Fluctuation Theorem for Arbitrary Open. Quantum Systems. Peter Hänggi, Michele Campisi, and Peter Talkner

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2 Acknowledgments Sekhar Burada Gert Ingold Eric Lutz Manolo Morillo Volkswagen Foundation

3 Tasaki-Crooks Theorem and Jarzynski Equality: State of the art Classical Isolated system Weak coupling Strong Coupling Isolated system Weak coupling Strong Coupling

4 What are fluctuation and work theorems about? Small systems: fluctuations may become comparable to average quantities. Can one infer thermal equilibrium properties from fluctuations in nonequilibrium processes?

5 Jarzynski s equality C. Jarzynski, PRL 78, 269 (1997) Z 1 (t )e βh(to) non-eq. H(t ) H(t f ) t e βw = e β F {H(t)} tf,t : protocol w: work performed F = F (t f ) F (t ), F (t) = β 1 ln Z(t) : average over realizations of the same protocol Jensen s inequality : F w Second Law

6 LETTERS DG), the use of Bennett s acceptance ratio method 2 makes it possible differs only NATURE Vol by one base-pair, September and (2) the 25 thermodynamic stabilizing to extract accurate estimates of DG using the CFT (see the Supplementary Information). three-helix junction of the 16S ribosomal RNA of Escherichia coli 12 effect of Mg 2þ ions on the RNA structure. The RNA we consider is a that binds the S15 protein. The secondary structure of this RNA is a satisfy the CFT, that is, common equation feature (1) (see in RNA the structures Supplementary that plays, in this case, a Information). We also notice crucial thatrole work in the distributions folding of the arecentral compatible domain of the 3S ribosomal with, and can be fitted to, subunit. gaussianfor distributions comparison, (data and tonot verify shown). the accuracy of the method, After subtracting the contribution we have pulled arisingthe from wildthe type entropy and aloss CzGdue togzc mutant (C754G to Thus the value of DG can be determined sible experimental once the pulling two distributions speeds (dissipatedtowork the stretching values lessof than the molecular G587C) of handles the three-helix attachedjunction. both sides of are known. In particular, the two 6k BT). distributions Under these cross conditions, at W ¼ the DG: unfolding theand hairpin refolding (DGwork handles ¼ 23.8 Figure k BT) 3 depicts and of the the unfolding extended and refolding singlestranded work distributions for distributions overlap over a sufficiently large PUðWÞ¼PRð2WÞ)W ¼ DG ð3þ range of work (ss)rna values ðdg to ssrna the ¼wild-type 23:7 ^ 1kBTÞ and mutant frommolecules the total(work work, values were binned into justify the use of the direct method to experimentally DG test equation ¼ 11:3 ^ :5 kbt, we about obtain 1 2 forequally the freespaced energyintervals). of unfolding For both molecules, the regardless of the pulling speed. (1). Although The workthe done simple on the identity molecules (3) during either pulling at zero force DG exp or ¼ 62:8distributions ^ 1:5kBT ¼ display 37:2 ^ 1a kcalmol very narrow 21 (at overlapping 258 C, region. In contrast already gives an estimate of DG, relaxation it is not is given necessarily by thevery areasprecise below the corresponding force in 1 mm Tris-HCl, ph 8.1, with1the mmhairpin EDTA), distribution, in excellent theagreement average dissipated work for the because it uses only the local extension behaviour curves of the (Fig. distribution 1). around unfolding pathway is now much larger in the range 2 4kBT W ¼ DG. Using the whole work distribution increases the precision of the free-energy estimate 19. In particular, as we show below, when the overlapping region of work values between the unfolding and refolding work distributions is too narrow (as may happen for large values of the average dissipated work, defined as kw disl ¼ kwl 2 DG), the use of Bennett s acceptance ratio method 2 makes it possible to extract accurate estimates of DG using the CFT (see the Supplementary Information). three-helix junction of theof16s longribosomal tails in the work RNAdistribution of Escherichia P U(W) coli 12 along the unfolding path effect of Mg 2þ ions on thefrom-equilibrium RNA structure. The conditions. RNA weour consider measurements is a reveal the presence We first experimentally test the validity of the CFT for a molecular that binds the S15 protein. The secondary structure of this RNA is a transition occurring near equilibrium. For this, we use a short common feature in RNA structures that plays, in this case, a interfering (si)rna hairpin that targets the messenger RNA of the crucial role in the folding of the central domain of the 3S ribosomal CD4 receptor of the human immunodeficiency virus (HIV) 11 and subunit. For comparison, and to verify the accuracy of the method, that unfolds Trap irreversibly bead but not too far from equilibrium at accessible experimental pulling speeds (dissipated work Trap values beadless than energy differences. Consider a system, kept in contact with we 19.) have The pulled JE indicates the wilda type practical and a CzG way togzc determine mutant free- (C754G to G587C) of the three-helix junction. 6k a bath at temperature T, whose equilibrium state is determined by a control parameter x. Initially, the control pa- BT). Under these conditions, the unfolding and refolding work Figure 3 depicts the unfolding and refolding work distributions for distributions overlap over a sufficiently large range of work values to the rameter wild-type is x and A and mutant the system molecules is in an (work equilibrium values were state binned A. into justify thelaser use of the direct method to experimentally test equation about trap The nonequilibrium 1 2 equally process spaced is intervals). obtained For by changing both molecules, x according to a given display protocol a very narrow x(t), from overlapping x A to some region. final value In contrast the (1). The work done on the molecules during either Handle pulling or distributions relaxation is given by the areas below the corresponding force with x B. In the general, hairpinthe distribution, final state the of the average system dissipated will not work be atfor the extension curves (Fig. 1). RNA unfolding equilibrium. pathway It will isequilibrate now much to larger in a state B the if range it is allowed 2 4kBT Unfolding and refolding work distributions at three molecule different and to further unfolding evolve with work the distribution control parameter shows a large fixed tailat and x B. strong pulling speeds are shown in Fig. 2. Irreversibility increases with the deviations The JE states fromthat gaussian behaviour. Thus, these molecules are ideal pulling speed and unfolding refolding work distributions become to test the validity of equation (1) in the far-from-equilibrium progressively Actuator more separated. Note, however, that the unfolding and DG W bead regime. As shown exp in the inset of exp Fig. 3, the plot, of the log(4) ratio of the refolding distributions cross at a value ofactuator the work DG ¼ the unfolding to the refolding kt B probabilities kt B NATURE Vol versus September total work 25 done on 11:3 ^ :5 kbt that does not depend on the pulling bead speed, as the molecule can be fitted to a straight line with a slope of 1.6, thus predicted by equation (3). Moreover, the work distributions also establishing where DG is the the validity free-energy of Figure the difference 2 CFT Test (see ofbetween the equation CFT using the (1)) equilibrium states A and B, and the angle brackets denote an an RNA under hairpin. farfrom-equilibrium conditions. Work distributions for RNA Our unfolding measurements (continuous reveal lines) the and presence refolding (dashed lines). We plot Figure 4. Testing the Jarzynski Figure 1 Force extension equality. A molecule curves. of The RNA stochasticity average of the taken unfolding over andan infinite negative work, number P R(2W), of nonequilibrium for refolding. Statistics: 13 pulls and three is attached to two beads refolding and subjected process is characterized to reversible by and a distributionof irreversible cycles of folding experiments oflong unfolding tails in orrepeated the refolding work distribution under molecules the (r protocol P¼ U(W) 1:5pNs along x(t). 21 ), the Frequently, 38 unfolding pulls and four path molecules (r ¼ 7:5pNs 21 ), work trajectories. and unfolding. Five unfolding A piezoelectric (orange) and refolding the JE (blue) is force recast in 7 the pulls form and three exp( W molecules dis)/k(r BT ¼ 2:pNs 1, 21 ), for a total of ten separate actuator controls the position extension of curves the bottom for the RNA bead, hairpin which, are shown (loading where rate W dis of 7.5W pn s 21 DG). is the experiments. dissipated Goodwork reproducibility along a given was obtained among molecules (see when moved, stretches The the blue RNA. area An under optical the curve trap represents formed by the worktrajectory. returned to the machine Supplementary Fig. S2). Work values were binned into about ten equally as the molecule switches from the unfolded to the folded two opposing lasers captures the top bead, and the change The state. exponential The RNA average spacedappearing intervals. Unfolding the and JE refolding implies distributions at different speeds sequence is shown as an inset. in momentum of light that exits the two-beam trap determines the force exerted 232 on the molecule connecting the two scopic systems, is the statement of the second law of ther- that W DGor, equivalently, show a common W dis, crossing which, around for macro- DG ¼ 11:3kBT. beads. The difference in positions of the bottom and top modynamics 25 in terms Nature of Publishing free energy Group and work. An important consequence of the JE is that, although on average beads gives the end-to-end length of the molecule. The blowup shows how the RNA molecule (green) is coupled W dis, the equality can only hold if there exist nonequilibrium trajectories with W dis. Those trajectories, some- with the two beads via molecular handles (blue). The handles end in chemical groups (red) that can be stuck to complementary groups (yellow) on the bead. The blowup is not times referred to as transient violations of the second law, represent work fluctuations that ensure the microscopic to scale: The diameter of the beads is around 3 nm, equations of motion are time-reversal invariant. The remarkable JE implies that one can determine the free- much greater than the 2-nm length of the RNA. energy difference between initial and final equilibrium ishingly rare with increasing system size. For large systems, the conventional second law emerges. connects those states, but also Figure via 3 a nonequilibrium, Free-energy recoveryirre- and test of the CFT for non-gaussian work states not just from a reversible or quasi-static process that versible process that connects distributions. them. The Experiments ability to were bypass carried out on the wild-type and mutant The Jarzynski equality reversible paths is of great practical S15 three-helix importance. junction without Mg 2þ. Unfolding (continuous lines) and D. refolding Colin (dashed lines) et work distributions. al. Nature Statistics: 9 pulls and two The various FTs that have been reported differ in the details of such considerations as whether the system s dy- Figure ing a 2 generalized Test of the theorem CFT using for anorange). RNA stochastic hairpin. Crossings microscopically Work between distributionsfor are indicated by black circles. In 1999, Gavin Crooks related molecules various (wild type, FTs purple); by deriv- 1,2 pulls and five molecules (mutant type, namics are stochastic or deterministic, whether the kinetic RNA reversible unfolding dynamics. (continuous 437, 5 lines) The Work and box refolding histograms below 231 (dashed were gives found (25) lines). details. to bewe reproducible plot among different molecules Figure 1 energy Force extension some curves. other The variable stochasticity is kept of theconstant, unfolding and and negative The past work, six Pyears R(2W), have for refolding. seen (error further Statistics: bars indicating consolidation, 13 pulls the range andof and three variability). Inset, test of the CFT for the refoldingwhether process isthe characterized system is byinitially a distribution prepared of unfolding equilibrium or refolding or molecules physicists (rnow ¼ 1:5pNs understand 21 ), 38 pulls that mutant. and neither four Data have molecules the been details linearly (r ¼of 7:5pNs interpolated just 21 ), between contiguous bins of the work trajectories. in a nonequilibrium Five unfolding steady (orange) state. and refolding A novel treatment (blue) force of dissipative curves for processes the RNA hairpin in nonequilibrium are shown (loading systems rate of 7.5 was pnintro- s 21 ). experiments. namics nor Good the somewhat reproducibility differing was obtained interpretations among molecules of en-(see The blueduced area under in 1997 the curve when represents Christopher the work Jarzynski returnedreported to the machine a non- Supplementary tropy production, Fig. S2). entropy Work values production were binned rate, into about dissipated ten 7 which pulls quantities and three molecules are maintained (r ¼ unfolding 2:pNs constant and 21 ), refolding for during a total work of the distributions. xx tendy- separate extension equally We first experimentally test the validity of the CFT for a molecular along the unfolding path while transition (absolute) occurring values near smaller equilibrium. than DG For this, we use a short occur more often along the interfering refolding path. (si)rna As hairpin can be that seentargets from the messenger RNA of Crooks equation (1), the CFT states that CD4 although receptor fluctuation Pof U(W), the human P R(2W) immunodeficiency depend virus (HIV) theorem 11 and on the pulling protocol, their ratio that unfolds depends irreversibly only on the but value not too of DG. far from equilibrium at acces- Unfolding and refolding work distributions with the result obtained using the Visual OMP from DNA software p tf,t (w) 21 DG mfold at three different p t,t f ( w) = e β( F w) ¼ 38 kcal mol 21 and the unfolding work distribution shows a large tail and strong pulling speeds are shown in Fig. 2. Irreversibility increases with the (at 258 C, in 1 mm NaCl). deviations from gaussian behaviour. Thus, these molecules are ideal pulling speed and unfolding refolding work distributions To extend the become experimental test of the validity of the CFT to the to test the validity of equation (1) in the far-from-equilibrium progressively more separated. Note, however, very-far-from-equilibrium that the unfolding and regime. regime As where shown the in work the inset distributions of Fig. 3, the are plot of the log ratio of the refolding distributions cross at a value no longer of the gaussian, work DGwe ¼ apply the G.E. the unfolding CFT to to determine: Crooks, the refolding (1) probabilities the difference PRE versus total work 6,2721 done on (1999) 11:3 ^ :5 kbt that does not depend oninthe folding pulling freespeed, energy asbetween the molecule an RNAcan molecule be fittedand to aastraight mutantline that with a slope of 1.6, thus predicted by equation (3). Moreover, the work differsdistributions only by onealso base-pair, establishing and (2) thethermodynamic validity of CFT stabilizing (see equation (1)) under far- Actuator Figure 4 Use of CFT to extract the stabilizing contribution free energy of the S15 three-helix junction (wild type). U (continuous lines) and refolding (dashed lines) work distri curves, 45 pulls and two molecules in Mg 2þ ; purple curve two molecules without Mg 2þ. Crossings between distributio by black circles. Work histograms are reproducible between (error bars indicating the range of variability). Inset, the sam logarithmic scale (axes labels as for the main panel) showin bars) the regions of work values where unfolding and refoldi are expected to cross each other by Bennett s acceptance ra (Supplementary Information).

7 Equilibrium versus nonequilibrium processes Isothermal quasistatic process: e βh(t ) /Z(t ) BATH β weak contact SYSTEM H(t ) xx xx xxquasixxstatic e βh(t f ) /Z(t f ) BATH β weak contact SYSTEM H(t f ) Non-equilibrium process: H(t ) H(t f ) F = w = w p tf,t (w) =? pdf of work t t f t

8 Definition of work A. Classical: w = tf t dt H(z(t), t) t = H(z(t f ), t f ) H(z(t ), t ) z(t): Trajectory in phase space z(t ): Starting point taken from Z 1 (t )e βh(z,t ) B. mechanical: w = e m (t f ) e n (t ) H(t)ϕ n,λ (t) = e n (t)ϕ n,λ (t) work is a random quantity due to the randomness inherent in the initial state ρ(t ) and in quantum mechanics.

9 Probability of work H(t)ϕ n,λ (t) = e n (t)ϕ n,λ (t) P n (t) = λ ϕ n,λ (t) ϕ n,λ (t) p n = Tr P n (t )ρ(t ) = probability of being at energy e n (t ) at t = t ρ n =P n (t )ρ(t )P n (t )/p n = state after measurement ρ n (t f ) =U tf,t ρ n U + t f,t p(m n) = TrP m (t f )ρ n (t f ) = conditional probability of getting to energy e m (t f )

10 Probability of work p tf,t (w) = δ(w [e m (t f ) e n (t )])p(m n)p n n,m

11 Characteristic function of work G tf,t (u) = dw e iuw p tf,t (w) = e iuem(t f ) e iuen(t) TrP m (t f )U tf,t ρ n U t + f,t p n m,n = Tre iuh(t f ) P m (t f )U tf,t e ih(t) ρ n U t + f,t p n m,n = Tre iuh H(t f ) e iuh(t ) ρ(t ) e iuh(t f ) e iuh(t ) t H H (t f ) = U t f,t H(t f )U tf,t, ρ(t ) = n P n (t )ρ(t )P n (t ), ρ(t ) = ρ(t ) [ρ(t ), H(t )]

12 Work is not an observable Note: G tf,t (u) is a correlation function. If work was an observable, i.e. if a hermitean operator W existed then the characteristic function would be of the form of an expectation value G W (u) = e iuw = Tre iuw ρ(t ) Hence, work is not an observable. P. Talkner, P. Hänggi, M. Morillo, Phys. Rev. E 77, (28) P.Talkner, E. Lutz, P. Hänggi, Phys. Rev. E 75, 512(R) (27)

13 Canonical initial state BATH β weak contact SYSTEM H(t ) t H(t ) H(t f ) t f t ρ(t ) = Z 1 (t )e βh(t ), Z(t ) = Tre βh(t ), ρ(t ) = ρ(t ) G c t f,t (β, u) = Z 1 (t )Tre iuh H(t f ) e iuh(t ) e βh(t )

14 Choose u = iβ e βw = dw e βw p tf,t (w) = Gt c f,t (iβ) = Tre βh H(t f ) e βh(t) Z 1 (t )e βh(t ) = Tre βh(t f ) /Z(t ) = Z(t f )/Z(t ) = e β F quantum Jarzynski equality

15 u u + iβ and time-reversal Z(t )Gt c f,t (u) = Tr U t + f,t e iuh(t f ) U tf,t e i( u+iβ)h(t ) = Tr e i( u+iβ)h(t f ) e βh(t f ) U t +,t f e i( u+iβ)h(t) U t,t f = Z(t f )Gt c,t f ( u + iβ) p tf,t (w) p t,t f ( w) = Z(t f ) Z(t ) eβw β( F w) = e Tasaki-Crooks theorem white H. Tasaki, cond-mat/9244. P. Talkner, P. Hänggi, J. Phys. A 4, F569 (27).

16 Microcanonical initial state ρ(t ) = ω 1 E (t )δ(h(t ) E), ω E (t ) = Tr δ(h(t E)) = e S(E,t )/k B ω E (t ): Density of states, S(E, t ): Entropy p mc t f,t (E, w) = ω 1 E (t )Tr δ(h H (t f ) E w)δ(h(t ) E) pt mc f,t (E, w) pt mc,t f (E + w, w) = ω E+w (t f ) = e [S(E+w,t f ) S(E,t )]/k B ω E (t ) P.Talkner, P. Hänggi, M. Morillo, Phys. Rev. E 77, (28)

17 Example Driven harmonic oscillator H(t) = ωa + a + f (t)a + f (t)a +, f (t) = for t < t G tf,t (u)=exp [ iu f (t f ) 2 ( ) ω + e iu ω 1 z 2 ] n= p n = TrP n (t )ρ(t ) = n ρ(t ) n, z = 1 tf ds ω t ( p n L n 4 z 2 sin 2 ωu ) 2 df (s) e iωs ds w = ω z 2 f (t f ) 2 ω, w 2 w 2 = 2( ω) 2 z 2 ( a + a ) w : independent of initial state; : average w.r.t. initial state ρ(t ) = n p n n n. P. Talkner, S. Burada, P. Hänggi, PRE 78, (28).

18 q mc r () 1 q mc r (3) z z p tf,t (w) = r Z q r δ[w ( ωr f (t f ) 2 /( ω) ) ] mc.i.s. n =, (a) r (b) r 3 4 q r mc (n) q c r( β) n = n = 1 n = r β =.5 β = 1. β = r mc.i.s. z = 2 β = β ω c.i.s. z = 2

19 q r cs (α) (a) α 2 =.1 α 2 = 1 α 2 = r coherent i.s. z = 2 q r cs (α) (b) z = 1 z = 2 z = r coherent i. s. α = 1 q r α = e αa+ α a p n = α 2n e α 2 n! canonical ensemble coherent state r coherent ( α ) versus canonical (β ω =.1), z = 2

Candidate experimental check of Jarzynski equality in the quantum regime Atom in Paul Trap: Harmonic Oscillator Prepare oscillator in thermal state Probe the

20 Candidate experimental check of Jarzynski equality in the quantum regime Atom in Paul Trap: Harmonic Oscillator Prepare oscillator in thermal state Probe the oscillator initial eigenstate Change trap stiffness Probe the oscillator final eigenstate Construct p tf,t (W ) G. Huber, F. Schmidt-Kaler, S. Deffner, E.Lutz, PRL (28)

21 : Weak Coupling H S (t) H SB H B H(t) = H S (t) + H B + H SB H S (t)p i,α (t) = e S i (t)p i,α (t), H B P i,α (t) = e B α P i,α (t), e S i (t f ) e S j (t ) = E = internal energy change e B α e B β = Q = exchanged heat ρ = i,α P i,α (t )ρ P i,α (t ) p tf,t (E, Q) = joint PDF for E and Q G E,Q t f,t (u, v) = Tre i(uhs H (t f ) vh B H (t f )) e i(uh S (t ) vh B ) ρ P. Talkner, M. Campisi, P. Hänggi, J.Stat.Mech. (29) P225

22 ρ = Z 1 (t )e β(hs (t )+H SB +H B ) therm. eq. at t [ β ] ρ 1 dβ e β (H S (t )+H B ) δh SB e β (H S (t )+H B ) ρ = Z 1 S (t )Z 1 (t )+H B ) B e β(hs ρ = i,α P i,α (t )ρ P i,α (t ) = ρ + O ((δh SB ) 2) G E,Q t f,t (u, v) = Tre i(uhs H (t f ) vh B H (t f )) e i(uh S (t ) vh B ) ρ

23 Crooks theorem for energy and heat Z S (t )G E,Q t f,t (u, v) = Z S (t f )G E,Q t,t f ( u + iβ, v iβ) p tf,t (E, Q) p t,t f ( E, Q) = Z S(t f ) Z S (t ) eβ(e Q) = e β( F S E+Q) w = E Q : work p Q,w t f,t (Q, w) p Q,w t,t f ( Q, w) = e β( F S w), p w t f,t (w) p w t,t f ( w) = e β( F S w) p tf,t (Q w) = p t,t f ( Q w), p tf,t (Q w) = pq,w t f,t (Q, w) p w t f,t (w)

24 Strong coupling: Treatment H(t) = H S (t) + H SB + H B the total system: p tf,t (w) p t,t f ( w) = Y (t f ) Y (t i ) eβw where: Y (t) = Tre β(h S (t)+h SB +H B ) and w = e m (t f ) e n (t ) e n (t)= instantaneous eigenvalues of total system

25 Free energy of a system strongly coupled to an environment Thermodynamic argument: F total system free energy F B bare bath free energy. F S = F F B With this form of free energy the three laws of thermodynamics are fulfilled. G.W. Ford, J.T. Lewis, R.F. O Connell, Phys. Rev. Lett. 55, 2273 (1985); P. Hänggi, G.L. Ingold, P. Talkner, New J. Phys. 1,1158 (28); G.L. Ingold, P. Hänggi, P. Talkner, Phys. Rev. E 79, (29).

26 Partition function Z S (t) = Y (t) Z B where Z B = Tr B e βh B

27 Main results p tf,t (w) p t,t f ( w) = eβw Y (t f ) Y (t ) = eβw Z S(t f ) Z S (t ) = eβ(w F S ) e βw = e β F S

28 Hamiltonian of Mean Force where also Z S (t) := Y (t) Z B = Tr S e βh (t) H (t) := 1 β ln Tr Be β(h S (t)+h SB +H B ) Tr B e βh B e βh (t) Z S (t) = Tr Be βh(t) Y (t) M. Campisi, P. Talkner, P. Hänggi, Phys. Rev. Lett. 12, 2141 (29).

29 Strong coupling: Example System: Two-level atom; bath : Harmonic oscillator H = ɛ ( 2 σ z + Ω a a + 1 ) ( + χσ z a a + 1 ) 2 2 H = ɛ 2 σ z + γ ɛ = ɛ + χ + 2 ( e βω ) β artanh sinh(βχ) 1 e βω cosh(βχ) γ = 1 ( 1 2e βω 2β ln cosh(βχ) + e 2βΩ ) (1 e βω ) 2 Z S = Tre βh S S = F S T F S = k b T ln Z S C S = T S S T M. Campisi, P. Talkner, P. Hänggi, J. Phys. A: Math. Theor (29)

30 Entropy and specific heat χ/ε Ω / ε 1 1 (a) χ/ε Ω / ε 1 1 (a) Ω/ɛ = 3 Ω / ε k B T/ε Ω / ε k B T/ε Ω / ε (b) 2 Ω / ε (b) χ/ε S S = χ/ε C S = Ω/ɛ = 1/3 S S <.6.4 C S <.5.2 Ω / ε k B T/ε Ω / ε k B T/ε

31 Summary Correlation function expression of characteristic function of work valid for all initial states of a closed system. Work is not an oservable. Canonical initial states. Crooks fluctuation theorem. Jarzynski s work theorem. Microcanonical state: Crooks type theorem yields microcanonical entropy changes. Weak coupling: theorems for energy and heat and work and heat Strong coupling: and work theorems

32 References P. Talkner, E. Lutz, P. Hänggi, Phys. Rev. E, 75, 512(R) (27). P. Talkner, P. Hänggi, J Phys. A 4, F569 (27). P. Talkner, P. Hänggi, M. Morillo, Phys. Rev. E 77, (28). P. Talkner, P.S. Burada, P. Hänggi, Phys. Rev. E 78, (28). P. Hänggi, G.-L. Ingold, P. Talkner, New J. Phys. 1, 1158 (28) P. Talkner, M. Campisi, P. Hänggi, J. Stat. Mech., P225 (29). G.-L. Ingold, P. Talkner, P. Hänggi, Phys. Rev. E 79, (29) M. Campisi, P. Talkner, P. Hänggi, Phys. Rev. Lett. 12, 2141 (29). M. Campisi, P. Talkner, P. Hänggi, J. Phys. A: Math. Theor (29)

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