Statistical work-energy theorems in deterministic dynamics
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] M a r Statistical work-energy theorems in deterministic dynamics
Chang Sub Kim ∗ Department of Physics, Chonnam National University, Gwangju 500-757, Republic of Korea
We theoretically explore the Bochkov-Kuzovlev-Jarzynski-Crooks work theorems in a finite systemsubject to external control, which is coupled to a heat reservoir. We first elaborate the mechan-ical energy-balance between the system and the surrounding reservoir and proceed to formulatethe statistical counterpart under the general nonequilibrium conditions. Consequently, a consis-tency condition is derived, underpinning the nonequilibrium equalities, both in the framework ofthe system-centric and nonautonomous Hamiltonian pictures and its utility is examined in a fewexamples. Also, we elucidate that the symmetric fluctuation associated with forward and backwardmanipulation of the nonequilibrium work is contingent on time-reversal invariance of the underlyingmesoscopic dynamics.
PACS numbers: 05.20.-y, 05.70.Ln, 82.37.Rs
I. INTRODUCTION
Nonequilibrium (NEQ) transformation is not charac-terized by a simple thermodynamic relation and even rel-evant variables to be specified are not given, in general, apriori. The Jarzynski identity rarely provides researcherswith a useful avenue to extract a definite, equilibrium in-formation of thermodynamic systems driven away fromequilibrium . The Crooks relation follows to describethe symmetric work-fluctuation in statistical dynamics ofthe systems . The two, widely known as the NEQ worktheorems are found to be intimately related to the ear-lier study by Bochkov and Kuzovlev . Aside from theirfunctionality , the NEQ work theorems deliver an insightinto the nature of the second law of thermodynamics inthe physical regime where indeterminacy matters beyondconventional bulk thermodynamics in the context of thedisparate fluctuation theorems (FTs) .The NEQ work theorems were framed for a thermo-dynamic system on which one cannot perform a workprecisely as instructed by a predetermined protocol .For instance, in single-molecule stretching experiments,one inevitably ends up with a stochastic signal in themeasured, force versus extension data . The free energy(FE) difference between two equilibrium states, suppos-edly undergone an irreversible transformation, is given byan equality , called the Jarzynski equality (JE). A dedi-cated feature is that there appears only a single tempera-ture in the JE, albeit the system is perturbed from initialequilibrium . Over the years, a great deal of researcheffort has been devoted to establish the NEQ formulationmore rigorously using the various mathematical methodsin the theoretical side and to test the idea in the real andcomputer experiments . Lately, research effort hasbroken a new ground to explore more intricate transitionbetween kinetic states of systems .In this paper we consider a finite system, maintainedin a heat reservoir, under controlled manipulation by anexternal agent. The statistical correlation of the systemwith the surrounding reservoir gives rise to a momentum-dependent thermostatting mechanism, which was not taken into account in the original Hamiltonian deriva-tion of the JE . The thermostatting damping forceis embraced explicitly in our formulation, together withthe time-dependent control force. The control force isgenerically macroscopic because it describes coupling ofan apparatus to its conjugate phase-observable of the sys-tem. Therefore, the entailed NEQ dynamics is mixed or mesoscopic on time scale in the sense that the coarse-grained damping force and the macroscopic external cou-pling conjoin to determine the microscopic time-evolutionof the system.The goal of our endeavor is to present a statistical-mechanical formulation of the NEQ work theorems inthe finite, dissipative systems governed by the proposed,mesoscopic dynamics. The main concerns are to inves-tigate the effect of the phase-volume contraction due tothermostatted dissipation and other nonpotential fields,as a consequence to derive the consistency criterion thatundergirds the NEQ work theorems, and to attentivelyclarify the subtlety of the notion of thermodynamic workin the NEQ measurements. We hope to add a further in-sight into our understanding of the fundamental relationbetween dissipation and irreversibility.The proposed, extended dynamics and correspondinggeneralized Liouville equation are partly based on theearly formalism by McLennan, for stead-states drivenby external nonconservative forces . The subsequentanalysis utilizes a formal solution to the modified Li-ouville equation, describing the NEQ distribution at aco-moving point in phase space, which is often called theKawasaki representation . The Kawasaki method pro-vides a closed expression for far-from equilibrium distri-butions and was previously applied to investigating thetransient FT in thermostatted fluid systems . Employ-ing the analytical representation for the NEQ ensembledensity, we show that the NEQ work theorems must betightened in a thermally open system by a certain con-sistency criterion.The JE is formulated for a unidirectional transforma-tion that its experimental or computational test may beintended in single, forward irreversible setups where thereverse work cannot be measured . Whereas,the Crooks work-fluctuation theorem (CWFT) is fur-nished for bidirectional, forward and backward measure-ments of a single small system . We show that thelatter, symmetric work theorem applies to the systemswhose dynamics is constrictively invariant under timereversal. The constrictive invariance revealed rests onthe mesoscopic symmetry of momentum-dependent dis-sipation. Lately, there is a growing interest in the roleof momentum, an odd-parity dynamical variable undertime inversion, associated with the entropy productionsin driven stochastic dynamics . We also notice a re-port which treats the microscopic reversibility of a Hamil-tonian system under an external perturbation, but unlikeour investigation, without dissipative mechanisms .The manipulating forces of the NEQ work measure-ments are usually installed as time-dependent parametersin the Hamiltonian . Such a nonautonomous description undesirably bears a delicacy in consistentlydefining the physical energy. This point was previouslycalled into question by others and elevated the ac-tive discussions about a proper definition of thermody-namic work among researchers . Recently, an exper-imental study carefully concerned how to correctly de-fine and measure thermodynamic work in small systems,which the authors found as a pertaining issue in pullingexperiments . In this paper, we explore both the nonau-tonomous picture and, what we term, the system-centric picture in a unified frame. In the latter picture the time-dependent forces responsible for external work are in-cluded in the equations of motion directly and the energyis defined via the bare Hamiltonian. Consequently, weexplicate the two different representations of work andilluminate which picture brings about the conventionalthermodynamic description.Our work exclusively concerns deterministic dynamicsthat stochastic dynamics is not in the scope of our inves-tigation. For the latter, we refer to a recent review whichcontains the latest issues in stochastic thermodynamicswith a complete list of references . The parallel NEQformulation in quantum dynamics, which is also beyondthe scope of the present article, can be found in the otherreview .This paper is organized as follows. In Sec. II we cointhe extended Hamilton equations of motion and the re-lated, modified Liouville equation, suitable for describ-ing NEQ work measurement in a thermally open sys-tem. Then, the mechanical work-energy theorems areestablished in Sec. III and the corresponding statisticalformulation is followed in Sec. IV delivering the NEQequalities with the upholding condition. In Sec. V thebidirectional, work-fluctuation theorems are consideredwith discussing the underlying symmetry of the gener-alized Liouville dynamics. In Sec. VI a few dynamicalsystems are examined to manifest the utility of the con-sistency criterion. Finally, a summary and a conclusionare provided in Sec. VII. II. EXTENDED HAMILTONIAN DYNAMICS
The mechanical influences on a physical systemare usually described by three types of forces: thecoordinate-dependent, momentum-dependent, and para-metric (time-dependent) forces. We recapitulate herehow these forces may enter the extended Hamilton equa-tions of motion which, in turn, constitute the generalizedensemble dynamics of the identically prepared systems inphase space.Let us suppose that a finite system under external con-trol is coupled to a heat reservoir. The total Hamiltonian H T can be written, in principle, as H T = H + H R + Φ (1)where H and H R are the Hamiltonian of the system andthe surrounding reservoir, respectively, and Φ denotes theboundary interaction between them. The bare Hamilto-nian of the system is specified by the generalized coordi-nates q and momenta p of the constituents of the system, H = H ( q, p ) . The Hamiltonian of the reservoir depends on the gen-eralized coordinates and momenta
X, Y of the reservoirdegrees of freedom as H R = H R ( X, Y ) . We shall assume that the interaction Φ depends only onthe coordinates of the system and the reservoir to bewritten as Φ = Φ( q, X ) . The total system must evolves in time obeying Hamil-tonian dynamics at the microscopic level. Accordingly,the NEQ Gibbs ensemble is described by the Liouvilleequation in the full phase space, ∂P∂t + { P, H T } = 0 (2)where P is the ensemble density of the total system and { P, H T } is the Poisson bracket . By encapsulating theunknown information on Φ as a statistical correlation ϕ between the system and the reservoir, one may writedown the total ensemble density in the form, P = ρρ R (1 + ϕ )where ρ R denotes the canonical equilibrium density of thereservoir at an absolute temperature T . The ensembledensity of the system ρ is a marginal density reducedfrom the total density via ρ ( q, p ; t ) = Z d Ω P ( q, p ; X, Y ; t ) , where d Ω ≡ dXdY , assuming proper normalization inthe full phase space.In practice, time-evolution of the ensemble density ofthe system alone is of concern, which can be obtainedby integrating Eq. (2) over the reservoir variables. Aftersome manipulation, the result is given as ∂ρ∂t + { ρ, H } = − X ∂∂p i ( ρ R i ) (3)where use has been made of the identity, R i = Z d Ω (cid:18) − ∂ Φ ∂q i (cid:19) ρ R (1 + ϕ ) (4)and the explicit expression of the Poisson bracket, { ρ, H } ≡ X (cid:18) ∂ρ∂q i ∂H∂p i − ∂H∂q i ∂ρ∂p i (cid:19) (5)with summation running over all degrees of freedom inthe system.When the system and the reservoir is statistically un-correlated, i.e. ϕ ≡ R i becomes only coordinate-dependent to give R i → − ∂∂q i V R ( q )where V R ≡ R d ΩΦ ρ R which may be absorbed into thesystem Hamiltonian, ˜ H ≡ H + V R . Then, Eq. (3) can be recast into the customary Liouvilleequation , ∂ρ∂t + { ρ, ˜ H } = 0 (6)where the effective Hamiltonian ˜ H generates the conser-vative dynamics: ˙ q i = ∂ ˜ H∂p i = { q i , ˜ H } , (7)˙ p i = − ∂ ˜ H∂q i = { p i , ˜ H } . (8)When the statistical correlation is taken into account,i.e. if ϕ = 0, the influence from the reservoir R i cannot berepresented as a potential force. In this case, the reservoirforce becomes momentum-dependent, in general, whichwe set R i ≡ D i ( q, p ; T ) (9)in which we have indicated explicitly the dependence of D on the reservoir temperature T in the canonical ensemble ρ R . Subsequently, Eq. (3) can be further manipulated togive ∂ρ∂t + X (cid:18) ˙ q i ∂ρ∂q i + ˙ p i ∂ρ∂p i (cid:19) = − Λ ρ. (10) The preceding equation describes a generalized Liouvilledynamics in the phase space of the system variablesalone. Note here that the second term on the left-hand-side (LHS) of Eq. (10) cannot be represented as the Pois-son bracket, { ρ, H } because the dynamic variables evolvenow according to ˙ q i = { q i , H } , ˙ p i = { p i , H } + D i which contain the effective, momentum-dependent force D arising from interaction between the system and thereservoir. The factor Λ on the right-hand-side (RHS) inEq. (10), which is identified to beΛ( q, p ; T ) = X ∂∂p i D i , provides a measure of contraction, or expansion, of phasevolume.Now that we have treated the coordinate-dependentand momentum-dependent forces, we consider the exter-nal time-dependent forces, frequently occurring when amechanical or electromagnetic control of the system is re-quired. In the Jarzynski scheme such an external controlis formally furnished with a time-dependent parameter,say λ ( t ), in the system Hamiltonian as H ( q, p ; λ ) ≡ H = H ( q, p ; λ ) . (11)In this picture the prescribed Hamiltonian H ( q, p ; λ ) de-velops non-autonomously in time because λ is dynami-cally independent of q and p , namely ∂λ ( t ) ∂q i = 0 = ∂λ ( t ) ∂p i . The dynamical variables in H obey the extended Hamil-ton equations of motion in the form,˙ q i = { q i , H} , (12)˙ p i = { p i , H} + D i ( q, p ; T ) , (13)which do not carry the manipulating forces. Here, weemphasize the temperature dependence in D i which fea-tures an essential difference between the deterministicHamiltonian dynamics for a thermally open system andstochastic dynamics for a Brownian motion. In the lat-ter the frictional force is phenomenologically treated asindependent of temperature.Another picture attainable is to prescribe the externalcontrol directly in the equations of motion and to definethe mechanical energy of the system as the instantaneousvalue of the bare Hamiltonian , H ( q, p ) = X (cid:26) p m + V ( q ) (cid:27) (14)where the potential energy V ( q ) includes both interactionamong constituents of the system and any other conser-vative external potentials. We term the latter picture the system-centric description in the sense that the energy ofthe system is specified solely by the system variables. Inorder to gain some insight into how time-dependent forcesenter the equations of motion, let us assume that the ex-ternal control may be isolated as a perturbation H ′ tothe bare Hamiltonian H . Then, the system Hamiltonianis written additively as H ( q, p ; λ ( t )) = H ( q, p ) + H ′ ( q, p ; λ ( t )) (15)where the perturbation term is not necessarily small.With the preceding recipe for H , Eqs. (12) and (13) gen-erate the extra terms V ex and G ex in the equations ofmotion, V exi ≡ ∂H ′ ∂p i (16)which contributes to time-development of the generalizedvelocity and G exi ≡ − ∂H ′ ∂q i (17)which describes the external, control force, acting on thedegree of freedom i in the open system.Finally, we propose the extended Hamilton equationsof motion in the system-centric picture as˙ q i = { q i , H } + V exi ( q, p ; r , t ) , (18)˙ p i = { p i , H } + D i ( q, p ; T ) + G exi ( q, p ; r , t ) . (19)The dissipative force D , stemming from the statisticalcorrelation between the system and the reservoir, playsthe role of a thermostat. Although we have identified theexternal fields, V ex and G ex via Eqs. (17) and (16), theyare not, in general, derivable from a Hamiltonian. Theyare required to take care of the coupling of the systemto the external, nonpotential fields at field point r in anopen system in the system-centric view. Equation (18)suggests that the canonical momentum p i be not relatedto the generalized velocity ˙ q i in the usual sense, but, un-der the extended dynamics, is given by p i = m ( ˙ q i − V exi ) . Evidently, the proposed, extended equations of mo-tion, Eqs. (12) and (13) in the nonautonomous pictureand Eqs. (18) and (19) in the system-centric picture,constitute a non-Hamiltonian dynamics due to the non-potential terms. Note also that the two Hamiltonians H and H must be identical when the perturbation is turnedoff. Some examples of such extended dynamics are con-sidered in Sec. VI. III. MECHANICAL WORK-ENERGYTHEOREMS
The deterministic state of an open system may be de-picted as a trajectory in phase space, governed either by the nonautonomous Eqs. (12) and (13) or by the system-centric Eqs. (18) and (19). The mechanical energy of thesystem is specified as an instantaneous value of Eq. (11)in the former or that of Eq. (14) in the latter, which isnot conservative in either case.We evaluate here how the energy of the system changesover a temporal interval τ , evolving under the extendeddynamics. It can be done in the system-centric picture bycarrying out the following manipulation of H ( q, p ) givenin Eq. (14),∆ H = Z τ dt ˙ H = Z τ dt X (cid:18) ∂H∂p i ˙ p i + ∂H∂q i ˙ q i (cid:19) . The required step is to substitute Eqs. (18) and (19) forthe time-rate of the dynamical variables in the above ex-pression. Subsequently, it can be seen that the conserva-tive dynamics is canceled out. Then, the following iden-tification from Eq. (14) is used in the remained terms, ∂H∂p i = p i m and ∂H∂q i = ∂V∂q i . Consequently, the induced change in the system-centricenergy is represented as∆ H = Z τ dt X p i m D i + W (20)where the first term on the RHS describes the energydissipation into the surroundings by the momentum-dependent force D . The expression W on the RHS ofEq. (20) represents the work done by the external fieldson the system, W ≡ Z τ dt X (cid:26) p i m G exi + ∂V∂q i V exi (cid:27) . (21)When the external velocity-field V ex is not coupled tothe system, it holds from Eq. (18) that the canonicalmomentum is related to the generalized velocity as usual,i.e. p i = m ˙ q i . The external work, then, is specified by thetime-dependent force G ex alone in its conventional form, W = X Z τ dt ˙ q i G exi . (22)Equation (20) constitutes the conventional, mechanical work-energy theorem in the integral representation , ex-tended to accommodate the various sort of non-potentialforces. It explains transformation of the mechanical en-ergy: The mechanical energy H increases with the ex-ternal work W performed on the system and decreasesby the energy-exchange interaction D with the surround-ings.The work-energy theorem may be envisaged with thenonautonomous Hamiltonian H ( p, q ; λ ), as well. The dy-namical variables in H obey the generalized equations ofmotion given in Eqs. (12) and (13), while the parameter λ ( t ) is manipulated according to a prescribed protocolover the period 0 ≤ t ≤ τ . The induced change in H isreadily evaluated as∆ H = Z τ dt ˙ H = Z τ dt X ˙ q i D i + W (23)where W is the parametric change of H ( p, q ; λ ), W = Z τ dt ˙ λ ∂ H ∂λ . (24)In passing to the second line in Eq. (23), the Hamilto-nian dynamics has been canceled out but the contributionfrom D i , the first term on the RHS. The second term W on the RHS of Eq. (23) represents the control work doneon the system by an external agent. Equation (23) is thedesired work-energy theorem pictured with the nonau-tonomous Hamiltonian H .Note that both work-energy theorems contain the samedissipation term; however, the definition of work ap-pears distinctively. The two descriptions do not pro-vide an equivalent measure to the mechanical energy ofthe system. To clarify how the change induced in thenonautonomous Hamiltonian ∆ H differs from that in thesystem-centric Hamiltonian ∆ H , we use the additive per-turbation model for the external manipulation, Eq. (15).The change in the perturbation term H ′ over the workperiod can be calculated via∆ H ′ = Z τ dt (cid:26)X (cid:18) ∂H ′ ∂q i ˙ q i + ∂H ′ ∂p i ˙ p i (cid:19) + ∂H ′ ∂t (cid:27) where ∂H ′ /∂p i and ∂H ′ /∂q i specify the external fields,Eqs. (16) and (17). Then, ∆ H is obtained by addingthe calculated ∆ H ′ to the energy change ∆ H specifiedin Eq. (20) as ∆ H = ∆ H + ∆ H ′ . The outcome has been shown to be exactly the same asthe one given in Eq. (23). In order to be more concrete,let us consider the linear coupling model, H ′ = − λ ( t ) G ( { q i } ) (25)where G ( { q i } ) is a phase-observable which is conjugateto the control parameter λ . The perturbation Hamil-tonian describes a mechanically forced interaction or adipole excitation in electromagnetic systems. The exter-nal fields associated with the perturbation are identifiedimmediately by Eqs. (16) and (17) as G exi = λ ∂G∂q i and V exi = 0 . Then, the work done on the system-centric Hamiltonian H is written via Eq. (21) as W = X Z G exi dq i . (26) On the other hand, it is given in the nonautonomouspicture via Eq. (24) as W = − X Z τ dt ˙ G exi q i = − X Z q i d G exi . (27)Thus, the two distinctive representations of externalwork have come to realization from the identical time-dependent force G ex : In the system-centric descriptionthe work W , Eq. (26) is represented as the integral ofthe forces over the displacements of coordinates. In con-trast, the work W , Eq. (27) is represented in the nonau-tonomous description as the negative integral of the co-ordinates over the variation of the forces. The former isreferred to as the exclusive work and the latter as theinclusive work by Jarzynski . By comparing two work-energy theorems, Eqs. (20) and (23), it follows that W = W + ∆ H ′ (28)where ∆ H ′ = − ∆ [ P q i G exi ] . The preceding Eq. (28)shows that the nonautonomous work W differs from thesystem-centric work W by the amount of the inducedenergy from the perturbation. Evidently, the energiesdefined by the two descriptions do not measure the sameamount of quantity in an identical setup.The nonautonomous description appears to carry apotential ambiguity in defining the physical energy-difference. The reason is that the time-dependent Hamil-tonian specifies the energy only up to an arbitrarytime-dependent factor without affecting the equations ofmotion . Consequently, the reference point of the en-ergy may not be the same in initial and final states. Weargue, however, that the explicit time-dependence of H through a coupling mechanism of a macroscopic appara-tus to the system [e.g. Eq. (25)], is not to be introducedarbitrarily but in a macroscopically controllable manner. IV. STATISTICAL WORK-ENERGYTHEOREMS
Having established the mechanical work-energy theo-rems, we now proceed to formulate their statistical coun-terparts. In a small system with a few degrees of freedomthe individual, trajectory-dependent work may be an ob-servable, however, fluctuation involved in the work mea-surement hinders the mechanical work-energy theoremfrom being useful. In performing work on a finite systeminstructed by a definite protocol, a myriad of trajectoriesparticipate due to insufficient information on the initialphase. In both cases, a statistical description is required.Here, we consider a finite system which is preparedinitially in equilibrium with a surrounding reservoir attemperature T and subsequently undergoes NEQ trans-formation manipulated by an external control. In theframework of classical statistical mechanics, later stageof the system is specified by a time-dependent ensembledensity in phase space . We conceive that the systemremains in contact with the heat reservoir, regardless ofthe coupling strength , which seems natural in an ex-perimental setup. A. Quasi-static average
The external, control work induces a change in the en-ergy of the system limited by the mechanical work-energytheorem. When the work is performed quasi-statically, itmay be assumed that the system remains in equilibriumwith the heat reservoir. Accordingly, the ensemble den-sity ρ ( t ) retains its canonical, equilibrium form ρ eq ateach instant over the work interval, 0 ≤ t ≤ τ .Here, we attempt directly to take statistical average ofthe work-energy balance over the quasi-equilibrium en-semble. First, we perform the average in the system-centric picture, Eq. (20), to evaluate h ∆ H i eq = h W i eq + h Z τ dt X ˙ q i D i i eq (29)where ρ eq is specified by the system-centric Hamiltonian,Eq. (14) at each instant. Under the quasi-static assump-tion, the average of ∆ H may be evaluated by switchingthe order of the ensemble average and the time-integralas h ∆ H i eq = h Z τ dt ˙ H i eq = Z τ d h H i eq . Then, it follows immediately that h ∆ H i eq = U ( τ ) − U (0) = 0 (30)where U is the internal energy defined to be U = Z dqdp ρ eq H ( q, p ) . The second equality in Eq. (30) arises because the func-tional form of H ( q, p ) remains the same at each instantand the dynamic variables span the entire phase space.Consequently, the internal energy remains to be constantin quasi-static processes. Similarly, the average of the ex-ternal work W in Eq. (29) can be performed by resortingto the explicit representation, Eq. (22), neglecting V ex here, as h W i eq = Z dqdpρ eq ( t ) X Z τ dt ˙ q i G exi = Z τ dt h X ˙ q i G exi i eq . The integrand in the preceding expression is the averagedpower, dW th dt = h X ˙ q i G exi i eq . Accordingly, one can write h W i eq = Z dW th ≡ W th (31)where dW th is the infinitesimal thermodynamic work done on the system which is not an exact differential.Lastly, the statistical average of the dissipation term onthe RHS of Eq. (29) may be evaluated as h Z τ dt X ˙ q i D i i eq = Z τ dt Z dqdpρ eq ( t ) X ˙ q i D i = − Z τ dt h Θ( q, p ; β ) i eq where Θ is the Rayleigh dissipation function defined tobe Θ ≡ − X D i ˙ q i . (32)Here, we identify the heat Q absorbed into the system,when its sign is positive, as Q ≡ − Z τ dt h Θ( q, p ; β, t ) i eq = Z T dS (33)where S represents the Clausius entropy of the system.Finally, by collecting the obtained expressions, Eqs. (30),(31), and (33) into Eq. (29), we reach the first law ofthermodynamics in the system-centric picture, W th + Q = 0 (34)which states that the work done on the system is preciselybalanced with the heat dissipated into the surroundings.The internal energy is unchanged in the system-centricdescription.Next, we carry on statistical average in the nonau-tonomous picture, Eq. (23), which takes h ∆ Hi eq = hWi eq + h Z τ dt X ˙ q i D i i eq (35)where ρ eq is specified in terms of the parametric Hamil-tonian, Eq. (11). Unlike Eq. (30), the change in the in-ternal energy is not zero in the time-dependent energylandscape, h ∆ Hi eq = U ( λ ( τ )) − U ( λ (0)) = ∆ U, (36)because U differs at each quasi-static instant, U ( λ ( t )) = Z dqdp ρ eq H ( p, q ; λ ( t )) . The average of the parametric work Eq. (24) can be ma-nipulated as hWi eq = h Z τ dt ˙ λ ∂ H ∂λ i eq = h Z dλ ∂ H ∂λ i eq = Z dλ ∂∂λ hHi eq . In passing to the third line in the preceding manipulation,use has been made of the fact that the parameter is nota coordinate in phase space. Then, the thermodynamicwork W th is represented as W th = Z dλ ∂U ( λ ) ∂λ . (37)The physical representation of the heat remains the samewith the system-centric picture, Eq. (33). Consequently,the first law takes the conventional form as∆ U = W th + Q (38)which includes the change in the internal energy for anopen system.Equation (38) can be recast in terms of the equilibriumHelmholtz FE, defined as F = U − T S , after a Legendretransformation, into the form ∆ F = W th − Z SdT which for an isothermal transformation reduces to∆ F = W th . (39)Note that all thermodynamic variables maintain theirquasi-equilibrium states over the work period and thatthe induced transformation is reversible , in the quasi-static processes. We also emphasize that the first lawmust hold not only in reversible processes but also in ir-reversible processes, albeit we have derived it only in thequasi-static limit. B. NEQ ensemble average
Under general NEQ conditions, the ensemble densityof the system does not keep up its quasi-equilibrium statecontinually over the work period. Instead, the NEQ en-semble density ρ ( t ) is governed by the generalized Liou-ville equation in phase space, Eq. (10) which is rewrittenhere for convenience as D t ρ ( t ) = − Λ ρ (40)where D t is the convective derivative along the phaseflow, D t ≡ ∂∂t + X (cid:18) ˙ q i ∂∂q i + ˙ p i ∂∂p i (cid:19) . The function Λ on the RHS of Eq. (40) is the compress-ibility factor defined as a divergence in phase space,Λ( q, p ; t ) = X (cid:18) ∂∂q i ˙ q i + ∂∂p i ˙ p i (cid:19) . (41)The dynamical variables q i and p i obey either Eqs. (12)and (13) or Eqs. (18) and (19), depending on the choice of the energy picture. Note that the second term in theconvective derivative D t ρ cannot be written as a Poissonbracket in the extended Hamilton dynamics.Although it is not tractable to solve Eq. (40), a formalsolution can be written by a direct integration in themoving frame with the phase fluid. The result is givenas ρ ( q, p ; t ) = U ( t ) ρ ( q, p ; 0) (42)where U is the time-evolution operator defined by U ( t ) = exp (cid:26) − Z t dt ′ Λ( q, p ; t ′ ) (cid:27) (43)and ρ ( q, p ; 0) represents an initial density at t = 0. Equa-tion (42) manifests that the compressibility function Λplays the role of a generator of convective time-translation in the present formulation. In the ordinary Liouvilleandynamics which preserves the phase volume, the NEQdensity becomes a constant of motion, i.e. D t ρ = 0.The initial equilibrium states are characterized by thesame density both in the system-centric and nonau-tonomous descriptions because the nonautonomousHamiltonian before turning on the work parameter isidentical to the system-centric Hamiltonian, H ( q, p ; 0) = H ( q , p ) . In both energy pictures the initial density is physicallyspecified as the canonical equilibrium distribution, ρ ( q, p ; 0) = ρ eq ( q , p )= exp { β ( F − H ( p , q )) } (44)where β = 1 / ( k B T ), k B and T being the Boltzmannconstant and the temperature of the heat reservoir, re-spectively, and F is the Helmholtz FE at initial stage.We now proceed to evaluate the statistical average ofthe mechanical work-energy theorem when the externalwork causes a fast change in the system during the workperiod τ . We shall consider the problem in the system-centric description first. Instead of directly taking theaverage of Eq. (20), however, we adopt the Jarzynski con-struction which defines the average of control work as aNEQ ensemble average of the weighted work-exponential in phase space. Technically, the average at t = τ may betaken equivalently either in the Schr¨odinger picture orin the Heisenberg picture in quantum mechanical terms,which is elaborated below. When the former is employed,the average is taken over the time-dependent distributionwhile the exponentiated work is considered a fixed-timephase function as h e − βW i = Z dqdpe − βW ρ ( q, p ; τ ) . (45)It is important here to recognize that the phase-spacemeasure dqdp is not invariant in the extended Hamil-tonian dynamics. Rather, the two generalized Liouvillemeasures dq dp and dqdp at different times t = 0 and τ ,respectively, are related to each other via the Jacobian J , dqdp = J ( τ ) dq dp . (46)The preceding Jacobian J itself obeys dynamics in theopposite sense to Eq. (40) D t J = Λ J . For the given initial condition, J (0) = 1, it can be for-mally integrated to give J ( τ ) = exp (cid:26)Z τ dt Λ( q, p ; t ) (cid:27) . (47)By inspecting that the Jacobian J differs from the time-evolution operator U only by the sign of the exponent,we attain that J and U evolve in the inverse sense toeach other, J ( τ ) U ( τ ) = 1 . (48)The preceding equation is the classical-mechanical uni-tarity condition which imposes the preservation of en-semble members in phase space, Z dqdpρ ( q, p ; t )= Z {J dq dp } {U ρ ( q, p ; 0) } = Z dq dp ρ ( q, p ; 0) . With help of Eqs. (42) and (46), Eq. (45) can be rewrit-ten as h e − βW i = Z dq dp e − βW ( τ ) ρ eq ( q, p ; 0) (49)where the subsequent rearrangement has been made of e R τ dt Λ( q,p ; t ) e − βW e − R τ dt Λ( q,p ; t ) = e − βW ( τ ) . Equation (49) is the Heisenberg representation of theNEQ ensemble average where the average is taken overthe initial equilibrium ensemble, whereas the work func-tion is interpreted to have evolved over the time-interval τ , limited by the mechanical work-energy theorem,Eq. (20). The NEQ averages may be viewed as a func-tional which maps the phase function of the work ex-ponential onto a scalar in phase space. We just provedthat the two pictures are identical in carrying out the NEaverages.Here, we carry on our calculation in the Schr¨odingerpicture to substitute the NEQ ensemble density, Eq. (42)at t = τ for ρ ( p, q ; τ ) in Eq. (45). Conforming to the me-chanical work-energy theorem Eq. (20), we replace thecontrol work W with the energy gain ∆ H minus the dis-sipated energy in Eq. (45) to cast the work exponentialinto e − βW = exp (cid:20) − β (cid:26) H ( q, p ; τ ) − H ( q , p ) − Z τ dt X p i m D i (cid:27)(cid:21) . (50)Then, it is straightforward to rearrange the integrand on the RHS of Eq. (45) to bring about h e − βW i = e βF Z dqdp n J − e β R τ dt P D i p i /m o e − βH ( q,p ; τ ) (51)where J − is the inverse Jacobian. Now, one can observethat if the expression in curly brackets in the integrandon the RHS of Eq. (51) reduces to unity, i.e. J − e β R τ dt P D i p i /m = 1 , (52) then Eq. (51) turns into h e − βW i = e βF e − βF ( τ ) where F ( τ ) is the Helmholtz FE at t = τ , F ( τ ) = − β − ln (cid:26)Z dqdpe − βH ( q,p ; τ ) (cid:27) . The value of the system-centric Hamiltonian varies withtime as a function of the dynamical variables, but thefunctional form of H ( q, p ) is fixed at each instant. Ac-cordingly, the resulting FE is the same at the initial andfinal equilibrium states with the identical temperature β , F ( τ ) = F (0). Consequently, the average of the exponen-tial work becomes h e − βW i = 1 (53)which is the Bochkov-Kuzovlev work relation . We now turn our attention to formulating the NEQwork theorem in the nonautonomous picture. To thisend, we only need to use the alternative representation ofthe mechanical work-energy theorem, Eq. (23), in defin-ing the work exponential Eq. (50). As previously men-tioned, the initial ensemble is identical to the system-centric case. Then, the statistical average Eq. (51) isreplaced by h e − β W i = e βF Z dqdp n J − e β R τ dt P D i p i /m o e − β H ( q,p ; λ ( τ )) . (54)It is evident that Eq. (54) becomes the proclaimedJarzynski equality , h e − β W i = e − β ∆ F (55)where ∆ F = F ( τ ) − F if the same condition given inEq. (52) meets. In this case, however, F ( τ ) = F be-cause the instantaneous FE ( F ) depends not only on thereservoir temperature but also on the control parameter λ , F ( β, λ ) = − β − ln (cid:26)Z dqdpe − β H ( q,p ; λ ) (cid:27) . The derived restraint Eq. (52) is physically satisfiedwhen Z τ dt Λ( q, p ; β, t ) + β Z τ dt Θ( q, p ; β, t ) ≡ D Eq. (9) in their definitions. The condition Eq. (56) hasbeen derived without invoking any specific models. It as-serts that the JE is loosened unless the non-vanishing Λand the scaled, dissipative power β Θ cancel exactly eachother out over the NEQ work performance in a thermallyopen system.The compressibility factor Λ, defined in Eq. (41), con-sists of three parts in the system-centric picture,Λ =
X (cid:26) ∂∂p i ( D i + G exi ) + ∂∂q i V exi (cid:27) ≡ Λ D + Λ G + Λ V , (57)where Λ D is the contribution from the dissipation D , Λ G from the control force G ex , and Λ V from the macroscopic velocity-coupling V ex . In the nonautonomous Hamilto-nian description, the compressibility factor takes only asingle term, Λ = Λ D . (58)Here, we discuss the physical implication of the enun-ciated condition given in Eq. (56). The finite system thatwe consider is assumed to remain in thermal contact witha single surrounding reservoir. Therefore, before turn-ing on or after turning off the work parameter λ shouldthe system come to equilibrium with the reservoir dueto boundary interaction of the system with the thermalreservoir. In this situation the generalized Liouville equa-tion Eq. (40) must admit the canonical ensemble density ρ eq as its solution, which yields X (cid:18) ˙ q i ∂ρ eq ∂q i + ˙ p i ∂ρ eq ∂p i (cid:19) = − Λ ρ eq (59)which constitutes the detailed balance in thermostatteddynamics. By directly substituting Eq. (44) for ρ eq and,then, by making use of the extended Hamilton equationsof motion excluding the external forces, one can showthat the above Eq. (59) is reduced toΛ = − β Θwhich evidently satisfies Eq. (56). Both Λ and Θ inEq. (56) are associated with the momentum-dependentforce D , Eq. (9). Therefore, Eq. (56) implies essentially aconsistency condition that the effective force D , originat-ing from the coarse specification of the boundary interac-tion by a macroscopic parameter β , must meet in order toassure the detailed balance condition. The temperature-dependence of D provides the NEQ dynamics of a ther-mally open system with a thermostatting mechanism.The JE, Eq. (55) is the desired, statistical work-energytheorem applying to general NEQ processes beyond thequasi-static limit. Appealing to Jensen’s inequality , h e − β W i ≥ e − β hWi , on average , compared to the FE in-crement, i.e. ∆ F ≤ hWi (60)where the equality holds for an isothermal, quasi-staticprocess, Eq. (39). In a small system with a few degrees offreedom it is not surprising to anticipate a statistical de-viation from Eq. (60) that such an individual mechanicalprocess as ∆
F ≥ W may occur occasionally. However,it would not occur in a finite system on any realistictime-scale because the observable is the averaged work, W th = hWi , not the individual realization of W , whichis subsumed in the second law of thermodynamics in itsgeneral form ∆ F ≤ W th − Z SdT. (61)It is suggestive to observe that Eq. (61) tends to Eq. (60)in an isothermal limit, implying that the validity of theJE may be restricted approximately to isothermal pro-cesses.Finally, we want to mention a suggestive report by oth-ers where it is shown that the excess of thermodynamicwork, W th − ∆ F over the work period τ is bounded frombelow by an information-theoretic measure . The mea-sure is quantified as the relative entropy between the ac-tual NEQ density ρ ( τ ) and the quasi-static equilibriumdensity ρ eq ( τ ). V. SYMMETRIC WORK FLUCTUATION: THEMESOSCOPIC REVERSIBILITY
Here, we explore the physical ground of the symmetricnature of the work FTs and its relation to the essen-tially one-way theorem of the JE. We shall first considerthe problem in the system-centric picture and continuallydescribe the companion result from the nonautonomouspicture.To this end, it is essential to deduce under what con-ditions the generalized Liouville dynamics, Eq. (40) gov-erned by the extended Hamilton equations of motion,Eqs. (18) and (19), may be invariant under time (mo-tion) reversal. The time-reversal operation, denoted by T : t → − t , is formalized by the following discrete trans-formation: q → q and p → − p. Conforming to them, we postulate that to every density ρ ( q, p ; t ) at instant t there corresponds a time-reverseddensity defined by T ρ ( q, p ; t ) T − = ρ ( q, − p ; − t ) . Then, by inspecting Eq. (40), one can verify that ρ ( q, − p ; − t ) is also solution to the generalized Liouville equation if the compressibility factor changes its sign un-der time-reversal, i.e. T Λ( q, p ; t ) T − = − Λ( q, p ; t ) . (62)The preceding Eq. (62) is the required condition whichmakes the generalized Liouville dynamics invariant un-der time-reversal. When it is satisfied, the time-reverseddensity propagates backward in time under the influenceof the propagator U as ρ ( q, − p ; − t ) = U ( t ) ρ ( q , − p ; 0) . (63)The concrete representation, Eq. (57) of Λ under the ex-tended Hamiltonian dynamics leads to the physical con-ditions to be imposed on the external fields, G exi ( − t ) = G exi ( t ) , (64) V exi ( − t ) = −V exi ( t ) , (65)and on the nonconservative thermostatting force, D ( − p ) = D ( p ) (66)which states that the momentum-dependent force mustbe even under inversion, p → − p ,The invariance condition Eq. (62) is special becausethe dynamics in generic NEQ work-measurements wouldbe asymmetric, i.e. irreversible, in general. We shallcall the exploited symmetry of the generalized Liouvilledynamics a dynamically mesoscopic reversibility in thesense that the external, non-potential couplings in theextended Hamilton equations of motion, Eqs. (18) and(19), are not microscopic but rather statistical in origin.In particular, we have recapitulated in Sec. II that in-sufficient knowledge about the interaction of the systemwith the surroundings is represented as the momentum-dependent forces D on the system. In below, we establishthat the NEQ work fluctuation theorems in fact reflectsthe mesoscopic reversibility of specially prepared dynam-ics.To proceed, let us denote the two equilibrium statesof the system by A and B , respectively, at both endsconnected by a pre-determined work protocol over theduration τ . Corresponding to the system prepared incanonical equilibrium ρ eq ( q, p ; A ) at t = 0, the number ofinitial micro-states in the range ( q , q + dq ) and ( p , p + dp ) would be proportional to ρ eq ( q, p ; A ) dq dp . Amongthese phase-space points, the number of initial micro-states belonging to a specific realization of forward work W F = W is restricted to ρ eq ( q, p ; A ) dq dp δ ( W F − W )where δ ( W F − W ) is the Dirac delta function. Thus, thework distribution g F in the forward process of performingwork by an amount of W may be written as g F ( W F = W ) = Z dq dp ρ eq ( q, p ; A ) δ ( W F − W ) (67)with normalization, Z g F ( W F ) dW F = 1 . h e − βW i ≡ Z dW F e − βW F g ( W F )= Z dq dp e − βW ρ eq ( q, p ; A )which is Eq. (49) in the Heisenberg picture.Similarly, one can construct the work distribution inthe reversed process, g R . Such a reversed procedure is notpermitted in general unless the forward work has beenperformed quasi-statically. For a fast work process, we assume that the invariance condition Eq. (62) is enforcedso that motion is still symmetric under time-reversal. Inthis case the control work is pretended to be carried outprecisely along backward trajectories by the amount of W R = T W F T − = − W. The reverse work-protocol sets up a new starting den-sity as the time-reversed, ending equilibrium state, ρ eq ( q, p ; B ), from the forward process and allows the sys-tem to evolve with abiding by Eq. (63). Then, by theequivalent arguments used in specifying Eq. (67) the re-verse work-distribution may be formulated as g R ( W R = − W ) = Z T [ dqdp ] T − T [ ρ eq ( q, p ; B )] T − δ ( W R − ( − W )) . (68)The generalized Liouville measure dqdp is invariant undertime-reversal, T [ dqdp ] T − = dqdp . The time-reversedequilibrium density at B is given by T [ ρ eq ( q, p ; B )] T − = e βF B e − β T [ H ( q,p ; B )] T − where the time-reversed Hamiltonian is limited by themechanical work-energy theorem Eq. (20) but in a tem-porally backward manner, T [ H ( A ) − H ( B )] T − = T W R T − + T "Z AB dt X p i m D i T − . The system-centric Hamiltonians are invariant undertime-reversal and the control work considered is re-versible with change of its sign as W R = − W F . In ad-dition, the dissipative work changes its sign under time-reversal by the imposed symmetry, Eq. (66) on the ther- mostatting force, which has been enforced by the invari-ance condition Eq. (62). Consequently, it turns out thatthe work-energy theorem applied in the backward senseis transformed, under time reversal, into the work-energytheorem in the forward direction, H ( q, p ; B ) = H ( q , p ; A ) + W F + Z BA dt X p i m D i . We just verified that the mechanical work-energy theo-rem, Eq. (20), also acts symmetrically under time inver-sion in a mesoscopically reversible system. By substi- tuting the last expression into Eq. (68), one can obtainthat g R ( W R ) = e βF B e − βW Z dqdpe − β { H ( q ,p ; A )+ R τ dt P D i p i /m } δ ( W F − W )= e βF B e − βW Z dq dp n J e − β R τ dt P D i p i /m o e − βH ( q ,p ; A ) δ ( W F − W ) (69)2where in the second step we have used the Jacobian re-lation, Eq. (46). Here, one can notice that the enclosedexpression in curly brackets in Eq. (69) is exactly whatappears in the enunciated consistency criterion for theNEQ work theorems, Eq. (52). For such a work mea-surement satisfying the criterion, Eq. (69) reduces to g R ( − W ) = e − βW g F ( W ) (70)which is the Bochkov-Kuzovlev version of the workFT .Next, we summarize the outcome from the nonau-tonomous picture, that we would have obtained insteadof Eq. (70) if we had formulated with the alternative me-chanical work-energy theorem, Eq. (23). After imposingthe consistency criterion, Eq. (52), on the companion ex-pression to Eq. (69), it can be shown straightforwardlythat the result takes the form, g R ( −W ) = e β (∆ F−W ) g F ( W ) (71)with ∆ F = F B −F A , which is the desired CWFT . Therequired time-reversal constraints in deriving Eq. (71) arethe invariance of the nonautonomous Hamiltonian, T H ( q, p ; λ ) T − = H ( q, p ; λ ) (72)and the already prescribed symmetry condition, Eq. (66).The two conditions guarantee that the NEQ work is re-versible, W R = −W F . The JE, Eq. (55), follows from theCWFT when both sides of Eq. (71) are integrated overthe full rage of work values W , assuming g F ( W ) and g R ( −W ) are properly normalized. However, the formeris more general in applicability than the latter because ithas been derived in Sec. IV B without requiring the time-reversal invariance of the underlying dynamics. Note thatin typical single small-system experiments of testing theJE, the system must be brought back to initial state af-ter completing the unidirectional work . It means thatthe dynamical reversibility of the system is still implicitlyimposed on the actual realization of the JE. The genuineirreversibility seems awaiting further to be explored. Weobserve researchers have put forth an effort lately to ex-tend the work FTs to account for irreversible transitionsbetween partial equilibrium states . VI. EXAMPLES
We consider here a few examples to demonstrate howthe consistency condition [Eq. (56)] for the NEQ work-energy theorems may be employed in actual NEQ dy-namics.
A. Isolated systems under time-dependent externalforces
As a simple situation, let us consider that only a time-dependent manipulation is put into action by an externalagent on an, otherwise, isolated system. In the system-centric picture, the system is describedby the extended Hamilton equations of motion,˙ q i = 1 m p i , ˙ p i = − ∂V∂q i + G exi , in the time-independent, energy landscape, Eq. (14).One can immediately see that the validity condition,Eq. (52) is satisfied because the control force does notcontribute to the phase-space compressibility,Λ G = X ∂∂p i G exi ( t ) = 0 . (73)The conclusion is unchanged even if there is an addi-tional dependence of the external force on the general-ized coordinates, G exi ( t ) = G exi ( q ; t ). Consequently, theBochkov-Kuzovlev equality, Eq. (53) holds trivially. Onthe other hand, in order to realize the symmetric workFT, Eq. (70) the external force must further comply withthe time-reversal symmetry, Eq. (64). When the meso-scopic reversibility is satisfied, the reverse work is thenegative of the forward work as W F = X Z τ dt ˙ q i ( t ) G exi ( t )= − X Z τ dt ˙ q i ( τ − t ) G exi ( τ − t )= − W R . (74)This was stated formally using the time-reversal operatorpreviously in Sec, V.In the nonautonomous Hamiltonian picture, Eq. (15),the external manipulation of the system must be builtinto the Hamiltonian as a time-dependent parame-ter. For instance, single-molecule pulling experimentsare often described by the phenomenological harmonicterm , H ′ = 12 k { G ( { q i } ) − λ ( t ) } (75)where k is the spring constant and G ( { q i } ) is the molec-ular extension which is a function of the generalized co-ordinates of all the atomic constituents. In this case, theparameter λ prescribes anchoring position of the pullingapparatus with the molecular system. It is apparent thatthe consistency criterion, Eq. (56) is satisfied becausethere is neither a dissipation nor a phase-volume contrac-tion. Accordingly, we predict that the JE, Eq. (55) mustwork straightly in such an experimental set-up, whereasthe CWFT, Eq.(71) requires the additional symmetrycondition, Eq. (72) to guarantee its applicability. B. Closed systems with thermostatted damping
In typical experiments, the system under investigationremains immersed in a heat reservoir that energy dissipa-tion is allowed with the surroundings. Accordingly, apart3from the manipulating force G ex , a dissipative mechanism D must be taken into account in the equations of motion,Eqs. (18) and (19),˙ q i = 1 m p i , ˙ p i = − ∂V∂q i + D i + G exi , where, for simplicity, we have set V ex = 0. Here, we con-sider that the dissipation is described by a phenomeno-logical linear force as D i = − γp i where the coefficient γ is assumed to be dynamically con-stant but to be dependent on the reservoir temperature, γ = γ ( β ). Since the external force G ex preserves thephase-volume of the system in carrying out the controlwork, Eq. (73), the consistency criterion, Eq. (56), is re-duced to requiring Z τ dt (Λ D + β Θ) ≡ . (76)One can calculate the Rayleigh dissipation function Θ,Eq. (32) to become Θ = 2 γK where K = P p / m is the kinetic energy of the system.Also, the compressibility factor Λ D from the dampingforce is given byΛ D = X ∂∂p i D i = − γf where f denotes the degrees of freedom in the system.Then, we find that Eq. (76) brings about the NEQ energyequipartition relation,1 τ Z τ dtK ≡ f β − (77)which suggests that the time-averaged kinetic-energy, h K i τ , over the work period τ be equal to the thermalenergy stored in the degrees of freedom f . The conditionEq. (77) becomes an identity in the equilibrium limit, τ → ∞ , but is a strong requirement for the unidirec-tional work theorem over the finite period.The applicability of the two-way work FT is furtherlimited by the symmetry requirement, Eqs. (64) and (66).Equation (66) is not satisfied in the present model be-cause the damping coefficient γ is assumed to be inde-pendent of momentum. Consequently, our theory pre-dicts that the thermodynamic work cannot be performedreversibly even if the time-dependent force G ex is sym-metric in time-inversion.The same consistency criterion, Eq. (76) must be metin the nonautonomous description. Consequently, theone-way JE would be effective in experiments where the condition of NEQ equipartition, Eq. (77) is enforced tobe satisfied. For a bidirectional set-up the symmetricCWFT is not promising because Eq. (66) is not satisfied,which is one of the symmetry conditions to be met. Theother condition from Eq. (72) does not affect the conclu-sion, of which explicit representation is not given in thepresent example. C. Computational algorithms of NEQ dynamics
Here, we examine NEQ molecular dynamics (MD) ofa planar Couette flow as a next, concrete example. Thefluid is assumed to be confined in spatial y direction, sub-ject to an external shear rate η along x , which is switchedon, say, at t = 0. In the steady-state the flow velocity ofthe system is specified by the linear profile, u x ( r ) = ηy, while other spatial components are zero, where r is thefield point. In the actual simulations a difficulty arisesthat the shearing work generates heat in the system,which is technically compensated by imposing a fictitiousthermostat . We consider here the Gaussian, iso-kineticthermostat condition that the kinetic energy in the co-moving frame with the flow is held as constant, X α,j p αj m ≡ f β − (78)where p αj /m is the peculiar velocity of α -th particlealong spatial j direction, j = x, y, z , given as p αj /m = ˙ q αj − u j ( r = q α ) . Then, the equations of motion must be modified toincorporate both the shearing and thermostat forces. Tothis end, we adopt the frequently used, SLLOD equationsof motion ,˙ q αj = 1 m p αj + X k q αk ∂u j ∂q αk (79)˙ p αj = − ∂V∂q αj − X k p αk ∂u j ∂p αk − γp αj . (80)where summation index k runs over the spatial degreeof freedom, x, y, z . The second terms on the RHSs ofEqs. (79) and (80) describe the coupling of the system tothe shear field. The third term on the RHS of Eq. (80)takes care of the fictitious, damping force associated withthe thermostat constraint, Eq. (78).The frictional coefficient γ can be specified by differ-entiating the iso-kinetic constraint, Eq. (78) with respectto time and by inserting ˙ p αj given in Eq. (80) into theoutcome. The result is given by γ ( p ; β ) = βmf X α,j (cid:26) p αj (cid:18) − ∂V∂q αj (cid:19) − ηp αx p αy (cid:27) (81)4which is evidently momentum-dependent and alsotemperature-dependent. By matching the SLLOD algo-rithm with the extended Hamilton equations of motion,Eqs. (18) and (19), one can identify the non-potentialterms as V exαj = X k q αk ∂u j ∂q αk → ηy α δ jx = u j ( q α ) , G exαj = − X k p αk ∂u j ∂q αk → − ηp αy δ jx , D αj = − γp αj , where y α is the y -coordinate of α -th particle, y α = q αy and δ jk is the Kronecker delta. Then, from Eq. (21)the work performed during the period τ by the external,shear field η on the system is represented as W = − η Z τ dt X α (cid:26) m p αx p αy − ∂V∂q αx y α (cid:27) (82)where the integrand is essentially the instantaneouspressure-tensor including the potential contribution. Thepreceding work W is the control work, associated withthe shearing, that enters into the work-energy theorems,Eqs. (20) and (53).To test how the Bochkov-Kuzovlev relation Eq. (53)may be fulfilled in the current system-centric picture wemust inspect the validity criterion, Eq. (56). There ap-pear three non-Hamiltonian sources which contribute tothe compressibility factor Λ in Eq. (57). It is a simplematter to calculate that the shearing fields do not affectphase volume,Λ G = X α,j ∂∂p αj ( − ηp αy δ jx ) = 0 , Λ V = X α,j ∂∂q αj ( ηy α δ jx ) = 0 . The preceding outcome manifests an interesting casethat a momentum-dependent force does not give rise tophase-space contraction. The remaining contribution inEq. (56) is from the thermostat force D to evaluate Z τ dt (Λ D + β Θ) = 0 . The above expression resembles Eq. (76), however, thedamping coefficient γ in D is now momentum-dependentvia Eq. (81) and the external velocity field V ex must bealso taken into account in evaluating the Rayleigh dissi-pation function Θ, defined in Eq. (32). The subsequentanalysis unfolds that the compressibility factor caused by D is given by Λ D = − f γ − X αj p αj ∂γ∂p αj (83)and that Θ is calculated to beΘ = β − γf + γη X α p αx y α . (84)When the above results for Λ D and Θ are substitutedinto the preceding condition, it follows that Z τ dt (Λ D + β Θ) = Z τ dt − X αj p αj ∂γ∂p αj + βγη X α p αx y α . (85)The outcome predicts that in order for the one-way NEQwork theorem to be operative the remaining contributionfrom the momentum-dependence of γ and the dissipationcaused by shearing η must sum up to vanish identically.In addition, even if the one-way theorem holds approxi-mately, the symmetric work FT is not likely so becausethe thermostatted shear flow does not possess the meso-scopic reversibility, Eq. (66). The effective damping coef-ficient, Eq. (81) does not possess a definite parity undermomentum-inversion.The situation is similar in other thermostat conditions.For instance, when the mechanical energy H is fixed (i.e.iso-energetic thermostat) instead of the kinetic energy K of the system in NEQ MD simulation, one can show thatthe thermostat coefficient still depends on momentum asfollowings, γ ( p ) = ˙ W ( q, p )2 K ( p ) (86)where ˙ W is the time-rate of the control work, specifiedin the current case as the integrand in Eq. (82).We have discussed the problem only in the system-centric picture because the coupling of the shear fieldto the SLLOD equations is not derivable from a nonau-tonomous Hamiltonian. Thermostatted MD may provide5a highly efficient and useful test bed of the NEQ equali-ties in deterministic many-body dynamics. VII. SUMMARY AND CONCLUSION
We have formulated the statistical work-energy theo-rems for a finite system immersed in a single heat reser-voir, under external manipulation, without appealing toa system–specific model. The major drawings from ourstudy are summarized here.We have proposed the extended dynamics to prescribedeterministic dynamics of a thermally open system underthe general NEQ conditions. The prescribed mesoscopicdynamics embraces the coordinate-dependent (conserva-tive), momentum-dependent (dissipative) forces, and thecoupling to external time-dependent (control) fields. Thedissipative force represents the statistical correlation ofthe system with the surrounding reservoir and thus playsthe role of a thermostat in our formulation.We have endeavored to formulate the extended dynam-ics both in the system-centric picture and in the nonau-tonomous Hamiltonian picture. In the former the me-chanical energy of the system is an instant value of thebare Hamiltonian, excluding the time-dependent pertur-bation, of which dynamical variables obey the extendedHamilton equations of motion [Eqs. (18) and (19)]. Theenergy is not conserved but complies with the mechan-ical work-energy theorem [Eq. (20)]. The work-energybalance was also formulated alternatively in the avenueof the nonautonomous Hamiltonian [Eq. (23)]. The re-sulting parametric work is represented as the negativeintegral of the coordinates over the variation of the exter-nal forces [Eq. (27)], differently from the system-centricdefinition of work as the integral of the forces over thedisplacements of coordinates [Eq. (26)].The mechanical work-energy theorems are exact butthey merely serve as a theoretical guidance due to theenormous degrees of freedom in the finite system (dueto fluctuation in small systems). In order to account forthe NEQ work measurement of the system controlled bythe external perturbation a statistical description musttake over. We have performed the statistical average ofthe mechanical work-energy theorems, adopting Jarzyn-ski’s mathematical recipe of the exponential work, overthe NEQ distribution of the identically prepared ensem-ble of the finite system. The NEQ phase-space densityobeys the generalized Liouville dynamics, of which gener-ator of convective time-development turns out to be thenon-vanishing compressibility factor. Consequently, wehave derived the Bochkov-Kuzovlev equality [Eq. (53)]in the system-centric description and the JE [Eq. (55)] inthe nonautonomous description. Our formulation is high-lighted by the physical criterion [Eq. (56)], being consis-tent with the detailed balance condition [Eq. (59)] for the generalized Liouville dynamics [Eq. 40)], of which satis-faction assures the NEQ equalities as rigorous theoremsin a thermally open system.The momentum-dependent, damping force renders theassociated NEQ thermodynamic process typically irre-versible. Nevertheless, the extended equations of motionmay be still symmetric under time reversal, conditionedon that the damping force is an even function of momen-tum and also that the other external fields are symmet-ric under time inversion. Such a constrictive symmetryis archived in the generalized Liouville dynamics if thecompressibility factor changes its sign under time reversal[Eq. (62)]. Then, the time-reversed ensemble density canpropagate backward in time under the same propagator.In such a system of possessing the mesoscopic reversibil-ity, we have shown that a NEQ work measurement maybe performed in the bidirectional manner with conform-ing to either the Bochkov-Kuzovlev WFT [Eq. (70)] orthe CWFT [Eq. (71)], depending on choice of the pic-ture. The symmetric CWFT yields the JE as a corollary,however, unlike the usual interpretation we note that thelatter is more general in the sense that it can be appliedto a time-asymmetric transformation.In conclusion, we have explored the NEQ work theo-rems by directly taking the NEQ ensemble average of themechanical energy balances. Consequently, a consistencycondition has been derived which tightens the Bochkov-Kuzovlev-Jarzynski-Crooks NEQ equalities to be legiti-mate in thermally open, finite systems. The conditionaffirms that the unidirectional work theorems for irre-versible transformation are contingent on that the con-tracted phase-volume from all involved nonconservativeforces must be precisely offset by the dissipated powerscaled by the equilibrium temperature over the work pe-riod, constituting the detailed balance in thermostat-ted, deterministic dynamics. The criterion is also im-plemented in the bidirectional work FTs, however, withthe additional symmetry requirement that the dynamicsof the system be invariant under time reversal even inthe presence of the dissipation and nonpotential manip-ulating forces. We hope that our unveiling provides re-searchers with a useful, theoretical appraisal of the NEQwork theorems in real or computer experiments.
Acknowledgments
The author is grateful to Gary P. Morriss for provid-ing a useful discussion, in particular about thermostattedNEQ dynamics.
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