Non-equilibrium nanothermodynamics J. Carrete, L. M. Varela, and L. J. Gallego Grupo de Nanomateriais e Materia Branda, Departamento de F´ısica da Materia Condensada, 8 0 Facultade de F´ısica, Universidade de Santiago de Compostela, 0 2 E-15782 Santiago de Compostela, Spain n a J Abstract 3 Entropy production for a system outside the thermodynamic limit is formulated using Hill’s ] h nanothermodynamics,inwhichamacroscopicensembleofsuchsystemsisconsidered. Theexternal c e m influence of the environment on the average nanosystem is connected to irreversible work with an - t explicit formula based on the Jarzynski equality. The entropy production retains its usual form as a t s a sum of products of fluxes and forces and Onsager’s symmetry principle is proven to hold for the . t a average nanosystem, if it is assumed to be valid for the macroscopic ensemble, by two methods. m - The first one provides expressions that relate the coefficients of the two systems. The second gives d n a general condition for a system under an external force to preserve Onsager’s symmetry. o c [ PACS numbers: 05.70.-a 3 v 1 7 7 4 . 0 1 7 0 : v i X r a 1 Non-equilibrium thermodynamics tries to understand macroscopic systems out of equilib- rium, and particularly those in steady states, without having to resort to the (unattainable) dynamic description of all their microscopic degrees of freedom, using instead the same few macroscopic variables as in the thermostatic case but allowing for situations where they depend on time. Interest on this field can be traced back to the works of Boltzmann on Thomson’s hypothesis about the division of a general process into a reversible and an irre- versible part [1]. However, it was not until 1931 that a truly systematic derivation of the thermodynamics of irreversible processes near equilibrium was attained by Onsager [2, 3] and later refined by Casimir [4]. Entropy production is perhaps the most important concept in non-equilibrium thermody- namics, totally absent from thermostatics. It is usual to look at it as a function of two sets of variables, the thermodynamic fluxes {φ } and forces {F }, defined in such a way that this i i production can be expressed as a sum of products of conjugates, ∆˙S = F φ , the fluxes i i i being zero at equilibrium. This expression is supplemented by a set ofPphenomenological relations which gives the fluxes as functions of the forces, these relations being such that the forces cancel at equilibrium. It is an experimental fact that there exists a neighbour- hood of equilibrium where the relations between the two sets of variables are linear, that is, φ = L F . i ij j j OnPsager’s main result [2, 3] is the symmetry of the phenomenological coefficients L = ij L , proven onthe basis of two general hypothesis: regression of fluctuations and microscopic ji dynamic reversibility. Systems under the effect of external magnetic fields or Coriolis forces are exceptions already known to Onsager and later treated by Casimir in Ref. 4. More recently, it has been shown that the second hypothesis can be dropped for certain models [5]. In the last decades, interest in nanoscopic systems has led to put them in the front line of science and technology. Important research from the point of view of statistical mechanics hasbeendoneduringthelastdecade, leadingtosuchnotableresultsastheJarzynskiequality [6] and the Evans [7] and Crooks [8] fluctuation theorems, which have been experimentally verified [9]. A good overview of these topics can be found in Ref. 10. Nevertheless, since the seminal work of Hill [11] in the early 60’s, not much theoretical efforts were dedicated to the strictly thermodynamic formalism in the nanoscale up to the first years of the XXI century, when the same author revisited his own work and renamed it as “nanothermodynamics” 2 [12, 13]. Hill generalized the equations for open systems introducing a term associated to the number of small systems in a macroscopic ensemble of them that explicitly takes into account the energetic contribution of surface and edge effects, system rotation and translation, etc., usually negligible for macroscopic systems. Inthiscontext, itisofgreatimportancetohaveatheoreticalframeworkfordescribing the operation of nanomachines. Therefore, some thermodynamic results have been tentatively extented to systems far from the thermodynamic limit. Particularly, regarding the theory of non-equilibrium fluctuations (of interest for the development of nanomotors), the validity of Onsager’s reciprocal relations is sometimes taken for granted [14]. It is thus desirable to put the non-equilibrium thermodynamics of small systems on firm theoretical foundations. The formulation of such a non-equilibrium nanothermodynamics is the main aim in this report. By analogy with Hill’s equilibrium theory, the number of nanosystems in an ensemble, which can be modified by production, destruction and transport, is introduced as a macroscopic variable that survives in the nanoscopic description. In order to study the thermostatics of a small system, Hill [11] started with a large number N of them, so that the ensemble itself was a system in the thermodynamic limit. The author developed a theory suitable for measurement devices that interact with many of the small systems in such a way that the relevant thermodynamic quantities are not those of an individual nanosystem, but their averages over a significant number of them, which in a homogeneous system will be equal to their average over all of the nanosystems. In other words, if each nanosystem is described by a set of extensive variables X(α) ν , and the α=1 ν state of the total system is characterized by X(α) ∪{N}, the qua(cid:8)ntities(cid:9)accessible to t α=1 measurement are X¯(α) := Xt(α) ν . The enntropyoof a nanosystem can likewise be defined N α=1 as S = St. n o N The total system obeys the usual set of thermostatic relations, particularly the Gibbs and Euler equations in entropic form, dS = y(α)dX(α) − εdN and S = y(α)X(α) − εN, t t T t t T α α with y(α) := ∂St and ε := −T P∂St . Obviously, in the dPescription of the ∂Xt(α) Xt(β6=α),N ∂N Xt(α) overall system(cid:16)εis si(cid:17)mply the chemical poten(cid:0)tiala(cid:1)ssociated with thenumber of nanosystems. However, in the thermodynamics of small systems it is called the subdivision potential, a new variable with no analogue in conventional thermodynamics. To formulate the Euler equation for the nanosystems it is enough to divide both terms in 3 theEuler equationbyN. Taking intoaccount thatdS = 1 [dS −SdN], theGibbsequation N t canalso beformulated and, subtracting thetwo equationsfor dS,aninhomogeneous pseudo- Gibbs-Duhem equation arises. Thus, the thermodynamic equations for the average small system are: ε S = y(α)X¯(α) − (1a) T α X dS = y(α)dX¯(α) (1b) α X ε −d = − X¯(α)dy(α). (1c) T (cid:16) (cid:17) Xα Comparing equations (1b) and (1a), it becomes apparent that S does not satisfy Euler’s theorem and thus it is not an homogeneous function of X¯(α) in the nanothermodynamic formalism. The thermostatics of a small system depen(cid:8)ds on(cid:9)its environment through ε. Thus, a small system has more degrees of freedom than its large counterpart. The addi- tional contribution to the entropy (or, equivalently, to the internal energy) comes from the aforementioned interface, edge, rotationand traslationeffects, which must become negligible as the size of the system is increased, if conventional thermodynamics is to be recovered. Recently, Ben-Amotz and Honig [15] have used the Jarzynski equality to give a general expression, dS = hd¯Wiχ(t) +k log exp −d¯W , for the entropy production of a system T B kBT χ(t) under a time-dependent constrainDt χ(t)(cid:16)in co(cid:17)nEtact with a thermostat at the (possibly also time-dependent) temperature T, averaged over the processes compatible with that constrain (d¯W is the elementary work associated to a particular process). If it is assumed that this kind of operation amounts to an average over the ensemble of nanosystems (a reasonable hypothesis since N is large) it is possible, using (1), to give an expression for the change in ε between times 0 and t during the process determined by χ(t), suitable for measurement 0 or simulation: ∆ε = T y(α)X(α) t0 χ(t) 0 α X (cid:12) (cid:12) −d¯W − hd¯Wi+k T log exp . (2) B k T Z (cid:20) (cid:28) (cid:18) B (cid:19)(cid:29)(cid:21) χ(t) The central part of this report is devoted to the application of Hill’s course of reasoning to a system out of equilibrium in the thermodynamic branch (linear regime). For simplicity, 4 only the case with two homogeneous macroscopic subsystems (A and B) will be considered. The results can be readily generalized to an arbitrary number of partitions or even to a continuous distribution, as long as large enough macroscopic differential volumes are taken in order to assure that they contain sufficient numbers of nanosystems. The method used is valid as long as differential calculus can describe the changes in the variables of the nanosystems to a good approximation (i.e. they are not too small). Suppose that the systems are separated by a diathermic, permeable and deformable wall and slightly out of equilibrium with each other. With the total system A ∪ B completely ν isolated, the total deformation variables X(α) = X(α) +X(α) are conserved. N = t At Bt α=1 N + N , however, can vary since it is pnerfectly conceivable othat the nanosystems (e.g. A B micelles) could split or merge even in a macroscopically isolated system. Conservation of N would be a reasonable assumption in two opposite limits: static nanosystems whose dissociation energy is so high that interactions with their environment cannot split them, and highly dynamic nanosystems which are continuously reorganizing, but in such a way that the fluctuations in the total number of systems are small compared to the average value. It is possible to choose the time derivatives of all extensive thermodynamic variables as fluxes. Expanding ∆S = S −S = ∆S +∆S to second order in these coordinates: t t t,eq At Bt 1 ∂y(α) ∆S = ∆X(α) y(α) + I ∆X(β) + t It " I 2 ∂X(β) It # I∈{A,B} α β It X X X ε ∂y(α) + ∆N − I + I ∆X(α) + I T ∂N It " I # I∈{A,B} α X X 1 ∂2S + It (∆N )2, (3) 2 ∂N2 I I I∈{A,B} X where ∆X denotes the deviation of X from its equilibrium value and all the derivatives are evaluated at equilibrium. Note that the bars have been dropped for notational simplicity. This implies that y(α) = y(α) for all α and ε ∆N = −ε ∆N and, given the conservation A B A A B B of X(α), it follows that ∆X(α) = −∆X(α). Taking this into account and differentiating the A B 5 previous expression with respect to time: ∂y(α) ∂y(α) ∆˙S = A ∆N − B ∆N + t A B ∂N ∂N " A B α X ∂y(α) ∂y(α) + A + B ∆X(β) ∆˙X(α)+ ∂X(β) ∂X(β)! At # At β At Bt X ∂y(α) ∂2S + A ∆X(α) + At∆N ∆˙N + ∂N At ∂N2 A A " α A A # X ∂y(α) ∂2S + − B ∆X(α) + Bt∆N ∆˙N . (4) ∂N At ∂N2 A B " α B B # X This expression has the form of a sum of products of fluxes and forces, the terms inside square brackets being the forces F(α) ν and F(N). To translate this expression into the α=1 nanoscopic language, the followin(cid:8)g equ(cid:9)alities must be used: d∆ N X(α) ∆˙X(α) = I I = N ∆˙X(α) +∆˙N X(α) (5a) It (cid:16) dt (cid:17) I It I It ∆˙S ∆˙N +∆˙N ∆˙S = t −S A B (5b) t N N2 ∆X(α) = N ∆X(α) +X(α)∆N −∆X(α)∆N , (5c) It I I I I I I giving the result ∆˙S = F(α)∆˙X(α) +F ∆˙N +F ∆˙N , (6a) A NA A NB B α X with N F(α) :=F(α) A (6b) t N F + F(α)X(α) NAt t A S F := α − t (6c) NA PN N2 F S F := NBt − t , (6d) NB N N2 whichmeansthattheentropyproductionoftheaveragesmallsystemcanalsobewrittenasa sum of products of fluxes and forces. As mentioned previously, there exists a neighbourhood of equilibrium in which a set of linear phenomenological relations between these variables holds. By means of equations (5) and (6), the macro and nanoscopic linear coefficients can 6 be related: N L(αβ) = A N L(αβ) +X(α)L(NAβ) + t N A A X(β)(cid:16) (cid:17) + A N L(αNA) +X(α)L(NANA) (7a) N A A 1 (cid:16) (cid:17) L(αNI) = N L(αNI) +X(α)L(NANI) (7b) t N A A 1 (cid:16) (cid:17) L(NIα) = N L(NIα) +X(α)L(NINA) (7c) t N A A L(N(cid:16)INJ) (cid:17) L(NINJ) = ; I,J ∈ {A,B}. (7d) t N The macroscopic system satisfies Onsager’s reciprocity by hypothesis, i.e. L(αNA) = L(NAα). t t Itiseasytosee, startingwiththelastequations oftheprevious blockandprogressively back- substituting, that the nanoscopic coefficients are also symmetric in this situation. These proportionality relations ensure that the second law of thermodynamics is obeyed by the average systems (although it can be transitorily violated by a small system) a topic also discused in Ref. 15. The quantity S is indeterminate in one additive constant; therefore, the component of t the fluxes proportional to it, arising from the terms in (6c) and (6d), must be zero. This last condition is equivalent to: L(αNA) = −L(αNB) (8a) L(NANA) = L(NBNB) = −L(NANB). (8b) These equalities allow for further interpretation of (7) using the change of variables (N ,N ) → (N,D), with D := N −N . The time derivative of N represents the creation A B A B of nanosystems per unit time, while at fixed N the time derivative of D corresponds to the transport of small systems. The two possible causes of variation of (N ,N ) are thus A B decoupled by this change. Furthermore, application of the Curie principle shows that vari- ation of D can only be coupled to vectorial fluxes such as those treated in this report, while variation of N can be coupled with chemical (scalar) processes. In particular, they cannot be coupled with each other, because of the different tensor rank of the forces involved. Equations (8) are equivalent to stating that L(αN) = L(αNA), L(DD) = L(NANA) and the rest of the phenomenological coefficients involving N or D are zero. L(DD) is proportional, t 7 with a factor 1 , to its nanometric equivalent. The remaining coefficients are: N L(NAα) +L(NBα) 2N L(Nα) = t t = L(Nα) (9a) t 2 N +D L(NN)X(α) L(αD) = t A . (9b) t N The last equation can be used as a definition of L(NN) in the particular case in which N is fixed. This result means that the contribution of a flux of nanosystems to that of an extensive variable X(α) is proportional to the amount X(α) of that variable that each At A nanosystem carries with it in its transit from A to B, as expected. The above results can be reproduced by an alternate method closer to the well- known macroscopic proof. This method rests on the hypothesis of regression of fluctu- ations, which states that the reaction of the system to a small deviation from equilib- rium caused by an external force is the same as if it was caused by a spontaneous fluc- ν tuation. Moreover, if Z(α) ν+1 := X(α) ∪ {N}, dynamic reversibility can be ex- α=1 t α=1 pressed as ∆Z(α)(t)∆(cid:8)Z(γ)((cid:9)t+τ) =n ∆Zo(α)(t+τ)∆Z(γ)(t) , which straightforwardly gives ∆˙Z(α(cid:10))∆Z(γ) = ∆Z(α)∆˙Z((cid:11)γ) . S(cid:10)ubstituting here a linea(cid:11)r development for a system in theDthermodynamEic lDimit, analogoEus to that of previous sections, ∆˙Z(α) = L(αβ)z(β), t β with z(β) := ∂St , and using the fact that Z(α)z(β) = −k δ(αβ) (easy toPprove for a ∂Z(β) B macroscopic system, see for instance Ref. 1) th(cid:10)e symme(cid:11)try of the phenomenological matrix follows immediately. However, for the nanosystem, the temporal evolution of the internal variables cannot be related only to their deviations from equilibrium, adopting instead the more general form [16]: ∂S ∆˙X(α) = L(αβ) +f(α)(t), (10) ∂X(β) β X where f(α)(t) represents a general force acting on the system. From Eq. (5c), ∆X(α) = ∆Xt(α)−Xt(,αe)q∆N. Approximating 1 ≃ 1 and taking into account that derivatives at constant N N Neq N are equal for the macroscopic andaverage system, it follows directly that ∆X(α) ∂S = ∂X(β) −kBδ(αβ). Thus, temporal reversibility implies that: (cid:10) (cid:11) N L(αβ) −L(βα) = f(α)∆X(β) −f(β)∆X(α) . (11) This result represents a general conditio(cid:10)n for Onsager’s symmet(cid:11)ry to hold when evolution of the system is conditioned by a general external force. In this particular case, as it has 8 previously been shown, the external forces f(α) are proportional to ∆N through constants L(αN), which makes the right hand side of the previous equation trivially equal to zero, thus proving the symmetry of the phenomenological submatrix involving only the internal coordinates. This second method cannot prove anything about the coefficients involving N, an external variable. To show their symmetry it is still necessary to relate them to their macroscopical counterparts. This formulation, however, connects more easily with the language of the fluctuation theorems. New cross-transport phenomena associated with the new degrees of freedom must appear in the nanosystems. As predicted by Eqs. (9), if the number of small systems is kept con- stant, thesephenomena willconsist simply intheexchange ofextensive variablestransported along with the nanosystems. Dynamic nanosystems, such as micelles, could be thus wor- thier of study. Since Hill’s equilibrium formalism has already been applied (and successfully compared to experimental data) to nanostructures such as nanosolids and nanowires [17], they could also be good candidates to find these cross-phenomena; for instance, in electric or thermal measurements. Acknowledgments ThisworkwassupportedbytheSpanishMinistryofEducationandScienceinconjunction with the European Regional Development Fund (Grants Nos. FIS2005-04239 and FIS2007- 66823-C02-02). J. Carrete wishes to thank the financial support of the Direcci´on Xeral de Ordenaci´on e Calidade do Sistema Universitario de Galicia, da Conseller´ıa de Educaci´on e Ordenaci´on Universitaria-Xunta de Galicia. [1] S. R. De Groot, Non-equilibrium Thermodynamics (Interscience Publishers, Inc., New York, 1962). [2] L. Onsager, Phys. Rev. 37, 405 (1931). [3] L. Onsager, Phys. Rev. 38, 2265 (1931). [4] H. B. G. Casimir, Rev. Mod. Phys. 17, 343 (1945). [5] D. Gabrielli, G. Jona-Lasinio, and C. Landim, Phys. Rev. Lett. 77, 1202 (1996). [6] C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997). 9 [7] D. J. Searles and D. J. Evans, Phys. Rev. E 60, 159 (1999). [8] G. E. Crooks, Phys. Rev. E 60, 2721 (1999). [9] G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles, and D. J. Evans, Phys. Rev. Lett. 89, 050601 (2002). [10] C. Bustamante, J. Liphardt, and F. Ritort, Phys. Today 58, 53 (2005). [11] T. L. Hill, Thermodynamics of Small Systems (W. A. Benjamin, Inc., New York, 1963, 1964). [12] T. L. Hill, Nano Letters 1, 273 (2001). [13] T. L. Hill and R. V. Chamberlin, Nano Letters 2, 609 (2002). [14] F. Ju¨licher, A. Ajdari, and J. Prost, Rev. Mod. Phys. 69, 1269 (1997). [15] D. Ben-Amotz and J. M. Honig, Phys. Rev. Lett. 98, 020602 (2006). [16] L. D. Landau and E. M. Lifshitz, Statistical Physics (Pergamon Press, Oxford, 1969). [17] P. Letellier, A. Mayaffre, and M. Turmine, Phys. Rev. B 76, 045428 (2007). 10