Stiffness of Probability Distributions of Work and Jarzynski Relation for Initial Microcanonical and Energy Eigenstates
aa r X i v : . [ qu a n t - ph ] J u l Stiffness of probability distributions of work and Jarzynski relation for initialmicrocanonical and energy eigenstates
Lars Knipschild, ∗ Andreas Engel, † and Jochen Gemmer ‡ Department of Physics, University of Osnabr¨uck, D-49069 Osnabr¨uck, Germany Institute of Physics, Carl von Ossietzky University of Oldenburg, D-26111 Oldenburg, Germany
We consider closed quantum systems (into which baths may be integrated) that are driven, i.e.,subject to time-dependent Hamiltonians. As a starting point we assume that, for systems initializedin microcanonical states at some energies, the resulting probability densities of work (work-PDFs)are largely independent of these specific initial energies. We show analytically that this assumptionof ”stiffness”, together with the assumption of an exponentially growing density of energy eigen-states, is sufficient but not necessary for the validity of the Jarzynski relation (JR) for the abovemicrocanonical initial states. This holds, even in the absence of microreversibility. To scrutinizethe connection between stiffness and the JR for microcanonical initial states, we perform numericalanalysis on systems comprising random matrices which may be tuned from stiff to nonstiff. In theseexamples we find the JR fulfilled in the presence of stiffness, and violated in its absence, whichindicates a very close connection between stiffness and the JR. Remarkably, in the limit of largesystems, we find the JR fulfilled, even for pure initial energy eigenstates. As this has no analoguein classical systems, we consider it a genuine quantum phenomenon.
I. INTRODUCTION
The long-standing question regarding whether, and inwhich way, closed finite quantum systems approach ther-mal equilibrium has recently gathered renewed attention.On the theoretical side thermalization and equilibrationhave been investigated e.g. for rather abstract settings[1–6] and also for more specific condensed-matter typesystems [7–10]. In these works major concepts are theeigenstate thermalization hypothesis (ETH) and typical-ity, both of which will also play certain roles in the paperat hand. The developments on experiments on ultra-coldatoms now allow for testing what have been merely the-oretical results before; see e.g. Ref. [11–13].Rather than just the existence of equilibration withinclosed quantum systems, lately the very peculiarities ofthe dynamical approach to equilibrium have moved tothe center of interest [11, 14]. Questions addressed inthis context include limits on relaxation time scales andagreement of unitary quantum dynamics of closed quan-tum systems with standard statistical relaxation princi-ples, such as Fokker-Planck equations [15–18], or moregeneral, standard stochastic processes [19, 20]. But alsothe emergence of universal non-equilibrium behavior in-volving work and driven systems is under discussion atpresent [21].To a large extent universal non-equilibrium behav-ior may be captured by fluctuation theorems, see e.g.Ref. [22] and references therein. The Jarzynski relation(JR), a general statement on work that has to be investedto drive processes also and especially far from equilib-rium, is a prime example of such a fluctuation theorem. ∗ [email protected] † [email protected] ‡ [email protected] Many derivations of the JR from various starting groundshave been presented. These include classical Hamilto-nian dynamics, stochastic dynamics such as Langevin ormaster equations, as well as quantum mechanical start-ing points [22–27]. However, all these derivations (ex-cept for Ref. [28]) assume that the system, that is actedon with some kind of “force”, is strictly in a Gibbsianequilibrium state before the process starts. (The no-tion of “the system” here routinely includes the bath.)Thus, this starting point differs significantly from theprogresses in the field of thermalization: There, the gen-eral features of thermodynamic relaxation are found toemerge entirely from the system itself without any neces-sity of evoking external baths or specifying initial statesin detail. Clearly, the preparation of a strictly Gibbsianinitial state requires the coupling to a (super-)bath priorto starting the process.This situation renders the question whether or not thestandard JR also holds for systems starting in other thanGibbsian states (e.g. micro-canonical states) rather ex-igent. Note that, other than for Gibbsain initial states,the answer to this question is expected to depend on spe-cific properties of the considered systems.In this context a property which we call ”stiffness ofwork-distributions” has been suggested as a key ingredi-ent for the validity of the JR for microcanonical initialstates in Ref. [28]. In this pioneering work the validity ofthe JR is proven for classical systems initialized in micro-canonical initial states given the systems feature stiffnessand microreversibility. Moreover, for a classical Lorentzgas stiffness and the validity of the JR for microcanonicalinitial states are numerically demonstrated. Furthermorethe JR was found to hold for micro-canonical initial statesfor some quantum spin-models exhibiting stiffness in Ref.[29] in a numerical study. The present work extends thisline of research in various directions: We examine the va-lidity of the JR not only for microcanonical initial statesbut also for initial pure energy eigenstates, the latter isconceptually beyond the scope of Ref. [28]. It is alsoimportant to note that stiffness is a sufficient but not anecessary condition for the validity of the JR, thus thepractical relevance of stiffness is challenged. The numeri-cal modelling in the paper at hand allows to address thispractical relevance by means of an investigation of thevalidity of the JR in the presence of stiffness, as well asin its absence. The latter is, to our best knowledge, sofar missing in the literature. Furthermore the results inthe current paper do not rely on microreversibility.The paper at hand is organized as follows: In Sec. IIwe introduce our basic hypothesis of probability densityfunction of work (work PDF’s) being largely independentof the respective energy for micro-canonical initial states.We call this property stiffness . The validity of the JR formicro-canonical initial states is shown to follow from thisassumption (together with the routinely applied assump-tion of an exponentially growing density of energy eigen-states). With an additional assumption on the systemdynamics which we call smoothness we derive the validityof the JR even for energy eigenstates. In Sec. III we in-troduce our modelling, which is partly based on randommatrices. In Sec. IV we provide numerical results formicro-canonical initial states indicating a very strong cor-respondence between the validity of the JR and stiffnessof the system dynamics. In Sec. V we numerically showthat also the aforementioned smoothness-assumption isfulfilled for our modelling in the limit of large systems.This completes the demonstration of the existence of aclass of systems which exhibit both, stiffness and smooth-ness and thus fulfill the JR even for energy eigenstates.We close with a discussion.
II. STIFFNESS AND SMOOTHNESS OF WORKPDF’S AND JARZYNSKI RELATION FORINITIAL MICROCANONICAL STATES ANDENERGY EIGENSTATES
The analysis at hand focuses exclusively on closed sys-tems. While it is physically appropriate to interpret theexamples in Sec. III in terms of “considered system” and“environment” or “bath”, we technically treat the sys-tem+environment compound regardless of the couplingstrength as one closed system. Thus, since there is no ex-ternal source or sink of heat, any energy change of the fullsystem is to be counted as work W (for an overview overdifferent perspectives, see e.g. Ref. [30].) The measure-ment of the inner energy is described by a two point pro-jective measurement scheme. In this respect we choosethe same starting point as employed in derivations of theJR as described, e.g. , in Ref. [31] and references therein.However, while in Ref. [31] the assumption of a canonical,Gibbsian initial state is of vital importance, we base ourconsideration on much larger classes of initial states ofthe full system. The central role which the assumptionof strictly Gibbsian state plays in the afore mentionedworks is replaced by the assumption of “stiffness” of the work-PDF’s (as introduced in in Eq. (13)).We consider a system described by a time-dependentHamiltonian H ( t ) during the time t ∈ [0 , T ], which in-duces a non-equilibrium process.The corresponding unitary time-propagation operator U is defined by: U := T exp − i Z T H ( t ′ ) dt ′ ! , (1)where T is the time-ordering operator and we tacitly set ~ = 1.Let | i i be the eigenstates of H (0) and | f i the eigen-states of H ( T ). Let further ǫ i and ǫ f be the correspond-ing eigenvalues, respectively. Starting from the initialstate | i i , p f ← i denotes the probability to make a transi-tion into | f i : p f ← i = Tr( | f i h f | U | i i h i | U † ) (2)The average over the work-PDFs h h ( W ) i W starting froman initial state ρ (0) can be calculated for an arbitraryfunction h ( W ) of the work W : h h ( W ) i W = X i,f Tr( ρ (0) | i i h i | ) p f ← i h ( ǫ f − ǫ i ) (3)Tr( ρ (0) | i i h i | ) is the probability to find the system afterthe first projective measurement in the initial state | i i and p f ← i is the probability to make a transition from | i i to the final state | f i . The work performed during thistransition is W = ǫ f − ǫ i .One can easily show that these transition-probabilities p f ← i are doubly stochastic: X i p f ← i = X f p f ← i = 1 (4)In general these transition-probabilities vary from eigen-state to eigenstate. We thus define the probability p F ← i to transition from an eigenstate | i i into an energy-interval E F : p F ← i = X f | ǫ f ∈ E F p f ← i , E n = [ nδ, ( n + 1) δ ] , n = I, F (5)Here, δ is to be chosen large compared to the level spacingof the full system, but small compared to the involvedenergy scales of E, W . Note that I and F are integersused to address the initial ( E I ) and final energy-intervals( E F ), respectively. This construction serves as a coarse-graining of the energy scale.In a similar way, we define the average probabilityto make a transition from an initial state | i i from theenergy-interval E I into an energy-interval E F : p F ← I = X i | ǫ i ∈ E I p F ← i Ω I (6)Ω I and Ω F denote the number of eigenstates of H (0) inthe interval E I and of H ( T ) in the interval E F , respec-tively.Ω n = Tr(Π n ) , Π I = X i | ǫ i ∈ E I | i i h i | , Π F = X f | ǫ f ∈ E F | f i h f | (7)Hence, p F ← I is the average over all p F ← i with ǫ i ∈ E I .Note that these transition-probabilities depend on thewidth δ of the final energy-interval. Closely related tothese transition-probabilities is the so-called work prob-ability density function (work-PDF), which describes theprobability to perform the work W = ( F − I ) δ startingfrom an initial energy E = Iδ . P E ( W ) = 1 δ p F ← I (8)The transition-probabilities and the work-PDFs are es-sentially the same, up to a constant rescaling-factor. Butin large systems these work-PDFs typically become inde-pendent of the concrete choice of δ [29].Starting from Eq. (3), the average over the work-PDFs h h ( W ) i W for a function h ( W ), which does not vary sig-nificantly on the scale of δ , can be calculated from p F ← I : h h ( W ) i W = X I,F
Tr( ρ (0)Π I ) p F ← I h ( ¯ E F − ¯ E I ) (9)¯ E n = nδ is an approximation of the energies in the initial( n = I ) interval E I and of the final ( n = F ) interval E F ,respectively.From Eq. 4 we derive the following properties of p F ← i and p F ← I : X i p F ← i = Ω F (10) X I Ω I p F ← I = Ω F (11)Up to now we only defined various quantities and de-rived general statements, but did not make any assump-tions. We now come to the derivation of the JR for micro-canonical initial states. To begin with, we define the lat-ter as ρ I mc (0) = Π I Ω − I (12)In order to derive the JR for micro-canonical initial statesto make two assumptions. First, we assume that theprobability to make a transition from a state from theenergy-interval E I into the energy-interval E F only de-pends on the difference of F and I : p F ← I = p ( F − I ) (13)We call this assumption stiffness . This assumption canbe also expressed in terms of work-PDFs P E ( W ). If these work-PDFs are independent of the initial energy E , thenEq. (13) is fulfilled.Our second assumption states that the densities ofstates (DOS) of the initial D ini ( E I ) := δ − Ω I and finalHamiltonian D fin ( E F ) := δ − Ω F grow exponentially: D ini ( ¯ E I ) = Z ini exp (cid:8) β ¯ E I (cid:9) ,D fin ( ¯ E F ) = Z fin exp (cid:8) β ¯ E F (cid:9) (14)Up to now β , Z ini and Z fin are just some positive realnumbers. In the discussion below (16) these numbersare interpreted in terms of standard statistical thermo-dynamics.Of course Eq. (13) and Eq. (14), are not expected tohold for all energies E . Here we only require that theserelations hold at least for an energy interval which is largeenough to comprise almost the entire work-PDF.To arrive at the JR for micro-canonical initial states,we start by calculating the average of exp {− βW } overthe work-PDFs according to Eq. (9). h exp {− βW }i W = X I ′ ,F Tr( ρ I mc (0)Π I ′ ) p F ← I ′ exp (cid:8) − β ( ¯ E F − ¯ E I ′ (cid:9) = X F p ( F − I ) exp (cid:8) − β ( ¯ E F − ¯ E I (cid:9) (15)In the last step we evaluated the sum over I ′ by usingTr(Ω − I Π I Π I ′ ) = δ I,I ′ and used the stiffness-assumptionEq. (13). By substituting F by F ′ + I − I ′ , while I ′ isthe new summation index and F ′ an arbitrary but fixedinteger, we get: h exp {− βW }i W == X I ′ p ( F ′ − I ′ ) exp (cid:8) − β ( ¯ E F ′ − ¯ E I ′ (cid:9) = 1Ω F ′ Z fin Z ini X I ′ p F ′ ← I ′ Ω I ′ = Z fin Z ini (16)In the second step we used that the DOS of the initialand the final Hamiltonian exponentially grow accordingto Eq. (14). In the last step we used Eq. (11).Eq. (16) formally is a JR for the work PDF’s obtainedby starting from microcanonical initial states, with thetemperature replaced by a parameter describing the ex-ponential growth of the DOS of the full system. As suchEq. (16) already represents the main result of the presentsection. Note that Eq. (16) holds for arbitrary processesand its r.h.s. only contains static, process-independentmodel parameters.Formally the JR could be fulfilled for microcanonicalinitial states, even if Eq. (13) and Eq. (14) do not hold.In this sense these assumptions are stronger than thevalidity of the JR, or to rephrase, these assumptions rep-resent sufficient but not necessary conditions. This pe-culiarity will be investigated in detail belowIn an analogous way we can derive Eq. (16) for initialenergy eigenstates ρ (0) = | i i h i | if we additionally assumethat p F ← i ≈ p F ← I (17)holds for all i ∈ N with ǫ i ∈ E I . This additional as-sumption means that the transition probabilities froman eigenstate | i i to an energy interval E F are smoothfunctions of the initial and final energy. We thereforecall it ”smoothness” . The validity of this assumption isinvestigated in Sec. V in a finite size scaling.In order to demonstrate even closer analogy of Eq. (16)with the standard JR, it remains to be explained in whichsense the r.h.s of Eq. (16) may be considered as the fa-miliar r.h.s of the standard JR, e − β ∆ F , where F is thefree energy. Such an identification would hold if − ln Z α β ? = F α . (18)In order to judge whether or not Eq. (18) is justified,consider the logarithm of Eq. (14),ln D α ( U ) = ln Z α + βU . (19)The index α ∈ { ini , fin } signals, whether the equationrefers to the initial or final Hamiltonian, respectively.Moreover, the discrete average Energies ¯ E I and ¯ E F arereplaced by the continous parameter U .If one identifies, along the lines of Boltzmann’s originalapproach, the entropy S α asln D α := S α (20)(where we tacitly set k B = 1), one may convert Eq. (19)into − ln Z α β = U − S α β . (21)Note that, in accordance with Eq. (14), ∂ U S α = β ,hence β has the meaning of inverse temperature, and ther.h.s. of Eq. (21) is, accordingly the free energy F asintroduced in standard textbooks on phenomenologicalthermodynamics. In this sense Eq. (18) indeed holds,which entails the rewriting of Eq. (16) in a form closerto the familiar one: h e − βW i E = e − β ∆ F , (22)where h· · · i E denotes the microcanonical expectationvalue corresponding to energy E . This concludes ourconsideration on the validity of a JR for microcanonicalinitial states under the assumption of stiff work-PDFs. III. MODELS AND DRIVING PROTOCOL
With the following numerical investigations we ascer-tain the pivotal relevance of stiff work-PDFs for the va-lidity of the JR for microcanonical initial states. We a H int System Bath DO S ~ e xp ( b E ) l H prot FIG. 1. Schematic structure of the numerical model. A two-level system is coupled via a random-interaction to a bathwith an exponentially growing DOS. The structure of the in-teraction takes influence on the resulting work-PDFs. therefore introduce a model that is partly based on ran-dom matrices. Within this model we can control thestiffness of the resulting work-PDFs via a single parame-ter ξ . This allows us to observe the influence of stiffnesson the JR for microcanonical initial states.We consider an isolated system comprising a rela-tively small subsystem (denoted as ”sys”, H sys ) and abigger part serving as heat bath (denoted by ”bath”, H bath ). Both parts may interact via H int . Finally, a time-dependent force periodically drives the system H prot .Concretely we choose the small subsystem to be a spinand the time dependent force to be a kind of microwavefield such that the whole model allows for an interpreta-tion in terms of a spin-resonance experiment with a finitelifetime of the spin excitation, see Fig. 1. A very similarmodel (spin-GORM model) has previously been used tostudy relaxation in finite environments [32]In detail the Hamiltonian of the full system reads: H ( t ) = H sys + H bath + αH int + λH prot ( t ) (23)The small subsystem is a simple two-level system, e.g.a spin- ⁄ -particle in a magnetic field B z . The Hamilto-nian of this subsystem is characterized as: H sys (cid:12)(cid:12) E sys j (cid:11) = E sys j (cid:12)(cid:12) E sys j (cid:11) , E sys1 | = ∓ B z (cid:12)(cid:12) E sys j (cid:11) obviously denote the eigenstates of H sys . We chose B z = 0 . H bath (cid:12)(cid:12) E bath j (cid:11) = E bath j (cid:12)(cid:12) E bath j (cid:11) E bath j = 1 β ln (cid:26) jN exp (cid:0) βE bathmax (cid:1) + (1 − jN ) exp (cid:0) βE bathmin (cid:1)(cid:27) (25)while N denotes the dimension of the bath. This def-inition yields an (strictly) exponentially growing DOSΩ bath ( E ) ∝ exp { βE } comprising energies from E bathmin = 0to E bathmax = 4 .
5. The constant β (which takes the role ofa temperature here) is chosen to 1. Note that for thismodel an exponentially growing DOS of the bath inducesan approximately exponentially growing DOS of the fullHamiltonian.As mentioned in the previous section, an exponentiallygrowing DOS is one of the conditions (Eq. (14)) used hereto derive the JR for micro-canonical initial states. As theDOS of many physical systems (spatially extended withshort range interactions, etc,) is well approximated byan exponential within not too large an energy range, thiscondition is routinely imposed in this context and repre-sents a natural cornerstone of the modelling. Note thatthis modelling corresponds to an ”ideal heat bath” i.e.,the temperature is always 1 /β , regardless of the actualbath energy E sys j We now define the interaction between the two partsof system. We introduce the following notation: (cid:12)(cid:12) E sys m , E bath n (cid:11) := | E sys m i ⊗ (cid:12)(cid:12) E bath n (cid:11) (26)Regarding this product basis we define the interaction-part. (cid:10) E sys m , E bath n (cid:12)(cid:12) H int (cid:12)(cid:12) E sys k , E bath l (cid:11) = (1 − δ mk ) · g ( E bath n + E bath l ) f ( | E bath n − E bath l | ) R nl (27) g ( ¯ E ) = exp (cid:26) − βξ ( ¯ E − E bathmax )4 (cid:27) f ( ω ) = exp (cid:26) − ω σ (cid:27) (28) R nl = R ln denote normally distributed random num-bers with zero mean and unit variance.To assess the rationale behind this modelling considerthe following.The interaction H int only allows transitions (for thenon-driven model, i.e., for λ = 0) between energet-ically similar bath-states. Direct transitions betweenstates with significantly different bath-energies are sup-pressed by the Gaussian function f ( ω ), i.e., their suppres-sion is controlled by the respective variance σ = 0 . γ of the z -component of the magnetization of thespin for some initial bath-energy E bath can be estimatedas γ ∝ exp (cid:8) β (1 − ξ ) E bath (cid:9) for our model. In a physicalsystem we would expect that γ depends on the temper-ature 1 /β of the bath, but not on its actual energy. For ξ = 1 the rate γ actually becomes independent of thebath energy E bath . We thus consider this the most phys-ical case.While it is not plain to be seen, it is an actual and mostimportant fact, that ξ also controls the stiffness of themodel. It turns out that stiff work-PDF’s arise preciselyat the above ”most physical” case ξ = 1. For smaller andlarger ξ stiffness is lost. For clarity of presentation we donot discuss the inner workings of this ”stiffness controlmechanism” here but simply present clear numerical ev-idence for its existence in App. A. Wp E =2.25 ( W ) x = 1.0 a = 0.40 l = 0.25 p E =2.25 ( W ) W x = 1.0 a = 0.20 l = 0.25 FIG. 2. Work-PDFs for two different bath-couplings α .For the weaker coupling one nicely sees two sharp peaks at W = ± B z , resulting from spin-flips induced by the resonantirradiation. For the stronger coupling the work-PDF is muchbroader. We finally introduce the time-dependent protocol ex-clusively acting on the ”sys”-part: H prot ( t ) = sin( ω prot t )( | E sys1 i h E sys2 | + h . c . ) (29)Thinking again of the system in terms of a spin- ⁄ -particle, the protocol describes a sinusoidally modulatedmagnetic field in the x -direction, as routinely used in spinresonance experiments. We choose ω prot = B z = 0 .
5, i.e.,the irradiation is on resonance. The duration of the pro-tocol is set to T = 3 . πω prot throughout this paper. IV. JARZYNSKI RELATION FORMICRO-CANONICAL INITIAL STATES ANDVARIOUS SYSTEM CONFIGURATIONS
We consider a micro-canonical ρ I mc (0) initial state fromthe center of the spectrum of the initial Hamiltonian H (0) with an energetic width of about δ ≈ . ρ I mc (0) = Ω − I Π I , I = j E δ k E = max( ǫ j ) + min( ǫ j )2 (30)The dimension of the bath is set to N = 4000.For this initial state we numerically check the JRfor three different stiffness parameters ξ = 0 . , . , . α = 0 , . , . , . . . , . λ = 0 , . , . , . . . , . D mc ( ξ, α, λ ) := Tr (cid:8) U ρ I mc (0) U + exp {− β ( H ( T ) − E ) } (cid:9) − exp {− β ∆ F } (31)Since we consider cyclic processes ∆ F is equal to zeroand exp {− β ∆ F } becomes equal to 1. If the JR holdsfor the considered set of parameters ( ξ , α and λ ), thecorresponding quantifier D ( ξ, α, λ ) vanishes.The results for the micro-canonical initial states aredisplayed in Fig. 3. Light green means that the JR isfulfilled, while other colors indicate deviations.In case of weak bath-couplings α or weak irradiation-strengths λ the JR is trivially fulfilled, even for micro-canonical initial states. For λ ≈ x = 0.6 x = 1.0 x = 2.0 al FIG. 3. D mc ( ξ, α, λ ) for various system configurations. Lightgreen (zero) indicates that the system complies with the JR,while other colors (non-zero values) quantify the deviationsfrom the JR. Apparently the JR is always fulfilled for ξ = 1,even for microcanonical initial states. adiabatic following and we thus expect to actually per-form zero work. For α ≈ β . So theprotocol acts on a system prepared in a Gibbsian state.For this scenario it is well-known that the JR holds. Wetherefore concentrate on the larger α s and λ s.For ξ = 1 . ξ = 0 . , . ξ = 0 . , .
0, we find deviations from the JR ”toboth sides” ( D mc ( ξ, α, λ ) positive as well as negative).These deviations appear to systematically depend on α and λ and are nonzero for most α, λ . However, there arefew combinations of α and λ for which the JR is fulfilled,see corresponding ”light green corridors” in Fig. 3.In App. B the dependence of the deviations D mc ( ξ, α, λ ) on the initial energy E is numerically in-vestigated in more detail. We find that at ξ = 1 theinitial energy plays a crucial role for the resulting devia-tions, but not so at ξ = 1 Especially at the ”light greencorridors” in Fig. 3, left and right panel, the JR is vio-lated for initial microcanonical states with energies otherthan E .These numerical finding suggests that the stiffness ofwork-PDFs is crucial for the validity of the JR for micro-canonical initial states. V. VALIDITY OF THE JARZYNSKI RELATIONFOR ENERGY EIGENSTATES AND FINITE SIZESCALING
Up to now we only investigated the validity of the JRfor micro-canonical initial states Eq. (30). We now turnto initial states being eigenstates of the initial Hamilto-nian H (0). We denote these initial states as Tr{ r (T) exp{- b (H(T) - E }} - 1 FIG. 4. Finite-size scaling of D es ( ξ, α, λ ) for eigenstates(from the center of the spectrum) of the respective initialHamiltonian H (0). Displayed are averages (symbols) andstandard-deviations (bars) for three different model param-eter sets: red: ( ξ = 1, α = 0 . λ = 0 .
25) blue: ( ξ = 0 . α = 0 . λ = 0 .
25) green: ( ξ = 2 . α = 0 . λ = 0 . N − . . ρ i es (0) = | i i h i | . (32)The energetic width of these states is δ = 0. In this sensethey are fundamentally different from micro-canonicalinitial states. But in this section we will demonstrate,that in the limit of large bath-dimension, both behavesimilar regarding the JR.Again, we use Eq. (31) to check whether the JR isfulfilled or not. We define the corresponding deviations D es ( ξ, α, λ ) completely analogous to the D mc ( ξ, α, λ ) (cf.Eq. (31)) but with ρ I mc (0) replaced by ρ i es (0) Note thatthe average of the D es ( ξ, α, λ ) over a pertinent range of i equals a corresponding D mc ( ξ, α, λ ). Thus the followingnumerical results (Fig. 4) do not only hold informationabout the sizes of the D es ( ξ, α, λ ) but also about the finitesize scaling of the D mc ( ξ, α, λ ).A systematic survey of the D es ( ξ, α, λ ), for all α, λ isnumerically very costly. We thus concentrate on caseswhere the violation of the JR is pronounced for ξ = 1i.e., α = 0 . λ = 0 .
25, cf. Fig. 3.Figure 4 shows statistical results on the D es ( ξ, α, λ ) forincreasing bath sizes N . (For clarity the results are dis-played over inverse bath size 1 /N .) Displayed are theaverages (diamonds) and standard deviations (vertical”error” bars) for a stiff system ξ = 1 and two nonstiffsystems ξ = 0 . ,
2. The statistics encompass 100 dif-ferent D es ( ξ, α, λ ) for adjacent i from the middle of therespective spectrum for each parameter set.The following principles may be inferred from Fig. 4:The averages appear to be independent of the system size N , thus the D mc ( ξ, α, λ ) are independent of the systemsize, hence Fig. 3 provides a representative picture alsofor other (larger) bath sizes than N = 4000. The stan-dard deviations of the D es ( ξ, α, λ ) decrease with bathsize, presumably as ∝ N − . as suggested by the tiltedparabolae.These findings strongly indicate that the JR is indeedfulfilled even for pure initial energy eigenstates for stiffsystems in the limit of large bath (total system) sizes.Note that in this case the statistical character of the cor-responding work-PDFs is entirely due to pure quantumuncertainties. Furthermore the JR appears to be alwaysviolated for pure initial energy eigenstates in the limit oflarge bath (total system) sizes if the system is nonstiff. VI. DISCUSSION
In this article we analytically show that the Jarzynskirelation holds also for a broad class of non-Gibbsian ini-tial states in quantum systems under certain conditions.For micro-canonical initial states these conditions are :An exponentially growing DOS of the initial and finalHamiltonian and stiff work-PDF’s i.e., work-PDF’s thatare independent of the initial energy. Moreover, numer-ics indicate that the converse also holds: systems that donot comply with the stiffness condition actually do vio-late the JR for micro-canonical initial states, independentof the size of the system.In order to analytically show the validity of the Jarzyn-ski relation for initial energy eigenstates we exploit an ad-ditional assumption on the work-PDF’s called ”smooth-ness”, which is expected to hold for large systems. Thisexpectation is supported by numerics for some examples,which shows that the Jarzynski relation is fulfilled in thelimit of large systems for systems that do exhibit smooth-ness, and violated for systems which do not.To conclude, there appears to be a very tight link be-tween the applicability of the Jarzynski relation and stiff-ness/smoothness for non-Gibbsian initial states which de-serves further exploration.
Acknowledgements:
This work has been funded by theDeutsche Forschungsgemeinschaft (DFG) - Grants No.397107022 (GE 1657/3-1), No. 355031190 - within theDFG Research Unit FOR 2692. Furthermore this re-search was supported in part by the National ScienceFoundation under Grant No. NSF PHY-1748958.
Appendix A: Stiffness of Work-PDFs
In the main text varied the model parameter ξ and justclaimed it would affect the stiffness of the work-PDFs. Inthis section we numerically check the actual influence ofthis model parameter on the work-PDFs.We therefore calculated the work-PDFs p E ( W ) forvarious model parameters. In Fig. 5 we exemplarilypresent the data for d = 4000, α = 0 . λ = 0 .
25 and ξ = 0 . , . , . H (0) with E ≈ . pE (0) x = 0.60.10 1 2 3 pE (0) x = 1.00.10 1 2 3 pE (0) x = 2.0 EEE
FIG. 5. For ξ = 1 . ξ = 0 . ξ = 2 . Fig. 5 shows the probabilities to perform zero work.For ξ = 1 . p E (0) appear to be approx-imately independent of E , while for ξ = 0 . ξ = 2 . d the work-PDFs be-come smoother, the slope for ξ = 0 . , . d . D ( x=2.0 , a=0.45 , l=0.15 )E E FIG. 6. Energy dependence of D ( ξ, α, λ ). Appendix B: Jarzynski Realtion for different initialenergies
In Sec. IV we considered deviations from the JR forvarious combinations of ξ , α and λ , but for a fixed initialenergy E and found that for some combinations of theseparameters the JR appeared to be fulfilled, even thoughcondition Eq. (13) is violated. We now consider the de-pendence of these deviations on the energy of the initialstate ρ (0) with the aforementioned parameters held con-stant. We consider micro-canonical initial states, definedaccording to Eq. (30), with various energies E . Theresulting deviations D ( ξ = 2 . , α = 0 . , λ = 0 .
15) aredisplayed in Fig (6).Note that for this parameter combination we foundthe JR fulfilled for the previously considered initial en-ergy E . The data suggests that there is only a smallenergy-range for which the JR is approximately fulfilledand E accidentally is within this region. The energy-dependence for other α and λ looks quite similar. Sowe can find specific micro-canonical initial states, whichcomply with the JR, even if condition Eq. (13) is notfulfilled. But since this is a feature of a very specificcombination of system and initial state we conclude thatthe JR is not fulfilled by this system and driving-protocolin general.In contrast, for ξ = 1 there is a wide region of initialenergies that fulfill the JR, which is a direct consequenceof the conditions Eq. (13) and Eq. (14). [1] Sandu Popescu, Anthony J. Short, and Andreas Win-ter, “Entanglement and the foundations of statistical me-chanics,” Nature Physics , 754–758 (2006).[2] Sheldon Goldstein, Joel L. 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