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On the Geometry and Entropy of Non-Hamiltonian Phase Space Alessandro Sergi∗ and Paolo V. Giaquinta Dipartimento di Fisica, Universitá degli Studi di Messina, Contrada Papardo 98166 Messina, Italy Weanalyzetheequilibriumstatisticalmechanicsofcanonical,non-canonicalandnon-Hamiltonian 7 equations of motion by throwing light into the peculiar geometric structure of phase space. Some 0 fundamentalissues regarding time translation and phasespace measure are clarified. Inparticular, 0 we emphasize that a phase space measure should be defined by means of the Jacobian of the 2 transformation between different types of coordinates since such a determinant is different from n zeroin thenon-canonicalcase evenifthephasespace compressibility isnull. Instead,theJacobian a determinantassociatedwithphasespaceflowsisunitywhenevernon-canonicalcoordinatesleadtoa J vanishingcompressibility,sothatitsuseinordertodefineameasuremaynotbealwayscorrect. To 4 betterillustrate thispoint,wederiveamathematical condition for definingnon-Hamiltonianphase 2 space flows with zero compressibility. The Jacobian determinant associated with time evolution in phasespaceisaltogetherusefulforanalyzingtimetranslationinvariance. Theproperdefinitionofa ] phasespacemeasureisparticularlyimportantwhendefiningtheentropyfunctionalinthecanonical, h non-canonical,andnon-Hamiltoniancases. Weshowhowtheuseofrelativeentropiescancircumvent c e some subtle problems that are encountered when dealing with continuous probability distributions m and phase space measures. Finally, a maximum (relative) entropy principle is formulated for non- canonical and non-Hamiltonian phase space flows. - t a t s February 2, 2008 the phase space compressibility. However, there are sys- . tems that, while being non-canonical, have a non-zero t a Jacobian and a vanishing compressibility. To show this, m we derive a condition in order to define non-canonical I. INTRODUCTION - and non-Hamiltonian systems with zero compressibility. d Then, we argue that the phase space measure should be n In this paper we address some general issues concern- o defined in terms of the Jacobianpertaining to the trans- ing the geometry of non-Hamiltonian phase space under c formationbetween different types of coordinates (for ex- [ the condition of thermodynamical equilibrium. In the ample, canonical vs. non-canonical coordinates). This field of classical molecular dynamics simulations, non- Jacobian should not be confused with another Jacobian 4 Hamiltonianformalisms havebeen developedin order to v determinant, that is associated with the transformation simulate numerically the effect of thermal and pressure 3 realizedbythe lawofmotion,andwhicharisesnaturally bathsonrelevantsubsystemsbymeansofafinitenumber 4 when addressing the time-translation property of statis- 3 of degrees of freedom [1, 2, 3, 4]. It is also worth men- tical mechanics in phase space. 1 tioningthatthenon-Hamiltonianapproachisthemethod 1 of choice for path integral [5] and ab-initio path integral Another approach to non-Hamiltonian statistical me- 5 molecular dynamics calculations [6]). More recently, it chanicshasbeenproposedinRefs.[13,14]. Itusesgener- 0 has been shown that the theories [7, 8] needed to de- alizedantisymmetricbracketstodefine non-Hamiltonian / t scribeinaconsistentwaythecouplingbetweenquantum equations of motion. A generalized Liouville equation a and classical degrees of freedom are non-Hamiltonian in fordistributionfunctionsinphasespacecanbewrittenin m their veryessence[9, 10]. Despite the evergrowingnum- suchawaythatitssolutionsnaturallyprovidethestatis- d- berofsuccessfulapplicationsofnon-Hamiltoniantheories tical weight of phase space. In addition, linear response n to molecular dynamics simulations, a clarification is de- theory[13,14]aswellasextensionsofthetheorytoquan- o sirable since classical non-Hamiltonian theories in phase tum and quantum-classical mechanics [8] can be easily c spacehavebeentypicallyformulatedbymeansoftwodis- formulated. Suchanapproachhasanalgebraicstructure : v tinct approaches. In one case, phase space is considered whose distinctive feature is the violation of the Jacobi i as a Riemann manifold, endowed with a metric tensor relation [15, 16, 17, 18]. It is important to note that X whose determinant is used to define the measure of the thefailureoftheJacobirelationimpliesthelackoftime- r volume element [11, 12]. Within this type of approach translationinvarianceofthealgebra[9]. Thismeansthat a it is not clear how to derive non-Hamiltonian equations if two arbitrary phase space observables obey a relation of motion but, once these are given, their statistical me- expressed by means of a generalized bracket at time t0, chanicsisdefinedbyintroducinganinvariantmeasureof then,ingeneral,suchtwoobservableswillnotsatisfythe phase space. As a matter of fact, the metric of phase same relation at time t6=t0. This leads to consequences space has been exclusively linked in Refs. [11, 12] to intheproperdefinitionoftheconstantsofmotion(forex- ample,thegeneralizedbracketoftwoconstantsofmotion isnotnecessarilyaconstantofmotion)andinthedefini- tion ofcorrelationfunctions. An example of the applica- ∗E-mail: [email protected] tion of a similar philosophy to address non-Hamiltonian 2 phase space can be found in Ezra’s work [19, 20], where non-Hamiltonian phase-space flows with zero com- the elegant formalism of differential forms and exterior pressibility. derivatives has been used. More recently, such an ap- (ii) Given that the compressibility is not a critical fea- proach has been re-expressed using the antisymmetric ture of non-Hamiltonian phase-space flows, we in- matrix structure introduced in Refs. [13, 14] in order to troduceamoreproperdefinitionofthephase-space devise measure-preserving algorithms for the numerical measurebymeansoftheJacobianofthecoordinate integrationofnon-Hamiltonianequations ofmotion[21]. transformation. Another issue that is particularly relevant for non- Hamiltonian statistical mechanics is the covariant def- (iii) We show that using the Jacobian determinant inition of the entropy functional. When dealing with J(t,t ) that is associated with the equations of 0 non-canonical and non-Hamiltonian systems with zero motion in order to define the phase-space measure compressibility,caremustbepayedbecause,asdiscussed can be misleading when the compressibility van- above, the conclusion that the measure is trivial may be ishes. We also show that this determinant is rele- erroneous [11, 12]. Instead, the phase space Jacobian, vantforthe invariancepropertiesofthe statistical- whichis different fromunity whenever non-canonicalco- mechanical averages with respect to time transla- ordinates are adopted, should be used to define the cor- tion. rect measure of phase space and the covariant entropy functional. However, such a functional, as already dis- (iv) We introduce a maximum-entropy approach for cussedinRefs.[22, 23,24], canbe definedwithout anya non-Hamiltonian systems. priori knowledge of the metric factor. The key point is (v) Weadopttheconceptofrelativeentropyinorderto that relative entropies are called for when dealing with ensure covariance in phase space. As a by-product continuous probability distributions. In fact, a relative ofthisanalysis,weprovethatRamshaw’scovariant entropy functional naturally accounts for unknown met- entropy [23] is actually a relative entropy. ric factors. Therefore,amaximumentropyprinciple can be introduced for non-canonical and non-Hamiltonian The paper is divided into two main sections. dynamics. Section II treats the peculiar geometry of phase space Since the (relative)entropyproductionis triviallynull and its influence on dynamics and statistical mechan- under the condition of thermodynamical equilibrium, a ics. In Sec. IIA, the canonical Hamiltonian phase space comment is called for. Indeed, it is worth underlining is briefly reviewed. The geometry of canonical phase that, while presenting results that aim at clarifying the space and its statistical mechanics are shortly discussed natureofnon-Hamiltonianequilibriumgeometry,wealso by means of Poissonbrackets. The invariance of equilib- intend to set the pre-conditions to cope with the non- rium statistical mechanics under time translation is dis- equilibrium case within an information theoretical per- cussed. Section IIB generalizes the result of Sec. IIA to spective (for a generalintroduction see Ref. [25] and ref- non-canonicalHamiltonianphasespaceflowsbyemploy- erences contained therein). Within such a framework, it ing non-canonicalbrackets. SectionIIC showsthat non- is not really important to describe the fine-grained com- Hamiltonianstatisticalmechanicsisnottime-translation plexity of phase space under non-equilibrium conditions: invariant. Conditionsinordertoobtainnon-Hamiltonian whatmattersistheabilitytoidentifythestatisticalcon- phase space flows with zero compressibility are also de- straints which allow one to perform calculations with a rived. Appendices A and B provide examples of simple predictive value. non-canonical and non-Hamiltonian dynamics with zero This approach is different from another approach to compressibility. non-equilibriumstatisticalmechanics[26]which,instead, Section III is devoted to a discussion of the relative aims at describing in a complete way the fine-grained entropy functional, of its covariance, and of a formula complexity of phase space out of equilibrium. This lat- for its production rate. In Sec. IIIA the covariant rela- ter approachhas led, in recent years, to some important tive entropy is illustrated and is used in order to derive achievements such as the discovery of the fractal nature a “maximum relative entropy” principle. Such results of phase space in non-equilibrium steady states [27] and are extended to the non-canonical and non-Hamiltonian ananalysisof some interestingstatistical models suchas casesinSec.IIIB. Aformulafortherateofproductionof Anasov’s systems [28]. However, these systems are dif- the relative entropy is derived in Sec. IIIC. Concluding ferent from those investigated by chemical physics (see remarks are given in Sec. IV. [29]forsomerecentexamples). However,webelievethat looking at the fine-grained nature of phase space or in- voking information theoretic techniques should not be II. GEOMETRY OF PHASE SPACE considered as mutually exclusive paths of investigation. Clarifying the above issues is not the only goal of this A. Canonical Hamiltonian Phase Space paper. For an easier reading of the paper, we anticipate the results we shall illustrate in the following: Let x = (q,p) denote a point in phase space, where q (i) We derive the conditions for obtaining a class of and p are generalized coordinates and momenta, respec- 3 tively. LetH(x)bethe(Hamiltonian)generalizedenergy is the corresponding operator along the p axis (carrying function of the system. We consider, here and in the infinitesimal changes of the generalized momenta). As following section, only the case in which H(x) is time- a matter of fact, it can be easily verified that Tˆ a(x) = q independent. Let 2n be the dimension of phase space. (∂a/∂q)δq and Tˆ a(x) = (∂a/∂p)δp. It is also easy to Then, the 2n×2n antisymmetric matrix Bs, or cosym- p verify that, because of the canonical relations in (8), plectic form [15, 16, 17, 18], reads as translations along the axis of different generalized coordinates, positions and momenta, commute. This means 0 1 Bs = . (1) that canonical phase space is flat even if generalized co- −1 0 ordinates are used and even if the Lagrangian manifold (cid:20) (cid:21) from which one builds phase space is a Riemann mani- Correspondingly,thecanonicalHamiltonianequationsof fold [16]. motion are given by In order to highlight the novel features which come into play in the formalism of the non-Hamiltonian case, 2n x˙ = Bs ∂H , (2) we think it useful to sketch briefly the equilibrium sta- i ij∂xj tistical mechanics of Hamiltonian canonicalsystems in a j=1 X form that can be easily generalized to the non-canonical for i = 1,...,2n. Equations (2) can be derived from a and non-Hamiltonian cases and that, by clearly distin- variational principle in phase space [30] applied to the guishing between the Jacobian J(x) and the Jacobian action written in symplectic form determinantJ˜(t,t0), isalsosuitedtodiscusstime translation invariance. 2n It is well known that the Liouville operator can be 1 A= dt 2x˙iBisjxj −H . (3) introducedusingPoissonbrackets,iLˆ ={...,H}Bs. The Z i,j=1 Liouvilleequationforthestatisticaldistributionfunction X   in phase space is written as [31]: The compressibility of phase space ∂ρ(x) =−iLˆρ(x). (11) 2n ∂x˙ 2n ∂Bs ∂H ∂t i ij κ = ∂x = ∂x ∂x (4) InEquation(11),thefunctionρ(x)=Jcanf(x)hasbeen i i j Xi=1 iX,j=1 introduced, where f(x) is the true distribution function iszerobecausethematrixBs isconstant. Poissonbrack- in phase space and Jcan =1 is the Jacobianof transfor- mationsbetweencanonicalcoordinates. Inthe canonical ets can be defined as case, the identity ρ(x) = f(x) trivially follows; however, 2n this notation will turn out to be convenient later. By ∂a ∂b {a(x),b(x)}Bs =iX,j=1∂xiBisj∂xj , (5) mpreoapnasgaotfortheexpL[iiotuLˆv]il=leeoxppe[rita{t.o.r.,oHne}Pc]anwhinotsreodauctcieonthies defined by where a(x) andb(x) are arbitraryphase space functions, a(x(t))=exp[itLˆ]a(x), (12) so that the equations of motion follow in the form where x without the time argument denotes its value at x˙i ={xi,H}Bs . (6) time zero. Statistical averages are calculated through The Jacobi relation ha(x)i= dxρ(x)a(x) , (13) {a,{b,c}Bs}Bs+{c,{a,b}Bs}Bs+{b,{c,a}Bs}Bs =0 (7) Z and correlation functions as is satisfied as an identity in canonical coordinates: hab(t)i= dxρ(x)a(x)exp[itLˆ]b(x). (14) {xi,xj}Bs =Bisj . (8) Z We recall that in both Eqs. (13) and (14) the averag- It is well-known that Poisson brackets can be used to ing over phase space coordinates is done by considering realize infinitesimal contact transformations [15, 16, 17]. the coordinates calculated at the initial time. Indeed, In particular the law of motion which carries x(0) into x(t) is another transformation of coordinates Tˆq ={...,p}Bsδq (9) dx(0)=J˜(t,0)dx(t), (15) is the operator realizing infinitesimal translations along where the Jacobian determinant satisfies the condition the q axis and ∂x(0) Tˆp =−{...,q}Bsδp (10) J˜(t,0)≡|∂x(t)|=1. (16) 4 At equilibrium where ρ (t)=e−iLtρ (t)=ρ =0, (17) 2n eq eq eq ∂z ∂z B = mBs k . (25) and mk ∂xi ij∂xj i,j=1 X ha(x)i = ha(x(t))i. (18) Equations of motion can also be obtained by means of The derivation of an analogous result for the correlation the variationalprinciple [30] which arises when applying functions ismoresubtle andwillpermitustomakelater the non-canonical transformation of coordinates to the important distinctions with the non-Hamiltonian case. symplecticexpressionoftheactiongiveninEq.(3). One Inthe Hamiltoniancanonicalcase,we getfromEq.(14): obtainsthefollowingformfortheactioninnon-canonical coordinates [30]: hab(t)i = dxρ (x)a(x)b(x(t)) eq 2n Z A= dt 1 ∂xi z˙ Bsx (z)−H′(z) , (26) = dx(t)ρeq(x(t)) e−itLa(x(t)) b(x(t)) Z 2i,j,m=1∂zm m ij j  X Z (cid:2) (cid:3)   = dx(t)ρ (x(t))a(x(t))eiLtb(x(t)). (19) on which the variation is to be performed on the z co- eq Z ordinates in order to obtain Eqs. (24). Poisson brackets We consider the special case in which become non-canonical brackets defined by b(x(0))={c(x(0)),d(x(0))}Bs , (20) {a′,b′}= 2n ∂a′(z)B (z)∂b′(z) , (27) ij where c(x) and d(x) are two arbitrary phase space func- ∂zi ∂zj i,j=1 tions. It is easy to prove, that if the Jacobi relation (7) X holds, then where a′(z) = a(x(z)). Non-canonical equations of mo- tioncanbeexpressedbymeansofthebracketinEq.(27) eitL{c(x(0)),d(x(0))}Bs ={c(x(t)),d(x(t))}Bs . (21) asz˙i ={zi,H′(z)}B. With alittle bitofalgebra,onecan Hence, it follows: verify that non-canonical brackets satisfy the Jacobi relation as an identity. The Jacobi relation leads to the ha (x) {c(x(t)),d(x(t))}Bsi following identity for B(z): = dx(t)ρeq(x(t))a(x(t))eitL{c(x(t)),d(x(t))}Bs 2n ∂B (z) ∂B (z) jk ij Z Sijk(z) = Bil +Bkl ∂z ∂z = dx(t)ρeq(x(t))a(x(t)){c(x(2t)),d(x(2t))}Bs Xl=1(cid:18) l l Z ∂Bki(z) = ha(x(t)){c(x(2t)),d(x(2t))}Bsi. (22) + Bjl ∂zl (cid:19)=0. (28) Equation (22) shows that, at equilibrium and in the The non-canonical brackets of phase space coordinates Hamiltonian canonicalcase, correlationfunctions are in- are given by variant under time translation. We remark that the Ja- cobi relation(7) is necessaryto derive this result so that {z ,z } =B (z). (29) itcanbeeasilyforeseenthatinthenon-Hamiltoniancase, i j B ij where the Jacobi relation no longer holds, this property Uponidentifying the phasespacepointasz ≡(ξ,ζ), one of time-correlationfunctions is no longer verified. can define the operators Tˆ = {...,ζ} δξ , (30) ξ B B. Non-canonical Hamiltonian Phase space Tˆ = −{...,ξ} δζ , (31) ζ B Let us consider a transformation of phase space coor- which realize infinitesimal translations along the axes ξ dinates z =z(x) such that the Jacobian and ζ. The non-canonical bracket relations of Eq. (29) ∂x implythattranslationsalongthephasespaceaxisingen- J =k k6=1. (23) eral do not commute or, in other words, phase space is ∂z curved. Notice we do not need to introduce a metric Coordinateszarecallednon-canonical. TheHamiltonian tensor. One just has to introduce parallel transport and transformsasascalarH(x(z))=H′(z)andtheequations an affine connection, which can be implicitly defined by of motion become [16, 18] means of the non-canonicalbracketsand of the infinitesimal translation operators Tˆ and Tˆ . 2n ∂H′(z) ξ ζ z˙ = B (z) , (24) Non-canonical phase spaces are obtained by means of m mk ∂zk anon-canonicaltransformationofcoordinatesappliedto k=1 X 5 canonical Hamiltonian systems. Suppose that there is a where wehave usedthe fact thatthe phase spaceflow in system with a non-canonical bracket satisfying the Ja- thexcoordinatesiscanonicalsothatk∂x(t)/∂x(0)k=1. cobi relation or its equivalent form given in Eq. (28). TheuseoftheJacobianprovidesthecorrectwayofdefin- Suppose also that detB 6= 0. Then, by means of Dar- ing the invariant measure because it can also be applied boux’s theorem the system can be put (at least locally) when there is no compressibility [30]. However, the cal- inacanonicalform. One wouldclassifysucha systemas culation of averages, correlation functions and linear re- Hamiltonian: following Refs. [16, 18], we think that this sponsetheorydoes notrequirethe invariantmeasureex- property follows from the validity of the Jacobi relation. plicitly. The Knowledge of ρ(z) is enough for statistical Inotherwords,ifthealgebraofbracketsisaLiealgebra, mechanics even in the non-Hamiltonian case and in the then the phase space is Hamiltonian [18]. presence of constraints, as shown in Refs. [8, 13, 14]. For non-canonical systems, statistical mechanical av- It is interesting to analyze the time-translationinvari- erages can be calculated as ance of non-canonical statistical mechanics. Under time evolution,the phasespace volumeelementtransformsas ha′(z)i= dzρ(z)a′(z(t)), (32) dz(0)=J˜(t,0)dz(0), (40) Z where the Jacobian determinant where ∂z(0) J˜(t,0)=| | (41) ρ(z)=J(z)f′(z), (33) ∂z(t) is different from unity if and only if the compressibil- the Jacobian J(z) being defined in Eq. (23). The Liou- ity κ is different from zero. As the simple example de- ville operator is defined by means of the non-canonical veloped in Appendix A shows, one can also have non- bracket canonical equations of motion with zero compressibility. iL′ ={...,H′(z)} , (34) However,forthesakeofcomparisonwiththepaperscited B inRef.[11],weshallexplicitlyconsiderthe caseinwhich while time propagationis given by κ6=0 so that J˜(t,0)=− tdt′κ(t′). In this case and at 0 equilibrium a′(z(t)) = exp[itL′]a′(z) R = exp[it{...,H′(z)}B]a′(z). (35) ha(z(0))i = dz(0)ρeq(z(0))a(z(0)) Z In non-canonical coordinates a compressibility = dz(t)J˜(t,0)ρ (z(0)) e−itLa(z(t)) . eq 2n ∂Bij(z)∂H′(z) Z (cid:2) (cid:3)(42) κ(z)= (36) ∂z ∂z i,j=1 i j The law of evolution of the Jacobian in Eq. (41) is: X d might, but not necessarily, be present (see Appendix A ln J˜(t,0) =−κ(t). (43) forasimpleexampleofanon-canonicalsystemwithzero dt (cid:16) (cid:17) compressibility). Integrating by parts Eq. (32), one ob- Now, let tains dw(x(t)) =κ(x(t)). (44) dt ha′(z)i= dz a′(z)exp[−t(iL′+κ(z))]ρ(z). (37) From Eq. (43) it can be seen that the function w, which Z is the primitive function ofκ, certainlyexists. Then, the Equation (37) implies that ρ(z) obeys the non-canonical Jacobian determinant can be written as Liouville equation J(t,0)=e−w(x(t))ew(x(0)) . (45) ∂ρ(z) = −(iL′+κ(z))ρ(z) The equilibrium distribution function follows as ∂t 2n ∂ ρeq(z(0))=Z−1e−w(0)δ(H(z(0))−C) (46) = (z˙ ρ(z)). (38) ∂z i Then, substituting Eqs. (46) and (45) into Eq. (42), we i i=1 X obtain: It is easy to see that dM(z)=J(z)dz provides the cor- ha(z(0))i rect invariant measure [30]: δ(H(z(0))−C) = dz(t)e−w(x(t)) e−itLa(z(t)) ∂x(t) ∂z(t) ∂x(t) ∂x(0) Z dz(t)k k = k kdz(0)k kk k Z ∂z(t) ∂z(0) ∂x(0) ∂z(t) δ(H(z(t))−C) (cid:2) (cid:3) = dz(t)e−w(x(t)) e−itLa(z(t)) , ∂x(0) Z = dz(0)k k, (39) Z ∂z(0) (cid:2) (cid:3)(47) 6 where the second equality follows because the Hamilto- because of Eq. (50). nian is conserved under time evolution. Then, upon in- We consider the lack of time translation invariance to tegrating by parts, one obtains: be the crucial feature of non-Hamiltonian statistical mechanics. ha(z(0))i = dz(t)ρ (t)a(z(t))=ha(z(t))i. (48) eq Z Equation(48) showsthat phasespaceaveragesaretime- 1. Non-Hamiltonian equations of motion with no phase translation invariantin the non-canonicalstatistical me- space compressibility chanicsofsystemsatequilibrium. Considernowthetime correlation function It is easy to realize that there are non-Hamiltonian ha{b(t),c(t)} i = dz(0)ρ (0)a(z(0)){b(t),c(t)} phase space flows (i.e., flows defined by brackets which B eq B do not satisfy the Jacobi relation) with zero compress- Z = dz(t)e−w(t)ew(0)e−w(0)δ(H(0)−C) ibility. We refer the reader to Appendix B for a trivial example. In order to show how this is possible, we Z × e−iLta(z(t)){b(t),c(t)} consider a particular sub-ensemble of non-Hamiltonian B phase space flows, viz., those flows that can be derived = dz(t)ρ (t)a(z(t))eiLt{b(t),c(t)} by means of a non-integrable scaling of time. Interest- eq B Z ingly, it was Nosé who originally considered this type of flowswhenheintroducedhiscelebratedthermostat[2,4]. = dz(t)ρ (t)a(z(t)){b(2t),c(2t)} eq B Nosé started with a canonical Hamiltonian system and Z = ha(t){b(2t),c(2t)} i, (49) thenperformedanon-canonicaltransformation,followed B byanon-integrablescalingoftime. Accordingly,wecon- where the last equality arises again from the validity of sider the non-integrable scaling of time the Jacobian relation (7). Equation (49) shows that, whenthebracketsatisfiestheJacobirelation,theequilib- dt=Φ(z)dτ , (52) rium statistical mechanics is invariant under time translation. We wouldlike to recallthat an algebraexpressed where τ is an auxiliary time variable. Such a scaling of by brackets satisfying the Jacobi relation is a Lie alge- dt is clearly non-integrable because, due to the depen- bra. Therefore,itisreasonabletoconsideraphasespace dence of dt on phase space coordinates, the integral dt theory observing such a property as Hamiltonian. dependsonthepathinphasespace. Ifweapplythisscal- R ingtoEq.(24),weobtainthefollowingnon-Hamiltonian equations: C. Non-Hamiltonian Phase Space dz 2n ∂H′(z) In Refs. [13, 14] it was shown how non-Hamiltonian i = Φ(z)B (z) ij equations of motion, brackets and statistical mechanics dτ ∂zj j=1 can be defined. One must simply keep the generalized X symplectic structure of the non-canonical equations in = 2n B˜ (z)∂H′(z) . (53) Eq. (24) and of the non-canonical bracket in Eq. (27) ij ∂z j anduse,inplaceofB(z),anantisymmetricmatrixB˜(z). Xj=1 ThematrixB˜(z)canbechosenarbitrarily,withthe only UsingtheantisymmetricmatrixBãsdefinedinEq.(53), constraint of being antisymmetric so that the Hamilto- one can introduce a non-Hamiltonian bracket nian is conserved. As a result the Jacobi relation may notbe satisfied[13]. When this happens,one ofthe con- 2n ∂a′ ∂b′ ditions of validity of the Darboux’s theorem fails so that {a′,b′} = B˜ (z) . (54) non-Hamiltonian phase space flows cannot be put into a B˜ ∂zi ij ∂zj i,j=1 canonicalform. ThefailureoftheJacobirelationimplies X eiLt{a(0),b(0)} 6={a(t),b(t)} . (50) This bracket does not satisfy the Jacobi relation so that B˜ B˜ the equations of motion (53) are non-Hamiltonian. In Hence, even if the non-Hamiltoniantheory has a vanish- this case,there is a non-vanishingtensor associatedwith ing compressibility,the equilibriumstatisticalmechanics the Jacobi relation: is not time translation invariant. To show this, we note that: 2n ∂B˜ (z) ∂B˜ (z) S˜ = B˜ (z) jk +B˜ (z) ij ijk il kl ∂z ∂z h{a(0),b(0)}B˜i = dz˜(0)ρ˜eq(0){a(0),b(0)}B˜ Xl=1 l l Z ∂B˜ (z) + B˜ (z) ki 6=0. (55) 6= dz˜(t)ρ˜eq(t){a(t),b(t)}B˜ (51) jl ∂zl ! Z 7 The compressibility of the non-Hamiltonian equa- However, as discussed in Refs. [22, 23, 24], the defini- tions (53) is given by tion given in Eq. (61) is not acceptable because it is not coordinate independent. 2n ∂(dz /dτ) 2n ∂B˜ ∂H′(z) When studying continuous probability distributions, i ij κ˜(z) = = oneshouldusetherelativeentropy,whichisameasureof ∂z ∂z ∂z i i j Xi=1 iX,j=1 the information[24] relativeto a state ofignorance(rep- 2n resentedbyagivendistributionfunction). Ifonedenotes ∂Φ(z)dz i = +Φ(z)κ(z), (56) thislatterdistributionbyµ(x),therelativeentropyreads ∂z dt i=1 i as X where dzi/dt andκ aregivenbythe non-canonicalequa- ρ(x) S [ρ]=−k dxρ(x)ln . (62) tions of motion prior to the non-integrable time-scaling. rel B µ(x) It is evident that whenever one chooses Φ(z) in such a Z (cid:18) (cid:19) way that For canonical Hamiltonian systems, a convenient distribution function with respectto whichone candefine the 2n ∂lnΦ(z)dzi relativeentropy,isgivenbythe distributionrepresenting =−κ(z), (57) ∂z dt the state of absolute ignorance, i.e., the uniform distri- i i=1 X bution. Then, one can set µ(x)=1, so that, in practice, non-Hamiltonian flows have vanishing compressibility Srel[ρ] in Eq. (62) coincides with the absolute entropy (see the trivial example given in Appendix B). Cor- S[ρ] in Eq. (61). However, it must be realized that a respondingly, when the scaling is chosen according to uniform distribution in canonical coordinates does not Eq. (57), the distribution function will obey a non- necessarily transform into another uniform distribution Hamiltonian equation without a compressibility: if more general coordinates (for example non-canonical) are used. Instead, the integralin Eq. (62) is well defined ∂ρ˜(z) and its value does not depend on a specific choice of co- = −{ρ˜,H′(z)} =−iL˜ρ˜(z). (58) ∂τ B˜ ordinates. For example, considering the transformation x→y, one has: III. RELATIVE ENTROPY dxρ(x)ln ρ(x) = dyρ′(y)ln ρ′(y) , (63) µ(x) µ′(y) Z (cid:18) (cid:19) Z (cid:18) (cid:19) A. The Hamiltonian case where dxρ(x)=dyρ′(y) and dxµ(x)=dyµ′(y). As is well known, the definition of the entropy func- The relative entropy Srel[ρ] in Eq. (62) is a measure tional for systems with continuous probability distribu- of missing information and can be used as the starting tion, needs a special care. We review, for the reader’s point of a maximum entropyprinciple in order to obtain convenience,somerelevantdefinitionswhichholdforthe the form of the least biased or maximum non-committal Hamiltoniancaseandwhichweplantogeneralizetonon- distributionfunctionρ(x). Tothisend,uponconsidering Hamiltonian systems in the following sections. thetwostatisticalconstraintshHi=Eand dxρ(x)=1, one is led to consider the quantity In order to be rigorous, one first assumes that phase R space can be divided into small cells of volume ∆(i) so that the coordinates in the i-th cell are denoted as x(i). I =S [ρ]+λ(E−hHi)+γ 1− dxρ(x) , (64) rel This way, phase space is effectively discretized and so is (cid:18) Z (cid:19) the distribution function: ρ(x(i)) ≡ ρ(i). The absolute information entropy can be defined as [24, 32]: where two Lagrangian multipliers (λ and γ) have been introduced. Upon maximizing I with respect to ρ(x), S[ρ]=−k ρ(i)lnρ(i) , (59) the following expression for ρ(x) is easily recovered: B i X λ ρ(x)=Z−1µ(x)exp − H(x) , (65) where kB is Boltzmann’s constant. The continuous limit (cid:20) kB (cid:21) where Z = dxµ(x)exp[−(λ/k )H(x)]. Equation (65) S[ρ]= lim −k ρ(i)lnρ(i) , (60) B ∆(i)→0 B ! generalizesthestandardmaximumentropyprinciple[32]. i R X to the case of the relative entropy of continuous proba- diverges. Upon subtracting the divergent contribution, bility distributions. In the canonical Hamiltonian case, −kln∆(i), one obtains the finite expression which has been treated in this section, µ(x) is trivially theuniformdistribution. However,fornon-canonicaland non-Hamiltonian phase space µ(x) plays a fundamental S[ρ]=−k dxρ(x)lnρ(x). (61) B role. Z 8 B. Non-canonical and non-Hamiltonian relative C. Relative entropy production entropy In order to simplify the notation, we rewrite the Liou- In the non-canonical and non-Hamiltonian case a Ja- ville equation as cobian enters the definition of the phase space volume element. Accordingly, one should write a coordinate- ∂ρ(x,t) =−∇·(ρx˙), (72) independent entropy functional as ∂t where we have introduced an obvious vectorial notation SJ = −kB dzJ(z)f′(z)lnf′(z) (toavoidindices)and∇=∂/∂xistheoperatorofdiffer- Z entiation with respect to phase space coordinates. From ρ(z) = −k dzρ(z)ln . (66) the relative entropy functional B J(z) Z (cid:18) (cid:19) Thisexpressionisrecoveredstartingfromtherelativeen- S =−kB dxρln γ−1ρ , (73) tropy. Anintrinsic(coordinate-free)formofEq.(66)has Z (cid:0) (cid:1) beengiveninRef.[20]. Inordertoseethis,onejustneeds the entropy production follows as to transform Eq. (62) into non-canonicalcoordinates: S˙ = −k dx −∇·(ρx˙)ln(γ−1ρ) µ(x)dx = 1·dx=J(z)dz (67) B f(x)dx = f′(z)J(z)dz =ρ(z)dz . (68) + γ γ−Z1∂ ρ(cid:2)−ργ−2∂ γ . (74) t t so that Eq. (66) naturally follows. If the Jacobian J(z) The term dx∂ ρ(cid:0)vanishes because o(cid:1)f(cid:3)the normalization t is not known, one can use any other distribution func- condition on ρ. Hence tion with respect to which the entropy is calculated, R i.e., m(x)dx = m′(z)J(z)dz. Let us define µ(z) = m′(z)J(z), in analogy with ρ(z) = f′(z)J(z). One can S˙ = −kB dx ρx˙ ·∇ln(γ−1ρ)−ργ−1∂tγ . (75) think of µ(z)as the solution ofa Liouville equationwith Z (cid:2) (cid:3) different interactions. For example, if iL′ = iL′ +iL′, The entropy production can then be put in the form 0 I one may define µ(z) as the solution of the equation ∇ρ ∇γ S˙ = k hγ−1∂ γi−k dxρx˙ · − .(76) B t B ∂µ(z) ρ γ =−(iL′ +κ )µ(z), (69) Z (cid:18) (cid:19) ∂t 0 0 We can show that Eq. (76) coincides with the formula with κ = κ0 +κI. The entropy determined by the ad- of the covariant entropy production formerly given by ditionalinteractions,representedbyiL′I, with respect to Ramshaw[23]when γ =ρf−1. To this end, upon noting the state where the term iL′I is absent, is given by that ρ(z) ρ−1∇ρ = f−1∇f +γ−1∇γ (77) S [ρ|µ]=−k dzρ(z)ln . (70) rel B µ(z) Z (cid:18) (cid:19) we obtain: Equation(70), with the correctinterpretationofthe dis- ∇ρ ∇γ tribution function µ(z), provides a coordinate-invariant dxρx˙ · − = − dxργ−1∇·(γx˙).(78) definition of the relative entropy which does not require ρ γ Z Z the explicit knowledge of the metric as well as of the Ja- (cid:2) (cid:3) In the general case, we obtain from Eq. (76) cobian. The maximum-entropy principle, as written in the previous section for the canonical case, also applies 1 ∂γ without major changes to the non-canonical and non- S˙ = kBh +x˙ ·∇γ i+kBhκi, (79) γ ∂t Hamiltonian dynamics. The functional to be maximized (cid:18) (cid:19) is and upon noting that I =S [ρ|µ]+λ(E−hH′(z)i)+γ 1− dzρ(z) , 1 rel hκi = dxρ (γ∇·x˙) . (80) (cid:18) Z (cid:19) γ Z which provides,by setting δI/δρ(z)=0, the generalized the relative entropy production can be written as canonical distribution in non-canonical coordinates S˙ =k hω(x,t)i, (81) λ B ρ(z)=Z−1µ(z)exp − H′(z) . (71) µ (cid:20) kB (cid:21) where The quantity Z = dzµ(z)exp[−(λ/k )H′(z)] is a 1 ∂γ µ B ω(x,t)= +∇·(γx˙) . (82) weighted partition function. γ ∂t R (cid:20) (cid:21) 9 Equation (81) is identical to the covariant entropy pro- dynamical ensembles. duction given by Ramshaw [23]. Ezra [20] has also pro- videdacoordinate-freeexpressionofEq.(81). Suchcom- Acknowledgment parisons are presented to validate our information theo- We thank Professor Gregory S. Ezra for his useful com- retical approach to non-Hamiltonian systems at equilib- ments and for a critical reading of the manuscript. rium. When ω = 0, i.e., if γ is a solution of the Liou- ville equation, then S˙ = 0. It is also clear that in an equilibrium ensemble (∂tρ = 0, ∂tf = 0, ∂tγ = 0) the APPENDIX A: A NON-CANONICAL SYSTEM production of (relative) entropy is trivially null. WITH ZERO COMPRESSIBILITY Consider the following simple Hamiltonian IV. CONCLUSIONS p2 p2 1 H= 1 + 2 + (q −q )2 , (A1) 1 2 2 2 2 Thegeometryofphasespaceispeculiar. Antisymmet- ric brackets, which define a Lie algebra (in the canon- whose canonical equations of motion can be easily writ- ical and non-canonical cases) or a non-Lie algebra (in tendown. Consider,instead,thefollowingnon-canonical the non-Hamiltonian case), can be used to connect, by transformation of coordinates x = (q1,q2,p1,p2) → z = meansofinfinitesimal“contact”transformations,nearby (ξ1,ξ2,π1,π2) defined by points over the manifold. A distinctive feature of non- q = ξ ξ−1 (A2) Hamiltonianequilibriumstatisticalmechanicsis thelack 1 1 2 q = ξ (A3) of time-translation invariance. This is mathematically 2 2 represented by the failure of the Jacobi relation. As a p1 = ξ2π1 (A4) matter of fact, whenever a bracket satisfies the Jacobi p = π . (A5) 2 2 relation, the equilibrium statistical mechanics is time translation invariant and the bracket realizes a Lie al- By using this transformation of coordinates onto the gebra. Wesurmisethatsuchtheoriesshouldbeclassified canonical equation of motion, one obtains the following as Hamiltonian. non-canonical equations of motion We argue that a non-vanishing phase space com- ξ˙ = ξ π ξ−1+ξ2π (A6) 1 1 2 2 2 1 pressibility is not a signature of non-canonical or non- ξ˙ = π (A7) Hamiltonian dynamics. There are cases (and it is worth 2 2 noting that Andersen’s constant pressure dynamics [1] π˙ = −π ξ−1π +ξ−1(ξ −ξ ξ−1) (A8) 1 2 2 1 2 2 1 2 is one of these) where the compressibility is zero but π˙ = −(ξ −ξ ξ−1). (A9) 2 2 1 2 the Jacobian is not unity and the dynamics is non- canonical (or non-Hamiltonian). In particular, we have The Hamiltonian in non-canonicalcoordinates is derived a condition for having a vanishing compressibil- π2 π2 1 ity in a restricted class of non-Hamiltonian phase space H′(z)=ξ22 21 + 22 + 2(ξ1ξ2−1−ξ2)2 . (A10) flows (those flows which are obtained by means of non- One can calculate canonical transformations of coordinates followed by a non-integrable scaling of time [2, 4]). In such cases, it is ∂H′(z) = ξ−1(ξ ξ−1−ξ ) (A11) obvious that the measure of phase space cannot be de- ∂ξ 2 1 2 2 1 rived from the phase space compressibility. Instead, one ∂H′(z) should use the Jacobian of the transformation between = ξ π2−(ξ ξ−1−ξ )(ξ ξ−2+1) (A12) ∂ξ 2 1 1 2 2 1 2 different types of phase space coordinates. However, it 2 ∂H′(z) is not necessary to know explicitly such a Jacobian for = aξ2π (A13) setting up a statistical mechanical theory. In fact, the ∂π1 2 1 distributionfunction andthe Liouvilleoperatorsaresuf- ∂H′(z) = π , (A14) ficient to this end. ∂π 2 2 We remark how, for continuous probability distribu- and write the equations in matrix form tions, one should use the relative entropy functional. The definition of the relative entropy is coordinate in- ξ˙ 0 0 1 ξ ξ−1 1 1 2 dependent and, measuring the state of ignorance rel- ξ˙ 0 0 0 1 ative to another given distribution, does not require π˙21  =  −1 0 0 −π1ξ2−1  the explicit knowledge of the Jacobian. A maximum- π˙  −ξ ξ−1 −1 π ξ−1 0  entropy principle, which applies to the relative entropy,  2   1 2 1 2      has been formulatedin the canonical,non-canonicaland ∂H′(z)/∂ξ 1 non-Hamiltonian case. Such a maximum-entropy princi- ∂H′(z)/∂ξ × 2 . (A15) ple may turn out to be relevant for applications in the ∂H′(z)/∂π  1 non-Hamiltonian dynamics of non-equilibrium thermo- ∂H′(z)/∂π  2    10 Equations (A6-A9) are obviously non-canonical, as it is We define the antisymmetric matrix clearlyseenbytheirmatrixformgiveninEq.(A15),and they have zero compressibility. Hence, metric theories cannotbe applied. The antisymmetric matrix appearing 0 0 ξ−1 −ξ−1ξ 2 2 1 in Eq. (A15) must be used to define the non-canonical 0 0 0 1 bracket,whichobviouslysatisfiestheJacobirelationand B =  −ξ−1 0 0 0  (B11) the Liouville equation.  ξ−12ξ −1 0 0   2 1    APPENDIX B: A NON-HAMILTONIAN SYSTEM WITH ZERO COMPRESSIBILITY whichshouldbeusedinordertodefinethenon-canonical bracket which satisfies the Jacobi relation. Now, if one wantstoapplyanon-integrablescalingoftimeinorderto Let us consider the Hamiltonian of Eq. (A1). We first obtain a non-Hamiltonian flow with zero compressibility obtainnon-canonicalequations of motionby considering κ˜, Eq. (57) can be used. Assuming the scaling function the transformation of coordinates Φ=Φ(ξ ), one obtains 2 q = ξ ξ (B1) 1 2 1 q = ξ (B2) 2 2 ∂Φ p = π (B3) π =ξ−1π , (B12) 1 1 ∂ξ 2 2 2 2 p = π . (B4) 2 2 The Hamiltonian becomes from which one readily finds Φ = ξ . Hence, the anti- 2 π2 π2 ξ2 symmetric matrix B˜ =ΦB is H′ = 1 + 2 + 2(ξ −1)2 . (B5) 1 2 2 2 The non-canonicalequations of motion are 0 0 1 −ξ 1 0 0 0 ξ ξ˙ = ξ−1π −ξ ξ−1π (B6) B˜ = 2 . (B13) 1 2 1 1 2 2 −1 0 0 0  ξ˙2 = π2 (B7)  ξ1 −ξ2 0 0  π˙ = −ξ (ξ −1) (B8) Non-Hamiltonian equations of motionare now defined 1 2 1 accordingto Eq. (53). Finally, it is not difficult to verify ξ˙ = −ξ (1−ξ ). (B9) 2 2 1 that the non-Hamiltonian bracket of Eq. (54), with B˜ definedin Eq.(B13), does notsatisfy the Jacobirelation Such non-canonicalequations have a compressibility and S˜ 6= 0. 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