Comment on: Inconsistency of the nonstandard definition of work
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] O c t Comment on: Inconsistency of the nonstandarddefinition of work
Luca Peliti
Dipartimento di Scienze Fisiche, Universit`a “Federico II”Complesso Monte S. Angelo, 80126 Napoli (Italy)
Abstract.
The objections raised by Vilar and Rubi [cond-mat arXiv:0707.3802v1]against the definition of the thermodynamical work appearing in Jarzynski’sequality are shown to be misleading and inconsistent.
Quid dices de primariis huius Gimnasii philosophis,qui aspidis pertinacia repleti, licet me ultro deditaopera millies offerente, nec Planetas, nec % , necperspicillum videre voluerunt? Verum ut ille aures,sic isti oculos, contra veritatis lucem obturarunt. Galileo (in a letter to Kepler, 1610)
1. Introduction
In a recent post, Vilar and Rubi (VR) [1] ascribe to Imparato and Peliti [2] the claimthat the standard definition of workWork = Force × Displacement , (1)should be unconditionally replaced by a ‘nonstandard’ definitiond W IP = − x d f, (2)in which the force and the displacement have their role interchanged, when consideringthe work performed by a force on a system. They argue that “this ‘nonstandard’definition of work is thermodynamically inconsistent at both the microscopic andmacroscopic scales and leads to non-physical results, including free energy changesthat depend on arbitrary parameters”. The dispute arose from the claim set forthby Vilar and Rubi in a previous post [3], in which it was argued that the connectionbetween the microscopic work W performed by a time-dependent force on a systemcannot be used to estimate free energy changes.In the present note I shall argue the following points, which are already clear toany honest reader of ref. [2]:(i) The expression (2), surprising as it is, is a straightforward consequence of thestandard definition of the thermodynamical work performed on a system, for thespecial case considered in [2], namely, when a uniform but time-varying force isapplied to a particle subject to a given potential; omment on: Inconsistency of the nonstandard definition of work bona fide energy differences which have observable consequences.I shall also argue that VR’s confusions stem from the fact that the thermodynamicalwork on a system represents the work done by the system one considers on theexternal bodies which act on it, rather than the work done on the system itself by theexternal bodies: a point stressed, e.g. , at the beginning of Gibbs’s founding book [4]on Statistical Mechanics, and that VR fail to appreciate.I shall first discuss these points in the context of equilibrium thermodynamics.Further points are relevant when considering manipulated systems, in particular smallsystems for which fluctuations are important.
2. Reversible work on the harmonic oscillator
Let us consider a simple thermodynamical system, i.e. , a one-dimensional oscillatorcharacterized by its mass m and spring constant k , kept at a fixed temperature T .The system is described by the hamiltonian H ( p, x ) = p m + 12 kx . (3)In the following we shall focus only on the displacement degree of freedom, namely x .Its equilibrium distribution is given by p eq ( x ) = e − kx / k B T Z , (4)where Z is given by Z = Z d x e − kx / k B T = p πk B T /k. (5)We shall now apply a uniform, but time-varying, force f ( t ) to the system. Wewish to evaluate the thermodynamical work performed on it, as the applied forcechanges from f = 0 to f , so slowly, that the system can be considered to remainat thermodynamical equilibrium at all times . This is called the reversible work inthermodynamics.Following the method described by J. W. Gibbs [4] and R. C. Tolman [5], oneproceeds as follows:1. One writes down the hamiltonian of the system in the presence of the appliedforce: H ( x, f ) = 12 kx − f ( x − γ ) . (6)Here γ is defined as the point in which the potential of the applied force vanishes.This point might depend on f , but we shall momentarily assume that it is fixed.It is determined by the actual device used to apply the constant force on thesystem, as discussed in the following. omment on: Inconsistency of the nonstandard definition of work
32. One applies either Gibbs’s equation (117) [4, p.45], or Tolman’s equa-tion (124.1) [5, p.542], to obtain the thermodynamical work d W associated witha small variation d f of the applied force:d W = (cid:28) ∂H∂f (cid:29) d f = − h ( x − γ ) i d f = − ( h x i − γ ) d f. (7)In this equation, h A i is the canonical average of the function A ( x ): h A i = 1 Z Z d x A ( x ) e − H ( x,f ) /k B T . (8)In our case, one obtains h x i = fk , (9)from which d W can be calculated via equation (7).3. One integrates the result with a variable force f ′ from the initial value f = 0 tothe final value f , obtaining∆ F = Z f * ∂H∂f (cid:12)(cid:12)(cid:12)(cid:12) f ′ + d f ′ = − Z f (cid:18) f ′ k − γ (cid:19) d f ′ = − f k + γf. (10)In this expression, ∆ F is the change in the Helmholtz free energy, F = E − T S .Since it is easy to see that in the present system the entropy S does not changeduring the manipulation, we can equate it with the change in the internal energy E . We have therefore∆ E = − f k + γf. (11)4. Since the average value of the applied potential is given by h U i = − f ( h x i − γ ) , by subtracting it from the above result, we obtain the change of the energy ofself-interaction of the spring∆ E int ( f ) = ∆ E − h U i = f k . (12)It would not be necessary to consider this elementary exercise in statisticalmechanics, were it not for the fact that in their recent post J. Vilar and M. Rubi [1](objecting to a similar derivation contained in [2]) have found that this resultis “inconsistent and unphysical both at the macroscopic and microscopic level.”Equation (7) is the one that VR incriminate. The two authors are chagrined bythe following facts:1. Let us first consider γ = 0. Then the free-energy change (10) is negative. Now,non-spontaneous processes should lead to positive free-energy changes. This is incontrast with previous results, including ones on macromolecules [6]. Moreoverthis result holds for any system described by the hamiltonian (6), including amacroscopic spring. This is in contrast with the results of elementary physics.2. Moreover, VR claim that the parameter γ does not have any physicalinterpretation, and that therefore in this result the free-energy change does notdepend on the actual physical system but rather on its mathematical description. omment on: Inconsistency of the nonstandard definition of work their ‘nonstandard’ definition of the thermodynamical workis preferable to Gibbs’s and Tolman’s one. They shun this burden by failing to noticeit. They should however agree that, if the potential energy of the interaction ofthe system with the bodies that provide the constant force is taken into account,both objections raised above disappear. VR proceed instead as if the expression (11)contained only the energy of interaction of the system with itself.I now show how the result (11) corresponds to the variation in the total energy(as defined in the above text by Gibbs) when the external force is applied by twophysically reasonable devices. I shall then discuss why the apparent paradox of point1. is such only in the minds of the authors of ref. [1] and their followers. But I nowwish to stress the point which probably lies at the heart of VR’s confusion, by quotingat length from Gibbs’s treatise.Returning to the case of the canonical distribution, we shall find otheranalogies with thermodynamics systems, if we suppose, as in the precedingchapters, ‡ that the potential energy ( ǫ q ) depends not only upon thecoordinates q . . . q n which determine the configuration of the system, butalso upon certain c¨oordinates a , a , etc. of bodies which we call external ,meaning by this simply that they are not to be regarded as forming any partof the system, although their positions affect the forces which act on thesystem. The forces exerted by the system on these bodies § will be representedby − dǫ q /da , − dǫ q /da , etc., while − dǫ q /dq . . . − dǫ q /dq n represent all theforces acting upon the bodies of the system, including those which dependupon the position of the external bodies, as well as those which depend onlyupon the configuration of the system itself. It will be understood that ǫ p depends only upon q , . . . q n , p , . . . p n , in other words, that the kinetic energyof the bodies which we call external forms no part of the kinetic energy ofthe system. It follows that we may write dǫda = dǫ q da = − A , (104)although a similar equation would not hold for differentiation relative to theinternal c¨oordinates.Thus Gibbs’s expression of the elementary reversible work dW = − X i A i da i , (13)(where, in Gibbs’s notation, the bar denotes the average over a canonical distribution)represents the average work done on the external bodies by the system (with changedsign), and therefore, in particular, does not vanish even if the coordinates of the systemdo not change over the time interval considered. ‡ See especially Chapter I, p. 4 (Note by JWG). § My italics (LP). omment on: Inconsistency of the nonstandard definition of work Electrostatic device
To illustrate this point, let us set up a device for applying a uniform but time-dependent force on our oscillator. We can use, for instance, the following electrostaticdevice. Let us assume that the mass of the oscillator carries a small charge q . Wetake two point-like bodies at infinity, one with the charge + Q and the other with thecharge − Q . We then let these two charged bodies come closer and closer to the origin(the equilibrium point of the oscillator), by letting the charge + Q be situated at thepoint − X + γ , and the charge − Q at the point X + γ . Thus the electric field actingon the oscillator at point x is given by E = Q πǫ (cid:20) x − X − γ ) + 1( x + X − γ ) (cid:21) = Q πǫ ( x − γ ) + X [( x − γ ) + X ] − x − γ ) X ≃ Q πǫ (cid:26) X + 3( x − γ ) X + 5( x − γ ) X + · · · (cid:27) (14)If X is large enough, then all terms beyond the first one are negligible, for the expectedexcursions of the oscillator from the origin. Then the force applied by the charge Q isgiven by f = qQ πǫ X . (15)Let us choose Q such that, even for the largest force f which we wish to apply, X is so large that the terms beyond the first in equation (14) are negligible. Thus bymoving the charges ± Q from infinity to ± X + γ , always symmetrically around thepoint γ , we can apply a uniform but time-varying force to our oscillator. It is nowclear that γ , far from being a fictitious parameter, corresponds to the location of thecenter of the device by which a uniform force is applied to the system we are studying.In order to change γ , external work must be supplied to the apparatus.Let us now evaluate the internal energy of the system as a function of X . Wehave E = (cid:28) kx + U ( x, X ) (cid:29) = 1 Z Z d x e − H ( x,X ) /k B T (cid:20) kx + qQ πǫ (cid:18) x + X − γ − X − x + γ (cid:19)(cid:21) . (16)The first term yields (cid:28) kx (cid:29) = 12 k h(cid:10) ( x − h x i ) (cid:11) + h x i i = 12 k B T + 12 f k . (17)The first term is given by the equipartition theorem, and the second by equation (9).One can expand the second term in powers of 1 /X , obtaining h U ( x, X ) i = − qQ πǫ (cid:20) X h ( x − γ ) i + 1 X (cid:10) ( x − γ ) (cid:11) + · · · (cid:21) . (18)Thus, if (cid:10) ( x − γ ) (cid:11) /X ≪
1, we have h U ( x, X ) i = − qQ πǫ h ( x − γ ) i X = − f k + γf, (19) omment on: Inconsistency of the nonstandard definition of work E = 12 k B T − f k + γf, (20)in agreement with equation (11). Gravity-field device
A simpler conceptual experiment can be set up imagining that the oscillator mass isconstrained to move along a line, which can be rotated in the vertical plane. Let m be the oscillator mass, g the acceleration of gravity, and let the hinge be placed at x = γ . If the line is now rotated clockwise by an angle θ , the oscillator mass will actedupon by a uniform force, directed towards increasing values of x , and of intensity mg sin θ . On the other hand, if the mass is at location x , its height with respect tothe horizontal line passing through the hinge is given by z = − ( x − γ ) sin θ . It is thena simple matter to evaluate the average of U ( x, θ ): h U ( x, θ ) i = mg h z i = − mg sin θ h x − γ i = − f k + γf. (21)Adding to it the average elastic energy k (cid:10) x (cid:11) we recover equation (11) again. Butit is amusing to verify that this result corresponds indeed to the work done by thesystem on the external device. Let us consider the line to be tilted by θ , and theposition of the oscillator to be x . Then the oscillator applies to the rectilinear guidea torque τ = mg cos θ ( x − γ ) . (22)As the angle changes by d θ , this torque executes on the guide a work τ d θ = mg ( x − γ ) d sin θ. The reversible elementary work made by the system on its environment is given bythe average of this expression, namely − d W rev = h τ i d θ = mg ( h x i − γ ) d sin θ, (23)where, according to equation (9), h x i = f /k = mg sin θ/k . The change in the internalenergy due to the transformation is given by d W rev , integrated between 0 and thefinal value of θ . It is easy to check that it yields again the result (11).When the rectilinear guide is tilted, the oscillator spring is stretched and its elasticenergy is increased. On the other hand, the potential energy of the mass in the gravityfield can either increase or decrease, and the resulting total energy change can be ofeither sign. If γ = 0, one has, for instance∆ E = − m g sin θ k = − f k . (24)VR claim that this result is inconsistent, because a negative free-energy change(which coincides in our case with the energy change) would imply that the process isspontaneous, and that the spring is unstable, in contradiction with elementary physics.They fail to notice, however, that, if the rectilinear guide is free to rotate around theorigin , the system is indeed unstable: the guide would rotate till it reaches a verticalstand, with the oscillator mass hanging on the spring. Thus, far from being unphysical,the result yields the correct prediction for the physical setup one is considering. Ofcourse, in an actual experiment, one would constrain the guide at a given angle θ , andthe oscillator will find equilibrium around a point h x i given by equation (9). omment on: Inconsistency of the nonstandard definition of work
3. Reversible and fluctuating work
The textbook definition of reversible work is the work performed when thethermodynamic transformation is so slow that the system can be considered to stay atthermodynamic equilibrium at all times. In this case, the trajectory average coincideswith the ensemble average, at least if equilibrium statistical mechanics holds. Thenthe performed work does not fluctuate , and one trivially has W rev = h W rev i = − k B T log D e − W rev /k B T E . (25)VR claim that this equality does not hold, presumably because in their mind thereversible work (which is a canonical average) fluctuates. On the other hand, a cleardistinction was made in ref. [2] between reversible and fluctuating work, a distinctionthat VR chose to ignore. For the benefit of the reader, I recall the definition ofthe infinitesimal fluctuating work on a system whose microscopic state is denotedby x = ( x i ), and described by the hamiltonian H ( x, µ ), depending on an externalparameter µ : d W = ∂H ( x, µ ) ∂µ d µ. (26)We then have, for a given infinitesimal change d µ ,d W rev = h d W i , (27)where the average is taken with respect to the canonical distribution with the givenvalue of µ . Notice that the fluctuating work does not depend on the change in themicroscopic state x of the system, but on the change of the external parameter µ ,because it represents the work done by the system on the external bodies that acton it. One can thus understand why it does not vanish if µ is suddenly changed:if, e.g. , we suddenly push the charges ± Q closer to the origin in the electrostaticdevice, we have to provide some work, part of which changes the interaction energyof the oscillator with the charges. By the same token, if we change γ , e.g. , by rigidlydisplacing the field-creating charges Q , we have to provide work on the system, evenif the oscillator’s mass does not move.The distribution of the fluctuating work exhibits a number of interestingproperties, among which the remarkable equality (cid:10) e − W/k B T (cid:11) = e − ∆ F/k B T derivedby Jarzynski [7], and which Hummer and Szabo [8] showed how to exploit in orderto obtain information on the equilibrium free-energy landscape from nonequilibriumexperiments. VR object to this development, claiming that the above definition of thefluctuating work is unphysical and inconsistent. We have just seen how nicely it fitswith equilibrium statistical mechanics, as defined by Gibbs and explained by Tolman.However, other quantities also exhibit remarkable distributions. Let us consider asystem described by the hamiltonian H ( x, µ ) = H ( x ) − µF ( x ) . (28)Then the fluctuating work defined above is given byd W = − F ( x ) d µ, (29)and satisfies Jarzynski’s equality. On the other hand, we can also define the work d W by d W = µ X i ∂F∂x i d x i , (30) omment on: Inconsistency of the nonstandard definition of work D e − W /k B T E = 1 . (31)However, it is true that it is difficult to exploit this identity in order to recoverinformation on an equilibrium quantity like ∆ F . Indeed, what VR have brilliantlyshown in [3] is that W cannot be used to reconstruct free-energy landscape, but theyfail to inform the reader of [1] that their arguments concern the use of W , leavingthe impression that their objections concern W and the use of the Jarzynski equality.Now there is no problem in applying the Jarzynski equality to W . The resulting ∆ F contains a contribution from the interaction between the system and the environmentwhich, contrary to VR’s statements, is easily subtracted off (see, e.g., the “histogrammethod” discussed in [11, 12]). It is the responsibility of the researcher to choose themost appropriate tools for one’s task. One should choose a spoon to eat one’s soupand a spade to dig a hole: VR appear to prescribe everybody to pick up the spoonand then they lament that it is not possible to dig holes.
4. Conclusions
We have seen that VR’s objections against ref. [2] stem from a biased and misleadingreading of it, and from their failure to appreciate some basic concepts in statisticalmechanics. I am at a loss to understand why as serious and competent physicists as VRcould fall in such blunders, unless their confusions arise from an aprioristic hostility tothe recent exciting developments in the statistical mechanics of manipulated systems.In this case, they would remind of Galileo’s colleagues, cited in the letter I have posted in limine , who refused to look in the telescope because it did not fit within their worldview. If it is so, let them be happy to encourage their followers to raise objectionsbased on even faultier arguments than their own [13]. I shall have no more to say ontheir subject.
Acknowledgments
I thank A. Imparato for many discussions and insights on this matter.
References [1] J. M. G. Vilar and J. M. Rubi, cond-mat arXiv:0707.3802v1 (2007).[2] A. Imparato and L. Peliti, cond-mat arXiv:0706.1134v1 (2007).[3] J. M. G. Vilar and J. M. Rubi, cond-mat arXiv:0704.0761v2 (2007).[4] J. W. Gibbs,
Elementary Principles in Statistical Mechanics (New Haven: Yale U. P, 1902),reprinted (Woodbridge CT: Ox Bow Press, 1981).[5] R. C. Tolman,
The Principles of Statistical Mechanics (Oxford: Oxford U. P., 1938), reprinted(New York: Dover, 1979).[6] J. Liphardt et al.,
Science , 1832 (2002).[7] C. Jarzynski, Phys. Rev. Lett. , 931 (1997).[8] G. Hummer, A. Szabo, Proc. Natl. Acad. Sci. USA , 3658 (2001).[9] C. Jarzynski, C. R. Physique , 495 (2007).[10] G. N. Bochkov and Yu. E. Kuzovlev, Physica A , 443 (1981).[11] O. Braun, A. Hanke, and U. Seifert, Phys. Rev. Lett., e.g.e.g.