First and Second Law of Thermodynamics at strong coupling
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec First and Second Law of Thermodynamics at strong coupling
Udo Seifert
II. Institut f¨ur Theoretische Physik, Universit¨at Stuttgart, 70550 Stuttgart, Germany
For a small driven system coupled strongly to a heat bath, internal energy and exchanged heatare identified such that they obey the usual additive form of the first law. By identifying thisexchanged heat with the entropy change of the bath, the total entropy production is shown to obeyan integral fluctuation theorem on the trajectory level implying the second law in the form of aClausius inequalilty on the ensemble level. In this Hamiltonian approach, the assumption of aninitially uncorrelated state is not required. The conditions under which the proposed identificationof heat is unique and experimentally accessible are clarified.
PACS numbers: 05.70.Ln
The thermodynamic analysis of a system coupled to awork source and a heat bath of temperature T typicallyrelies crucially on the assumption that the coupling to thebath is weak. If this condition is not met, partitioningthe work W spent in a process into dissipated heat Q and an increase in internal energy of the system ∆ E inthe form of a first law W = ∆ E + Q (1)leaves the question open whether at all, and, if so, inwhich form the interaction between system and bath iscontained in the two terms on the right hand side. Thesame issue arises in the second law when it is written inthe additive form as a Clausius inequality∆ S tot = ∆ S + Q /T ≥ S tot in one of thesystem ∆ S and one of the bath given by the heat dividedby T .Work is arguably the least problematic of the five quan-tities appearing above since it can easily be identifiedeven in the presence of strong coupling. By treating sys-tem and bath including the interaction as one big closedsystem that evolves under a time-dependent Hamilto-nian, the change of the latter from an initial state toa final one represents work. If the initial state is drawnfrom a canonical ensemble for the whole system, work isknown to obey exact relations like the Jarzynski equal-ity [1] and the Crooks relation [2] even in the presenceof strong coupling [3] as, inter alia , many single moleculeexperiments over the last decade have demonstrated con-vincingly [4–9]. Since typical work values of even a fewhundred k B T become tiny when divided by the num-ber of molecules in the solution in contact with the bio-molecule, the change in the interaction between bath andmolecule is not necessarily negligible compared to thosein internal energy of the molecule. The success of theseexperiments therefore rests partially on the fact that theirinterpretation does not require splitting the work into in-ternal energy and heat for these strongly coupled system.On the other hand, for driven solid state devices, recent progess in ultra-sensitive calorimetry should soon makeheat exchange directly accessible experimentally [10, 11].Exploring the role of strong coupling for equilibrium thermodynamics has a long history going back, in theclassical case, at least to Kirkwood’s concept of a po-tential of mean force [12, 13], see, e.g., Ref. [14] fora recent analysis. For quantum systems, the role ofstrong coupling has been discussed in particular in thecontext of damped harmonic oscillators for quite sometime [15, 16] with a recent emphasis on apparent anoma-lies like a negative specific heat [17]. How to formulate aconsistent thermodynamics for a strongly coupled systemunder non-equilibrium conditions, like relaxation afteran initial quench or genuine time-dependent driving, hasfound more attention lately for quantum systems thanfor classical ones. Various approaches and schemes arediscussed [18–33] without arguably reaching a consensusyet on how to identify, beyond work, the terms in (1) and(2) uniquely.Crucial aspects surface similarly in both frameworks,classical and quantum. One common subtle issue con-cerns entropy production since treating the full system asclosed, which works so nicely for an identification of work,implies on the other hand that the total change of Gibbs,or Shannon, entropy (classically), or of the von Neumannentropy in the quantum case, remains strictly constanteven under time-dependent driving. A positive entropyproduction results, however, if one ignores the correla-tions between system and bath, see, e.g., Ref. [22]. Eventhen, however, the identification of heat is not uniqueas, e.g., the comparison of two schemes for a simple re-laxation for quantum Brownian motion has shown [27].Moreover, in these approaches, one often assumes thatinitially system and bath are individually equilibrated asif there was no interaction. For most bio-molecular sys-tems in aqueous solution, however, such an assumptionis certainly rather unrealistic.In this paper, we present an approach that allows toidentify the terms appearing in the additive forms of thefirst and the second law consistently for driven classi-cal systems strongly coupled to a heat bath without re-quiring an initially uncorrelated state. In the limit ofweak coupling, these quantities will become the estab-lished ones. A particular virtue of this approach is thatthe terms appearing in (1) and (2) can be inferred frommeasurements involving only observables of the system.As reference for the driven case, and to establish no-tation, we first recall the equilibrium situation, see, e.g.,[14, 25]. For a system coupled to a heat bath, the totalHamiltonian reads H tot ( ξ, λ ) = H s ( ξ s , λ ) + H b ( ξ b ) + H i ( ξ ) , (3)comprising, in this order, system, bath, and interactionHamiltonian. A micro state in the full phase space iswritten as ξ ≡ ( ξ s , ξ b ) where ξ s and ξ b denote microstates in the phase space of system and bath, respec-tively. The control parameter λ , which will later be usedto drive the system, does neither affect the bath nor theinteraction part of the Hamiltonian. While in a Hamil-tonian approach it may look more natural to consider amicrocanonical equilibrium for the full system, for techni-cal reasons that will become clear later we rather choosea canonical equilibrium for the total system at inversetemperature β . Then the probability to find the systempart in a state ξ s is given by p eq ( ξ s | λ ) = exp[ − β ( H ( ξ s , λ ) − F ( λ ))] . (4)Here, H ( ξ s , λ ) ≡ H s ( ξ s , λ ) − β − ln h exp[ − βH i ( ξ )] i b . (5)is an effective Hamiltonian, or, in the jargon of physicalchemistry, a potential of mean force. It involves a canon-ical average over the pure bath (at fixed ξ s ) denoted inthe following by h ... i b ≡ Z dξ b ... exp[ − β ( H b ( ξ b ) − F b )] , (6)where F b is the free energy of the pure bath. The λ -dependent free energy of the system is defined throughexp[ − β F ( λ )] ≡ Z dξ s exp[ − β H ( ξ s , λ )] . (7)Still in equilibrium, this free energy implies through thestandard relation S = β ∂ β F for the entropy of the sys-tem S ( λ ) = Z dξ s p eq ( ξ s )[ − ln p eq ( ξ s ) + β ∂ β H ( ξ s , λ )] , (8)setting Boltzmann’s constant to 1 throughout. Likewise,the internal energy E = F + S /β becomes E ( λ ) = Z dξ s p eq ( ξ s ) E ( ξ s , λ ) (9)with E ( ξ s , λ ) ≡ H ( ξ s , λ ) + β∂ β H ( ξ s , λ ) . (10) In the weak coupling limit, the three energy functions H s , H , and E converge.The additional contribution ∼ ∂ β H ( ξ s , λ ) beyond whatone might have expected naively for entropy and internalenergy takes into account that due to the finite interac-tion the bath is correlated with the microstate ξ s of thesystem. In fact, with the standard canonical equilibriumfor the total system obeying in obvious notation the re-lation F tot = E tot − S tot /β and that for the pure bathwith F b = E b −S b /β the above identified thermodynamicquantities of the system fulfill X = X tot − X b (11)for X = F , E , S . This additive relation indicates thatin this approach the interaction is fully accounted forthrough modification of the quantities refering to the sys-tem.We now drive the system for a time t through a time-dependent control parameter λ τ , with 0 ≤ τ ≤ t . Thetotal system comprising the system proper, the heat bathand the interaction is assumed to be closed. An initialphase point ξ then evolves in time deterministically into ξ t . The corresponding mapping ξ t = ξ t ( ξ ) has Jacobian1 due to Liouville’s theorem. We first keep a trajectory-based approach [34, 35] in which all quantities become afunction of the initial phase point ξ .The work spent in the driving is the total energy dif-ference w ( ξ ) ≡ ∆ H tot ( ξ, λ ) ≡ H tot ( ξ t , λ t ) − H tot ( ξ , λ ) . (12)Here, and in the following, ∆ operating on a quantity im-plies the difference of this quantity between final and ini-tial value. Hamiltonian dynamics implies that this workcan also be written as w ( ξ ) = Z t dτ ∂ λ H s ( ξ τ s , λ τ ) ∂ τ λ (13)which is the form used in stochastic energetics [34, 35]. Infact, one could replace here ∂ λ H s by either ∂ λ H or ∂ λ E without changing the subsequent results since all threederivatives are the same.As a key step in the present approach, motivated by(9), internal energy of the system along a driven tra-jectory ξ τ is identified as E ( ξ τ s , λ ), independent of thespecific (and in any case unknown) value of the instan-taneous bath coordinates ξ τ b . As we will show below,thus a consistent thermodynamic scheme arises. This as-signment of internal energy implies the identification ofdissipated heat as q ( ξ ) = w ( ξ ) − ∆ E ( ξ s , λ ) = ∆[ H tot ( ξ, λ ) − E ( ξ s , λ )] . (14)It is instructive to show more explicitly how the inter-action modifies the standard forms of the terms in thefirst law. Writing H i ( ξ ) = h H i ( ξ ) i b + δH i ( ξ ) (15)we separate the mean interaction, at fixed system co-ordinate ξ s , from its fluctuations δH i ( ξ ). Similarly, theenergy of the bath is split according to H b ( ξ b ) = h H b ( ξ b ) i b + δH b ( ξ b ) . (16)With (5) and (10), the change in internal energy thenbecomes after little algebra∆ E ( ξ s , λ ) = ∆[ H s ( ξ s , λ ) + h H i ( ξ ) i b + (17)+ h δH i ( ξ ) B i ( ξ ) i b + h δH b ( ξ b ) B i ( ξ ) i b ]where B i ( ξ ) ≡ exp[ − βδH i ( ξ )] / h exp[ − βδH i ( ξ )] i b . (18)Thus the average interaction is fully attributed to the in-ternal energy, which, however, also picks up two morecontributions from the fluctuations. Correspondingly,the heat (14) becomes q ( ξ ) = ∆[ H b ( ξ b ) + δH i ( ξ ) (19) −h δH i ( ξ ) B i ( ξ ) i b − h δH b ( ξ b ) B i ( ξ ) i b ] . Beyond the standard expression of dissipated heat, whichis the change in energy of the bath ∆ H b ( ξ b ), the first twofurther contributions depend on how much the interac-tion fluctuates for a fixed system state ξ s . The last con-tribution depends on correlations of the interaction withfluctuations of the bath. In the weak coupling limit, theseadditional contributions vanish since the interaction be-comes negligible.The first law is thus obeyed on the trajectory level byconstruction. It will remain valid on the ensemble levelafter averaging with, in principle, any initial distribution p ( ξ ). As physically sensible initial distributions we willchoose from now on p ( ξ ) = p ( ξ s ) p eq2 ( ξ b | ξ s , λ ) (20)where p eq2 ( ξ b | ξ s , λ ) ≡ exp[ − β ( H i ( ξ ) + H b ( ξ b ) − F b )] h exp[ − βH i ( ξ )] i b . (21)The initial distribution of the system p ( ξ s ) is arbitrary.For technical reasons, we require that it does not vanishanywhere on the phase space of the system. The bath isassumed to be equilibrated initially for any system state ξ s . In the following, averages with this initial distributionwill be denoted by h ... i . Note that with the option of aninitially non-equilibrated system part relaxation towardsequilibrium at constant control parameter, e.g., after aquench of the system, is covered by this framework aswell. If p ( ξ s ) is the equilibrium distribution (4), thenthe initial distribution (20) corresponds to the canonicalequilibrium in the full phase space.We now turn to checking the consistency of the pro-posed identification of heat with the additive form of the second law. As a technical tool, we will use the trivial butpowerful identity, or integral fluctuation theorem (IFT),1 = Z dξ t ρ ( ξ t ) = h exp[ln[ ρ ( ξ t ( ξ )) /p ( ξ )]] i . (22)Liouville’s theorem ensures that this IFT is valid for anynormalized function ρ ( ξ ) provided the initial distribution p ( ξ ) vanishes nowhere on the full phase space. By choos-ing the legitimate factorized form ρ ( ξ ) = p t ( ξ s ) p eq2 ( ξ b | ξ s , λ t ) (23)where p τ ( ξ s ) is the true marginal distribution for ξ s attime τ , the IFT (22) becomes after trivial algebra1 = h exp[ − (∆ s ( ξ ) + βq ( ξ ))] i (24)where the average is over the initial distribution (20).Here, the change in system entropy along the trajectoryis∆ s ( ξ ) ≡ − ln p t ( ξ t s ) + ln p ( ξ ) + ∆ β ∂ β H ( ξ s , λ ) . (25)The first two terms amount to the change in stochas-tic entropy familiar from stochastic thermodynamics [36].The third contribution, called intrinsic entropy in a re-lated context [37], has the same physical origin as dis-cussed above in equilibrium. If we now identify, as usual,the entropy change of the bath on the trajectory levelwith the exchanged heat (times β ), the exponent in (24)becomes the total entropy production,∆ s tot ( ξ ) ≡ ∆ s ( ξ ) + βq ( ξ ) , (26)which thus obeys an IFT h exp[ − ∆ s tot ( ξ )] i = 1 . (27)Even though this IFT looks like the one derived ear-lier using a stochastic dynamics [36], one should notethat here it follows from a Hamiltonian dynamics for astrongly coupled driven system.The second law (2) for the calligraphic capitalizedquantities that denote the averages with respect to theinitial distribution (20) follows trivially from Jensen’s in-equality applied to (27). On a mathematical level, wehave thus shown that if internal energy, heat, and thetwo contributions to total entropy production are identi-fied as suggested here, the additive form of the first andsecond law are valid in the presence of strong coupling.Can heat and the other quantities be measured in anexperiment where one has access to the trajectory of thedegrees of freedom of the system ξ τ s but, of course, notto the bath coordinates? Equilibration at fixed λ yields H ( ξ s , λ ) from measuring the corresponding equilibriumdistribution (4). Repeating these measurements at aslightly different temperature will lead to ∂ β H ( ξ s , λ ) andthus to the internal energy E ( ξ s , λ ) through (10). Forthe driven system, the work is accessible from observingthe trajectory ξ τ s using (13) since the λ -dependence of H s ( ξ s , λ ) is controlled in an experiment. Hence, the heatcan be inferred from evaluating (14). Finally, the changein system entropy follows from measuring the marginaldistributions p t ( ξ s ) and p ( ξ s ). Thus, all quantities are,at least in principle, measurable experimentally from tra-jectories ξ τ s without ever having to measure a bath de-gree of freedom. The ensemble quantities appearing in(1) and (2) then follow from averaging the trajectory-resolved measurements.A few further aspects, implications and perspectivesare worth noting. First, is this assignment of heat, or,equivalently, internal energy unique? On a formal level,there seems to be freedom. Replacing internal energy,heat and change in system entropy on the trajectorylevel according to E ( ξ s , λ ) → E ( ξ s , λ ) + χ ( ξ s , λ ) , q → q − ∆ χ ( ξ s , λ ), and ∆ s → ∆ s + β ∆ χ ( ξ s , λ ), respectively,with an arbitrary system state function χ ( ξ s , λ ), whichvanishes in the weak coupling limit, leaves the first law(14) and the IFT (27) invariant. In fact, the choice χ ( ξ s , λ ) = H ( ξ s , λ ) − E ( ξ s , λ ) amounts to what has beendiscussed in Ref. [27] under the label ”poised”. The cru-cial point, however, is that any choice χ ( ξ s , λ ) = 0 willspoil the thermodynamic relation S = β ∂ β F , or, equiv-alently, d E | λ = T d S | λ , when applied on the ensemblelevel to equilibrium. As long as one requires these latterrelations for assigning the label ”thermodynamically con-sistent” only the present scheme with χ ( ξ s , λ ) ≡ t . Us-ing the latter in (22) should therefore be interpreted asa mathematical convenience for deriving the IFT (24)rather than as a statement about the true distribution.As an aside, note that substituting the canonical distri-bution of the full system at λ t for ρ ( ξ ) into (22) yieldsthe strongly coupled Jarzynski equality [3] for an initiallyequilibrated system in one line.Third, equality in the second law usually requires aquasistatic process. In our approach, the second law (2)follows from the IFT (27). Any IFT requires for a sat-uration of the corresponding inequality that the under-lying distribution for the exponent is delta-like. Thusequality in (2) holds if and only if ∆ s tot ( ξ ) vanishesidentically for all initial micro states ξ . Ultimately, thisrequirement implies that the distribution for the full sys-tem starts and remains canonical throughout the pro-cess. In this respect, the strong coupling case does notdiffer from weak coupling. In fact, from a more phys-ical perspective, one would expect that a moderate orstrong coupling should facilitate equilibration and hence the realization of quasistatic conditions even more thanthe common idealized weak coupling case does.Fourth, so far, we have not split the total volume intoone of the system and one of the bath which would giverise to a pressure term. It would be interesting to explorewhich modifications arise from such a perspective in thecase of strong coupling [38].Finally, since the main part of this paper dealt withclassical systems, it is worth emphasizing that the presentscheme suggests, by analogy with (10), as an internalenergy operator for the system in the quantum caseˆ E ≡ − (1 + β∂ β )[ β − ln Tr b exp[ − β ( ˆ H s + ˆ H i + ˆ H b − F b )]] , (28)where hats denote operators, the trace is over the bathdegrees of freedom and exp( − βF b ) ≡ Tr b exp( − β ˆ H b ).In general, this operator ˆ E will be a quite complicatedfunction of temperature and the parameters of the totalHamiltonian. The change in internal energy then follows,in principle, from two point measurements of ˆ E at τ = 0and τ = t . Since, in general, [ ˆ E, ˆ H tot ] = 0, work as givenby the difference in total energy can not be measuredsimultaneously. Hence, heat as the difference of workand internal energy is not accessible through this routein the quantum case.In conclusion, for a classical driven system stronglycoupled to a heat bath not only work but also inter-nal energy, dissipated heat and entropy production canbe identified on the level of a trajectory of the system.Total entropy production obeys an integral fluctuationtheorem implying, on the ensemble level, a consistentinterpretation of the second law as a Clausius inequal-ity. For an experimental realization, the heat accompa-nying conformational changes of mechanically manipu-lated bio-molecules should be accessible experimentallythrough measurements at two different temperatures assuggested here. While the theory is not confined to thisparticular class, with such experiments these moleculescould turn out to become one paradigm for studying heatexchange in small driven strongly coupled systems.Acknowledgments: I thank S. Goldt and P. 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