continuously from β ( t − ) > β ( t + ) <
0. Therefore,by working with the inverse temperature, we demonstrated that our conclusions remain true even for exotic negativetemperature states in the bath.Second, we comment on the assumption that β ( t ) needs to be differentiable. In fact, since unitary dynamicsgenerated by the Liouville-von Neumann equation are differentiable, there are good reasons to expect that the effective(inverse) temperature of the bath defined via Eq. (3) also changes in a differentiable way. Importantly, since therelation dS B [ β ( t )] = β ( t ) dE B [ β ( t )] has to hold only under the integral, our central result (4) remains valid if there isa finite number of times for which β ( t ) is not differentiable, but still continuous. We indeed observe this behaviourin Sec. (B), where β ( t ) shows (non-differentiable) spikes without invalidating Eq. (4). Finally, cases where β ( t ) isnot even continuous can be only generated by singular cases such as, e.g., a Hamiltonian with a time-dependencedescribed by a Dirac delta function. Strictly speaking, these models are, of course, unphysical. Nevertheless, weinvestigate this case in detail in Sec. (C) for the swap engine. Indeed, the swap engine models the swap operation ashappening instantaneously, which generates a discontinuous evolution of β ( t ). Despite this feature, we find that thecentral inequality Σ (cid:48) − Σ ≥ (cid:48) − Σ and D { π B [ β ( τ )] | π B [ β (0)] } quickly becomes negligibly small. (B) Details of the Landauer erasure protocol We first review the bounds of Refs. [30–32] before specifying the details of our numerical studies.
Bounds to the dissipated heat
The bound of Ref. [30] is very simple and reads − Q ( τ ) ≥ − T (0)∆ S S ( τ ) + 2 T (0)∆ S S ( τ )ln ( d −
1) + 4 , (21)where d is the dimension of the bath Hilbert space.The bound of Ref. [31] is based on the fact that the dynamics of the bath can be written as ρ B ( t ) = (cid:80) α A α ( t ) ρ B (0) A † α ( t ), where the bath operators A α ( t ) are commonly called Kraus operators. Defining A ( t ) ≡ (cid:80) α A α ( t ) A † α ( t ), the bound reads − Q ( t ) ≥ − T (0) ln tr B { A ( t ) ρ B (0) } . (22) − h e a t b o und s / k B T ( ) N = 1 N = 3 N = 5 t . . . k B T ( t ) / ¯ h Ω t t Figure 3.
First row:
As a function of dimensionless time Ω t for N = 1 (left), N = 3 (centre) and N = 5 (right) spins inthe bath, plot of the dissipated heat − Q ( t ) and Landauer’s original lower bound − T (0)∆ S S ( t ) (thick solid black lines) and thebound (21) [30] (dotted turquoise line), the bound (24) [32] (dash-dotted purple line) and our bound (6) (dashed magenta line).Note that we used a logarithmic scale for better visibility. Second row:
Again as a function of time for N = 1 (left), N = 3(centre) and N = 5 (right) spins in the bath, plot of the time-evolution of the (dimensionless) nonequilibrium temperature T ( t ). Numerical parameters: Ω = Ω = 1, J = J = 1, T (0) = 1 / We now demonstrate that this bound is zero for the case of Landauer erasure, i.e., whenever the system startsin a maximally mixed state. To this end, we note that the operators A α ( t ) are microscopically defined as A α ( t ) = (cid:112) λ j (cid:104) s k | U SB ( t ) | s j (cid:105) , where U SB ( t ) is the global system bath unitary and the initial state of the system was decomposedas ρ S (0) = (cid:80) j λ j | s j (cid:105)(cid:104) s j | . Thus, we see that the index α is actually a double-index α = ( j, k ). Now, if ρ S (0) is amaximally mixed state, then λ j = 1 / √ d S , where d S is the dimension of the system Hilbert space. Consequently, A ( t ) = (cid:88) α A α ( t ) A † α ( t ) = (cid:88) j,k d S (cid:104) s k | U SB ( t ) | s j (cid:105)(cid:104) s j | U † SB ( t ) | s k (cid:105) = (cid:88) k d S (cid:104) s k | U SB ( t ) U † SB ( t ) | s k (cid:105) = (cid:88) k d S = 1 , (23)where 1 denotes the identity operator in the bath space. Thus, − ln tr B { A ( t ) ρ B (0) } = − ln(1) = 0.Next, we turn to the bound from Ref. [32]. Interestingly, this bound also employs the definition (3) of temperature.We introduce the two functions f [ β ( t )] ≡ tr B { H B π B [ β ( t )] } − tr B { H B π B [ β (0)] } and g [ β ( t )] ≡ S B [ β ( t )] − S B [ β (0)],which determine the energy and entropy change with respect to the fictitious equilibrum state of the bath at inversetemperature β ( t ) starting from β (0). It was then found that − Q ( t ) ≥ ( f ◦ g − )[ − ∆ S S ( t )] . (24)Here, ( f ◦ g − )( x ) ≡ f [ g − ( x )] denotes the concatenation of two functions. Model 1: Spin coupled to a spin chain
The total system-bath Hamiltonian H S + H B + V is specified by H S = Ω σ Sz , H B = Ω N (cid:88) j =1 σ ( j ) z + J N − (cid:88) j =1 ( σ ( j ) x σ ( j +1) x + σ ( j ) y σ ( j +1) y ) , V = J ( σ Sx σ (1) x + σ Sy σ (1) y ) . (25)Here, σ x,y,z denotes the usual Pauli matrix, N the number of spins in the bath and Ω , Ω , J and J are real-valuedparameters. The time-evolution of the system-bath state is implemented by exact numerical integration withoutfurther approximations (apart from choosing a time-step-size δt ). t − . . . h e a t b o und s / k B T ( ) N = 1 t N = 3 t N = 5 Figure 4. As a function of dimensionless time Ω t for N = 1 (left), N = 3 (centre) and N = 5 (right) spins in the bath, plot ofthe dissipated heat − Q ( t ) and Landauer’s original lower bound − T (0)∆ S S ( t ) (thick solid black lines) and the bound (21) [30](dotted turquoise line), the bound (22) [31] (dash-dotted purple line) and our bound (6) (dashed magenta line). Parametersare as in Fig. 3 except of a different temperature, T (0) = 2, and a different initial system state (an excited state). We start by giving additional numerical results concering the Landauer erasure protocol, which starts with a fullymixed state ρ S (0) = ( | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ) /
2, where | (cid:105) ( | (cid:105) ) denotes the ground (excited) state of the system. Figure 3 showsthe bounds from Refs. [29, 30, 32] and our bound (6) on the total heat dissipation for N = 1 , , N = 1 we observe non-differentiable ‘spikes’ in the evolution of the temperature T ( t ). Nevertheless, in unison withour last comment in Sec. (A), we find perfect agreement with our central result (4) (not shown explicitly).Next, we provide additional numerical results when the system starts in a pure state, which we take to be theexcited state. In this case, we expect the bound from Ref. [31] to be useful and, indeed, Fig. 4 demonstrates this. Model 2: Spin coupled to a random matrix environment
This model was investigated in detail in Sec. IV of Ref. [20]. It also consists of a spin with Hamiltonian H S = (cid:15) | (cid:105)(cid:104) | + (cid:15) | (cid:105)(cid:104) | in contact with a bath with Hamiltonian H B = (cid:80) j E j | j (cid:105)(cid:104) j | . The bath Hamiltonian is split into twoenergy bands E and E . Each band E i is of width δ centered around (cid:15) i and contains a number of V i eigenenergies,which are randomly sampled from the interval [ (cid:15) i − δ/ , (cid:15) i + δ/ (cid:15) − (cid:15) ≥ δ such thatthe two bands do not overlap. For the numerics, we set V = 10 and leave V ≡ V as a free parameter. Finally, theinteraction Hamiltonian is V = λσ x ⊗ B, B = (cid:88) E j ∈E (cid:88) E k ∈E c ( E j , E k ) | E j (cid:105)(cid:104) E k | + h.c. (26)Here, c ( E j , E k ) is a matrix of independent and identically distributed complex random numbers with zero mean andvariance a . The dynamics are generated by formally integrating the exact Schr¨odinger dynamics for one realizationof the coupling matrix c ( E j , E k ), but we observe that the qualitative features of the dynamics do not change fordifferent realizations. Despite the bath dimension d = V + V = 10 + V is relatively small, the randomness in thecoupling helps to mimic a more realistic (i.e., large) heat bath. Numerical results are shown in Fig. 5 for the case ofLandauer erasure. (C) The swap engine with a repeated interactions bath We consider a two-level system with Hamiltonian H S = ∆ S | (cid:105)(cid:104) | , where | (cid:105) denotes the excited state (and | (cid:105) theground state) as our ‘working medium’.The cold bath is assumed to be an ideal weakly coupled bath, which simply prepares the system in a canonicalensemble at temperature T C in each cycle. We denote this state as π S ( T C ) ≡ e − H S /T C / Z S ( t C ).The hot bath is made up of N non-interacting qubits. Each qubit is described by the Hamiltonian H H = ∆ H | (cid:105)(cid:104) | and initialized in the state π H [ T H (0)] = e − H H /T H (0) / Z H [ T H (0)]. At regular time-intervals kτ cycle , the k ’th qubit of − − h e a t b o und s / k B T ( ) V = 10 V = 50 V = 100 (cid:15) − (cid:15) ) t/ ¯ h k B T ( t ) / ( (cid:15) − (cid:15) ) (cid:15) − (cid:15) ) t/ ¯ h (cid:15) − (cid:15) ) t/ ¯ h Figure 5.
First row:
As a function of dimensionless time ( (cid:15) − (cid:15) ) t/ (cid:126) for V = 10 (left), V = 50) (centre) and V = 100 (right)levels in the upper band of the bath, plot of the dissipated heat − Q ( t ) and Landauer’s original lower bound − T (0)∆ S S ( t )(thick solid black lines) and the bound (21) [30] (dotted turquoise line), the bound (24) [32] (dash-dotted purple line) and ourbound (6) (dashed magenta line). Note that we used a logarithmic scale for better visibility. Second row:
Again as a functionof dimensionless time for V = 10 (left), V = 50) (centre) and V = 100 (right) levels in the upper band of the bath, plot of thetime-evolution of the (dimensionless) nonequilibrium temperature T ( t ). Numerical parameters: (cid:15) − (cid:15) = 1, δ = 1, λ = 0 . T = 1 / the hot bath interacts with the system. This interaction is assumed to be fast enough such that the effect of thecold bath can be neglected to lowest order. Furthermore, we assume the time τ cycle between two interaction to belarge enough such that the system had enough time to relax to the thermal state π S ( T C ) in between two interactions.Finally, we assume that the interaction between the k ’th bath qubit and the system implements a swap operationdescribed via the unitary operator U swap | k, l (cid:105) = | l, k (cid:105) . Here, | k, l (cid:105) = | k (cid:105) S ⊗ | l (cid:105) H denotes the system qubit in state k and the bath qubit in state l ( k, l ∈ { , } ).We are now in a position, where we can analyze an arbitrary cycle from a thermodynamic perspective. We startwith the swap operation. The total work equals the change in internal energy of the system and the bath qubit: W = tr SB { ( H S + H B )[ U swap π S ( T C ) ⊗ π H [ T H (0)] U † swap − π S ( T C ) ⊗ π H [ T H (0)]] } . (27)Since there is no work performed on the system during the thermalization processes, we want that this work isnegative. A quick calculation reveals the explicit expression W = (∆ S − ∆ H ) (cid:20) e − ∆ H /T H (0) Z H [ T (0)] − e − ∆ S /T C Z S ( T C ) (cid:21) = (∆ S − ∆ H ) e ∆ S /T C − e ∆ H /T H (0) ( e ∆ H /T H (0) + 1)( e ∆ S /T C + 1) . (28)The condition for work extraction, W <
0, follows as T H (0) T C > ∆ H ∆ S > . (29)From Eq. (27) we can also easily calculate the change in system and bath energy during the swap operation, whichbecomes ∆ E S = ∆ S (cid:20) e − ∆ H /T H (0) Z H [ T (0)] − e − ∆ S /T C Z S ( T C ) (cid:21) , ∆ E H = − ∆ H ∆ S ∆ E S = − Q H . (30)Here, we equated the change in bath energy with minus the heat flow into it in accordance with our framework aboveand we observe that W = ∆ E S − Q H , which is the first law during the swap operation.During the subsequent equilibration step, the system relaxes back to its initial state by dissipating an amount ofheat Q C = − ∆ E S into the cold bath. This concludes the cycle. Note that the system entropy and energy does notchange over an entire cycle, a property which we also used in the main text.0 N − entropy production /k B & relative entropy N − − − − absolute/relative error Figure 6.
Left:
Plot of Σ (cid:48) ( N ) − Σ( N ) (pink crosses) and ND { π B [ T H ( Nτ )] | π B [ T H (0)] } (black circles) as a function of N . Right:
Plot of e abs ( N ) (pink crosses) and e rel ( N ) (black circles) as a function of N . Note that in both plots results are shownon a double-logarithmic scale. Numerical parameters are as for Fig. 2 in the main text: ∆ S = 1, ∆ H = 3 / T H (0) = 1, T C = 1 / Now, the total amount of heat flown and the work after n interactions simply become Q tot H = nQ H , Q tot C = nQ C , W tot = nW = − Q tot C − Q tot H . (31)Furthermore, the two different notions of entropy production become after n interactionsΣ = − nQ C T C − n (cid:88) k =1 Q H T H ( kτ ) ≥ , Σ (cid:48) = − nQ C T C − nQ H T H (0) ≥ , (32)which is plotted in Fig. 2(b). Furthermore, the time-dependent temperature of the hot bath T H ( kτ ), which is plottedin the inset of Fig. 2(b), can be obtained by solving k ∆ H e ∆ S /T C + 1 + ( N − k ) ∆ H e ∆ H /T H (0) + 1 = N ∆ H e ∆ H /T H ( kτ ) + 1 , (33)where N denotes the total number of qubits in the hot bath. Note that for k = N the temperature of the finite bathis exactly T H ( N τ ) = T C ∆ H / ∆ S . According to Eq. (29), this means that we can no longer extract work from thesystem after we have used up all N qubits. The plot of the efficiencies η and η (cid:48) , Fig. 2(c), immediately follows fromthe above considerations.Finally, we return to the second comment made at the end of Sec. (A) of the SM. There, we found that ourcentral result (4) can break down if the bath temperature does not change in a differentiable way, which is thecase here. Clearly, this behaviour is the more pronounced, the smaller the number of qubits in the cold bath.In the extreme case of a bath with a single qubit, the temperature would instantaneously jump from T H (0) to T H ( τ ) = T C ∆ H / ∆ S < T H (0). Once more, we repeat that this behaviour is unphysical . In reality, the implementationof the swap operation takes a finite time and then T H ( t ) would change continuously.Out of curiosity, we ask, however, what happens to our central result (4) for the idealized swap engine consideredhere. The answer is shown in Fig. 6, where we plot the following quantities. First, we plot Σ (cid:48) ( N ) − Σ( N ) as afunction of the number of qubits N in the hot bath after N cycles have been completed (i.e., after each bath qubithas interacted with the system). As one can see (left plot in Fig. 6), the inequality Σ (cid:48) ( N ) − Σ( N ) ≥ T H (0) and the associated final equilibrium state at T H ( τ ). Since there are no correlations between the qubits in thehot bath, this quantity becomes N D { π B [ T H ( τ )] | π B [ T H (0)] } , where π B ( T ) is the canonical ensemble at temperature T of a single qubit. We observe that, in unison with our central result (4), the disagreement with Σ (cid:48) ( N ) − Σ( N )is very small already for moderate N . To quantify this difference more precisely, we also display the absolute andrelative error in the right plot of Fig. 6: e abs ( N ) ≡ Σ (cid:48) ( N ) − Σ( N ) − N D { π B [ T H ( N τ )] | π B [ T H (0)] } , (34) e rel ( N ) ≡ Σ (cid:48) ( N ) − Σ( N ) − N D { π B [ T H ( N τ )] | π B [ T H (0)] } Σ (cid:48) ( N ) − Σ( N ) . (35)While the absolute error stays constant as a function of N , the relative error decreases as 1 /N because the ‘amountof discontinuity’ in T H ( t ) gets smaller with increasing N . This result is intuitive and reassuring.1 (D) Nanoscopic heat engines beyond the steady state regime If the system has not yet reached a steady state, Eqs. (7) and (8) of the main text generalize toΣ = ∆ S S ( τ ) − (cid:90) ¯ dQ C ( t ) T C ( t ) − (cid:90) ¯ dQ H ( t ) T H ( t ) ≥ , (36)Σ (cid:48) = ∆ S S ( τ ) − Q C ( τ ) T C (0) − Q H ( τ ) T H (0) ≥ . (37)Using the first law ∆ U S = Q C + Q H + W , we can rewrite both expressions asΣ (cid:48) = W ( τ ) T C (0) − ∆ F S ( τ ) T C (0) + Q H ( τ ) T C (0) − Q H ( τ ) T H (0) ≥ . (38)Here, we have introduced the change in nonequilibrium free energy ∆ F S ( τ ) = ∆ U S ( τ ) − T C (0)∆ S S ( τ ) with respectto the reference temperature T C (0) of the initial cold bath. Now, there are different possibilities to split Σ (cid:48) into an A -and B -term depending on which resources we want to convert to each other. However, independent of how we splitΣ (cid:48) = A + B , we can always write Σ = A + B + Σ − Σ (cid:48) . (39)Therefore, we obtain the efficiencies η (cid:48) = − AB , η = − AB + Σ − Σ (cid:48) . (40)Since 0 < B + Σ − Σ (cid:48) ≤ B , we obtain in general the conclusion that η (cid:48) /η ≤
1. Thus, for any process (not just heatengines) the true efficiency according to the second law (1) is larger than the efficiency inferred from Eq. (2), whichwas previously asserted to be the second law [2, 3]. (E) Efficiencies and second law for far-from-equilibrium baths
In general, a finite bath might be driven out of equilibrium during a thermodynamic process such that it is nolonger well described by a time-dependent temperature (although this does not seem to be the case in most currentexperiments [4–9, 11]). In order to describe the bath more accurately, we then need, however, more information. Here,we assume this information to be available in terms of the probability distribution of the bath energies p ( e B , t ) at time t (we only consider one bath here, but the argument generalizes to multiple baths). Here, e B does not necessarily referto an eigenenergy of the bath Hamiltonian. Instead, p ( e B , t ) can describe some coarse-grained probability distributionas long as the error is small enough such that, for instance, the average bath energy is well approximated bytr B { H B ρ B ( t ) } ≈ (cid:88) e B e B p ( e B , t ) . (41)Since e B does not necessarily refer to a single energy eigenstate, we denote by V ( e B ) all microstates which give rise tomeasurement outcome e B . Note, if p ( e B , t ) = δ e B ,e (cid:48) B for some e (cid:48) B , then the state of the bath equals the conventionalmicrocanonical ensemble.We now define the following notion of entropy, which is known as observational entropy (for a recent review of thisconcept, which goes back to Boltzmann, Gibbs, von Neumann and Wigner, see Ref. [36]): S E B obs ( t ) ≡ (cid:88) e B p ( e B , t )[ − ln p ( e B , t ) + ln V ( e B )] . (42)As demonstrated in Refs. [19, 20], the second law can then be generalized to˜Σ = ∆ S S ( τ ) + S E B obs ( τ ) − S B [ T ( τ )] − (cid:90) ¯ dQ ( t ) T ( t ) ≥ . (43)2Since the Gibbs state maximizes entropy for a fixed energy, we infer that S E B obs ( τ ) − S B [ T ( τ )] <
0. Hence, we obtainthe central result 0 ≤ ˜Σ ≤ Σ ≤ Σ (cid:48) . (44)Thus, ˜Σ gives an even tighter bound than the conventional Clausius inequality Σ ≥ η based on ˜Σ will satisfy1 ≥ ˜ η ≥ η ≥ η (cid:48) . (45)Thus, if we assume additional information and control about nonequilibrium features of the bath, efficiencies can beeven higher.Of course, additional information about nonequilibrium features could be also available in other forms. For instance,if we additionally know (parts of) the system-bath correlation quantified by the mutual information, an even tightersecond law emerges [19, 20]. On the other hand, acquiring information experimentally or theoretically is costly too.At the end, the second law (and thermodynamics in general) is about a tradeoff between knowing the essential features andand