Integral Fluctuation Theorem for Microcanonical and Pure States
IIntegral Fluctuation Theorem for Microcanonical and Pure States
Robin Heveling, ∗ Jiaozi Wang, † and Jochen Gemmer ‡ Department of Physics, University of Osnabr¨uck, D-49069 Osnabr¨uck, Germany
We present a derivation of the integral fluctuation theorem (IFT) for isolated quantum systemsbased on some natural assumptions on transition probabilities. Under these assumptions of “stiff-ness” and “smoothness” the IFT immediately follows for microcanonical and pure quantum states.We numerically check the IFT as well as the validity of our assumptions by analyzing two exemplarysystems. We have been informed by T. Sagawa et al. that he and his co-workers found comparablenumerical results and are preparing a corresponding paper, which should be available on the sameday as the present text. We recommend reading their submission.
I. INTRODUCTION
The second law of thermodynamics states that inisolated systems the total entropy can only increase.However, the second law is merely of statistical nature,i.e. there may be exceedingly rare but possible processesin which the entropy does indeed decrease. As systemsizes become smaller, violations to the second lawbecome more prominent. These deviations are notrandom, but obey themselves rigid rules, which are oftensummarized under the name of fluctuation theorems.Fluctuation theorems formulate and, in some sense,generalize the second law of thermodynamics by relatingthe entropy production of processes, which may takethe system arbitrarily far away from equilibrium, toproperties of the equilibrated system in a quantitativemanner (however, they make no statement about thesystem’s route to equilibrium). Just as for the secondlaw, the underlying mechanisms which render thesetheorems valid or invalid are still under discussion. Inthis work, we will show that the validity of the IFT formicrocanonical and pure quantum states follows fromnatural assumptions on transition probabilities we call“stiffness” and “smoothness” [1, 2]. In essence, stiffnessstates that transition probabilities are largely indepen-dent of the initial energies. Furthermore, smoothnessstates that individual transition probabilities are closeto the average transition probability in some respectiveenergy interval.In Sect. II, we recapitulate the formulation of the IFTfor a general system-bath setup. Thereafter, in Sect. III,we formulate a coarse-grained version of the IFT. In Sect.IV, we introduce the notion of stiffness and show the va-lidity of the coarse-grained IFT follows from the assump-tion of stiffness. Following, in Sect. V, we introduce thenotion of smoothness. It is presented how the assumptionof smoothness connects the microscopic and the coarse-grained IFT. In Sect. VI, we substantiate our theoreticalconsiderations by analyzing two specific numeric exam-ples. In Sect. VII follows a brief conclusion. ∗ [email protected] † [email protected] ‡ [email protected] II. INTEGRAL FLUCTUATION THEOREMFOR COMPOSITE SYSTEMS
The purpose of this preliminary section is to formulatethe integral fluctuation theorem for a generic system-bath setup with total time-independent Hamiltonian H = H sys + H bath + H int , (1)where H sys is the system Hamiltonian and H bath is thebath Hamiltonian. System and bath are allowed to inter-act via an interaction term H int . The composite systemis initialized in a product state ρ (0) = ρ sys (0) ⊗ ρ bath (0) , (2)i.e. system and bath are initially uncorrelated and thenbrought into contact at t = 0 via H int . The time evolu-tion operator U ( t ) = exp( − i Ht ) propagates the compos-ite system unitarily in time (for the remainder of this textwe set (cid:126) to unity). We assume that initial system stateand initial bath state are both diagonal in the eigenbasesof their respective Hamiltonians, such that Eq. (2) maybe rewritten as ρ (0) = (cid:88) k,b P k ini W b ini | k, b (cid:105)(cid:104) k, b | , (3)where | k, b (cid:105) = | ε k sys (cid:105) ⊗ | ε b bath (cid:105) are products of eigenstatesof system and bath Hamiltonians. The quantity P k ini is the initial weight distribution over the energy eigen-states of the system and, respectively, W b ini is the weightdistribution over the energy eigenstates of the bath.A central operator of interest is the entropy productionoperator σ ( t ) = − log ρ sys ( t ) + βH bath , (4)where β = 1 /T is the inverse temperature of the bath( k B = 1) and ρ sys ( t ) = Tr bath { ρ ( t ) } is the reduced den-sity operator of the system at time t . The operator σ ( t )is explicitly time-dependent due to the first term. Theeigenvalues of σ ( t ) are given by σ j,a ( t ) = − log P j sys ( t ) + βε a bath (5) a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b nd the eigenstates by | σ j,a ( t ) (cid:105) . We consider an ensem-ble average of differences in projective measurement out-comes at the initial time t = 0 and some final time t (de-noted by double brackets (cid:104)(cid:104) • (cid:105)(cid:105) ). Given some arbitrarybut nicely behaved function f , the ensemble average of f ( σ ( t )) is defined by (cid:104)(cid:104) f (∆ σ ) (cid:105)(cid:105) = (cid:88) j,k,a,b f ( σ j,a ( t ) − σ k,b (0)) (6) × P k ini W b ini R ( jk, ab ) , where R ( jk, ab ; t ) = |(cid:104) σ j,a ( t ) | U ( t ) | σ k,b (0) (cid:105)| is the proba-bility to transition from an initial state | σ k,b (0) (cid:105) to a finalstate | σ j,a ( t ) (cid:105) . For brevity, we drop the explicit time de-pendence of the quantity in double brackets and in theargument of the transition probabilities. For f = id,Eq. (6) yields the average entropy production (cid:104)(cid:104) ∆ σ (cid:105)(cid:105) ,which can be written as a standard quantum mechanicalexpectation value (cid:104)(cid:104) ∆ σ (cid:105)(cid:105) = (cid:104) σ ( t ) − σ (0) (cid:105) = ∆ S sys + β ∆ U bath , (7)where the first term is the change in von Neumann en-tropy of the system and the second term the heat emittedfrom the bath. An important quantity related to the av-erage entropy production is (cid:104)(cid:104) e − ∆ σ (cid:105)(cid:105) , which is defined byEq. (6) by setting f ( x ) = exp( − x ). This quantity (here-after referred to as the IFT quantity) may be used tojudge if the integral fluctuation theorem (cid:104)(cid:104) e − ∆ σ (cid:105)(cid:105) = 1 (8)holds (if the initial energy distribution of the bath iscanonical, Eq. (8) is known to hold exactly). The IFTimplies the second law of thermodynamics in the sensethat the average entropy production (between some ini-tial and some final point in time) is positive. This canbe obtained by plugging Eq. (8) into the Jensen inequal-ity e (cid:104)(cid:104) x (cid:105)(cid:105) ≤ (cid:104)(cid:104) e x (cid:105)(cid:105) yielding (cid:104)(cid:104) ∆ σ (cid:105)(cid:105) ≥
0. Applying the twopoint measurement scheme [3] to the IFT quantity, i.e.averaging the exponentials of differences in measurementoutcomes of σ ( t ), yields (cid:104)(cid:104) e − ∆ σ (cid:105)(cid:105) = (cid:88) j,k,a,b P j fin P k ini e − β ( ε a bath − ε b bath ) P k ini W b ini R ( jk, ab )= (cid:88) j,k,a,b P j fin W b ini e − β ( ε a bath − ε b bath ) R ( jk, ab ) , (9)where P j fin = P j sys ( t ). Importantly, even though the P k ini ’scancel out at the second equality sign, the sum over k must still be kept (otherwise the IFT does not evenhold for canonical initial states). For system-bath setupstreatable with exact diagonalization techniques, Eq. (9)constitutes a convenient formula to check whether theIFT holds. III. COARSE-GRAINING THE INTEGRALFLUCTUATION THEOREM
In this section, we will derive a “coarse-grained” ver-sion of the IFT similar to Eq. (9). In general, the IFTaddresses a time-dependent process described by someHamiltonian H ( t ). Here, we set the initial Hamiltonian H (0) = H ini and final Hamiltonian H ( T ) = H fin equalto the uncoupled Hamiltonian H unc = H sys + H bath . Theprotocol then reads as follows, between the initial time t = 0 and final time t = T the interaction H int is instan-taneously switched on, which induces transitions betweenthe eigenstates of the uncoupled Hamiltonian. In the fol-lowing, we drop the subscripts “ini” and “fin” since initialand final Hamiltonian coincide. The eigenvalue equationsfor the initial and the final Hamiltonian read H unc | k, b (cid:105) = ε k,b | k, b (cid:105) (10)with eigenvalues ε k,b = ε k sys + ε b bath . The eigenstates are | k, b (cid:105) = | ε k sys (cid:105) ⊗ | ε b bath (cid:105) . The density of states (DOS) ofthe bath is given byΩ bath ( E bath ) = (cid:88) b δ ( E bath − ε b bath ) . (11)This description will now be extended to finite energyresolutions. We assume that the system only comprisesa few energy levels, e.g. a single spin. Therefore,we will resort to dividing just the energy scale of thebath into bins of finite size. To this end, we willintroduce the bin size ∆ (not to be confused with thedelta used in the notation for the ensemble average)and divide the energy scale of the bath into intervalsaccording to E B bath = [( B − / , ( B + 1 / B = ..., − , − , , , , ... . The midpoint of the energyinterval E B bath of width ∆ is B ∆. The bin size ∆ shouldbe small compared to the energy scale of the bath,but large compared to its level spacing. As of yet,individual eigenvalues were denoted by an “ ε ”. Now,energy intervals are denoted by an “E” and enumeratedby capitalized indices.The probability R ( jk, Ab ) to transition from an initialeigenstate | k, b (cid:105) to any state | j, a (cid:105) with ε a bath ∈ E A bath , i.e.to a range of bath eigenstates, is obtained by R ( jk, Ab ) = (cid:88) a : ε a bath ∈ E A bath R ( jk, ab ) . (12)In a similar fashion, we define the average probability totransition from an initial state | k, b (cid:105) with ε b bath ∈ E B bath to a final energy interval E A bath by R ( jk, AB ) = 1Ω B bath (cid:88) b : ε b bath ∈ E B bath R ( jk, Ab ) , (13)where Ω B bath = (cid:90) ( B +1 / B − / Ω bath ( E bath ) d E bath (14)2s the number of bath energy eigenvalues ε b bath withinthe energy interval E B bath . The coarse-grained transitionprobabilities satisfy (cid:88) j,A R ( jk, AB ) = 1 . (15)In addition, we need the concept of microreversibility,which will serve as starting point for our derivation. Inshort, a certain setup is microreversible, if the relevantobservables and the Hamiltonian are real in the workingbasis. In the case of microreversibility, considerationsalong the lines of microcanonical fluctuation theoremsyield R ( jk, AB ) (cid:101) R ( jk, AB ) = Ω A bath Ω B bath , (16)where the tilde indicates the average probability of atransition of a time reversed process described by thetime evolution operator (cid:101) U ( t ) of a backwards protocol im-plemented by (cid:101) H = H ( T − t ). In the case at hand we havethat (cid:101) U ( t ) = U ( t ). We start our derivation by assumingmicroreversibility, i.e. that Eq. (16) is fulfilled. Ad-ditionally assuming an exponentially growing density ofstates of the bath we arrive at R ( jk, AB ) (cid:101) R ( jk, AB ) = e β ∆ A e β ∆ B , (17)where (cid:101) R ( jk, AB ) = R ( kj, BA ) (18)are the transition probabilities of the backwards motion.This relation only holds, if, in addition to microre-versibility, the protocol is symmetric in time, which isthe case here.By algebraic manipulation of Eq. (17), multiplying with P j fin W B ini and summing over j, k, A, B we get that (cid:88) j,k,A,B P j fin W B ini e − β ∆( A − B ) R ( jk, AB ) (19)= (cid:88) j,k,A,B P j fin W B ini (cid:101) R ( jk, AB ) . The l.h.s. of Eq. (19) looks similar to the r.h.s. ofEq. (9), such that we define the coarse-grained (“c.g.”)version of the IFT as (cid:104)(cid:104) e − ∆ σ (cid:105)(cid:105) c.g. = (cid:88) j,k,A,B P j fin W B ini e − β ∆( A − B ) R ( jk, AB ) . (20) Indeed, Eq. (20) may be interpreted as a “coarse-grained” or “microcanonical” version of (cid:104)(cid:104) e − ∆ σ (cid:105)(cid:105) inthe sense that: i . energy changes in the bath arenow counted on the level of energy intervals ratherthan individual energy eigenvalues, ii . the transitionprobabilities now apply to transitions of microcanonicalinitial states restricted to the initial interval E B bath tothe final energy interval E A bath rather than to transitionsbetween individual eigenstates and iii . the initialprobabilities for the bath are now probabilities to findthe bath in the respective energy interval rather thanin the corresponding eigenstate. One may be inclinedto think that Eq. (19) in general becomes Eq. (9) inthe limit of small energy intervals. But this sentiment isflawed, since Eq. (16) and thus Eq. (19) rely on notionsthat ultimately break down in the limit of small energyintervals. IV. VALIDITY OF THE COARSE-GRAINEDINTEGRAL FLUCTUATION THEOREM VIASTIFFNESS
In this section we will define the property of stiffnessfor coarse-grained transition probabilities, which we willutilize to show that (cid:104)(cid:104) e − ∆ σ (cid:105)(cid:105) c.g. = 1 holds for all possible W B ini . We call a transition probability from an initialenergy interval E B bath to a final energy interval E A bath stiff (for given j, k ), if we have that R ( jk, AB ) = R ( jk, A − B ) , (21)i.e. the probability to transition from the initial energyinterval E B bath to a state within the final energy interval E A bath is only a function of the difference in energies. Theabove definition of stiffness implies that (cid:88) A R ( jk, AB ) = (cid:88) B R ( jk, AB ) . (22)Now, plugging the occupation probability of an initialmicrocanonical bath state W B ini = δ B,B (cid:48) (which, in thiscase, completely “fills” one energy interval) into the r.h.s.of Eq. (19) yields (cid:104)(cid:104) e − ∆ σ (cid:105)(cid:105) c.g. = (cid:88) j,k,A,B P j fin δ B,B (cid:48) (cid:101) R ( jk, AB ) (23)= (cid:88) j,k,A P j fin (cid:101) R ( jk, AB (cid:48) )= (cid:88) j,k,A P j fin R ( kj, B (cid:48) A )= (cid:88) j,k,B (cid:48) P j fin R ( kj, B (cid:48) A )= (cid:88) j P j fin (cid:88) k,B (cid:48) R ( kj, B (cid:48) A )= (cid:88) j P j fin = 1 .
3o go to the second line the sum over B was evaluated,at the third equality sign Eq. (18) was used, at thefourth equality sign the stiffness property, i.e. Eq. (22),was applied, then the sums were factorized and Eq.(15) was employed to reach the last line. Thus, thecoarse-grained IFT is shown for microcanonical initialstates under the assumption of stiffness.Note that we started from Eq. (16), which is knownto be true for microreversible setups. Even if microre-versibility is broken, e.g. by magnetic fields, Eq. (16)may still hold and serve as a starting point for ourderivation. We conjecture that, even if Eq. (16) is vio-lated, the IFT will still be fulfilled in systems featuringstiffness. In that sense, Eq. (16) is a convenient tool toderive the desired result, but not the essential ingredient. V. LINK BETWEEN COARSE-GRAINED ANDMICROSCOPIC INTEGRAL FLUCTUATIONTHEOREM VIA SMOOTHNESS
In this section we will define the property of smoothnessof transition probabilities and show that under the as-sumption of smoothness, Eq. (20) actually becomes anarbitrarily good approximation of Eq. (9). This holdseven if the bath initially only occupies a single energyeigenstate. In general, the probability to transition fromsome initial energy eigenstate to some final energy inter-val of course differs from the average transition proba-bility to go from the initial energy interval (in which theinitial energy lies) to the final energy interval. We denotethe difference of these two quantities by r according to R ( jk, Ab ) = R ( jk, AB ) + r ( jk, Ab ) . (24)We call a set of transition probabilities from an initialstate | k, b (cid:105) to a final energy interval E A bath smooth (forgiven j, k ), if we have that r ( jk, Ab ) ≈ ε b bath ∈ E B bath , i.e. all transition probabilities to gofrom a state with initial energy ε b bath within the initialinterval E B bath to the final energy interval E A bath are closeto the average value of transition probabilities withinthat initial energy interval.To continue we need two more ingredients. Firstly, wewill use that the exponential factor exp( − β ( ε a bath − ε b bath ))only changes negligibly when replacing the distance be-tween individual energy eigenvalues with the distance be-tween energy intervals, namely e − β ( ε a bath − ε b bath ) ≈ e − β ∆( A − B ) . (26)This can be achieved by making the bin size ∆ sufficientlysmall or by increasing the temperature. Secondly, we usethat the initial weight in an energy interval of the bath W B ini is obtained by summing all individual initial weightsof eigenstates with energies in that initial energy interval,i.e. (cid:88) b : ε b bath ∈ E B bath W b ini = W B ini . (27)The starting point is the r.h.s. of Eq. (9). In thederivation below, firstly, the sums over a and b are splitaccording to the belonging of ε a bath and ε b bath to theirrespective energy intervals E A bath and E B bath . Then, Eq.(26) is employed such that the exponential factor can bepulled to the front. Next, we recognize the sum over a from Eq. (12). To go to the next line, we plug in Eq.(24) and abbreviate all terms linear in r as O ( r ). Then,Eq. (27) was used. Finally, we employ the assumptionof smoothness, i.e. Eq. (25), which allows to neglectterms linear in r .Thus, it can be seen that the two versions of theIFT indeed coincide if the transition probabilities aresufficiently smooth. To repeat, the initial energy intervalcan not be arbitrarily small, such that the IFT forsingle energy eigenstates does not follow immediately.However, the validity of the IFT follows under theassumption of smoothness.4 (cid:104) e − ∆ σ (cid:105)(cid:105) = (cid:88) j,k,a,b P j fin W b ini e − β ( ε a bath − ε b bath ) R ( jk, ab ) (28)= (cid:88) j,k,A,B (cid:88) a : ε a bath ∈ E A bath (cid:88) b : ε b bath ∈ E B bath P j fin W b ini e − β ( ε a bath − ε b bath ) R ( jk, ab ) ≈ (cid:88) j,k,A,B P j fin e − β ∆( A − B ) (cid:88) b : ε b bath ∈ E B bath W b ini (cid:88) a : ε a bath ∈ E A bath R ( jk, ab )= (cid:88) j,k,A,B P j fin e − β ∆( A − B ) (cid:88) b : ε b bath ∈ E B bath W b ini R ( jk, Ab )= O ( r ) + (cid:88) j,k,A,B P j fin e − β ∆( A − B ) (cid:88) b : ε b bath ∈ E B bath W b ini R ( jk, AB )= O ( r ) + (cid:88) j,k,A,B P j fin W B ini e − β ∆( A − B ) R ( jk, AB ) r → −→ (cid:104)(cid:104) e − ∆ σ (cid:105)(cid:105) c.g. VI. NUMERICAL VERIFICATION OF THE IFT
In this section, we will present a numerical analysison two exemplary systems, a hardcore boson modeland and a transverse Ising model with defects. Wewill numerically check whether the IFT holds for bothsystems. Furthermore, we will investigate to what extentstiffness and smoothness are fulfilled/violated.As mentioned in Sect. II, the initial state of thecomposite system will be a product state of a systemstate and an energy eigenstate of the bath correspondingto some inverse temperature β . In the following, weemploy a microcanonical definition of temperature. Forsome given energy E , the inverse temperature will bedetermined by the exponent of an exponential fit of theDOS in the direct vicinity of E .The first setup of interest is a hardcore boson model,which was also the subject in Ref. [4]. The system justconsists of a single site and the bath is a quadratically(4 ×
4) shaped lattice, yielding a total of 17 sites. The sys-tem interacts with one corner of the bath. The relevantHamiltonians read H sys = ωn H int = − γ (cid:48) ( c † c + c † c ) (29) H bath = ω (cid:88) i n i − γ (cid:88) (cid:104) i,j (cid:105) ( c † i c j + c † j c i ) + g (cid:88) (cid:104) i,j (cid:105) n i n j , where c j ( c † j ) are annihilation (creation) operators of aboson on site j and n j = c † j c j is the occupation numberon site j . The double sums (cid:104) i, j (cid:105) run over horizontallyand vertically neighboring sites with open boundaries.Note that, due to symmetry of the interaction term, theeigenstates of the entropy production operator | σ k,b ( t ) (cid:105) are equal to the eigenstates | k, b (cid:105) of the uncoupled Hamil-tonian. The parameters take values γ = 1 . g = 0 . ω = 10 .
0, while the interaction strength γ (cid:48) is varied. Theinitial bath energy is ε b bath = 29 .
286 corresponding to aninverse temperature of β = 0 . ρ (0) = | , b (cid:105)(cid:104) , b | , (30)i.e. initially the system site is occupied with probabilityone. Following Ref. [4], we restrict the dynamics to aspecific particle sector, here N = 4. Since the system siteis initially occupied with probability one, there can onlybe three particles left in the bath, i.e. (cid:80) i (cid:104) b | n i | b (cid:105) =3, where the sum runs over all bath sites. The onsitepotential is the same on every site, thus, it does not affectthe occupation dynamics on any site.
20 25 30 35 40 45 50050100150
FIG. 1. Histograms of bath energy eigenvalues in the N = 3and N = 4 particle sectors fitted with Gaussians. The quan-tity Ω E bath indicates the number of eigenvalues within a binof size ∆ = 1 . E . The energetically accessibleparts are well approximated by a single exponential functionwith exponent βE .
20 40 60 80 1000.850.900.951.001.05
FIG. 2. The IFT quantity (cid:104)(cid:104) e − ∆ σ (cid:105)(cid:105) plotted over time t for thehardcore boson model. There are small deviations of about0 .
05 from the desired value of one. For the strongest interac-tion strength γ (cid:48) = 4 . However, it changes the entropy production. For ω =10 . N = 3 and N = 4 particle sectors can both be well ap-proximated by a single exponential function, cf. Fig. 1.This property was crucial for our derivation of the coarse-grained IFT, cf. Eq. (17). (Note that the N = 3 particlesector is relevant due to the sum over k in Eq. (9)).The temporal behavior of the IFT quantity is depictedin Fig. 2 for various interaction strengths. The IFTquantity moderately fluctuates around the desired valueof one with deviations of about 0 .
05. For the strongestinteraction γ (cid:48) = 4 . FIG. 3. The coarse-grained IFT quantity (cid:104)(cid:104) e − ∆ σ (cid:105)(cid:105) c.g. plottedover time t for the hardcore boson model. The initial bathstate is microcanonical, comprising 50 eigenstates neighboring ε b bath . Deviations from one are visibly smaller than in Fig. 2. FIG. 4. The IFT quantity (cid:104)(cid:104) e − ∆ σ (cid:105)(cid:105) plotted as a function oftime for the Ising model. For weak to moderate interactionstrengths the IFT is practically fulfilled. The second setup under consideration is the transverseIsing model with defects. The system consists of a singlesite and the bath is a chain of length L = 14, yieldinga total of 15 sites. The system, bath and interactionHamiltonians are given by H sys = ωσ x H int = λ σ z σ z (31) H bath = g L − (cid:88) i =1 σ zi σ zi +1 + h L (cid:88) i =1 σ xi + µ ( h σ z + h σ z ) , where σ x,y,zi denote Pauli matrices on site i . Again, dueto symmetry, the eigenstates of the entropy productionoperator are equal to the eigenstates of the uncoupledHamiltonian. The parameters are chosen as ω = 1 . g = 1 . h = 1 . µ = 0 . h = 1 .
11 and h = 1 .
61, whilethe interaction strength λ is varied. This choice of pa-rameters ensures that the bath exhibits chaotic behavior.The initial bath energy is ε b bath = − .
569 correspondingto an inverse temperature β = 0 . ρ (0) = |↓ , b (cid:105)(cid:104)↓ , b | . (32)The temporal behavior of the IFT quantity is shownin Fig. 4 for various interaction strengths. For weak( λ = 0 .
3) to moderate ( λ ≤ .
0) interactions the IFT ispractically fulfilled. For the strongest interaction λ = 3 . . λ = 0 . λ = 3 .
0. For bothcases we set k = ↓ and calculate transition probabilitiesfor both j = ↓ , ↑ . We inspect two different initial energyintervals of the bath. The first interval E is centeredaround ε b bath with bin size ∆ = 0 . E (cid:48) is shifted by an amount Ξ relative to E ,i.e. E (cid:48) = E + Ξ. The shift Ξ is equal to the arithmeticmean of standard deviations of the uncoupled Hamilto-nian with respect to n b bath eigenstates, where n b is thenumber of eigenstates in the energy interval E .Ξ = 1 n b (cid:88) b (cid:113) (cid:104) ψ ↓ ,bt | H | ψ ↓ ,bt (cid:105) − (cid:104) ψ ↓ ,bt | H unc | ψ ↓ ,bt (cid:105) (33)Here ψ ↓ ,bt = exp( − i Ht ) | ↓ , b (cid:105) is the final state of thecomposite system at the final time t = 100. For λ = 0 . λ = 3 .
0) we have Ξ = 0 .
702 (Ξ = 3 . j = ↑ , ↓ and initial bath energies E , E (cid:48) yield four different curves for transition prob-abilities, which can be viewed in Fig. 5 for λ = 0 . λ = 3 .
0. In the case of λ = 0 . j = ↑ as well as for j = ↓ exactly coincide for the different initial bath energies,indicating that stiffness exists for these combinations.In Fig. 6 the situation is different as there are cleardiscrepancies for both j = ↑ and j = ↓ . This violationof stiffness may lead to the deviation for λ = 3 . - - FIG. 5. Transition probabilities for λ = 0 . j = ↑ , ↓ andinitial energy intervals E , E (cid:48) . The transition probabilitiescoincide for the two different initial energies, which indicatesthe existence of stiffness. - - FIG. 6. Transition probabilities for λ = 3 . j = ↑ , ↓ and ini-tial energy intervals E , E (cid:48) . There are noticeable deviationsbetween the transition probabilities for both initial energies,which indicates that stiffness is violated to some extent. We consider the averaged quantity in Eq. (34), where,again, we set k = ↓ , the bin size to ∆ = 0 .
125 andthe initial bath energy interval to E as above. Weaverage over j = ↑ , ↓ (thus, the 2 in the denominator),over final energy intervals A with N A = 120 and over n b = 50 bath eigenstates with energies neighboring ε b bath .(For clarity, if convenient, we write the energy intervaldenoted by E B bath as the argument of a transition prob-ability R instead of the capitalized index B . Additionalcommas and a semicolon are then used to improve read-ability.) δ = (cid:115) N A n b (cid:88) j,A,b | R ( j ↓ , Ab ) − R ( j ↓ ; A, E ) | (34) - - - FIG. 7. The averaged quantity δ logarithmically plotted overthe bath size L . The dashed black line serves as a guide tothe eye. As the bath size increases, deviations of individualtransition probabilities from the average decrease. δ approximately scales as ∝ / √ L . Note that δ is smallest for λ = 3 .
0. Thus,we conclude that the deviation of the IFT quantity fromone for λ = 3 . VII. CONCLUSION
In this article we presented a way to derive the inte-gral fluctuation theorem for microcanonical and purequantum states under the assumption that the transitionprobabilities fulfill the properties of stiffness and smooth-ness. We numerically checked the validity of the IFT fortwo exemplary systems. Furthermore, the existence of stiffness and smoothness was directly scrutinized. Thisnumerical analysis supports the idea that stiffness andsmoothness are critical mechanisms enabling the valid-ity of the IFT. In further work we plan to present amore thorough numerical analysis, checking stiffness andsmoothness for a wide range of models and parameters.
ACKNOWLEDGMENTS
We thank T. Sagawa and E. Iyoda for extensive andfruitful discussions. As mentioned in the abstract, werecommend reading their submission to the arXiv aboutthe origins of the IFT. This work was supported by theDeutsche Forschungsgemeinschaft (DFG) within the Re-search Unit FOR 2692 under Grant No. 397107022 [1] D. Schmidtke, L. Knipschild, M. Campisi, R. Steinigeweg,and J. Gemmer, “Stiffness of probability distributionsof work and Jarzynski relation for non-Gibbsian initialstates,” (2018).[2] L. Knipschild, A. Engel, and J. Gemmer, “Stiffness ofprobability distributions of work and Jarzynski relation for initial microcanonical and energy eigenstates,” (2020).[3] M. Esposito, U. Harbola, and S. Mukamel, “Nonequi-librium fluctuations, fluctuation theorems, and countingstatistics in quantum systems,” (2009).[4] E. Iyoda, K. Kaneko, and T. Sagawa, “Fluctuation theo-rem for many-body pure quantum states,” (2017).[1] D. Schmidtke, L. Knipschild, M. Campisi, R. Steinigeweg,and J. Gemmer, “Stiffness of probability distributionsof work and Jarzynski relation for non-Gibbsian initialstates,” (2018).[2] L. Knipschild, A. Engel, and J. Gemmer, “Stiffness ofprobability distributions of work and Jarzynski relation for initial microcanonical and energy eigenstates,” (2020).[3] M. Esposito, U. Harbola, and S. Mukamel, “Nonequi-librium fluctuations, fluctuation theorems, and countingstatistics in quantum systems,” (2009).[4] E. Iyoda, K. Kaneko, and T. Sagawa, “Fluctuation theo-rem for many-body pure quantum states,” (2017).