Advances in Physics, 2002, Vol. 51, No. 7, 1529±1585 The Fluctuation Theorem Denis J. Evans* Research School of Chemistry, Australian National University, Canberra, ACT 0200 Australia and Debra J. Searles School of Science, Gri(cid:129) th University, Brisbane, Qld 4111 Australia [Received 1 February 2002; revised 8 April 2002; accepted 9 May 2002] Abstract The question ofhowreversiblemicroscopicequations ofmotion can lead to irreversiblemacroscopicbehaviourhasbeenoneofthecentralissuesinstatistical mechanics for more than a century. The basic issues were known to Gibbs. Boltzmann conducted a very public debate with Loschmidt and others without asatisfactory resolution. In recent decades there has been noreal change in the situation.In1993wediscoveredarelation,subsequentlyknownastheFluctuation Theorem (FT), which gives an analytical expression for the probability of observingSecondLawviolatingdynamical¯uctuationsinthermostatteddissipa- tivenon-equilibriumsystems.Therelationwasderivedheuristicallyandappliedto thespecialcaseofdissipativenon-equilibriumsystemssubjecttoconstantenergy `thermostatting’.TheserestrictionsmeantthatthefullimportanceoftheTheorem was not immediately apparent. Within a few years, derivations of the Theorem wereimprovedbutithasonlybeeninthelastfewofyearsthatthegeneralityofthe Theorem hasbeen appreciated.Wenowknowthat theSecond LawofThermo- dynamics can be derived assuming ergodicity at equilibrium, and causality. We taketheassumptionofcausalitytobeaxiomatic.Itiscausalitywhichultimatelyis responsibleforbreakingtimereversalsymmetryandwhichleadstothepossibility ofirreversiblemacroscopicbehaviour. The Fluctuation Theorem does much more than merely prove that in large systems observed for long periods of time, the Second Law is overwhelmingly likelytobevalid.TheFluctuationTheoremquanti®estheprobabilityofobserving Second Law violations in small systems observed for a short time. Unlike the Boltzmann equation, the FT is completely consistent with Loschmidt’s observa- tionthatfortimereversibledynamics,everydynamicalphasespacetrajectoryand its conjugate timereversed `anti-trajectory’, areboth solutions ofthe underlying equations of motion. Indeed the standard proofs of the FT explicitly consider conjugatepairsofphasespacetrajectories.Quantitativepredictionsmadebythe Fluctuation Theorem regarding the probability of Second Law violations have beencon®rmedexperimentally,bothusingmoleculardynamicscomputersimula- tionandveryrecentlyinlaboratoryexperiments. Contents page 1. Introduction 1530 1.1. Overview 1530 *To whom correspondence should be addressed. e-mail: [email protected] Advancesin PhysicsISSN0001±8732print/ISSN1460±6976online#2002Taylor& FrancisLtd http://www.tandf.co.uk/journals DOI:10.1080/00018730210155133 1530 D. J. Evans and D. J Searles 1.2. Reversibledynamicalsystems 1534 1.3. Example:SLLOD equations for planar Couette ¯ow 1538 1.4. Lyapunovinstability 1539 2. Liouville derivation ofFT 1541 2.1. The transient ofFT 1541 2.2. The steady state FT and ergodicity 1545 3. Lyapunovderivation ofFT 1546 4. Applications 1553 4.1. Isothermal systems 1553 4.2. Isothermal±isobaric systems 1555 4.3. Free relaxation in Hamiltonian systems 1556 4.4. FT for arbitraryphase functions 1559 4.5. Integrated FT 1561 5. Green±Kubo relations 1562 6. Causality 1564 6.1. Introduction 1564 6.2. Causal and anticausalconstitutive relations 1565 6.3. Green±Kubo relations for the causal and anticausal linear response functions 1566 6.4. Example:the Maxwell modelofviscosity 1568 6.5. Phase space trajectories for ergostatted shear¯ow 1570 6.6. Simulation results 1572 7. Experimental con®rmation 1574 8. Conclusion 1579 Acknowledgements 1584 References 1584 1. Introduction 1.1. Overview Linear irreversible thermodynamics is a macroscopic theory that combines Navier±Stokes hydrodynamics, equilibrium thermodynamics and Maxwell’s postu- late of local thermodynamic equilibrium. The resulting theory predicts in the near equilibrium regime, where local thermodynamic equilibrium is expected to be valid, thattherewillbea`spontaneousproductionofentropy’innon-equilibriumsystems. This spontaneous production of entropy is characterized by the entropy source strength, ¼, which gives the rate of spontaneous production of entropy per unit volume. Using these assumptions it is straightforward to show [1] that dr¼ r;t dr J r;t X r;t >0; 1:1 i i … †ˆ … † … † … † … … ±X ² whereJ r;t isoneoftheNavier±Stokeshydrodynamic¯uxes(e.g.thestresstensor, i … † heat¯uxvector,...)atpositionrandtimetandX isthethermodynamicforcewhich i isconjugate to J r;t (e.g.strainratetensordividedby theabsolute temperatureor i … † the gradient of the reciprocal of the absolute temperature,...respectively). As discussed in reference [1], equation (1.1) is a consequence of exact conservation laws,theSecondLawofThermodynamicsandthepostulateoflocalthermodynamic equilibrium. The conservation laws (of energy, mass and momentum) can be taken as given. The postulate of local thermodynamic equilibrium can be justi®ed by assuming The Fluctuation Theorem 1531 analyticity of thermodynamic state functions arbitrarily close to equilibrium. y Assuming analyticity, then local thermodynamic equilibrium is obtained from a ®rstorderexpansionofthermodynamicpropertiesintheirreversible¯uxes X .We i f g take this `postulate’ as highly plausibleÐespecially on physical grounds. However, the rationalization of the Second Law of Thermodynamics is a di(cid:128) erentissue.Thequestionofhow irreversiblemacroscopicbehaviour,assummar- ized by the Second Law of Thermodynamics, can be derived from reversible microscopicequationsof motionhasremainedunresolvedeversincethefoundation of thermodynamics. In their 1912 Encyclopaedia article [3] the Ehrenfests made the comment:Boltzmanndidnotfullysucceedinprovingthetendencyoftheworldtogoto a ®nal equilibrium state ... The very important irreversibility of all observable processes can be ®tted into the picture: The period of time in which we live happens tobe aperiodinwhichtheH-functionofthepartoftheworldaccessibletoobservation decreases. This coincidence is not really an accident, it is a precondition for the existenceoflife.Theviewthatirreversibility isaresultof ourspecialplaceinspace± timeisstillwidelyheld[4].InthepresentReviewwewillargueforanalternative,less anthropomorphic, point of view. In this Review we shall discuss a theorem that has come to be known as the Fluctuation Theorem (FT). This `Theorem’ is in fact a group of closely related Fluctuation Theorems. One of these theorems states that in a time reversible, thermostatted, ergodic dynamical system, if S…t†ˆ¡› J…t†FeV ˆ VdV ¼…r;t†=kB is the total (extensive) irreversible entropy production rate, where V is the system „ volume, F an external dissipative ®eld, J is the dissipative ¯ux, and › 1=k T e B ˆ where T is the absolute temperature of the thermal reservoir coupled to the system and k is Boltzmann’s constant, then in a non-equilibrium steady state the B ¯uctuations in the time averaged irreversible entropy production SS·t ²…1=t† 0tdsS…s†, satisfy the relation: „ lim 1 ln p…SS·t ˆA† A: 1:2 t!1 t p…SS·t ˆ¡A†ˆ … † The notationp SS·t A denotestheprobability thatthevalueof SS·t liesinthe range A to A dA a…nd pˆSS·t† A denotes the corresponding probability SS·t lies in the ‡ … ˆ¡ † range A to A dA. The equation is valid for external ®elds, F , of arbitrary e ¡ ¡ ¡ magnitude. When the dissipative ®eld is weak, the derivation of (1.2) constitutes a proof of the fundamental equation of linear irreversible thermodynamics, namely equation (1.1). Loschmidt objected to Boltzmann’s `proof’ of the Second Law, on the grounds thatbecausedynamicsistimereversible,foreveryphasespacetrajectorythereexists aconjugatetimereversedantitrajectory [5] which is also asolution of theequations of motion. If the initial phase space distribution is symmetric under time reversal z symmetry(whichisthecaseforalltheusualstatisticalmechanicalensembles)thenit was then argued that the Boltzmann H-function (essentially the negative of the {See:CommentsontheEntropyofNonequilibriumSteadyStatesbyD.J.EvansandL.Rondoni, Festschrift for J.R.Dorfman [2]. Apparently,iftheinstantaneousvelocitiesof allof the elementsof anygivensystem are reversed, z the total course of the incidents must generally be reversedfor every given system. Loschmidt, reference [5], page 139. 1532 D. J. Evans and D. J Searles dilutegasentropy),couldnotdecreasemonotonicallyaspredictedbytheBoltzmann H-theorem. However, Loschmidt’s observation does not deny the possibility of deriving the Second Law. One of the proofs of the Fluctuation Theorem given here, explicitly considers bundles of conjugate trajectory and antitrajectory pairs. Indeed the existence of conjugate bundles of trajectory and antitrajectory segments is central to the proof. By considering the measure of the initial phases from which these conjugate bundles originate, we derive a Fluctuation Theorem which con®rms that for large systems, or for systems observed for long times, the Second Law of Thermodynamicsislikelytobesatis®edwithoverwhelming(exponential)likelihood. The Fluctuation Theorem is really best regarded as a set of closely related theorems. One reason for this is that the theorem deals with ¯uctuations, and since one expects the statistics of ¯uctuations to be di(cid:128) erent in di(cid:128) erent statistical mechanical ensembles, there is a need for a set of di(cid:128) erent, but related theorems. Asecond reason forthe diversityof thissetof theorems isthatsometheoremsrefer tonon-equilibriumsteadystate¯uctuations,e.g.(1.2),whileothersrefertotransient ¯uctuations.Iftransient¯uctuationsareconsidered,thetimeaveragesarecomputed fora®nitetimefromazerotimewheretheinitialdistributionfunctionisassumedto be known: for example it could be one of the equilibrium distribution functions of statistical mechanics. Even when the time averages are computed in the steady state, they could be computed for an ensemble of experiments that started from a known, ergodically consistent, distribution in the (long distant) past or, if the system is ergodic, time averages could be computed at di(cid:128) erent times during the course of a single very long phase space trajectory . As we shall see, the Steady State Fluctuation y Theorems (SSFT) are asymptotic, being valid in the limit of long averaging times, while the corresponding Transient Fluctuation Theorems (TFT) are exact for arbitrary averaging times. The TFT can therefore be written, p SS·t A =p SS·t A exp At; t>0. ‰ … ˆ †Š ‰ … ˆ¡ †Šˆ ‰ Š 8 We can illustrate the SSFT expressed in equation (1.2) very simply. Suppose we considerashearingsystemwithaconstantpositivestrainrate,® @u =@y,whereu x x ² isthestreamingvelocityinthex-direction.Supposefurtherthatthesystemisof®xed volumeandisincontactwithaheatbathata®xedtemperatureT.Timeaveragesof the xy-element of the pressure tensor, PP· , are proportional to the negative of the xy;t time-averaged entropy production. A histogram of the ¯uctuations in the time- averagedpressuretensorelementcouldbeexpectedasshownin®gure1.1.Inaccord with the Second Law, the mean value for PP· is negative. The distribution is xy;t approximately Gaussian. As the number of particles increases or as the averaging time increases we expect that the variance of the histogram would decrease. For the parameters studied in this example, the wings of the distribution ensure that there is a signi®cant probability of ®nding data for which the time averaged entropyproduction isnegative. The SSFT gives a mathematical relationshipforthe ratio of peak heights of pairs of data points which are symmetrically distributed about zero on the x-axis, as shown in ®gure 1.1. The SSFT says that it becomes exponentially likely that the value of the time-averaged entropy production will be positive rather than negative. Further, the argument of this exponential grows {The equivalence of these two averages is the de®nition of an ergodic system. The Fluctuation Theorem 1533 Figure1.1. A histogram showing ¯uctuations in the time-averaged shear stress for asystem undergoing Couette ¯ow. linearly with system size and with the duration of the averaging time. In either the large system or long time limit the SSFT predicts that the Second Law will hold absolutely and that the probability of Second Law violations will be zero. If ... denotes an average over all ¯uctuations in which the time-integrated h iSS·t>0 entropy production is positive, then one can show that from the transient form of equation (1.2), that µpp……SSSS··tt ><00††¶ˆhexp…¡SS·tt†iSS·t<0 ˆhexp…¡SS·tt†i¡SS·t1>0 >1 …1:3† gives the ratio of probabilities thatfor a ®nite system observed fora ®nite time, the SecondLawwillbesatis®edratherthanviolated(seesection4.5).Theratioincreases approximately exponentially with increased time of observation, t, or with system size (since S isextensive). [There isa corresponding steadystate form of (1.3)which is valid asymptotically, in the limit of long averaging times.] We will refer to the varioustransientorsteadystateformsof(1.3)astransientorsteadystate,Integrated Fluctuation Theorems (IFTs). The Fluctuation Theorems are important for a number of reasons: (1) they quantify probabilities of violating the Second Law of Thermo- dynamics; (2) they are veri®able in a laboratory; (3) the SSFT can be used to derive the Green±Kubo and Einstein relations for linear transport coe(cid:129) cients; (4) they are valid in the nonlinear regime, far from equilibrium, where Green± Kubo relations fail; (5) local versions of the theorems are valid; 1534 D. J. Evans and D. J Searles (6) stochastic versions of the theorems have been derived [6±11]; (7) TFT and SSFT can be derived using the traditional methods of non- equilibrium statistical mechanics and applied to ensembles of transient or steady state trajectories; (8) the Sinai±Ruelle±Bowen (SRB) measure from the modern theory of dynamical systems can be used to derive an SSFT for a single very long dynamical trajectory characteristic of an isochoric, constant energy steady state; (9) FTs can be derived which apply exactly to transient trajectory segments while SSFTs can be derived which apply asymptotically (t ) to non- !1 equilibrium steady states; (10) FTs can be derived for dissipative systems under a variety of thermodynamic constraints (e.g. thermostatted, ergostatted or unthermos- tatted, constant volume or constant pressure), and (11) a TFT can be derived which proves that an ensemble of non-dissipative purely Hamiltonian systems will with overwhelming likelihood, relax from anyarbitrary initial (non-equilibrium) distribution towards theappropriate equilibrium distribution. Point (11). is the analogue of Boltzmann’s H-theorem and can be thought of as a proof of Le Chatelier’s Principle [12, 13]. In this Review we will concentrate on the ensemble versions of the TFT and SSFT.AdetailedaccountoftheapplicationoftheSRBmeasuretothestatisticsofa single dynamical trajectory hasbeengivenelsewhereby GallavottiandCohen(GC) [14,15].However, itistrueto saythatforthis morestrictlydynamical derivationof theSSFTstherearemanyunansweredquestions.Forexample,essentiallynothingis known of the application of the SRB measure and GC methods to dynamical trajectories which are characteristic of systems under various macroscopic thermo- dynamic constraints (e.g. constant temperature or pressure). All the known results seemtobeapplicableonlytoisochoric,constantenergysystems.Alsoanhypothesis which is essential to the GC proof of the SSFT, the so-called chaotic hypothesis, is little understood in terms of how it applies to dynamical systems that occur in nature.FThavealsobeendevelopedforgeneralMarkovprocessesbyLebowitzand Spohn[7] and aderivationof FTusingtheGibbsformalismhasbeenconsideredin detail by Maes and co-workers [8±10]. 1.2. Reversible dynamical systems A typical experiment of interest is conveniently summarized by the following example. Consideran electricalconductor (a moltensaltforexample)subjectat say t 0, to an applied electric ®eld, E. We wish to understand the behaviour of this ˆ system from an atomic or molecular point of view. We assume that classical mechanics gives an adequate description of the dynamics. Experimentally we can onlycontrolasmallnumberofvariableswhichspecifytheinitialstateofthesystem. We might only be able to control the initial temperature T 0 , the initial volume … † V 0 and the number of atoms in the system, N, which we assume to be constant. … † The microscopic state of the system is represented by a phase space vector of the coordinates and momenta of all the particles, in an exceedingly high dimensional spaceÐphase spaceÐ q ;q ;...;q ;p ;...;p q;p C where q;p are the f 1 2 N 1 Ng²… †² i i position and conjugate momentum of particle i. There are a huge number of initial The Fluctuation Theorem 1535 microstates C 0 ,thatareconsistentwiththeinitial macroscopicspeci®cationofthe … † system T 0 ;V 0 ;N . … … † … † † We couldstudythemacroscopic behaviour of the macroscopic system by taking just one of the huge number of microstates that satisfy the macroscopic conditions, and then solving the equations of motion for this single microscopic trajectory. However, we would have to take care that our microscopic trajectory C t , was a … † typical trajectoryand that it did notbehave in an exceptional way. Thebestwayof understanding the macroscopic system would be to select a set of N initial phases C C 0 ; j 1;...;N and compute the time dependent properties of the macro- j C f … † ˆ g scopic system by taking a time dependent average A t of a phase function A C h … †i … † over the ensemble of time evolved phases NC A t A C t =N : j C h … †iˆ … … †† j 1 Xˆ Indeed, repeating the experiment with initial states that are consistent with the speci®ed initial conditions is often what an experimentalist attempts to do in the laboratory.Althoughtheconceptof ensembleaveragingseemsnaturalandintuitive to experimental scientists, the use of ensembles has caused some problems and misunderstandings from a more purely mathematical viewpoint. Ensemblesarewellknowntoequilibriumstatisticalmechanics,theconceptbeing ®rst introduced by Maxwell. The use of ensembles in non-equilibrium statistical mechanicsislesswidelyknownandunderstood. Forourexperimentitwilloftenbe y convenient to choose the initial ensemble which is represented by the set of phases C 0 ; j 1;...;N ,tobeoneof thestandardensemblesofequilibriumstatistical j C f … † ˆ g mechanics.However,sometimeswemaywishtovarythissomewhat.Inanycase,in all the examples we will consider, the initial ensemble of phase vectors will be characterizedby aknown initial N-particledistributionfunction,f C ;t ,whichgives … † the probability, f C ;t dC , that a member of the ensemble is within some small … † neighbourhood dC of a phase C at time t, after the experiment began. The electric ®eld does work on thesystem causing an electriccurrent, I, to ¯ow. We expect that at an arbitrary time t after the ®eld has been applied, the ensemble averagedcurrent I t will be in the direction of the ®eld; thatthe work performed h … †i on the system by the ®eld will generate heatÐOhmic heating, I t · E; and that h … †i there will be a `spontaneous production of entropy’ S t I t · E=T t . It will h … †iˆh … † … †i frequently be the case that the electrical conductor will be in contact with a heat reservoirwhich®xesthetemperatureofthesystemsothatT t T 0 T; t.The … †ˆ … †ˆ 8 particles in this system constitute a typical time reversible dynamical system. We are interested in an number of problems suggested by this experiment: (1) How dowereconcilethe`spontaneousproductionof entropy’,withthetime reversibility of the microscopic equations of motion? (2) For agiveninitial phase C 0 whichgeneratessome time dependentcurrent j … † I t ,canwegenerateLoschmidt’sconjugateantitrajectory whichhasatime- j … † reversed electric current? (3) Is there anything we can say about the deviations of the behaviour of individual ensemble members, from the average behaviour? {Forfurtherbackgroundinformationonnon-equilibriumstatisticalmechanicsseereference[16]. 1536 D. J. Evans and D. J Searles In general, it is convenient to consider equations of motion for an N-particle system, of the form, p qq_ i C C · F i ˆm‡ i… † e 1:4 9 … † pp_i ˆFi…q†‡Di…C †· Fe¡Si¬…C †pi;= where F is the dissipative external ®eld that couples to ;the system via the phase e functions C…C † and D…C †, Fi…q†ˆ¡@F…q†=@qi is the interatomic force on particle i (and F…q† is the interparticle potential energy), and the last term ¡Si¬…C †pi is a deterministictimereversiblethermostatusedto addorremoveheatfromthesystem [16].ThethermostatmultiplierischosenusingGauss’sPrincipleofLeastConstraint [16], to ®x some thermodynamic constraint (e.g. temperature or energy). The thermostat employs a switch, S, which controls how many and which particles i are thermostatted. Themodelsystemcouldbequiterealisticwithonlysome particles subjectto the external ®eld. For example, some ¯uid particles might be charged in an electrical conduction experiment, while other particles may bechemicallydistinct, being solid at the temperatures and densities under consideration. Furthermore these particles may form the thermal boundaries or walls which thermostat and `contain’ the electrically charged particles ¯uid particles inside a conduction cell. In this case S 1onlyforwallparticlesandS 0forallthe¯uidparticles.Thiswouldprovide i i ˆ ˆ a realistic model of electrical conduction. In other cases we might consider a homogeneous thermostat where S 1; i. It i ˆ 8 is worth pointing out that as described, equations (1.4) are time reversible and heat canbebothabsorbedandgivenoutby thethermostat.However, inaccordwiththe Second Law of Thermodynamics, in dissipative dynamics the ensemble averaged value of the thermostat multiplier is positive at all times, no matter how short, ¬ t >0; t>0. h … †i 8 Oneshouldnotconfusea real thermostatcomposedofaverylarge(inprinciple, in®nite)numberof particleswiththepurelymathematicalÐalbeitconvenientÐterm ¬. In writing equation (1.4) it is assumed that the momenta p are peculiar (i.e. i measuredrelativetothelocalstreamingvelocityofthe¯uidorwall).Thethermostat multiplier may be chosen, for instance, to ®x the internal energy of the system H0 ² p2i=2m‡1=2 F…q† ; i:XSiˆ0 µ Xj ¶ in which case we speak of ergostatted dynamics, or we can constrain the peculiar kinetic energy of the wall particles K p2=2m d N k T =2; 1:5 W ² i ˆ C W B w … † SXiˆ1 with N S, in which case we speak of isothermal dynamics. The quantity T W i W ˆ de®ned by this relation is called the kinetic temperature of the wall, and d is the C P Cartesian dimension of the system. For homogeneously thermostatted systems, T W becomes the kinetic temperature of the whole system and N becomes just the W number of particles N, in the whole system. For ergostatted dynamics, the thermostat multiplier, ¬, is chosen as the instantaneous solution to the equation, The Fluctuation Theorem 1537 HH_ C J C V· F 2K C ¬ C 0; 1:6 0 e W … †²¡ … † ¡ … † … †ˆ … † where J is the dissipative ¯ux due to F de®ned as e p HH_ad JV· F i · D F · C · F ; 1:7 0 ²¡ e ²¡ m i¡ i i e … † µ ¶ X HH_ad istheadiabatictimederivativeoftheinternalenergyandV isthevolumeofthe 0 system. Equation (1.6) is a statement of the First Law of Thermodynamics for an ergostatted non-equilibrium system. The energy removed from (or added to) the system by the ergostat must be balanced instantaneously by the work done on (or removed from) the system by the external dissipative ®eld, F . For ergostatted e dynamicswesolve(1.6)fortheergostatmultiplierandsubstitutethisphasefunction into the equations of motion. For thermostatted dynamics we solve an equation which is analogous to (1.6) but which ensures that the kinetic temperature of the walls or system, is ®xed [16]. The equationsof motion (1.4) are reversible wherethe thermostat multiplier is de®ned in this way. One might object that our analysis is compromised by our use of these arti®cial (time reversible) thermostats. However, the thermostat can be made arbitrarily remote from the system of physical interest [17]. If this is the case, the system cannot `know’ the precise details of how entropy was removed at such a remote distance. This means that the results obtained for the system using our simple mathematical thermostat must be the same as those we would infer for the same systemsurrounded(atadistance)byarealphysicalthermostat(saywithahugeheat capacity). These mathematical thermostats may be unrealistic, however in the ®nal analysis they are very convenient but ultimately irrelevant devices. Using conventional thermodynamics, the total rate of entropy absorbed (or released!) by the ergostat is the energy absorbed by the ergostat divided by its absolute temperature, S t 2KW C ¬ C =TW t dCNWkB¬ t J t V· Fe=TW t : 1:8 … †ˆ … † … † … †ˆ … †ˆ¡ … † … † … † The entropy ¯owing into the ergostat results from a continuous generation of entropy in the dissipative system. Theexactequationof motionfortheN-particledistribution functionisthe time reversible Liouville equation @f C ;t @ … † · CC_f C ;t ; 1:9 @t ˆ¡@C ‰ … †Š … † which can be written in Lagrangian form, df C ;t d … † f C ;t · CC_ L C f C ;t : 1:10 dt ˆ¡ … † dC ²¡ … † … † … † Thisequationsimplystatesthatthetimereversibleequationsofmotionconservethe numberof ensemblemembers,N .Thepresenceof thethermostatisre¯ectedinthe C phase space compression factor, L C @CC_ · =@C , which is to ®rst order in N, … †² L dCNW¬.Againonemightwonderaboutthedistinction betweenHamiltonian ˆ¡ dynamicsofrealisticsystems,wherethephasespacecompressionfactorisidentically zero and arti®cial ergostatted dynamics where it is non-zero. However, as Tolman pointed out [18], in a purely Hamiltonian system, the neglect of `irrelevant’ degrees of freedom (as in thermostats or for example by neglecting solvent degrees of freedom in a colloidal or Brownian system) inevitably results in a non-zero phase 1538 D. J. Evans and D. J Searles space compression factor for the remaining `relevant’ degrees of freedom. Equation (1.8) shows that there is an exact relationship between the entropy absorbed by an ergostat and the phase space compression in the (relevant) system. 1.3. Example: SLLOD equations for planar Couette ¯ow A very important dynamical system is the standard model for planar Couette ¯owÐthe so-called SLLOD equations for shear ¯ow. Consider N particles under shear.Inthissystemtheexternal®eldistheshearrate,@u =@y ® t (they-gradient x ˆ … † of the x-streaming velocity), and the xy-element of the pressure tensor, P , is the xy dissipative¯ux,J[16].Theequationsofmotionfortheparticlesaregivenbythethe so-called thermostatted SLLOD equations, qq_ p=m i®y; pp_ F i®p ¬p: 1:11 i ˆ i ‡ i i ˆ i¡ yi ¡ i … † Here, i is a unit vector in the positive x-direction. At arbitrary strain rates these equationsgive an exactdescriptionofadiabatic(i.e. unthermostatted)Couette¯ow. This is because the adiabatic SLLOD equations for a step function strain rate @ux t =@y ® t ®Y t , are equivalent to Newton’s equations after the impulsive … † ˆ … †ˆ … † impositionof alinearvelocity gradientat t 0(i.e. dq 0 =dt dq 0 =dt i®y) ˆ i… ‡† ˆ i… ¡† ‡ i [16]. There is thusa remarkablesubtletyin the SLLOD equations of motion. If one starts at t 0 , with a canonical ensemble of systems then at t 0 , the SLLOD ¡ ‡ ˆ ˆ equations of motion transform this initial ensemble into the local equilibrium ensembleforplanarCouette¯owatashearrate®.TheadiabaticSLLODequations therefore give an exact description of a boundary driven thermal transport process, although the shear rate appears in the equations of motion as a ®ctitious (i.e. unnatural) external ®eld. This was ®rst pointed out by Evans and Morriss in 1984 [19]. AtlowReynoldsnumber,theSLLODmomenta,p,arepeculiarmomentaand¬ i is determined using Gauss’s Principle of Least Constraint to keep the internal energy, H0 ˆSp2i=2m‡F…q†, ®xed [16]. Thus, for a system subject to pair interactions y N 1 N ¡ F q ¿ qij ; … †ˆ … † i 1 j>i Xˆ X N N N ¬ ® p p =m 1=2 x F p2=m ˆ¡ xi yi ¡ ij yij i µ i 1 i;j ¶¿ i 1 Xˆ X Xˆ N P ®V p2=m P ®V=2K p ; 1:12 ²¡ xy i ˆ¡ xy … † … † ¿ i 1 Xˆ where F is the y-component of the intermolecular force exerted on particle i by j yij andx x x.Thecorrespondingisokineticformforthethermostatmultiplieris, ij j i ² ¡ N N F · p ® p p =m i i¡ xi yi ¬ Xi µXiˆ1 ¶: 1:13 ˆ N … † p2=m i i 1 Xˆ {We limit ourselves to pair interactions only forreasons of simplicity.