Role of External Flow and Frame Invariance in Stochastic Thermodynamics
aa r X i v : . [ c ond - m a t . s o f t ] D ec Role of External Flow and Frame Invariance in Stochastic Thermodynamics
Thomas Speck, Jakob Mehl, and Udo Seifert II. Institut f¨ur Theoretische Physik, Universit¨at Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany
For configurational changes of soft matter systems affected or caused by external hydrodynamicflow, we identify applied work, exchanged heat, and entropy change on the level of a single trajectory.These expressions guarantee invariance of stochastic thermodynamics under a change of frame ofreference. As criterion for equilibrium vs. nonequilibrium, zero vs. nonzero applied work replacesdetailed balance vs. nonvanishing currents, since both latter criteria are shown to depend on theframe of reference. Our results are illustrated quantitatively by calculating the large deviationfunction for the entropy production of a dumbbell in shear flow.
PACS numbers: 82.70.-y, 05.40.-a
Thermodynamic notions like applied work, dissipatedheat, and entropy have been used successfully to an-alyze processes in which single colloidal particles orbiomolecules are manipulated externally [1]. Various ex-act relations have been shown to constrain the distribu-tion function arising from the ever present thermal fluctu-ations on this scale [2, 3, 4, 5]. So far, the systematic con-ceptual work has been focused on cases where the drivingcrucial to generate a nonequilibrium situation arises froma time-dependent potential expressing the effect of a mov-ing laser trap, micropipet, or AFM tip. In these cases,the identification of external work, internal energy andhence dissipated heat is straightforward. Many experi-ments on a variety of systems have proven the consistencyand potency of such an extension of thermodynamic no-tions to the micro or nano world [6, 7, 8, 9, 10, 11].As another source of nonequilibrium, external hydro-dynamic flow arguably is the most common and beststudied case in soft matter systems. On the level of sin-gle objects like a polymer, vesicle, or red-blood cell, itcan cause dramatic shape transitions (for recent exam-ples see, e.g., Refs. [12, 13, 14, 15, 16]). The theoret-ical analysis of these phenomena is typically based onequations of motion like the Langevin equation for sin-gle objects or distribution functions and their projectionto few-body correlation functions for bulk systems likecolloidal suspensions [17, 18, 19]. Concepts like work orentropy production, however, have played no systematicrole in describing such phenomena yet. Having in mindthe conceptual advantage achieved for time-dependentpotentials by using such notions, the question arises quitenaturally whether a similar analysis for nonequilibriumphenomena in soft matter systems caused by externalflow is possible. Indeed, Turitsyn et al. [20] calculated theentropy production for linear equations of motion and il-lustrated it for a dumbbell in shear flow. However, theiridentification of entropy production was not related tothe dissipated heat leading to a fundamentally differentdependence on the external flow.Conceptually, there is an even deeper issue hidden be-hind the proper treatment of external flow. Its presenceor absence depends on the frame of reference as the pres- ence or absence of currents does. Since the latter are usu-ally taken as an indicator for nonequilibrium, our analysiswill question the traditional role of both detailed balanceand nonvanishing currents as criteria for equilibrium andnonequilibrium, respectively. They will be replaced by aframe-independent identification of applied work.For an almost trivial but revealing paradigmatic case,consider a colloidal particle dragged through a viscousfluid along the trajectory x ( t ) by a harmonic laser trapof strength k moving with velocity u [6, 8]. In the labo-ratory frame, the particle is moving in a time-dependentpotential U ( x, t ) = ( k/ x − u t ) . By applying thestandard definition of work [2] as the external change ofthe potential energy, the applied power˙ w = ∂ t U (1)becomes ˙ w = − u k ( x − u t ). Changing to the comovingframe with y ≡ x − u t , the potential U ( y ) = ( k/ y be-comes time-independent and therefore one would naivelyfind ˙ w = 0. Of course, the work should be independentof the chosen frame of reference. Moreover, since in thecomoving frame detailed balance holds and no currentin the y variable occurs, one would usually consider thesystem to be in equilibrium in this representation. Thephysical origin of the apparent contradictions is the factthat in the comoving frame the particle experiences asteady flow of the fluid with velocity − u . Due to fric-tion, this flow pushes the particle against the force of thepotential thus spending work. The standard definition ofwork fails in the presence of flow since the flow advectsthe particle, which is not accounted for in Eq. (1). Modi-fying this expression will affect also the expression for theheat dissipated into the fluid as well as the expression forthe entropy production.We will now derive the refined expressions for workand heat in the presence of a, possibly time-dependent,external flow u ( r , t ). For complete generality and futureapplications, we consider a soft matter system like poly-mers, membranes, or a colloidal suspension composed of N particles with positions Γ ≡ { r , . . . , r N } . The indexdenotes the particle number and in the following we sumover same indices. The system has a total energy U (Γ , t )which is the sum of an internal energy due to particleinteractions and a possible contribution due to externalpotentials. In addition to direct interactions, we allow forhydrodynamic interactions which are of great importancefor soft matter systems.In 1997, Sekimoto formulated the first law of thermo-dynamics δq = δw − d U along a single stochastic tra-jectory [21]. We follow this route and start by identi-fying three possible sources of work w , i.e., of changesof energy caused externally. First, the potential energy U (Γ , t ) can be time-dependent if it is manipulated exter-nally. Second, it is convenient to allow for direct noncon-servative forces f k that spend work through displacingthe particles. Third, the external flow drives the sys-tem and therefore changes its energy, which must betaken into account as work as well. The advection ofparticles is described through the convective derivativeD t ≡ ∂ t + u ( r k ) · ∇ k . Through both replacing the par-tial derivative in Eq. (1) with the convective derivativeand measuring particle displacements with respect to theflow, we obtain the new definition˙ w ≡ D t U + f k · [˙ r k − u ( r k )] (2)for the work increment δw = ˙ w d t which replaces Eq. (1).The first law then leads to the heat production rate˙ q ≡ ˙ w − d U/ d t = [˙ r k − u ( r k )] · [ −∇ k U + f k ] . (3)The sign of heat is convention, here we take it to bepositive if energy is dissipated into the surrounding fluid.The expression (3) shows that no heat is dissipated whenparticles move along a fluid trajectory, ˙ r k = u ( r k ). Inthis case, the applied work corresponds to the change intotal energy.A change of frame necessarily changes the flow there-fore leaving the expressions for work (2) and heat (3) in-variant. Explicitly, we can allow time-dependent orthog-onal transformations and an arbitrary time-dependentshift of the origin. Coming back to the introductory ex-ample, we see that for a stationary trap in the comovingframe with flow u = − u , the applied power followingEq. (2) now reads ˙ w = − u ky in agreement with therate obtained for a moving trap and resting fluid. Finally,note that the definitions (2) and (3) are independent ofwhether or not hydrodynamic interactions are induced.These will affect the dynamics but they do not enter thedefinitions of work and heat explicitly.We now turn to entropy production. A trajec-tory dependent stochastic entropy is defined as s ( t ) ≡− ln ψ (Γ( t ) , t ) through the distribution function ψ (Γ , t )of the N particle positions [5]. The temperature of thesurrounding fluid is T and throughout the paper, weset Boltzmann’s constant to unity. Calculating the to-tal time derivative ˙ s ≡ d s/ d t , we obtain the equation of motion˙ s = ∂ t s − ˙ r k · ∇ k ln ψ = D t s + [˙ r k − u ( r k )] · F k /T − [˙ r k − u ( r k )] · [ −∇ k U + f k ] /T, (4)where we have separated the heat production rate (3)and introduced the total effective force F k ≡ −∇ k ( U + T ln ψ ) + f k . (5)Apart from the gradient of the potential energy U (Γ , t )and nonconservative forces f k that cannot be written asgradient of a potential, a “thermodynamic” force con-tributes to F k arising from the stochastic interactionsbetween system and the surrounding fluid [17]. In theabsence of external flows and nonconservative forces, de-tailed balance must hold. The thermodynamic forcethen ensures that ∇ k ( U + T ln ψ ) = 0 leads to the cor-rect Gibbs-Boltzmann equilibrium distribution ψ eq ∼ exp( − U/T ). In Eq. (4), the last term is the heat (3)divided by the temperature of the fluid T . We interpretthis term as the entropy produced in the fluid throughClausius’ formula ∆ s m = q/T . The total entropy pro-duction rate ˙ s tot ≡ ˙ s + ˙ s m then becomes˙ s tot = D t s + [˙ r k − u ( r k )] · F k /T. (6)The mean total entropy production rate follows as T h ˙ s tot i = h [˙ r k − u ( r k )] · F k i (7)since the mean of the convective derivative is zero forincompressible fluids ( ∇ · u = 0) and vanishing boundaryterms.So far we did not resort to a specific dynamics. How-ever, in order to both prove positivity of Eq. (7) and giveit a more familiar appearance known from the theoryof polymer dynamics [17], we turn to the Smoluchowskiequation [17, 18] ∂ t ψ + ∇ k · ( v k ψ ) = 0 (8)governing the evolution of the distribution function ψ (Γ , t ). Any deviation of a particle’s local mean velocity v k ≡ u ( r k ) + µ kl F l (9)from the velocity of the external flow u ( r ), which dragsthe particles due to friction, has to be caused by the totalforce F k . In this model, hydrodynamic interactions nowenter through a dependence of the symmetric mobilitymatrices µ kl (Γ) on the particle positions Γ. We demandthe inverse matrices defined through µ km µ − ml = δ kl toexist. Specific expressions for µ kl are obtained from ei-ther the Oseen or Rotne-Prager tensor [18].The local mean velocity v k (Γ , t ) is the average of theactual stochastic velocity ˙ r k over the subset of trajecto-ries passing through a given configuration Γ. This allowsus to perform the mean in Eq. (7) in two steps: First,we average over stochastic trajectories crossing a specificpoint Γ in configuration space, which amounts to replac-ing ˙ r k by v k . Second, we average over the distribution ψ (Γ , t ) of the point Γ. Finally, we use Eq. (9) to replacethe total force F k . The resulting quadratic expression T h ˙ s tot i = h [ v k − u ( r k )] · µ − kl [ v l − u ( r l )] i > F k = 0.On the trajectory level, we have identified heat andentropy production using physical arguments. There is amore formal way of identifying entropy by starting fromthe weight P [Γ( t ) | Γ ] ∼ exp {− S [Γ( t ) | Γ ] /T } for a tra-jectory Γ( t ) starting in Γ(0) = Γ involving the action S . The entropy produced in the surrounding mediumis usually identified with the part of the action that isasymmetric with respect to time-reversal [22]. For ex-plicit expressions for both the path probability and theaction in the presence of hydrodynamic interactions asneeded in our case, we refer to Ref. [23].We can deduce two expressions for the asymmetricpart of the action since in the presence of external flow,we have two choices how to define the operation “timereversal”. First, we could formally treat the flow as anonconservative force. It would then be invariant withrespect to time reversal and we would obtain˙ s ∗ m = ˙ r k · [ −∇ k U + f k + µ − kl u ( r l )] /T. (11)This expression has been identified with the entropy pro-duction rate in Ref. [20]. Second, we can extend theoperation of time reversal to the particles of the fluid,which is physically more appropriate. Reversing the ve-locity of the fluid particles then effectively amounts tothe change of the sign of the external flow velocity u ( r ).This leads to ˙ s m = ˙ q/T involving the heat productionrate from Eq. (3). We thus recover the physically moti-vated definition of the heat (3) as the asymmetric part ofthe action under time reversal only if the flow is reversedas well [20].For a discussion of the crucial difference betweenEqs. (11) and (3) in a specific system, consider a dumb-bell in a flow u ( r ) = κ r with traceless matrix κ = κ S + κ A , which can be split up into a symmetric part κ S and a skew-symmetric part κ A . The dumbbell consistsof two particles at positions r and r with displacement r ≡ r − r and potential energy U ( r ) = ( k/ r . Theapplied power (2) then becomes˙ w = u ( r ) · ∇ U = k r · κ S r , (12)i.e., only the symmetric part κ S , which reflects an elonga-tion flow component, contributes to the work. Physically, the elongation flow drags the two particles apart due tofriction and therefore spends permanently work againstthe elastic force keeping the particles together. Neglect-ing the boundary term ∆ U/t ∼ s m ∼ ( k/T ) r · κ S r . Note that here only the dis-placement r enters. This is not true if the expression (11)is interpreted as entropy production rate, leading to thetime extensive part˙ s ∗ m ∼ ( T µ ) − [2 ˙ R · κ A R + (1 / r · κ A r ] (13)with bare mobility µ . Here, also the center of mass R ≡ ( r + r ) / κ A in contrast to Eq. (12).We will now calculate the large deviation function ofthe medium entropy production rate for the dumbbell intwo-dimensional shear flow, i.e., all entries of the matrix κ are now zero except κ xy = ˙ γ where ˙ γ is the strainrate [18]. The large deviation function h ( σ ) ≡ lim t →∞ − t ln p (∆ s m , t ) (14)quantifies the asymptotic fluctuations σ = ∆ s m / ( h ˙ s m i t )of the entropy production in the limit of large observa-tion times t with mean production rate h ˙ s m i . Insteadof obtaining the function h ( σ ) from the time-dependentprobability distribution p (∆ s m , t ) directly, we will calcu-late its Legendre transform α ( λ ) as the lowest eigenvalueof the operator ˆ L λ = ˆ L − λ ˙ s m . The operator ˆ L λ gov-erns the evolution of the generating function [24, 25, 26].The production rate depends only on the displacement r ,whose dynamics is determined through the Smoluchowskioperator [17] ˆ L = τ − ∇ · [ r + ( T /k ) ∇ ] − κ r · ∇ . Forthis illustration, we neglect hydrodynamic interactionsby using mobility matrices µ kl = µ δ kl defining thetime scale τ − ≡ µ k .For the lowest eigenfunction of the eigenvalue equationˆ L λ ψ ( r , λ ) = − α ( λ ) ψ ( r , λ ) , (15)we use the ansatz ψ = exp[ − ( k/ T ) r · C λ r ] with a sym-metric matrix C λ . We are lead to this ansatz since weknow that the stationary distribution ( λ = 0) of r isa Gaussian [27], where C becomes the inverse covari-ance matrix. Inserting this ansatz into Eq.(15) results in α ( λ ) = τ − tr( C λ − ), where C λ is the solution of thequadratic matrix equation( C λ + D ) T ( C λ + D ) = S λ , S λ = 14 (cid:18) γ (2 λ − γ (2 λ −
1) 1 + ˜ γ (cid:19) , D = 12 (cid:18) − γ − (cid:19) . -1.2-0.8-0.4 0 0.4 0.8 1.2-0.25 0 0.25 0.5 0.75 1 1.25 α ( λ ) λ Eq. (17)Eq. (16) 0 0.2 0.4 0.6 0.8 1-1 0 1 2 3 4 5 h ( σ ) [ m ean r a t e ] σ FIG. 1: Left: Comparison of the eigenvalue α ( λ ) fromEq. (16) (solid) with the solution (17) obtained in Ref. [20](dashed) based on Eq. (11). (Parameters for both curves are˜ γ = 3 and τ = 1.) The mean entropy production rate is thesame for both expressions as indicated by the matching slopeat λ = 0. Right: The corresponding large deviation functions h ( σ ) obtained from the Legendre transform of α ( λ ). The only parameter left is the dimensionless strain rate˜ γ ≡ ˙ γτ . We only need the trace of the matrix C λ , which,after some tedious calculations, can be expressed astr C λ = q tr S λ + 2 p det S λ − ˜ γ / − tr D leading to the solution α ( λ ) = τ − (cid:20)q + p γ λ (1 − λ ) − (cid:21) . (16)This function is defined within the interval λ − λ λ + with branch points λ ± = (1 ± p γ − ). Such branchpoints imply linear asymptotes for h ( σ ) and thereforeexponential tails for the distribution p (∆ s m ) [31]. On thelevel of the function α ( λ ), the fluctuation theorem [5,24, 28, 29] is expressed through the symmetry α ( λ ) = α (1 − λ ), which is fulfilled by Eq. (16). In Fig. 1, wecompare the function (16) with the solution α ∗ ( λ ) = τ − hp γ λ (1 − λ ) − i (17)obtained in Ref. [20] based on the entropy productionrate in Eq. (13) with fixed center of mass ( ˙ R = 0), whichalso fulfills a fluctuation theorem. Note that our expres-sion for h ( σ ) based on Eq. (16) predicts substantiallylarger fluctuations especially for trajectories with largerthan mean entropy production.The dumbbell just discussed also provides a counter-example to the standard definition of nonequilibrium. Inpure rotational flow with skew-symmetric matrix κ = κ A = (cid:18) γ/ − ˙ γ/ (cid:19) , there is a nonvanishing current manifested through theever tumbling dumbbell superficially indicating nonequi-librium. On the other hand, the length | r | undergoes onlyequilibrium fluctuations. For all practical purposes, thisis an equilibrium system. In fact, the applied power (2) vanishes. Combining this result with the moving trapcase discussed at the beginning, we conclude that in thepresence of flow the proper frame-invariant criterion fordistinguishing equilibrium from genuine nonequilibriumis whether or not the applied power obeys ˙ w = 0.Summarizing further, we have shown that consider-ing external flow in the expression for the work (2) [andtherefore in the derived quantities like dissipated heat (3)and entropy production (6)] guarantees frame invarianceof stochastic thermodynamics. While formally two ex-pressions for the asymmetric part of the path probabilityunder time reversal are possible which both fulfill thefluctuation theorem, only one leads to a consistent iden-tification of thermodynamic quantities for the dynamicsof soft matter systems in flow.We thank R. Finken and R. K. P. Zia for inspiringdiscussions. [1] C. Bustamante, J. Liphardt, and F. Ritort, Physics To-day , 43 (2005).[2] C. Jarzynski, Phys. Rev. Lett. , 2690 (1997).[3] G. E. Crooks, Phys. Rev. E , 2721 (1999).[4] T. Hatano and S. Sasa, Phys. Rev. Lett. , 3463 (2001).[5] U. Seifert, Phys. Rev. Lett. , 040602 (2005).[6] G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles, andD. J. Evans, Phys. Rev. Lett. , 050601 (2002).[7] J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco Jr, andC. Bustamante, Science , 1832 (2002).[8] E. H. Trepagnier, C. Jarzynski, F. Ritort, G. E. Crooks,C. J. Bustamante, and J. Liphardt, Proc. Natl. Acad.Sci. U.S.A. , 15038 (2004).[9] F. Douarche, S. Joubaud, N. B. Garnier, A. Petrosyan,and S. Ciliberto, Phys. Rev. Lett. , 140603 (2006).[10] V. Blickle, T. Speck, L. Helden, U. Seifert, andC. Bechinger, Phys. Rev. Lett. , 070603 (2006).[11] D. Collin, F. Ritort, C. Jarzynski, S. Smith, I. Tinoco,and C. Bustamante, Nature , 231 (2005).[12] R. G. Winkler, Phys. Rev. Lett. , 128301 (2006).[13] C. Misbah, Phys. Rev. Lett. , 028104 (2006).[14] H. Noguchi and G. Gompper, Phys. Rev. Lett. ,128103 (2007).[15] M. Abkarian, M. Faivre, and A. Viallat, Phys. Rev. Lett. , 188302 (2007).[16] V. Kantsler and V. Steinberg, Phys. Rev. Lett. ,036001 (2006).[17] M. Doi and S. F. Edwards, The Theory of Polymer Dy-namics (Clarendon Press, Oxford, 1986).[18] J. K. G. Dhont,
An Introduction to Dynamics of Colloids (Elsevier, Amsterdam, 1996).[19] J. M. Brader, T. Voigtmann, M. E. Cates, and M. Fuchs,Phys. Rev. Lett. , 058301 (2007).[20] K. Turitsyn, M. Chertkov, V. Chernyak, and A. Puliafito,Phys. Rev. Lett. , 180603 (2007).[21] K. Sekimoto, J. Phys. Soc. Jpn. , 1234 (1997).[22] C. Maes, S´em. Poincar´e , 29 (2003).[23] C. Maes, K. Netocny, and B. Wynants (2007),arXiv:0708.0489.[24] J. L. Lebowitz and H. Spohn, J. Stat. Phys. , 333 (1999).[25] P. Visco, J. Stat. Mech.: Theor. Exp. p. P06006 (2006).[26] A. Imparato and L. Peliti, J. Stat. Mech.: Theor. Exp. p.L02001 (2007).[27] J. Johnson, Macromolecules , 103 (1987).[28] D. J. Evans, E. G. D. Cohen, and G. P. Morriss, Phys.Rev. Lett. , 2401 (1993).[29] G. Gallavotti and E. G. D. Cohen, Phys. Rev. Lett. ,2694 (1995). [30] R. van Zon and E. G. D. Cohen, Phys. Rev. E , 46102(2003).[31] An unbounded potential U in connection with exponen-tial tails of p (∆ s mm