Viscosity, Reversibillity, Chaotic Hypothesis, Fluctuation Theorem and Lyapunov Pairing
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b Viscosity, Reversibillity, Chaotic Hypothesis,Fluctuation Theorem and Lyapunov PairingGiovanni Gallavotti
February 23, 2021
Abstract : Incompressible fluid equations are studiedwith UV cut-off and in periodic boundary conditions.Properties of the resulting ODEs holding uniformly inthe cut-off are considered and, in particular, are conjec-tured to be equivalent to properties of other time reversibleequations. Reversible equations with the same regulariza-tion and describing equivalently the fluid, and the fluc-tuations of large classes of observables, are examined inthe context of the “Chaotic Hypothesis”, “Axiom C” andthe “Fluctuation Theorem”.
I. ON THE EQUATIONS
The incompressible Euler equation, denoted E , in aperiodic container T d = [0 , π ] d , d = 2 ,
3, for a smoothvelocity field u ( x ) , x ∈ T d is: ˙u ( x ) = − ( u e ( x ) · ∂ e x ) u ( x ) − ∂ x P ( x ) , ∂ x · u ( x ) = 0 , Z T d dx u ( x ) = (1.1)where P = − P di,j =1 ∆ − ( ∂ i u j ∂ j u i ) is the ’pressure’ and∆=Laplace operator.It is also useful to consider the E equations from the“Lagrangian viewpoint”: a configuration of the fluid isdescribed by assigning the dispacement x = q ξ of a fluidelement, from the reference position ξ ∈ T d , and thevelocity ˙ q ξ of the same fluid element. So the state ofthe fluid is ( q , ˙q ) where q is a smooth map of T d toitself and ˙q is a smooth vector field on T d with 0 aver-age. Denote F the space of the dynamical configurations( q , ˙q ) ∈ Dif ( T d ) × Lin ( T d ) = F where Dif ( T d ) is theset of diffeomorphisms of T d and Lin ( T d ) the space ofthe vector fields on T d .Actually we concentrate on the subspace of ( q , ˙q ) ∈ ( SDif ( T d ) × SLin ( T d )) def = S F ⊂ F where the evolu-tion of an incompressible fluid takes place:
SDif ( T d ) be-ing the volume preserving diffeomorphisms and SLin ( T d )the 0 -divergence vector fields.A ( q , ˙q ) = { q ξ , ˙ q ξ } ξ ∈T d ∈ F should be regarded as aset of Lagrangian coordinates labeled by ξ ∈ T d . Andthe equations Eq.(1.1) can be derived from a Hamil-tonian in canonical coordinates ( q , p ) ∈ F which is quadratic in p and which generates motions in F evolv-ing leaving S F invariant. Therefore the motion in F is a“geodesic motion” ( i.e. a motion generated by a Hamil-tonian quadratic in the momenta).A key remark is that the motions that follow initialdata in S F remain, as long as the evolution is defined and smooth , in S F , [3, 43], i.e. S F is an invariant surfacein F . And the equations of motion that H generates canbe written (using incompressibility of ( q , p ) ∈ S F ) as:˙ p ξ = − ∂ q ξ Q ( q , p ) ξ , ˙ q ξ = p ξ (1.2)where Q ξ is q -dependent and quadratic in p , see Ap-pendix A.Since ˙ p ξ = ∂ t p ξ + ( p e ξ · ∂ e p ξ ), setting x = q ξ , u ( x ) = p ξ and P ( x ) = ∂ q ξ Q ( q , p ) ξ , the equations become:˙ q ξ = p ξ , ∂ t u ( x ) + (( u e · ∂ e ) u )( x ) = − ∂P ( x ) (1.3)with ∂ · u = 0, and P as above. The Lagrangian form of Euler’s equations, Eq.(1.2) or (1.3), will be called E ∗ .See Appendix A.The above “geodesic” formulation of E , E ∗ will be usedto exhibit symmetry properties of Euler’s equation whichmay be relevant also for the IN ( irreversible Navier-Stokes ) equations: ∂ t u ( x ) + (( u e · ∂ e ) u )( x ) = ν ∆ u ( x ) − ∂P ( x ) + f ( x ) (1.4)with the conditions ∂ · u = 0, R T d u = 0. II. ULTRAVIOLET REGULARIZATION
Here we study the regularized version, see below, of E or IN, Eq.(1.1),(1.4), obtained by requiring that theFourier’s transform u k of u does not vanish only formodes k with components ≤ N .We shall focus on properties of the solutions which holduniformly in the cut-off N : the space of such u ’s with 0divergence ( ∂ · u = 0) and 0 average ( R T d u ( x ) = 0) willbe denoted C N .Therefore the equation in dimension d = 2 , u β, k = u β, − k , β =1 , . . . , d, k ∈ Z d , | k β | ≤ N : thus the number of real co-ordinates is N = 4 N ( N + 1) in 2D and N = 2(4 N +6 N + 3 N ) in 3D and N will be the dimension of thephase space C N .For instance in 3D choose, for each k = 0, two unitvectors e β ( k ) = − e β ( − k ) , β = 1 ,
2, mutually orthogonaland orthogonal to k ; data are combined to form a velocityfield: u ( x ) = X < | k |≤ N u k e − i k · x , k = ( k β ) β =1 , , u k = X β =1 , i u β, k e β ( k ) , k · e β ( k ) = 0 (2.1)with | k | = max j | k j | , u − k ,j = u k ,j . This might be a very short time. D β ,β ,β k , k , k = − ( e β ( k ) · k )( e β ( k ) · e β ( k )). In-troduce also forcing f = P k ,β if k ,β e β ( k ) e − i k · x and vis-cosity in the form − ν k u k .The IN equations Eq.(1.4) become, if k def = k + k + k and the sum is restricted to | k | , | k | , | k | ≤ N :˙ u β, k = X β ,β k = k k D β ,β ,β k , k , k u β , k u β , k − ν k u β, k + f β, k (2.2)which will define the regularized IN equation .The 2D case is similar but simpler: no need for thelabels β , and e ( k ) can be taken k ⊥ || k || .The coefficients D β ,β ,β k , k , k can be used to check that if ν = 0 , f = then for all u ∈ C N : ddt Z T d u ( x ) d x = 0 , ddt Z T d u ( x ) · ( ∂ ∧ u ( x )) d x = 0 (2.3)As is well known, the first of Eq.(2.3) leads to the a priori , N -independent, bounds for the solutions of the E and INequations: || u X,N ( t ) || ≤ max( E , ( F ν ) ) , X = E , IN (2.4)satisfied (for all X) by solutions t → u X,N ( t ) def = S X,Nt u ,in terms of E = || u (0) || = P β, k | u β, k | and F = || f || .From now on the cut-off N will be kept constant andthe solution of the equations will be denoted simply S t u dropping the X, N as superscript of the solution map S t .Only when not clear from the context a superscript E orIN or a label N will be added to clarify whether referenceis made to the evolution, or to its properties, following E or IN equation with cut-off N .By scaling, the equation can and will be written in afully dimensionless form in which || f || = 1.The Jacobian of the Euler flow S E , N t u with UV cut-off N is more easily written, without using the Fourier’stransform representation of u , directly from Eq.(1.1) and,see Appendix B, is the sum of the following convolutionoperator on ( ϕ j ( x )) dj =1 = ϕ ∈ L ( T d ) × R d : ∂ ˙ u i ( x ) ∂u j ( y ) = −P δ ( x − y ) ∂ y j u i ( y ) , i, j = 1 , .., d, (2.5) plus an antisymmetric operator on the same space; here P is the orthogonal projection, in the L ( T d ) × R d metric,on the divergenceless fields ϕ . The operator acts on the fields ϕ with 0 divergence(this is used in deriving Eq.(2.5) to discard contributions The symmetries of D arise from the identities R (( u e · ∂ e ) u ) · u =0 and R ( u · ∂ u e ) · ( ∂ e ∧ u e ) = 0, by integration by parts. Other projections could be used: this is convenient to followthe analysis in [33]. that vanish on the divergenceless fields ϕ ): and in the endits symmetric part is P times the multiplication operator,on 0-divergence fields ( ϕ j ( x )) dj =1 = ϕ ∈ L ( T d ) × R d , by: W i,j ( x ) = 12 (cid:16) ∂ x j u i ( x ) + ∂ x i u j ( x ) (cid:17) (2.6) i.e. P times the operator ( J ϕ ) i ( x ) = P j W i,j ( x ) ϕ j ( x ).Introducing also viscosity (and forcing, which how-ever does not contribute) Eq.(2.6) immediately leadsto express the symmetric part of the Jacobian of theregularized IN, irreversible Navier-Stokes, as P times J νi,j = νδ i,j ∆ + W i,j .Defining, for d = 2 , w ( x ) = d − d P di,j =1 W i,j ( x ) the inequality J ν ≤ ν ∆ + w ( x ), derived in [28, 33] for thenonregularized IN equation, remains valid for the regu-larized one and leads to the bound, [28, 33]: Theorem: the sum of the averages of the first p eigen-values of the (Schr¨odinger operator) νD + w ( x ) yields anupper bound to the sum of the first p Lyapunov exponents(of any invariant distribution on F ) of the flow S t . III. REVERSIBLE EQUATIONS
The theory of nonequilibrium fluctuations has led tostudying phenomena via equations considered equivalent(at least for some of the purposes of interest) to the “fun-damental” ones.Thus new non-Newtonian forces have been added tosystems of particles claiming that the values of importantquantities would have the same values as those impliedby the fundamental equations, even in cases in which themodification was drastic: with the advantage, in severalcases, of greatly facilitating simulations, [11, 25, 30].At the same time the idea that modification of theequations would not affect, at least in some importantcases, most of their predictions arose in other domains:it appeared for instance, in [39], to show that the Navier-Stokes (IN above) equation could be modified, into newreversible equations, still remaining consistent with se-lected predictions of the Obukov-Kolmogorov theory.In [14, 15] an attempt was presented to link empiricalequivalence observations to the well established theory ofthe equivalence of ensembles in Statistical Mechanics. And a paradigmatic example was the NS incompressible Lyapunov exponents depend on the invariant distributionused to select data: here they will be defined as the timeaverages of the eigenvalues of the symmetric part of the Ja-cobian of the evolution equation, [33]. The u -dependent nonaveraged eigenvalues will be called local Lyapunov exponents. A naive version would be to claim that modifications of equa-tions describing given phenomena will not alter ’many other’properties of their solutions if the modifications have the ef-fect that properties known to hold, by emprical or theoretical i.e. with a force withFourier’s coefficients non zero only for modes | k | < K f for some K f ). In this case new equation proposed was: ˙u = − ( u e · ∂ e ) u + α ( u )∆ u + f − ∂ P (3.1)with the multiplier α ( u ) so defined that a “global” quan-tity becomes a constant of motion: for instance the en-ergy E ( u ) = P k ,i | u k ,i | or the enstrophy D ( u ) = P k ,i k | u k ,i | .In Statistical Mechanics global conserved quantitiesdefine the ensembles , which are collections of station-ary probability distributions on phase space giving thestatistical fluctuations of observables in the ’equilibriumstates’.The main property being that the “local” observableshave in each state properties independent on the specialglobal quantity that defines a given state, at least in somelimiting situation (like in the “thermodynamic limit”, inwhich the container volume → ∞ ).Distinction between local and global observables is es-sential: in particle systems global quantities can be thetotal energy (microcanonical ensemble) or the total ki-netic energy (isokinetic ensemble) or the total potentialenergy or the average value of certain observables (likethe kinetic energy, in the canonical ensemble).Local observables, in such systems, are observables O V ( q , p ) whose value depends on the configuration ofpositions and velocities of particles located, at the timeof observation, in a region V of finite size compared tothe total volume V of the system. And local observables, in most systems and in station-ary states, evolve exhibiting statistical properties of thevalues of O V which have a limit as V → ∞ , for all V , i.e. become independent of the “volume cut-off V ”. Local and global observables arise often also in connec-tion with the theory of many systems whose evolution iscontrolled by differential equations.In the next section the example of the fluid equations,always in presence of a UV cut-off N , in Eq.(1.4),(2.2))will be analyzed choosing viscosity force as ν ∆ u or α ( u )∆ u as in Eq.(3.1) with:(1) α ( u ) = P k f k · u k P k k | u k | (2) α ( u ) = Λ( u ) + P k k f k · u k P k k | u k | (3.2) analysis, are a priori verified: of course the question is ’whichare the other properties?’ and ’are they interesting?’. Always to be thought as ≫ V . Necessary in almost all cases because of lack of existence-uniqueness of solutions of the equations of motion in infinitevolume, just as in the IN equations in 3D with infinite cut-off. where, with D introduced in Eq.(2.2):Λ( u ) = X k + k + k =0 D i,j,r k , k , k k u k ,i u k ,j u k ,r (3.3)With the choice (1) the equation Eq.(3.1) generates evo-lutions conserving exactly the energy E ( u ), considered in[40], while with the choice (2) evolution conserves exactly the enstrophy D ( u ), considered in [18, 19]. But remarkthat Λ ≡ IV. ENSEMBLES
In the case of the fluid equations in Eq.(1.4) andEq.(3.1) define:
Local observables: are functions of the velovity fields u which depend on the Fourier’s modes u k with | k | < K with K ≪ N , i.e. of finite size compared to the maxi-mum value N (UV cut-off ) used to make the equationsmeaningful It can be said that local observables refer to measure-ments that can be effected looking at large scale proper-ties of the fluid.While in Statistical Mechanics locality refers to eventsin regions in position space small with respect to the vol-ume cut-off V , in fluid mechanics locality refers to eventsmeasurable in regions in Fourier’s space small comparedto the ultraviolet cut-off N . Hence locality has a physicalmeaning when the aim of the theory is to study proper-ties of “large scale” observables ( i.e. expressible in termsof Fourier’s components u k of the velocity fields u with | k | − of the order of the linear size of the container).Hereafter consider Eq.(3.1) and Eq.(1.4) with f fixedand with only few Fourier’s components non zero, say | k | < K f with K f fixed, and k f k = 1: such f will becalled a “large scale forcing”.Having named (2.2) “irreversible” IN, consistently theEq.(3.2) will be named “reversible” RE in case (1), or“reversible” RN in case (2). Properties of RE,RN are:(1) they generate reversible evolutions u → S t u : i.e. if I u = − u is the “time reversal” then IS t = S − t I .(2) RE evolutions conserve exactly energy E = E ( u ) andRN conserve exactly enstrophy D = D ( u ).Stationary distributions are usually associated withchaotic evolutions: therefore the multipliers α ( u ( t )) inEq.(3.1) should show, at large E or D , chaotic fluctua-tions and behave effectively as a constants: this leads toseveral “equivalence conjectures”. The latter value N , “ultraviolet cut-off”, is certainly necessary in3D E , [8], just to make sense of the equations, and “might” benecessary in 3D IN, [12]. N as a reminder thatall quantities considered so far were defined in presenceof a UV cut-off N and to discuss variations of N .Collect the invariant ( i.e. stationary) distributions forIN,RE,RN and denote the collections E INN , E REN , E RNN re-spectively: we call each such collection an ensemble .The stationary distributions are parameterized by theviscosity ν in E IN or by the energy E in the E RE or bythe enstrophy D in E IN .Denoting as µ IN,νN , µ
RE,EN , µ
RN,DN the stationary distri-butions, respectively, in the ensembles E INN E REN , E RNN , weshall try to establish a correspondence between the ele-ments µ IN,νN , µ RE,EN , µ RN,DN so that corresponding distri-butions can be called “equivalent” in the sense discussedbelow.To fix the ideas we focus first on the correspondencebetween the distributions in E INN and E RNN : the simplestsituation arises when above equations, for each ν small or D large, admit a unique stable invariant distribution, i.e. a unique “natural stationary distribution” in the senseof [34, 35, 37], a key concept whose relevance has beenstressed since [32].For an observable O ( u ) define h O i IN,νN def = µ IN,νN ( O ), h O i RN,DN def = µ RN,DN ( O ) the respective time averages of O ( u ( t )) observed under the ( N -regularized) IN and RN evolutions.Define also the work per unit time done by the forcing: L ( u ) = Z T d f ( x ) · u ( x ) dx (2 π ) d = X k f k · u k , (4.1)So the average work per unit time in the stationary stateswith parameters ν or D of the ensembles E INN , E RNN is, re-spectively, h L i IN,νN ≡ µ IN,νN ( L ) or h L i RN,DN ≡ µ RN,DN ( L ).Given µ RN,DN , µ
IN,νN : define µ IN,νN to be correspondent to µ RN,DN , denote this by µ IN,νN ∼ µ RN,DN , if the timeaverage of the work per unit time is equal in the twodistributions: h L i IN,νN = h L i RN,DN (4.2)The natural distributions, see footnote9, are associatedwith chaotic evolutions: therefore the multipliers α ( u ( t ))should show, at large D , chaotic fluctuations and behaveeffectively as constants equal to their average.Hence the proposal, [14, 15]: for an observable O ( u )define h O i Nν def = µ IN,νN ( O ), h O i ND def = µ RN,D ( O ) the respec-tive time averages of O ( u ( t )) observed under the IN andRN evolutions; then: I.e. almost all initial data selected via a probability with con-tinuous density ρ ( u ) d u on the N -dimensional phase space C N ,as in Sec.II, assign the same statistics to the time-fluctuationsof the observables. Conjecture 1:
Under the equivalence condition Eq.(4.2),“equal dissipation”, if O ( u ) is an observable, then: lim ν → h O i Nν = lim ν → h O i NE (4.3)The collection of stationary distributions µ ∈ E IN canbe assimilated to the distributions of Statistical Mechan-ics canonical ensemble and the distributions µ ∈ E RN canbe assimilated to the distributions of the microcanonicalensemble. The regularization N plays the role of the vol-ume and the friction ν that of temperature, the enstrophythat of energy.So there is ´ some ´ similarity between the equilibriumstates equivalence in Statistical Mechanics and the equiv-alence proposed by the conjecture 1 about averages ob-served following the two different evolutions IN and RN ,under the condition of equal dissipation. V. ENSEMBLES IN FLUIDS ANDSTATISTICAL MECHANICS
However the need to consider the limit as ν → V → ∞ of the vol-ume of the system container and restricting the observ-ables O to be local .In the conjecture in Sec.IV, instead, the observables are unrestricted and the role of the volume V is played by thecut-off N . Clearly for a full analogy equivalence shouldhold for ν fixed as N → ∞ , provided the observables aresuitably restricted. To see what has to be understood to try to establisha closer connection between the theory of the ensemblesin Statistical Mechanics and the proposed fluid equationsequivalence the key remark is that the conjectured equiv-alence is based on the chaoticity of the evolution, whichis ensured by the ν → homogeniza-tion phenomena: see [16, 17, 24, 26] for fluid equationsor [5, 17, 22]. Thus the conjecture in Sec.IV althoughquite unsatisfactory, as pointed out, seems to hold in itsgenerality, [22, 24]. Far more interesting would be to dispose of the condi-tion ν → For instance in Statistical Mechanics microcanonical andcanonical ensembles are equivalent unless, of course, one isinterested in the fluctuations of the (global) observable ’totalenergy’. models ) that gen-erate motions apt to describe many of the features foundin the observations. One of the first examples is in thederivation of the (compressible) Navier-Stokes equationsin [29].A model can even fail to respect one or more of thefundamental laws or symmetries: like the time reversalsymmetry breaking which accounts phenomenologicallyfor dissipation. This has never been considered a vio-lation of the basic principles: it has been always clearthat it was simply due to the procedure followed in thederivations.Then the idea arises that there could (should ?) ex-ist models representing the same phenomena at the samelevel of accuracy and preserving some of the propertiesthat other models do not respect, but which are proper-ties on which there is a minor interest in the context onwhich one is working, [18, 19].The case of the Navier-Stokes equation has been pro-posed as an example of the possibility of describing anincompressible fluid via a reversible equation, withoutthe need (as in conjecture 1 above) of taking the limit ν → Under the equal dissipation conditionEq.(4.2) and if O is a local observable, as defined inSec.IV, then lim N →∞ h O i Nν = lim N →∞ h O i NE (5.1) for all ν > . The conjecture 2 therefore adds to conjecture 1 therestriction that the observables O must be local and re-places the equivalence condition ν → condition N → ∞ (keeping the equal dissipation).The ensemble E IN,νN is analogous to the canonical en-semble with ν as temperature while E IN,DN is analogousto the microcanonical ensemble with the enstrophy D asthe energy and N → ∞ corresponds to V → ∞ , i.e. to the thermodynamic limit necessary for all local observ-ables to show the same statistics.The analogy with Statistical Mechanics is now ’essen-tially’ complete (however see Sec.VI below) and providesan example of use of the ’thermodynamic limit’ amongthe ideas emerging in nonequilibrium theory, [1, 36]. VI. EQUIVALENCE AND PHASETRANSITIONS
Conjecture 2 of Sec.V leaves a gap in the strict anal-ogy between Fluid Mechanics and Statistical Mechanicsensembles. Is there an analogue of the phase transitions?So far we have considered the ensembles E IN,ν , E RN,D assuming that for each pair of ν, D the equations IN and RN admit just one “natural” stationary distributioncontrolling the fluctuations of the (local) observables.However it is possible that initial data chosen with adistribution density ρ ( u ) > u with positive probability:this case would be met if the evolution admitted severalattracting sets in the phase space C N .If so, label the “indecomposable” invariant distribu-tions by µ θ ∈ E IN,νN , θ = 1 , , . . . , q ν,N . Likewiselabel the “indecomposable” invariant distributions by µ θ ∈ E RN,DN , θ = 1 , , . . . , p D,N . Each µ θ will be called a“pure phase“.For simplicity we assume that q ν , p D < ∞ and say thatat the values ν or D there are q ν or p D “pure phases”.Then, keeping in mind the theory of phase transitinsin Statistical Mechanics, conjecture 2 should be modifiedas: If under the equivalence condition between ν and D ,Eq.(4.2), there are q ν,N respectively p D,N pure phases,then q ν,N , p D,N have the same limit q ≥ as N → ∞ ,and it is possible to establish a ←→ correspondence be-tween the µ j ∈ E RN,νN and the µ j ∈ E RN,DN such thatthe distribution of the local observables become, in corre-sponding µ ’s and in the limit N → ∞ , the same. If one thinks to the ferromagnetic Ising model in voume V at low temperature then there are two indecomposablepure phases in which the total magnetization or just itsaverage is fixed to some m = ± m ∗ = 0, [23], whether theboundary conditions are periodic or free or whether thedynamics is of Glauber type or other. Make correspon-dent the phases with the same m then the local observ-ables ( i.e. the observables O which depend only on thespins located in a fixed region) have fluctuation with thesame statistics in the thermodynamic limit, V → ∞ . Indecomposable means that for each θ with probabiity 1 withrespect to µ θ initial data generate precisely µ θ itself: synoni-mous of ergodic. RN and deveoped in[40]. VII. CHAOTIC HYPOTHESIS ANDREVERSIBILITY
In a general evolution equation ˙ x = g ( x ) , x ∈ bR n gen-erating motions t → S t x which lead to an attracting set A on which they are chaotic ( i.e. have positive Lyapunovexponents) the “chaotic hypothesis” is:Chaotic hypothesis (CH): The attracting sets can beconsidered smooth surfaces on which the motion is anAnosov flow, [13, 21]. The assumption implies the exsistence of a unique sta-tionary probability distribution µ on A which is a naturaldistribution in the sense that it gives the statistical prop-erties of the motions of almost all initial data chosen inthe vicinity of A with a probability with density ρ ( x ) > RN or RE mightoffer insights.Imagine to fix the UV cut-off N and that for some ν the evolution appears to generate trajectories of IN thatvisit densely the entire phase space. We expect that to bethe case at small ν , at fixed N : and for ν = 0 ergodicity isexpected to hold. As ν increases the system develops anattracting set which, if the CH holds, should still be thefull phase space (a consequence of the structural stabilityof Anosov systems ).For such value of ν let D be the average enstrophy: weconsider the RN evoution of initial data with enstrophy This is a closed set A such that all initial data x close enoughto A are such that the distance d ( S t x, A ) −−−→ t →∞ Anosov evolutions are smooth flows on bounded smooth sur-faces A such that at every point x the evolution is hyperbolic( i.e. in a system of coordinates following S t x as t varies the S t x is a hyperbolic fixed point); furthermore any open set U ⊂ A is such S t U covers any prefixed point x ∈ A for in-finitely many t > t and for all t (“motion of most pointscovers densely A ”, “recurrence”), [4, 27]. Structural stability means here that small pertubations ofAnosov systems are still Anosov systems.[2, 38, 41]. D ( u ) = D . The phase space “contracts” at a rate σ ( u ), i.e. . if u β, k = u r,β, k + iu i,β, k , β = 1 ,
2, see (2.1), at arate equal (by Liouville’s theorem) to: − σ ( u ) = − ∗ X k ,β (cid:16) ∂ ˙ u r, k ,β ∂u r, k ,β + ∂ ˙ u i, k ,β ∂u i, k ,β (cid:17) (7.1)where P ∗ k denotes summation over the k so that onlyone k between ± k contributes (the contribution is inde-pendent on which one is selected).Let F = P ∗ k k f k u k , E = P ∗ k k | u k | , E = P ∗ k k | u k | , K = P ∗ k k , then: − σ ( u ) = 2 (cid:16) K − E ( u ) E ( u ) (cid:17) α ( u ) + F ( u ) E ( u ) (7.2)which has the same expression in dimension 2 , α is of course different).If CH holds the “Fluctuation theorem”, FT, can be ap-plied and the result is that it implies a simple predictionon the non local observable p = 1 t Z t σ ( u ( t ′ )) σ + dt ′ (7.3)where σ + is the average value of σ ( u ( t )). The fluctua-tions of p in the stationary distribution µ RN,DN have theprobability that p ∈ [ a, b ] is exp ( t max p ∈ [ a,b ] s ( p ) + o ( t ))and the “large deviations rate” s ( p ) has the symmetryproperty, [13, 20, 21]: s ( p ) − s ( − p ) = p t σ + (7.4)which follows combining CH and the time reversibility. The observable σ ( u ) can be considered also as an ob-servable for the IN evolution . Although it is non localit has been tested in a few cases to see whether it nev-ertheless obeys the same fluctuation relation Eq.(7.4) incorresponding distributions, see [18] for a positive result.But .... VIII. ATTRACTORS AND SMALL SCALES
However the assumption that at an enstrophy value D the stationary distribution µ RN,DN arises from an evolu-tion which leads to an attracting set invariant under timereversal is too strong.Certainly it does not cover the cases in which the UVcut-off N is large enough and the u k components arestrongly damped for | k | large (as implied by the equiva-lence conjecture).Hence if N is large the attracting set A will shrink andits time reversal image I A will become different from A :a spontaneous breaking of time reversal .The consequence is that the FT cannot be applied tothe observable σ ( u ), not even if the CH is assumed in thereversible RN equation.6Nevertheless FT could be applied, under the CH, tothe motion on A if the time reversal I could be replacedby another map e I which leaves A invariant and on A the e IS t = S − t e I holds. Because by CH A is a surface onwhich the evolution is of Anosov type.In this case the fluctuation relation will be applied nolonger to σ ( u ), but to the sum σ A of the local Lyapunovexponents relative to the motion on A : clearly the neg-ative exponents pertaining to the attraction to A shouldnot be counted.Hence the question under which conditions a time re-versal for the motions on A exists is preliminary to thesecond question of how to identify the Lyapunov expo-nents of the motions on A .Considering the RN equations with UV cut-off N andfixed enstrophy D . Suppose that for small N ( i.e. atstrong regularization) the motions invade densely thephase space C N : i.e. the attracting set A coincides with C N . Increasing N arrives a N c beyond which the (aver-age) viscosity affects the components u k with large k sothat A becomes smaller than C N .So the evolution is reversible for all N , but for N largeits restriction to the attracting set A is not.In [6] the question of existence, as a “remnant” of theglobal symmetry I , of a time reversal e I operating on A has been examined and a geometric property, named Ax-iom C property, leading to the existence of e I was identi-fied and shown to have a “structural stability” property(as demanded to properties of physical relevance). Thedefinition and main properties of Axiom C are describedin Appendix D.A scenario for the application to IN,RN (and moregeneral) equations in which time reversal is a symmetrybut A does not coincide with the full phase space can bethe following.Assume that Axiom C holds for RN, hence there is amap e I : A → A such that e IS t = S − t e I : to apply FTthe problem still remains of identifying the phase spacecontraction σ A , i.e. the local Lyapunov exponents whichcontribute to the phase space contraction on the surface A .In studying the Lyapunov spectrum for IN, RN thefollowing “pairing symmetry” has been tested and ap-proximately verified in a few
2D simulations and for afew values of the ensembles parameters ν, D .If the N local Lyapunov exponents are arranged inde-creasing order and their time averages are λ ≥ λ ≥ The local exponents are defined as the eigenvalues of the sym-metric part of the Jacobian of the motion on A : their sumdefines the contraction (or expansion) of the surface elementsof A . i.e. small pertubations of systems with the axiom C propertystill have the property, [6]. Persistence under perturbationsis clearly essential in most Physics theories, [31]. . . . , ≥ λ N − , then( λ k + λ N − − k ) = n + O ( k − ) , k = 0 , . . . , N n < λ k turned out to have,for each k , in IN and RN very different fluctuations butremarkably the same average in corresponding distribu-tions µ IN,νN and µ RN,DN : quite unexpected a result be-cause the λ k are not local observables. See figs.7,8 and,respectively, figs.5,6 in [18, 19]. Relations like Eq.(8.1)are called “pairing rules”.So among the N / λ k , λ N − − k there may be,depending on the values of ν, D , pairs in which both el-ements are < i.e. there may be n ∗ ≤ N / N / − n ∗ negative pairs.A natural interpretation of the above pairing rule isthat the pairs of exponents < A while the other n ∗ pairsare associated with the chaotic motion on the attractingset; the phase space contraction on A would then be σ A ( u ) = P n ∗ k =0 ( λ k ( u ) + λ N − − k ( u )).The interest of the above remarks is that if CH, axiomC and pairing are satisfied and if the O ( k − ) in Eq.(8.1)can be neglected the consequent relation: σ A ( u ) = 2 n ∗ N σ ( u ) (8.2)can be used to define the phase space contraction on A .The advantage is that σ A is measurable simply by mea-suring σ ( u ) from the equations of motion using Eq.(8.2).Then, applying FT, the relation Eq.(7.4) is simplychanged into: s ( p ) − s ( − p ) = p t n ∗ N σ + (8.3)in the case of the RN evolution.Furthermore if the equivalence conjecture can be ex-tended to the non local observables σ, σ A then the fluctu-ation relation gives a prediction on fluctuations of bothIN and RN and, if n ∗ < N /
2, a test of the Axiom C.The above scenario, proposed first in [7, p.445] andleading to the formulation of Axiom C, does not seemto have been tested, not even for simple test examplesand it is certainly interesting if it can be confirmed insome instances: the only attempt to check Eq.(8.3) dealt,[18], with cases in which n ∗ = N /
2. Hence it does notdeal with the most interesting part of the above sce-nario and in particular it does not test the Axiom C:however it did yield the result that the fluctuation rela-tion holds in equivalent distributions, i.e. the observable σ ( u ), Eq.(7.2), satisfies the same Eq.(7.4) even in theirreversible evolution IN.There are cases in which the phase space contractioncan be identified with entropy creation: this is impor-tant as the entropy production is accessible, in a labo-ratory experiment, to measurements of heat and work7exchanges with the surroundings: however it is very dif-ficult to perform complete analysis of such energy ex-changes and among the many experimental works veryfew convincingly discuss the problem. IX. OTHER ENSEMBLES
In statistical Mechanics there are several equivalent en-sembles. The same should hold for the fluids consideredabove. For instance we could compare IN with the equa-tion that will be called RE given by Eq.(3.1) with a ( u )given by the first of Eq.(3.2).The RE is reversible and conserves the global quantity E ( u ), energy, instead of enstrophy. The ensemble is nowthe collection of the stationary states µ RE,EN ∈ E
REN .The equivalence condition is still the equality of theenergy dissipation, i.e. of the average work done per unittime by the forcing: hence Eq.(4.2) is modified as h L i IN,νN = h L i RE,EN (9.1)and the analysis of the previous sections can be repeated.Care has to be exercized because the condition Eq.(4.2)is not the same as E = hEi IN,νN (unlike the correspondingcase of the RN equations). , This implies that a first test of the conjecture in thecase of RN is obtained by fixing ν and computing theaverage enstrophy D and checking that if ν, D correspondin the sense of Eq.(4.2): h αD i RN,DN = ν hDi IN,νN , i . e . h α i RN,N = ν (9.2)where D ( u ) denotes, as above, the enstrophy. While forRE, if ν and E correspond in the sense of Eq.(9.1), theanalogous test is to check h α Di RE,E hDi
RN,ν = ν (9.3)The above relations have been tested in several cases,with particular care and a few positive results for theequivalence between IN and RN; only in very few casesfor the IN and RE equivalence. If the RN equation is multiplied side by side by u β, k andthe result is summed ove k , β , it immediately follows thatboth conjectures imply D = hDi IN,νN . And conversely the lat-ter equality implies the equal dissipation property, Eq.(4.2),hence the condition for equivalence. The ensemble E REN , in which the global quantity conserved isthe energy rather than the enstrophy, has been considered indetail in [40] where a different kind of very interesting phasetransition phenomena occurring in the RE equations is stud-ied. In the limiting case in which ν → E = hEi IN,νN instead of theequal average dissipation, as in Eq.(9.1), is not appreciable.
Appendix A: Euler flow is geodesic
Here some details on the Hamiltonian representationEq.(1.2) for the Euler flow are presented, listing again forthe reader ´ s convenience, the conventions set in Sec.I.It has to be kept in mind that in analytic mechanics thecanonical coordinates for n -degrees of freedom systemsare given as strings of 2 n variables ( { p i , q i } ni =1 ): particle i is located at position q i and has momentum p i .In a Lagrangian description of a fluid, coordinates willbe ( q , ˙q ) = ( { q ξ , ˙ q ξ } ξ ∈T d ) with ( q , ˙q ) consisting in a dif-feomorphism q : ξ → q ξ in the space Dif ( T d ) of C ∞ diffeomorphisms of T d and ˙q ∈ Lin ( T d ) where Lin ( T d )is the space of the C ∞ vector fields ’tangent’ to q : ina pair ( q , ˙q ) the vector ˙ q ξ ∈ R d is considered a vectorapplied at the point q ξ .Hence, given q , ˙q , the derivative ∂ q ξ,i ˙ q ξ,i is defined aswell as the divergence (div ˙q ) ξdef = P di =1 ∂ q ξ,i ˙ q ξ,i of ˙ q ξ .The space of the pairs ( q , ˙q ) will be called F and thepoints of T d become labels of a fluid element located atthe point q ξ with velocity ˙ q ξ .More formally ( q , ˙q ) ∈ Dif ( T d ) × Lin ( T d ) def = F where Dif ( T d ) is the space of the C ∞ diffeomorphisms of T d and Lin ( T d ) the space of the C ∞ vector fields with 0average: for each ( q , ˙q ) the vector ˙ q ξ ∈ R d is consideredapplied to the point q ξ . and (div ˙q ) ξ = P di =1 ∂ q ξ ˙ q ξ .Actually we concentrate on the subspace of ( q , ˙q ) ∈ SDif ( T d ) × ( SLin ( T d )) def = S F ⊂ F where the evolutionof an incompressible fluid takes place:
SDif ( T d ) beingthe volume preserving diffeomphisms and SLin ( T d ) the0 -divergence vector fields, i.e. for each such pair ( q , ˙q ) itis (div ˙q ) ξ = 0.If the positions q ξ are moved the variation of ˙ q ξ is pro-portional to ∂ ˙ q ξ ∂q η ; the Lagrangian is: L ( ˙q , q ) = Z T d (cid:16) ˙ q ξ − Q ( ˙q , q ) ξ (cid:17) dξ (A.1)where Q is the quadratic form on F : − π Z T d dγ d X r,s =1 (cid:16) ′′ | q ξ − q γ | ′′ (cid:17) ∂ ˙ q γ,r ∂q γ,s ∂ ˙ q γ,s ∂q γ,r (A.2)and − (4 π ) − / ′′ | x − y | ′′ symbolizes the Green’s func-tion for the Laplacian on T : so that ∆ q ξ Q ξ =( ∂ e p ) ξ ( ∂ p e ) ξ .Addition of Q corresponds to a force which has theproperty that it keeps data in S F inside S F as long asthey evolve smoothly in time: this is checked in the fol-lowing. i.e. formally the summation over images y + 2 π n , n ∈ Z d ; whichmakes sense if the kernel is applied to a smooth function with 0average. L leads to the canonical variables via: p ξ,idef = ˙ q ξ,i − π Z T d dλ Z T d dγ { ′′ | q λ − q γ | ′′ }· δ i,r ∂δ ( q ξ − q γ ) ∂q γ,s ∂ ˙ q γ,s ∂q γ,r def = ˙ q ξ,i + A ( q , ˙q ) ξ,i (A.3)where in the last equality (defining A ) p ξ,i = ˙ q ξ,i and A = 0 hold if q ∈ SDif ( T d ) and ˙q ∈ SLin ( T d ), i.e. if P ds =1 ∂ ˙ q γ,s ∂q γ,s = 0: so that the diffeomorphism q is in-compressible, and also ˙ q is divergenceless, div ˙q = 0, andthe double integral vanishes because R dλ { ′′ | q λ − q γ | ′′ } − is a constant and integration by parts over γ becomespossible as dγ = dq γ if q ∈ SDif ( T d ).Defining the linear operator A − ( q , p ), obtained byinverting p = ˙q + A ( q , ˙q ), the equation for ˙p is readilyobtained, at least if ( p , q ) ∈ S F :˙ q ξ = p ξ , ˙ p ξ = ∂ q ξ L ( ˙q , q ) = ∂ q ξ Q ( ˙q , q ) (A.4)and the Hamiltonian: H ( p , q ) def = Z T d ( p ξ · ˙ q ξ − L ( p − A − ( q , p ) , q ) ξ ) dξ (A.5)yields canonical equations, which for data in S F are :˙ q ξ = p ξ , ˙ p ξ = − ∂ q ξ P ξ ( q , p ) P ξdef = Z T d dγ d X r,s =1 (cid:16) − (4 π ) − ′′ | q ξ − q γ | ′′ (cid:17) ∂p γ,r ∂q γ,s ∂p γ,s ∂q γ,r (A.6)which hold only if ( p , q ) ∈ S F , while in F the equationswould be more involved (but uninteresting for the presentpurposes) although still Hamiltonian.The equations can be written for data in S F setting x = q ξ , u ( x ) = p ξ , P ( x ) = P ξ . Then dp ξ,i dt = ∂ t p ξ,i + P j p ξ,j ∂ q ξ,j p ξ,i and: ∂ · u ( x ) = 0 , P ( x ) = − d X r,s =1 ∆ − ( ∂ s u r ( x ) ∂ r u s ( x )) ∂ t q ξ = u ( q ξ ) , ∂ t u ( x ) = − u e ( x ) · ∂ e u ( x ) − ∂ P ( x )+ f ( x ) + ν ∆ u ( x ) (A.7)where the terms in the third line are added in the casethere is forcing and NS viscosity: the ν ∆ u will be re-placed in the next Appendix C by Ekman’s viscosity − νu and the equation thus modified will be used to exhibit aspecial symmetry arising in this case.Therefore the Eq.(A.7) coincide with the Navier Stokesequations and their solutions will remain in S F s long asthey remain smooth: for data not in S F only solutionslocal in time can be envisaged and the equations wouldbe more involved. Representing a constrained motion as a special case of un-
Appendix B: Euler ´ s equation Jacobian The Jacobian is obtained by taking suitable functionalderivatives of the transport term and the pressure term,before applying the projection operator P in Eq.(2.4).The contribution of the transport term is (before apply-ing P ): ∂ ˙ u i ( x ) ∂u j ( y ) = − δ ( x − y ) ∂ x j u i ( x ) − u k ( x ) δ i,j ∂ x k δ ( x − y ) (B.1)where the second term is an antisymmetric operator in L ( T d ) × R d . The contribution from the pressure termis ∂ ∂ x i P ( x ) ∂u j ( y ) = − ∂ x i Z T d dz ∆ − ( x − z ) · ∂ z k δ ( z − y ) ∂ z j u k ( z )=2( ∂ x i ∂ y k ∆ − ( x − y )) ∂ y j u k ( y ) (B.2)and in both Eq.(B.1),(B.2) summation over k is intended.The latter operator does not contribute to the Jacobianbecause acting on a divergenceless field yields 0; thereforethe symmetric part of the Jacobian is the multiplicationoperator, in L ( T d ) × R d , by: W i,j ( x ) = −
12 ( ∂ x j u i ( x ) + ∂ x j u i ( x )) (B.3)followed by the orthogonal projection P on the subspaceof the divergenceless fields SLin ( T d ) ⊂ L ( T d ) × R d ,Sec.I. Appendix C: The pairing symmetry in E ∗ Symmetry on the Lyapunov spectrum for the fluidequations is not really surprising, at least not in the caseof the equations obtained from Euler ´ s equations in La-grangian form when forcing and viscosity of the form − ν u constrained motion subject suitable extra forces follows a fa-miliar prototype. A point mass constrained on a circle ofradius R , centered at the origin O ∈ R , can be seen as apoint subject to a centripetal force evolving under the La-grangian L = ˙q − ( | q |− R ) R ˙q . This leads to p = ˙q θ with θ = (1 − | q |− RR ) and, for the Hamiltonian, H = p θ , to theequations ˙q = p θ − , ˙p = − p θ R q | q | . Thus it appears thatthe phase space R is analogous to F , the maps of the cir-cle ξ → s , mapping the arc ξ to the arc s , correspond with SDif and the vectors tangent to the circle are analogous to
SLin . The motion is in general a geodesic motion, as longas it is defined ( i.e. as long as | q | 6 = 0 , ∞ ), which for datainitially on the circle and initial velocity tangent to it is auniform rotation. On such motions the Hamiltonian value is p as θ = 1, alike the corresponding vanishing of Q on S F in Eq.(1.3),(A.5). ´ s viscosity) are added. The ultimate reason isthat the Euler flow is a geodesic flow as discussed in Sec.I and Appendix A. A heuristic analysis follows.It is convenient to review the method discovered in [10].Let H ( p , q ) ∈ R n be a n -degrees of freedom Hamilto-nian and to its equations a friction − ν p is introducedturning the equations of motion into ˙p = − ∂ q H ( p , q ) − ν p , ˙q = ∂ p H ( p , q ).The Jacobian matrix J ν ( i.e. the linearization of theflow in phase space ( p , q ) ∈ R n ) is a symplectic matrixup to the contribution from the friction force: J ν = J +the contribution from the friction force.The latter is a diagonal matrix with the first n ele-ments = − ν and the rest = 0: hence it is the sum of adiagonal matrix with the first half entries equal to − ν and the others equal to + ν plus a second diagonal ma-trix with all the n elements = − ν .Hence the Jacobian matrix of the equations with fric-tion is recognized to be the sum of the symplectic Jaco-bian matrix of the Hamiltonian H ( p , q ) + ν p · q and ofthe constant diagonal matrix − ν : namely J ν = J − ν .Therefore the eigenvalues will be equal to − ν plusthose of a symplectic matrix: such eigenvalues arise inpairs of opposite sign. The same pairing will remaintrue for products of Jacobians J ν ( u i ) and therefore theLyapunov exponents will arise in pairs with sum exactly= − ν .In Sec.I the equations E ∗ in F defined by Eq.(1.3) aregeodesic flows (Appendix A) which evolve in F leavinginvariant the manifold S F ⊂ F . The Jacobian is there-fore symplectic even if restricted to S F .Hence if viscosity and forcing are added, with forcedepending only on q ξ and friction expressed as − νp ξ (Ekman ´ s viscosity), obtaining dissipative equations thatcan be called IE ∗ ( i.e. Lagrangian Euler’s with Ekman’sviscosity), the pairing to − ν is valid by above argumenton the geodesic flow.The argument in [9] can also be applied to RE ∗ , i.e. to the equation obtained by adding to E ∗ the forcing anda reversible viscosity − α ( p ) p ξ with, if x = q ξ , u ( x ) = p ξ ,and α ( u ) given by item (1) in Eq.(3.1).The RE ∗ equation, in 2D and 3D, implies energy con-servation: therefore there is a 0 Lyapunov exponent thatcan be added to the 0 exponent associated with the flowdirection.Discarding the latter 0 exponents the important newidea added in [9] to the above summarized work [10] willimply that the Lyapunov exponents of the Lagrangianmotion of RE ∗ on S F are paired to the average of α ( u )(if they exist, i.e. in 2D). In [9] the force is supposed conservative: in RE ∗ the forceis solenoidal (divergenceless). Yet this does not affect theresult because what really matters is that the Jacobian of theequation with 0 viscosity is a symplectic matrix; and this isimplied by the Hamiltonian nature of the evolution. Note The analysis does not apply to the 2
N − IE ∗ , RE ∗ will verify a pairing symmetry by reflection atthe level − ν or to the average of α respectively. But thisneither applies to the Navier Stokes equations, becausethe viscosity in not a force proportional to u , nor to IE ∗ in 3D because there is no global existence uniqueness re-sult on the flow S t . About the regularized E ∗ , IE ∗ , RE ∗ equations pairing might just be approximate.See [15, Sec.6] for heuristic considerations about possi-ble extensions of the above remarks to the NS equationsconsidered as equations for the velocity field only. Aparticular intriguing question is: is there a relation be-tween the Lyapunov spectrum of E ∗ , Eq.(1.2), and thecorresponding spectrum of the simple Euler equation E ,Eq.(1.1) ?For instance is it possible that the E ∗ (Lagrangian Eu-ler equations, Eq.(1.2)) has a Lyapunov spectrum whichis just the one of E (the Euler equations E , Eq.(1.1)) witheach exponent counted twice, [19] ? If so the pairing inthe simple Euler equations E with forcing and Ekmanviscosity added would hold at the level − ν and in thecorresponding reversible equations at the level of the av-erage of − α ( u ). Appendix D: The Axiom C.
To describe the main features of the Axiom C, [6], con-sider first the simpler case of a reversible diffeomorphism S , i.e. such that there is a diffeomorphism, I such that IS = S − I, I = 1. Imagine that the attracting set A differs from its time reversal image I A = R and that CHholds.The tangent space at a generic point z is supposed tobe smoothly decomposed as T u ( z ) ⊕ T s ( z ) ⊕ T m ( z ). If z ∈A or z ∈ R then T s ( z ) , T u ( z ) coincide with the tangent,at z , to the stable manifold of S on A or R respectively;furthermore for each ball U δ ( x ) ⊂ A , of radius δ , considerthe manifolds W i ( x ) ∩ U δ ( x ) , i = u, s , and assume thatthey can be continued into smooth manifolds W + , W − everywhere tangent T s ⊕ T m and T u ⊕ T m and whichintersect R in a single point e x = P x if δ is small enough:thus defining P as a map between A , R .Finally, as the labels s,u suggest, the vectors in T s , T u uniformly contract exponentially as time tends to + ∞ or −∞ respectively, while vectors in T m contract ex-ponentially as t → ±∞ ( i.e. in both directions, being ´ squeezed ´ on A and R ). that in the above preliminary analysis of the method in [10]the pairing would also follow, and for the same reason, if aLorentz or Coriolis type of force were added, like ϕ ( q ) ∧ ˙ q with ϕ ( q ) an incompressible field, i.e. ϕ ( q ) = ∂ ∧ A ( q ). S t can be described similarly byimagining that T m contains also the neutral direction ddt S t u and contracts transversally to it. In this contextAxiom C adds to the Axiom B, [42], the assumption ofthe existence of a repeller R intersected by the manifoldsemerging from A .The latter property permits to establish the map P , thus allowing to define the composition e I = P I , actingas a time reversal on A and R , because the invarianceof the manifolds implies that on A ∪ R it is
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