Viscosity, Reversibillity, Chaotic Hypothesis,

Fluctuation Theorem and Lyapunov Pairing

Giovanni Gallavotti

Abstract: Incompressible fluid equations are studied with UV cut-off and in
periodic boundary conditions. Properties of the resulting ODEs holding
uniformly in the cut-off are considered and, in particular, are conjectured to
be equivalent to properties of other time reversible equations. Reversible
equations with the same regularization and describing equivalently the fluid,
and the fluctuations of large classes of observables, are examined in the
context of the “Chaotic Hypothesis”, “Axiom C” and the “Fluctuation Theorem”.

##  I On the equations

The incompressible Euler equation, denoted $\bf E$, in a periodic container
${\mathcal{T}}^{d}=[0,2\pi]^{d},\,d=2,3$, for a smooth velocity field ${{\bf
u}}(x),x\in{\mathcal{T}}^{d}$ is:

$\displaystyle{{}{\bf\dot{{\bf u}}}(x)=\kern-2.84526pt-({{{\bf
u}}_{\kern-3.0pt\lower 7.0pt\hbox{$\widetilde{}$}}\kern
3.0pt}(x)\cdot{{\mbox{\boldmath$\partial$}}_{\kern-3.0pt\lower
7.0pt\hbox{$\widetilde{}$}}\kern 3.0pt}_{\raise 5.69054pt\hbox{$x$}}){{\bf
u}}(x)\kern-2.84526pt-{\mbox{\boldmath$\partial$}}_{x}P(x),}$
---
$\displaystyle{{}{\mbox{\boldmath$\partial$}}_{x}\cdot{{\bf
u}}(x)=0,\quad\int_{{\mathcal{T}}^{d}}dx\,{{\bf u}}(x)={\bf 0}}$
(1.1)

where
$P=-\sum_{i,j=1}^{d}\Delta^{-1}({\partial}_{i}u_{j}\,{\partial}_{j}u_{i})$ is
the ’pressure’ and $\Delta$=Laplace operator.

It is also useful to consider the $\bf E$ equations from the “Lagrangian
viewpoint”: a configuration of the fluid is described by assigning the
dispacement $x=q_{\xi}$ of a fluid element, from the reference position
$\xi\in{\mathcal{T}}^{d}$, and the velocity $\dot{q}_{\xi}$ of the same fluid
element. So the state of the fluid is $({{\bf q}},{\bf\dot{q}})$ where ${{\bf
q}}$ is a smooth map of ${\mathcal{T}}^{d}$ to itself and ${\bf\dot{q}}$ is a
smooth vector field on ${\mathcal{T}}^{d}$ with $0$ average. Denote
${\mathcal{F}}$ the space of the dynamical configurations $({{\bf
q}},{\bf\dot{q}})\in Dif({\mathcal{T}}^{d})\times
Lin({\mathcal{T}}^{d})={\mathcal{F}}$ where $Dif({\mathcal{T}}^{d})$ is the
set of diffeomorphisms of ${\mathcal{T}}^{d}$ and $Lin({\mathcal{T}}^{d})$ the
space of the vector fields on ${\mathcal{T}}^{d}$.

Actually we concentrate on the subspace of $({{\bf
q}},{\bf\dot{q}})\in(SDif({\mathcal{T}}^{d})\times
SLin({\mathcal{T}}^{d})){\buildrel
def\over{=}}S{\mathcal{F}}\subset{\mathcal{F}}$ where the evolution of an
incompressible fluid takes place: $SDif({\mathcal{T}}^{d})$ being the volume
preserving diffeomorphisms and $SLin({\mathcal{T}}^{d})$ the $0$-divergence
vector fields.

A $({{\bf q}},{\bf\dot{{\bf
q}}})=\\{q_{\xi},\dot{q}_{\xi}\\}_{\xi\in{\mathcal{T}}^{d}}\in{\mathcal{F}}$
should be regarded as a set of Lagrangian coordinates labeled by
$\xi\in{\mathcal{T}}^{d}$. And the equations Eq.(1.1) can be derived from a
Hamiltonian in canonical coordinates $({{\bf q}},{{\bf p}})\in{\mathcal{F}}$
which is quadratic in ${{\bf p}}$ and which generates motions in
${\mathcal{F}}$ evolving leaving $S{\mathcal{F}}$ invariant. Therefore the
motion in ${\mathcal{F}}$ is a “geodesic motion” (i.e. a motion generated by a
Hamiltonian quadratic in the momenta).

A key remark is that the motions that follow initial data in $S{\mathcal{F}}$
remain, as long as the evolution is defined and smooth,111This might be a very
short time. in $S{\mathcal{F}}$, [3, 43], i.e. $S{\mathcal{F}}$ is an
invariant surface in ${\mathcal{F}}$. And the equations of motion that $H$
generates can be written (using incompressibility of $({{\bf q}},{{\bf p}})\in
S{\mathcal{F}}$) as:

$\displaystyle{\dot{p}_{\xi}=-{\mbox{\boldmath$\partial$}}_{\scriptstyle
q_{\xi}}\kern-2.84526ptQ({{\bf q}},{{\bf
p}})_{\xi},\quad\dot{q}_{\xi}=p_{\xi}}$
---
(1.2)

where $Q_{\xi}$ is ${{\bf q}}$-dependent and quadratic in ${{\bf p}}$, see
Appendix A.

Since $\dot{p}_{\xi}={\partial}_{t}p_{\xi}+({p_{\kern-3.0pt\lower
7.0pt\hbox{$\widetilde{}$}}\kern 3.0pt}_{\raise
5.69054pt\hbox{$\xi$}}\cdot{{\partial}_{\kern-3.0pt\lower
7.0pt\hbox{$\widetilde{}$}}\kern 3.0pt}p_{\xi})$, setting
$x=q_{\xi},u(x)=p_{\xi}$ and $P(x)={\mbox{\boldmath$\partial$}}_{\scriptstyle
q_{\xi}}\kern-2.84526ptQ({{\bf q}},{{\bf p}})_{\xi}$, the equations become:

$\displaystyle{{}\dot{q}_{\xi}=p_{\xi},\quad{\partial}_{t}u(x)+(({u_{\kern-3.0pt\lower
7.0pt\hbox{$\widetilde{}$}}\kern 3.0pt}\cdot{{\partial}_{\kern-3.0pt\lower
7.0pt\hbox{$\widetilde{}$}}\kern 3.0pt})u)(x)=-{\partial}P(x)}$
---
(1.3)

with ${\mbox{\boldmath$\partial$}}\cdot{{\bf u}}=0$, and $P$ as above. The
Lagrangian form of Euler’s equations, Eq.(1.2) or (1.3), will be called $\bf
E^{*}$. See Appendix A.

The above “geodesic” formulation of $\bf E,E^{*}$ will be used to exhibit
symmetry properties of Euler’s equation which may be relevant also for the IN
(irreversible Navier-Stokes) equations:

${\partial}_{t}{{\bf u}}(x)+(({{{\bf u}}_{\kern-3.0pt\lower
7.0pt\hbox{$\widetilde{}$}}\kern 3.0pt}\cdot{{\partial}_{\kern-3.0pt\lower
7.0pt\hbox{$\widetilde{}$}}\kern 3.0pt}){{\bf u}})(x)=\nu\Delta{{\bf
u}}(x)-{\partial}P(x)+{{\bf f}}(x)$ (1.4)

with the conditions ${\mbox{\boldmath$\partial$}}\cdot{{\bf u}}=0$,
$\int_{{\mathcal{T}}^{d}}{{\bf u}}=0$.

##  II Ultraviolet regularization

Here we study the regularized version, see below, of $\bf E$ or IN,
Eq.(1.1),(1.4), obtained by requiring that the Fourier’s transform ${{\bf
u}}_{{\bf k}}$ of ${{\bf u}}$ does not vanish only for modes ${{\bf k}}$ with
components $\leq N$.

We shall focus on properties of the solutions which hold uniformly in the cut-
off $N$: the space of such ${{\bf u}}$’s with $0$ divergence
(${\mbox{\boldmath$\partial$}}\cdot{{\bf u}}=0$) and $0$ average (
$\int_{{\mathcal{T}}^{d}}{{\bf u}}(x)=0$) will be denoted ${\mathcal{C}}_{N}$.

Therefore the equation in dimension $d=2,3$ is expressed in terms of complex
scalars $u_{\beta,{{\bf k}}}={\overline{u}}_{\beta,-{{\bf
k}}},\,\beta=1,\ldots,d,\,{{\bf k}}\in Z^{d},\,|{{\bf k}}_{\beta}|\leq N$:
thus the number of real coordinates is ${\mathcal{N}}=4N(N+1)$ in 2D and
${\mathcal{N}}=2(4N^{3}+6N^{2}+3N)$ in 3D and ${\mathcal{N}}$ will be the
dimension of the phase space ${\mathcal{C}}_{N}$.

For instance in 3D choose, for each ${{\bf k}}\neq 0$, two unit vectors ${\bf
e}_{\beta}({{\bf k}})=-{\bf e}_{\beta}(-{{\bf k}}),\,\beta=1,2$, mutually
orthogonal and orthogonal to ${{\bf k}}$; data are combined to form a velocity
field:

$\displaystyle{{}{{\bf u}}({{\bf x}})=\sum_{0<|{{\bf k}}|\leq N}{{\bf
u}}_{{\bf k}}e^{-i{{\bf k}}\cdot{{\bf x}}},\ \ {{\bf
k}}=(k_{\beta})_{\beta=1,2,3}}$
---
$\displaystyle{{}{{\bf u}}_{{\bf k}}=\sum_{\beta=1,2}i\,u_{\beta,{{\bf
k}}}\,{\bf e}_{\beta}({{\bf k}}),\qquad{{\bf k}}\cdot{\bf e}_{\beta}({{\bf
k}})=0}$
(2.1)

with $|{{\bf k}}|=\max_{j}|k_{j}|,u_{-{{\bf k}},j}={\overline{u}}_{{{\bf
k}},j}$.

Define $D_{{{\bf k}}_{1},{{\bf k}}_{2},{{\bf
k}}}^{\beta_{1},\beta_{2},\beta}=-({\bf e}_{\beta_{1}}({{\bf
k}}_{1})\cdot{{\bf k}}_{2})({\bf e}_{\beta_{2}}({{\bf k}}_{2})\cdot{\bf
e}_{\beta}({{\bf k}}))$. Introduce also forcing ${{\bf f}}=\sum_{{{\bf
k}},\beta}if_{{{\bf k}},\beta}{\bf e}_{\beta}({{\bf k}})e^{-i{{\bf
k}}\cdot{{\bf x}}}$ and viscosity in the form $-\nu{{\bf k}}^{2}{{\bf
u}}_{{\bf k}}$.

The IN equations Eq.(1.4) become, if ${{\bf k}}^{2}{\buildrel
def\over{=}}k_{1}^{2}+k_{2}^{2}+k_{3}^{2}$ and the sum is restricted to
$|{{\bf k}}|,|{{\bf k}}_{1}|,|{{\bf k}}_{2}|\leq N$:

$\kern-5.69054pt\dot{u}_{\beta,{{\bf
k}}}=\kern-5.69054pt\sum_{\beta_{1},\beta_{2}\atop{{\bf k}}={{\bf
k}}_{1}+{{\bf k}}_{2}}\kern-5.69054ptD_{{{\bf k}}_{1},{{\bf k}}_{2},{{\bf
k}}}^{\beta_{1},\beta_{2},\beta}\,\,u_{\beta_{1},{{\bf
k}}_{1}}u_{\beta_{2},{{\bf k}}_{2}}-\nu{{\bf k}}^{2}u_{\beta,{{\bf
k}}}+f_{\beta,{{\bf k}}}$ (2.2)

which will define the regularized IN equation.

The 2D case is similar but simpler: no need for the labels $\beta$, and ${\bf
e}({{\bf k}})$ can be taken $\frac{{{\bf k}}^{\perp}}{||{{\bf k}}||}$.

The coefficients $D_{{{\bf k}}_{1},{{\bf k}}_{2},{{\bf
k}}_{3}}^{\beta_{1},\beta_{2},\beta_{2}}$ can be used to check that if
$\nu=0,{{\bf f}}={\bf 0}$ then for all ${{\bf u}}\in{\mathcal{C}}_{N}$:222The
symmetries of $D$ arise from the identities $\int(({{{\bf
u}}_{\kern-3.0pt\lower 7.0pt\hbox{$\widetilde{}$}}\kern
3.0pt}\cdot{{\mbox{\boldmath$\partial$}}_{\kern-3.0pt\lower
7.0pt\hbox{$\widetilde{}$}}\kern 3.0pt}){{\bf u}})\cdot{{\bf u}}=0$ and
$\int({{\bf u}}\cdot{\mbox{\boldmath$\partial$}}{{{\bf u}}_{\kern-3.0pt\lower
7.0pt\hbox{$\widetilde{}$}}\kern
3.0pt})\cdot({{\mbox{\boldmath$\partial$}}_{\kern-3.0pt\lower
7.0pt\hbox{$\widetilde{}$}}\kern 3.0pt}\wedge{{{\bf u}}_{\kern-3.0pt\lower
7.0pt\hbox{$\widetilde{}$}}\kern 3.0pt})=0$, by integration by parts.

$\frac{d}{dt}\int_{{\mathcal{T}}^{d}}{{\bf u}}({{\bf x}})^{2}d{{\bf x}}=0,\
\frac{d}{dt}\int_{{\mathcal{T}}^{d}}\kern-8.53581pt{{\bf u}}({{\bf
x}})\cdot({\mbox{\boldmath$\partial$}}\wedge{{\bf u}}({{\bf x}}))d{{\bf x}}=0$
(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 $\bf E$ and IN equations:

$\kern-8.53581pt||{{\bf
u}}^{X,N}(t)||^{2}_{2}\leq\max(E_{0},(\frac{F_{0}}{\nu})^{2}),\ X={\bf E},IN$
(2.4)

satisfied (for all X) by solutions $t\to{{\bf u}}^{X,N}(t){\buildrel
def\over{=}}S^{X,N}_{t}{{\bf u}}$, in terms of $E_{0}=||{{\bf
u}}(0)||_{2}^{2}=\sum_{\beta,{{\bf k}}}|u_{\beta,{{\bf k}}}|^{2}$ and
$F_{0}=||{{\bf f}}||_{2}$.

From now on the cut-off N will be kept constant and the solution of the
equations will be denoted simply $S_{t}{{\bf u}}$ dropping the $X,N$ as
superscript of the solution map $S_{t}$. Only when not clear from the context
a superscript $\bf E$ or IN or a label $N$ will be added to clarify whether
reference is made to the evolution, or to its properties, following $\bf E$ or
IN equation with cut-off $N$.

By scaling, the equation can and will be written in a fully dimensionless form
in which $||{{\bf f}}||_{2}=1$.

The Jacobian of the Euler flow $S^{{\bf E},{\rm N}}_{t}{{\bf u}}$ with UV cut-
off $N$ is more easily written, without using the Fourier’s transform
representation of ${{\bf u}}$, directly from Eq.(1.1) and, see Appendix B, is
the sum of the following convolution operator on
$(\varphi_{j}(x))_{j=1}^{d}={\mbox{\boldmath$\varphi$}}\in
L_{2}({\mathcal{T}}^{d})\times R^{d}$:

$\frac{{\partial}\dot{u}_{i}(x)}{{\partial}u_{j}(y)}=-{\cal
P}\delta(x-y){\partial}_{y_{j}}u_{i}(y),\ i,j=1,..,d,$ (2.5)

plus an antisymmetric operator on the same space; here ${\cal P}$ is the
orthogonal projection, in the $L_{2}(T^{d})\times R^{d}$ metric, on the
divergenceless fields $\varphi$. 333Other projections could be used: this is
convenient to follow the analysis in [33].

The operator acts on the fields $\varphi$ with $0$ divergence (this is used in
deriving Eq.(2.5) to discard contributions that vanish on the divergenceless
fields $\varphi$): and in the end its symmetric part is $\cal P$ times the
multiplication operator, on $0$-divergence fields
$(\varphi_{j}(x))_{j=1}^{d}={\mbox{\boldmath$\varphi$}}\in
L_{2}({\mathcal{T}}^{d})\times R^{d}$, by:

$W_{i,j}(x)=\frac{1}{2}\Big{(}{\partial}_{x_{j}}u_{i}(x)+{\partial}_{x_{i}}u_{j}(x)\Big{)}$
(2.6)

i.e. $\cal P$ times the operator
$(J{\mbox{\boldmath$\varphi$}})_{i}(x)=\sum_{j}W_{i,j}(x)\varphi_{j}(x)$.

Introducing also viscosity (and forcing, which however does not contribute)
Eq.(2.6) immediately leads to express the symmetric part of the Jacobian of
the regularized IN, irreversible Navier-Stokes, as $\cal P$ times
$J_{i,j}^{\nu}=\nu\delta_{i,j}\Delta+W_{i,j}$.

Defining, for $d=2,3$, $w(x)^{2}=\frac{d-1}{4d}\sum_{i,j=1}^{d}W_{i,j}(x)^{2}$
the inequality $J^{\nu}\leq\nu\Delta+w(x)$, derived in [33, 28] for the
nonregularized IN equation, remains valid for the regularized one and leads to
the bound, [33, 28]:

Theorem: the sum of the averages of the first $p$ eigenvalues of the
(Schrödinger operator) $\nu D+w(x)$ yields an upper bound to the sum of the
first $p$ Lyapunov exponents (of any invariant distribution on
${\mathcal{F}}$) of the flow $S_{t}$. 444Lyapunov exponents depend on the
invariant distribution used to select data: here they will be defined as the
time averages of the eigenvalues of the symmetric part of the Jacobian of the
evolution equation, [33]. The $u$-dependent non averaged eigenvalues will be
called local Lyapunov exponents.

##  III Reversible equations

The theory of nonequilibrium fluctuations has led to studying phenomena via
equations considered equivalent (at least for some of the purposes of
interest) to the “fundamental” ones.

Thus new non-Newtonian forces have been added to systems of particles claiming
that the values of important quantities would have the same values as those
implied by the fundamental equations, even in cases in which the modification
was drastic: with the advantage, in several cases, of greatly facilitating
simulations, [30, 11, 25].

At the same time the idea that modification of the equations would not affect,
at least in some important cases, 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 new reversible equations, still
remaining consistent with selected predictions of the Obukov-Kolmogorov
theory.

In [14, 15] an attempt was presented to link empirical equivalence
observations to the well established theory of the equivalence of ensembles in
Statistical Mechanics. 555A naive version would be to claim that modifications
of equations describing given phenomena will not alter ’many other’ properties
of their solutions if the modifications have the effect that properties known
to hold, by emprical or theoretical analysis, are a priori verified: of course
the question is ’which are the other properties?’ and ’are they interesting?’.
And a paradigmatic example was the NS incompressible fluid in the simple case
of periodic boundary conditions and forcing acting at large scale (i.e. with a
force with Fourier’s coefficients non zero only for modes $|{{\bf k}}|<K_{f}$
for some $K_{f}$). In this case new equation proposed was:

${\bf\dot{{\bf u}}}=-({{{\bf u}}_{\kern-3.0pt\lower
7.0pt\hbox{$\widetilde{}$}}\kern
3.0pt}\cdot{{\mbox{\boldmath$\partial$}}_{\kern-3.0pt\lower
7.0pt\hbox{$\widetilde{}$}}\kern 3.0pt}){{\bf u}}+\alpha({{\bf u}})\Delta{{\bf
u}}+{{\bf f}}-{\mbox{\boldmath$\partial$}}P$ (3.1)

with the multiplier $\alpha({{\bf u}})$ so defined that a “global” quantity
becomes a constant of motion: for instance the energy ${\mathcal{E}}({{\bf
u}})=\sum_{{{\bf k}},i}\frac{1}{2}|u_{{{\bf k}},i}|^{2}$ or the enstrophy
${\mathcal{D}}({{\bf u}})=\sum_{{{\bf k}},i}{{\bf k}}^{2}|u_{{{\bf
k}},i}|^{2}$.

In Statistical Mechanics global conserved quantities define the ensembles,
which are collections of stationary probability distributions on phase space
giving the statistical fluctuations of observables in the ’equilibrium
states’.

The main property being that the “local” observables have in each state
properties independent on the special global quantity that defines a given
state, at least in some limiting situation (like in the “thermodynamic limit”,
in which the container volume $\to\infty$).

Distinction between local and global observables is essential: in particle
systems global quantities can be the total energy (microcanonical ensemble) or
the total kinetic energy (isokinetic ensemble) or the total potential energy
or the average value of certain observables (like the kinetic energy, in the
canonical ensemble).

Local observables, in such systems, are observables $O_{V_{0}}({\bf q},{\bf
p})$ whose value depends on the configuration of positions and velocities of
particles located, at the time of observation, in a region $V_{0}$ of finite
size compared to the total volume $V$ of the system.666Always to be thought as
$\gg V_{0}$.

And local observables, in most systems and in stationary states, evolve
exhibiting statistical properties of the values of $O_{V_{0}}$ which have a
limit as $V\to\infty$, for all $V_{0}$, i.e. become independent of the “volume
cut-off $V$”.777Necessary in almost all cases because of lack of existence-
uniqueness of solutions of the equations of motion in infinite volume, just as
in the IN equations in 3D with infinite cut-off.

Local and global observables arise often also in connection with the theory of
many systems whose evolution is controlled 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 $\nu\Delta{{\bf u}}$ or $\alpha({{\bf u}})\Delta{{\bf u}}$ as in Eq.(3.1)
with:

$\displaystyle{(1)}$ | $\displaystyle{{}\quad\alpha({{\bf u}})=\frac{\sum_{{\bf k}}{{\bf f}}_{{\bf k}}\cdot{\overline{{\bf u}}}_{{\bf k}}}{\sum_{{\bf k}}{{\bf k}}^{2}|{{\bf u}}_{{\bf k}}|^{2}}}$
---|---
$\displaystyle{(2)}$ | $\displaystyle{{}\quad\alpha({{\bf u}})=\frac{\Lambda({{\bf u}})+\sum_{{\bf k}}{{\bf k}}^{2}{{\bf f}}_{{\bf k}}\cdot{\overline{{\bf u}}}_{{\bf k}}}{\sum_{{\bf k}}{{\bf k}}^{4}|{{\bf u}}_{{\bf k}}|^{2}}}$
(3.2)

where, with $D$ introduced in Eq.(2.2):

$\Lambda({{\bf u}})=\sum_{{{\bf k}}_{1}+{{\bf k}}_{2}+{{\bf
k}}_{3}=0}D^{i,j,r}_{{{\bf k}}_{1},{{\bf k}}_{2},{{\bf k}}_{3}}{{\bf
k}}_{3}^{2}\,u_{{{\bf k}}_{1},i}u_{{{\bf k}}_{2},j}u_{{{\bf k}}_{3},r}$ (3.3)

With the choice (1) the equation Eq.(3.1) generates evolutions conserving
exactly the energy ${\mathcal{E}}({{\bf u}})$, considered in [40], while with
the choice (2) evolution conserves exactly the enstrophy ${\mathcal{D}}({{\bf
u}})$, considered in [19, 18]. But remark that $\Lambda\equiv 0$ in 2D,
implied by Eq.(2.3).

##  IV Ensembles

In the case of the fluid equations in Eq.(1.4) and Eq.(3.1) define: Local
observables: are functions of the velovity fields ${{\bf u}}$ which depend on
the Fourier’s modes ${{\bf u}}_{{\bf k}}$ with $|{{\bf k}}|<{\overline{K}}$
with ${\overline{K}}\ll N$, i.e. of finite size compared to the maximum value
$N$ (UV cut-off) used to make the equations meaningful 888The latter value
$N$, “ultraviolet cut-off”, is certainly necessary in 3D $\bf E$, [8], just to
make sense of the equations, and “might” be necessary in 3D IN, [12].

It can be said that local observables refer to measurements that can be
effected looking at large scale properties of the fluid.

While in Statistical Mechanics locality refers to events in regions in
position space small with respect to the volume cut-off $V$, in fluid
mechanics locality refers to events measurable in regions in Fourier’s space
small compared to the ultraviolet cut-off $N$. Hence locality has a physical
meaning when the aim of the theory is to study properties of “large scale”
observables (i.e. expressible in terms of Fourier’s components ${{\bf
u}}_{{\bf k}}$ of the velocity fields ${{\bf u}}$ with $|{{\bf k}}|^{-1}$ of
the order of the linear size of the container).

Hereafter consider Eq.(3.1) and Eq.(1.4) with ${{\bf f}}$ fixed and with only
few Fourier’s components non zero, say $|{{\bf k}}|<K_{f}$ with $K_{f}$ fixed,
and $\|{{\bf f}}\|_{2}=1$: such ${{\bf f}}$ will be called a “large scale
forcing”.

Having named (2.2) “irreversible” IN, consistently the Eq.(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 ${{\bf u}}\to S_{t}{{\bf u}}$: i.e. if
$I{{\bf u}}=-{{\bf u}}$ is the “time reversal” then $IS_{t}=S_{-t}I$.
(2) RE evolutions conserve exactly energy $E={\mathcal{E}}({{\bf u}})$ and RN
conserve exactly enstrophy $D={\mathcal{D}}({{\bf u}})$.

Stationary distributions are usually associated with chaotic evolutions:
therefore the multipliers $\alpha({{\bf u}}(t))$ in Eq.(3.1) should show, at
large $E$ or $D$, chaotic fluctuations and behave effectively as a constants:
this leads to several “equivalence conjectures”.

For clarity we reintroduce a label $N$ as a reminder that all quantities
considered so far were defined in presence of a UV cut-off $N$ and to discuss
variations of $N$.

Collect the invariant (i.e. stationary) distributions for IN,RE,RN and denote
the collections
${\mathcal{E}}^{IN}_{N},{\mathcal{E}}^{RE}_{N},{\mathcal{E}}^{RN}_{N}$
respectively: we call each such collection an ensemble.

The stationary distributions are parameterized by the viscosity $\nu$ in
${\mathcal{E}}^{IN}$ or by the energy $E$ in the ${\mathcal{E}}^{RE}$ or by
the enstrophy $D$ in ${\mathcal{E}}^{IN}$.

Denoting as $\mu^{IN,\nu}_{N},\mu^{RE,E}_{N},\mu^{RN,D}_{N}$ the stationary
distributions, respectively, in the ensembles
${\mathcal{E}}^{IN}_{N}{\mathcal{E}}^{RE}_{N},{\mathcal{E}}^{RN}_{N}$, we
shall try to establish a correspondence between the elements
$\mu^{IN,\nu}_{N}$, $\mu^{RE,E}_{N}$, $\mu^{RN,D}_{N}$ so that corresponding
distributions can be called “equivalent” in the sense discussed below.

To fix the ideas we focus first on the correspondence between the
distributions in ${\mathcal{E}}^{IN}_{N}$ and ${\mathcal{E}}^{RN}_{N}$: the
simplest situation arises when above equations, for each $\nu$ small or $D$
large, admit a unique stable invariant distribution, i.e. a unique “natural
stationary distribution” in the sense of [34, 35, 37],999I.e. almost all
initial data selected via a probability with continuous density $\rho({{\bf
u}})d{{\bf u}}$ on the ${\mathcal{N}}$-dimensional phase space
${\mathcal{C}}_{N}$, as in Sec.II, assign the same statistics to the time-
fluctuations of the observables. a key concept whose relevance has been
stressed since [32].

For an observable $O({{\bf u}})$ define $\langle
O\rangle^{IN,\nu}_{N}{\buildrel def\over{=}}\mu^{IN,\nu}_{N}(O)$, $\langle
O\rangle^{RN,D}_{N}{\buildrel def\over{=}}\mu^{RN,D}_{N}(O)$ the respective
time averages of $O({{\bf u}}(t))$ observed under the ($N$-regularized) $IN$
and $RN$ evolutions.

Define also the work per unit time done by the forcing:

$L({{\bf u}})=\int_{{\mathcal{T}}^{d}}{{\bf f}}(x)\cdot{{\bf
u}}(x)\frac{dx}{(2\pi)^{d}}=\sum_{{\bf k}}{{\bf f}}_{{\bf
k}}\cdot{\overline{{\bf u}}}_{{\bf k}},$ (4.1)

So the average work per unit time in the stationary states with parameters
$\nu$ or $D$ of the ensembles ${\mathcal{E}}^{IN}_{N},{\mathcal{E}}^{RN}_{N}$
is, respectively, $\langle L\rangle^{IN,\nu}_{N}\equiv\mu^{IN,\nu}_{N}(L)$ or
$\langle L\rangle^{RN,D}_{N}\equiv\mu^{RN,D}_{N}(L)$.

Given $\mu^{RN,D}_{N},\mu^{IN,\nu}_{N}$: define $\mu^{IN,\nu}_{N}$ to be
correspondent to $\mu^{RN,D}_{N}$, denote this by
$\mu^{IN,\nu}_{N}\sim\mu^{RN,D}_{N}$, if the time average of the enstrophy  is
equal in the two distributions:

$\langle{\mathcal{D}}\rangle^{IN,\nu}_{N}=D$ (4.2)

The natural distributions, see footnote9, are associated with chaotic
evolutions: therefore the multipliers $\alpha({{\bf u}}(t))$ should show, at
large $D$, chaotic fluctuations and behave effectively as constants equal to
their average.

Hence the proposal, [14, 15]: for an observable $O({{\bf u}})$ define $\langle
O\rangle^{N}_{\nu}{\buildrel def\over{=}}\mu^{IN,\nu}_{N}(O)$, $\langle
O\rangle^{N}_{D}{\buildrel def\over{=}}\mu^{RN,D}(O)$ the respective time
averages of $O({{\bf u}}(t))$ observed under the IN and RN evolutions; then:

Conjecture 1: Under the equivalence condition Eq.(4.2), equal average
enstrophy, if $O({{\bf u}})$ is an observable, then:

$\lim_{\nu\to 0}\langle O\rangle^{N}_{\nu}=\lim_{\nu\to 0}\langle
O\rangle^{N}_{D}$ (4.3)

The collection of stationary distributions $\mu\in{\mathcal{E}}^{IN}$ can be
assimilated to the distributions of Statistical Mechanics canonical ensemble
and the distributions $\mu\in{\mathcal{E}}^{RN}$ can be assimilated to the
distributions of the microcanonical ensemble. The regularization $N$ plays the
role of the volume and the friction $\nu$ that of temperature, the enstrophy
that of energy.

So there is ´some´ similarity between the equilibrium states equivalence in
Statistical Mechanics and the equivalence proposed by the conjecture 1 about
averages observed following the two different evolutions $IN$ and $RN$, under
the condition of equal average enstrophy.

##  V Ensembles in Fluids and Statistical Mechanics

However the need to consider the limit as $\nu\to 0$ in Eq.(4.2) limits
strongly the analogy: the Statistical Mechanics theory of equivalence of the
ensembles requires considering the thermodynamic limit $V\to\infty$ of the
volume of the system container and restricting the observables $O$ to be
local.

In the conjecture in Sec.IV, instead, the observables are unrestricted and the
role of the volume $V$ is played by the cut-off $N$. Clearly for a full
analogy equivalence should hold for $\nu$ fixed as $N\to\infty$, provided the
observables are suitably restricted.101010For instance in Statistical
Mechanics microcanonical and canonical ensembles are equivalent unless, of
course, one is interested in the fluctuations of the (global) observable
’total energy’.

To see what has to be understood to try to establish a closer connection
between the theory of the ensembles in Statistical Mechanics and the proposed
fluid equations equivalence the key remark is that the conjectured equivalence
is based on the chaoticity of the evolution, which is ensured by the $\nu\to
0$ condition in Eq.(4.2).

So the same argument can simply be extended to many other equations in which
the size of a parameter controls the increasingly “chaotic” motion of a
system. Examples of this phenomenon have been explicitly considered adding new
examples to a wide literature of homogenization phenomena: see [24, 16, 17,
26] for fluid equations or [22, 17, 5]. Thus the conjecture in Sec.IV although
quite unsatisfactory, as pointed out, seems to hold in its generality, [24,
22].

Far more interesting would be to dispose of the condition $\nu\to 0$ and to
realize a stronger analogy with Statistical Mechanics. The idea is that some,
by far not all, equations describing macroscopic phenomena arise as scalingn
limits of fundamental equations governing evolutions of systems of particles
interacting via forces veryfying all principles and symmetries of Physics:
staying within classical systems among these are Newton’s laws, time reversal
and parity and charge symmetry…

The evolutions, at so fundamental a level, are certainly chaotic and the
ergodic hypothesis epitomizes this property: from them, via approximations
and/or heuristic arguments, arise simplified equations (models) that generate
motions apt to describe many of the features found in the observations. One of
the first examples is in the derivation of the (compressible) Navier-Stokes
equations in [29].

A model can even fail to respect one or more of the fundamental laws or
symmetries: like the time reversal symmetry breaking which accounts
phenomenologically for dissipation. This has never been considered a violation
of the basic principles: it has been always clear that it was simply due to
the procedure followed in the derivations.

Then the idea arises that there could (should ?) exist models representing the
same phenomena at the same level of accuracy and preserving some of the
properties that other models do not respect, but which are properties on which
there is a minor interest in the context on which one is working, [19, 18].

The case of the Navier-Stokes equation has been proposed as an example of the
possibility of describing an incompressible fluid via a reversible equation,
without the need (as in conjecture 1 above) of taking the limit $\nu\to 0$ but
paying the price of restricting attention to a suitable (large) family of
observables.

In the NS case the equations of motion are irreversible but they arise from a
fundamental microscopic representation which is reversible and chaotic. If, as
in most experimental studies, interest is on properties of “large scale” then
it is natural to extend the conjecture 1 to the NS evolution restricting
attention to the case in which the macroscopic forces act at large scale and
whose results have to be observed also on large scale, [19, 18].

This can be formalized into the:

Conjecture 2: Under the equal average enstrophy Eq.(4.2) and if $O$ is a local
observable, as defined in Sec.IV, then

$\lim_{N\to\infty}\langle O\rangle^{N}_{\nu}=\lim_{N\to\infty}\langle
O\rangle^{N}_{D}$ (5.1)

for all $\nu>0$.

The conjecture 2 therefore adds to conjecture 1 the restriction that the
observables $O$ must be local and replaces the equivalence condition $\nu\to
0$ with the condition $N\to\infty$ (keeping equal average enstrophy).

The ensemble ${\mathcal{E}}^{IN,\nu}_{N}$ is analogous to the canonical
ensemble with $\nu$ as temperature while ${\mathcal{E}}^{IN,D}_{N}$ is
analogous to the microcanonical ensemble with the enstrophy $D$ as the energy
and $N\to\infty$ corresponds to $V\to\infty$, i.e. to the thermodynamic limit
necessary for all local observables to show the same statistics.

The analogy with Statistical Mechanics is now ’essentially’ complete (however
see Sec.VI below) and provides an example of use of the ’thermodynamic limit’
among the ideas emerging in nonequilibrium theory, [36, 1].

##  VI Equivalence and phase transitions

Conjecture 2 of Sec.V leaves a gap in the strict analogy between Fluid
Mechanics and Statistical Mechanics ensembles. Is there an analogue of the
phase transitions?

So far we have considered the ensembles
${\mathcal{E}}^{IN,\nu},{\mathcal{E}}^{RN,D}$ assuming that for each pair of
$\nu,D$ the equations $IN$ and $RN$ admit just one “natural” stationary
distribution controlling the fluctuations of the (local) observables.

However it is possible that initial data chosen with a distribution density
$\rho({{\bf u}})>0$ generate a statistics which still depends on the initial
${{\bf u}}$ with positive probability: this case would be met if the evolution
admitted several attracting sets in the phase space
${\mathcal{C}}_{\mathcal{N}}$.

If so, label the “indecomposable” invariant distributions by
$\mu_{\theta}\in{\mathcal{E}}^{IN,\nu}_{N}$,
$\theta=1,2,\ldots,q_{\nu,N}$.111111Indecomposable means that for each
$\theta$ with probabiity $1$ with respect to $\mu_{\theta}$ initial data
generate precisely $\mu_{\theta}$ itself: synonimous of ergodic. Likewise
label the “indecomposable” invariant distributions by
$\mu_{\theta}\in{\mathcal{E}}^{RN,D}_{N}$, $\theta=1,2,\ldots,p_{D,N}$. Each
$\mu_{\theta}$ will be called a “pure phase“.

For simplicity we assume that $q_{\nu},p_{D}<\infty$ and say that at the
values $\nu$ or $D$ there are $q_{\nu}$ or $p_{D}$ “pure phases”.

Then, keeping in mind the theory of phase transitins in Statistical Mechanics,
conjecture 2 should be modified as:

If under the equivalence condition between $\nu$ and $D$, Eq.(4.2), there are
$q_{\nu,N}$ respectively $p_{D,N}$ pure phases, then $q_{\nu,N},p_{D,N}$ have
the same limit $q\geq 1$ as $N\to\infty$, and it is possible to establish a
$1\,{\kern-1.0pt\leftarrow\kern-5.0pt\to\kern-1.0pt}\,1$ correspondence
between the $\mu_{j}\in{\mathcal{E}}^{RN,\nu}_{N}$ and the
$\mu_{j}\in{\mathcal{E}}^{RN,D}_{N}$ such that the distribution of the local
observables become, in corresponding $\mu$’s and in the limit $N\to\infty$,
the same.

If one thinks to the ferromagnetic Ising model in voume $V$ at low temperature
then there are two indecomposable pure phases in which the total magnetization
or just its average is fixed to some $m=\pm m^{*}\neq 0$, [23], whether the
boundary conditions are periodic or free or whether the dynamics is of Glauber
type or other. Make correspondent the phases with the same $m$ then the local
observables (i.e. the observables $O$ which depend only on the spins located
in a fixed region) have fluctuation with the same statistics in the
thermodynamic limit, $V\to\infty$.

See comments following Eq.(9.1) for other analogies with phase transitions
arising in $RN$ and deveoped in [40].

##  VII Chaotic hypothesis and reversibility

In a general evolution equation $\dot{x}=g(x),x\in bR^{n}$ generating motions
$t\to S_{t}x$ which lead to an attracting set121212This is a closed set
${\mathcal{A}}$ such that all initial data $x$ close enough to ${\mathcal{A}}$
are such that the distance
$d(S_{t}x,{\mathcal{A}})\,\vtop{\halign{#\cr\rightarrowfill\crcr\kern-1.0pt\nointerlineskip\cr\hskip
3.0pt${\scriptstyle t\to\infty}$\hskip 3.0pt\crcr}}\,0$. ${\mathcal{A}}$ on
which they are chaotic (i.e. have positive Lyapunov exponents) the “chaotic
hypothesis” is:

Chaotic hypothesis (CH): The attracting sets can be considered smooth surfaces
on which the motion is an Anosov flow, [21, 13].131313Anosov evolutions are
smooth flows on bounded smooth surfaces ${\mathcal{A}}$ 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 $U\subset{\mathcal{A}}$ is such $S_{t}U$ covers any
prefixed point $x\in{\mathcal{A}}$ for infinitely many $t>t_{0}$ and for all
$t_{0}$ (“motion of most points covers densely ${\mathcal{A}}$”,
“recurrence”), [4, 27].

The assumption implies the exsistence of a unique stationary probability
distribution $\mu$ on ${\mathcal{A}}$ which is a natural distribution in the
sense that it gives the statistical properties of the motions of almost all
initial data chosen in the vicinity of ${\mathcal{A}}$ with a probability with
density $\rho(x)>0$.

This assumption is a generalization of the ergodic hypothesis in equilibrium
problems, and examples which do not satisfy it are easy to find. Still it is
an assumption that has been proposed to be applicable to most systems
undergoing chaotic motions, in analogy with the ergodic hypothesis in
equilibrium thermodynamics, [36].

The real problem is to show that it not only has the merit of providing a
conceptual extension of ideas at the basis of equilibrium Statistical
Mechanics to nonequilibrium and Fluid Mechanics but it has also predictive
power on new observations.

The simplest applications of the CH deal with reversible evolutions; hence the
equations $RN$ or $RE$ might offer insights.

Imagine to fix the UV cut-off $N$ and that for some $\nu$ the evolution
appears to generate trajectories of $IN$ that visit densely the entire phase
space. We expect that to be the case at small $\nu$, at fixed $N$: and for
$\nu=0$ ergodicity is expected to hold. As $\nu$ increases the system develops
an attracting set which, if the CH holds, should still be the full phase space
(a consequence of the structural stability of Anosov systems141414Structural
stability means here that small pertubations of Anosov systems are still
Anosov systems.[2, 41, 38].).

For such value of $\nu$ let $D$ be the average enstrophy: we consider the $RN$
evoution of initial data with enstrophy ${\mathcal{D}}({{\bf u}})=D$. The
phase space “contracts” at a rate $\sigma({{\bf u}})$, i.e. . if
$u_{\beta,{{\bf k}}}=u_{r,\beta,{{\bf k}}}+iu_{i,\beta,{{\bf
k}}},\,\beta=1,2$, see (2.1), at a rate equal (by Liouville’s theorem) to:

$-\sigma({{\bf u}})=-\sum_{{{\bf
k}},\beta}^{*}\Big{(}\frac{{\partial}\dot{u}_{r,{{\bf
k}},\beta}}{{\partial}u_{r,{{\bf k}},\beta}}+\frac{{\partial}\dot{u}_{i,{{\bf
k}},\beta}}{{\partial}u_{i,{{\bf k}},\beta}}\Big{)}$ (7.1)

where $\sum^{*}_{{\bf k}}$ denotes summation over the ${{\bf k}}$ so that only
one ${{\bf k}}$ between $\pm{{\bf k}}$ contributes (the contribution is
independent on which one is selected).

Let $F_{4}=\sum^{*}_{{\bf k}}{{\bf k}}^{4}{{\bf f}}_{{\bf
k}}{\overline{u}}_{{\bf k}}$, $E_{6}=\sum_{{\bf k}}^{*}{{\bf k}}^{6}|{{\bf
u}}_{{\bf k}}|^{2}$, $E_{4}=\sum^{*}_{{\bf k}}{{\bf k}}^{4}|{{\bf u}}_{{\bf
k}}|^{2}$, $K_{2}=\sum^{*}_{{\bf k}}{{\bf k}}^{2}$, then:

$-\sigma({{\bf u}})=2\Big{(}K_{2}-\frac{E_{6}({{\bf u}})}{E_{4}({{\bf
u}})}\Big{)}\,\alpha({{\bf u}})+\frac{F_{4}({{\bf u}})}{E_{4}({{\bf u}})}$
(7.2)

which has the same expression in dimension $2,3$ (but the expression of
$\alpha$ is of course different).

If CH holds the “Fluctuation theorem”, FT, can be applied and the result is
that it implies a simple prediction on the non local observable

$p=\frac{1}{t}\int_{0}^{t}\frac{\sigma({{\bf
u}}(t^{\prime}))}{\sigma_{+}}dt^{\prime}$ (7.3)

where $\sigma_{+}$ is the average value of $\sigma({{\bf u}}(t))$. The
fluctuations of $p$ in the stationary distribution $\mu^{RN,D}_{N}$ have the
probability that $p\in[a,b]$ is $\exp\,(t\max_{p\in[a,b]}s(p)+o(t))$ and the
“large deviations rate” $s(p)$ has the symmetry property, [20, 13, 21]:

$s(p)-s(-p)=p\,t\,\sigma_{+}$ (7.4)

which follows combining CH and the time reversibility.

The observable $\sigma({{\bf u}})$ can be considered also as an observable for
the IN evolution. Although it is non local it has been tested in a few cases
to see whether it nevertheless obeys the same fluctuation relation Eq.(7.4) in
corresponding 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 $\mu^{RN,D}_{N}$ arises from an evolution which leads to an
attracting set invariant under time reversal is too strong.

Certainly it does not cover the cases in which the UV cut-off $N$ is large
enough and the ${{\bf u}}_{{\bf k}}$ components are strongly damped for
$|{{\bf k}}|$ large (as implied by the equivalence conjecture).

Hence if $N$ is large the attracting set ${\mathcal{A}}$ will shrink and its
time reversal image $I{\mathcal{A}}$ will become different from
${\mathcal{A}}$: a spontaneous breaking of time reversal.

The consequence is that the FT cannot be applied to the observable
$\sigma({{\bf u}})$, not even if the CH is assumed in the reversible RN
equation.

Nevertheless FT could be applied, under the CH, to the motion on
${\mathcal{A}}$ if the time reversal $I$ could be replaced by another map
$\widetilde{I}$ which leaves ${\mathcal{A}}$ invariant and on ${\mathcal{A}}$
the $\widetilde{I}S_{t}=S_{-t}\widetilde{I}$ holds. Because by CH
${\mathcal{A}}$ is a surface on which the evolution is of Anosov type.

In this case the fluctuation relation will be applied no longer to
$\sigma({{\bf u}})$, but to the sum $\sigma_{\mathcal{A}}$ of the local
Lyapunov exponents 151515The local exponents are defined as the eigenvalues of
the symmetric part of the Jacobian of the motion on ${\mathcal{A}}$: their sum
defines the contraction (or expansion) of the surface elements of
${\mathcal{A}}$. relative to the motion on ${\mathcal{A}}$: clearly the
negative exponents pertaining to the attraction to ${\mathcal{A}}$ should not
be counted.

Hence the question under which conditions a time reversal for the motions on
${\mathcal{A}}$ exists is preliminary to the second question of how to
identify the Lyapunov exponents of the motions on ${\mathcal{A}}$.

Considering the RN equations with UV cut-off $N$ and fixed enstrophy $D$.
Suppose that for small $N$ (i.e. at strong regularization) the motions invade
densely the phase space ${\mathcal{C}}_{N}$: i.e. the attracting set
${\mathcal{A}}$ coincides with ${\mathcal{C}}_{N}$. Increasing $N$ arrives a
$N_{c}$ beyond which the (average) viscosity affects the components $u_{{\bf
k}}$ with large ${{\bf k}}$ so that ${\mathcal{A}}$ becomes smaller than
${\mathcal{C}}_{N}$.

So the evolution is reversible for all $N$, but for $N$ large its restriction
to the attracting set ${\mathcal{A}}$ is not.

In [6] the question of existence, as a “remnant” of the global symmetry $I$,
of a time reversal $\widetilde{I}$ operating on ${\mathcal{A}}$ has been
examined and a geometric property, named Axiom C property, leading to the
existence of $\widetilde{I}$ was identified and shown to have a “structural
stability” property (as demanded to properties of physical
relevance).161616i.e. small pertubations of systems with the axiom C property
still have the property, [6]. Persistence under perturbations is clearly
essential in most Physics theories, [31]. The definition and main properties
of Axiom C are described in Appendix D.

A scenario for the application to IN,RN (and more general) equations in which
time reversal is a symmetry but ${\mathcal{A}}$ does not coincide with the
full phase space can be the following.

Assume that Axiom C holds for RN, hence there is a map
$\widetilde{I}:{\mathcal{A}}\to{\mathcal{A}}$ such that
$\widetilde{I}S_{t}=S_{-t}\widetilde{I}$: to apply FT the problem still
remains of identifying the phase space contraction $\sigma_{\mathcal{A}}$,
i.e. the local Lyapunov exponents which contribute to the phase space
contraction on the surface ${\mathcal{A}}$.

In studying the Lyapunov spectrum for $IN,RN$ the following “pairing symmetry”
has been tested and approximately verified in a few 2D simulations and for a
few values of the ensembles parameters $\nu,D$.

If the ${\mathcal{N}}$ local Lyapunov exponents are arranged indecreasing
order and their time averages are
$\lambda_{0}\geq\lambda_{1}\geq\ldots,\geq\lambda_{{\mathcal{N}}-1}$, then

$(\lambda_{k}+\lambda_{{\mathcal{N}}-1-k})=n+O(k^{-1}),\
k=0,\ldots,\frac{{\mathcal{N}}}{2}$ (8.1)

and the constant $n<0$ and the $\lambda_{k}$ turned out to have, for each $k$,
in IN and RN very different fluctuations but remarkably the same average in
corresponding distributions $\mu^{IN,\nu}_{N}$ and $\mu^{RN,D}_{N}$: quite
unexpected a result because the $\lambda_{k}$ are not local observables. See
figs.7,8 and, respectively, figs.5,6 in [19, 18]. Relations like Eq.(8.1) are
called “pairing rules”.

So among the ${\mathcal{N}}/2$ averages
$\lambda_{k},\lambda_{{\mathcal{N}}-1-k}$ there may be, depending on the
values of $\nu,D$, pairs in which both elements are $<0$: i.e. there may be
$n^{*}\leq{\mathcal{N}}/2$ pairs of opposite sign and ${\mathcal{N}}/2-n^{*}$
negative pairs.

A natural interpretation of the above pairing rule is that the pairs of
exponents $<0$ represent the exponents controlling the approach to
${\mathcal{A}}$ while the other $n^{*}$ pairs are associated with the chaotic
motion on the attracting set; the phase space contraction on ${\mathcal{A}}$
would then be $\sigma_{\mathcal{A}}({{\bf
u}})=\sum_{k=0}^{n^{*}}(\lambda_{k}({{\bf
u}})+\lambda_{{\mathcal{N}}-1-k}({{\bf u}}))$.

The interest of the above remarks is that if CH, axiom C and pairing are
satisfied and if the $O(k^{-1})$ in Eq.(8.1) can be neglected the consequent
relation:

$\sigma_{\mathcal{A}}({{\bf u}})=\frac{2n^{*}}{{\mathcal{N}}}\sigma({{\bf
u}})$ (8.2)

can be used to define the phase space contraction on ${\mathcal{A}}$. The
advantage is that $\sigma_{\mathcal{A}}$ is measurable simply by measuring
$\sigma({{\bf u}})$ from the equations of motion using Eq.(8.2).

Then, applying FT, the relation Eq.(7.4) is simply changed into:

$s(p)-s(-p)=p\,t\,\frac{2n^{*}}{{\mathcal{N}}}\,\sigma_{+}$ (8.3)

in the case of the RN evolution.

Furthermore if the equivalence conjecture can be extended to the non local
observables $\sigma,\sigma_{\mathcal{A}}$ then the fluctuation relation gives
a prediction on fluctuations of both IN and RN and, if
$n^{*}<{\mathcal{N}}/2$, a test of the Axiom C.

The above scenario, proposed first in [7, p.445] and leading to the
formulation of Axiom C, does not seem to have been tested, not even for simple
test examples and it is certainly interesting if it can be confirmed in some
instances: the only attempt to check Eq.(8.3) dealt, [18], with cases in which
$n^{*}={\mathcal{N}}/2$. Hence it does not deal with the most interesting part
of the above scenario and in particular it does not test the Axiom C: however
it did yield the result that the fluctuation relation holds in equivalent
distributions, i.e. the observable $\sigma({{\bf u}})$, Eq.(7.2), satisfies
the same Eq.(7.4) even in the irreversible evolution IN.

There are cases in which the phase space contraction can be identified with
entropy creation: this is important as the entropy production is accessible,
in a laboratory experiment, to measurements of heat and work exchanges with
the surroundings: however it is very difficult to perform complete analysis of
such energy exchanges and among the many experimental works very few
convincingly discuss the problem.

##  IX Other ensembles

In statistical Mechanics there are several equivalent ensembles. The same
should hold for the fluids considered above. For instance we could compare IN
with the equation that will be called RE given by Eq.(3.1) with $a({{\bf u}})$
given by the first of Eq.(3.2).

The RE is reversible and conserves the global quantity ${\mathcal{E}}({{\bf
u}})$, energy, instead of enstrophy. The ensemble is now the collection of the
stationary states $\mu^{RE,E}_{N}\in{\mathcal{E}}^{RE}_{N}$.

The equivalence condition is equality of the average energy, hence Eq.(4.2) is
modified as

$\langle{\mathcal{E}}\rangle^{IN,\nu}_{N}=E$ (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
$E=\langle{\mathcal{E}}\rangle^{IN,\nu}_{N}$ (unlike the corresponding case of
the RN equations).171717If the RN equation is multiplied side by side by
${\overline{u}}_{\beta,{{\bf k}}}$ and the result is summed ove ${{\bf
k}},\beta$, it immediately follows that both conjectures imply
$D=\langle{\mathcal{D}}\rangle^{IN,\nu}_{N}$. And conversely the latter
equality implies the equal dissipation property, Eq.(4.2), hence the condition
for equivalence.,181818The ensemble ${\mathcal{E}}^{RE}_{N}$, in which the
global quantity conserved is the energy rather than the enstrophy, has been
considered in detail in [40] where a different kind of very interesting phase
transition phenomena occurring in the RE equations is studied. In the limiting
case in which $\nu\to 0$ as well as in the analysis of the transition in [40]
it is likely that the difference bewteen imposing the condition
$E=\langle{\mathcal{E}}\rangle^{IN,\nu}_{N}$ instead of the equal average
enstrophy, as in Eq.(9.1), is not appreciable.

This implies that a first test of the conjecture in the case of RN is obtained
by fixing $\nu$ and computing the average enstrophy $D$ and checking that if
$\nu,D$ correspond in the sense of Eq.(4.2):

$\langle\alpha\rangle^{RN,D}_{N}\,D=\nu\langle{\mathcal{D}}\rangle^{IN,\nu}_{N},\quad{\it
i.e.\ }\ \langle\alpha\rangle^{RN,N}=\nu$ (9.2)

where ${\mathcal{D}}({{\bf u}})$ denotes, as above, the enstrophy. While for
RE, if $\nu$ and $E$ correspond in the sense of Eq.(9.1), the analogous test
is to check

$\frac{\langle\alpha{\mathcal{D}}\rangle^{RE,E}}{\langle{\mathcal{D}}\rangle^{RN,\nu}}=\nu$
(9.3)

The above relations have been tested in several cases, with particular care
and a few positive results for the equivalence between IN and RN in 2D; only
in very few cases for the IN and RE equivalence.

##  Appendix A Euler flow is geodesic

Here some details on the Hamiltonian representation Eq.(1.2) for the Euler
flow are presented, listing again for the reader´s convenience, the
conventions set in Sec.I.

It has to be kept in mind that in analytic mechanics the canonical coordinates
for $n$-degrees of freedom systems are given as strings of $2n$ variables
$(\\{p_{i},q_{i}\\}_{i=1}^{n})$: particle $i$ is located at position $q_{i}$
and has momentum $p_{i}$.

In a Lagrangian description of a fluid, coordinates will be $({{\bf
q}},{\bf\dot{{\bf
q}}})=(\\{q_{\xi},\dot{q}_{\xi}\\}_{\xi\in{\mathcal{T}}^{d}})$ with $({{\bf
q}},{\bf\dot{{\bf q}}})$ consisting in a diffeomorphism ${{\bf q}}\,:\,\xi\to
q_{\xi}$ in the space $Dif({\mathcal{T}}^{d})$ of ${\mathcal{C}}^{\infty}$
diffeomorphisms of ${\mathcal{T}}^{d}$ and ${\bf\dot{{\bf q}}}\in
Lin({\mathcal{T}}^{d})$ where $Lin({\mathcal{T}}^{d})$ is the space of the
${\mathcal{C}}^{\infty}$ vector fields ’tangent’ to ${{\bf q}}$: in a pair
$({{\bf q}},{\bf\dot{{\bf q}}})$ the vector $\dot{q}_{\xi}\in R^{d}$ is
considered a vector applied at the point $q_{\xi}$.

Hence, given ${{\bf q}},{\bf\dot{{\bf q}}}$, the derivative
${\partial}_{q_{\xi,i}}\dot{q}_{\xi,i}$ is defined as well as the divergence
$({\rm div}{\bf\dot{{\bf q}}})_{\xi}{\buildrel
def\over{=}}\sum_{i=1}^{d}{\partial}_{q_{\xi,i}}\dot{q}_{\xi,i}$ of
$\dot{q}_{\xi}$.

The space of the pairs $({{\bf q}},{\bf\dot{{\bf q}}})$ will be called
${\mathcal{F}}$ and the points of ${\mathcal{T}}^{d}$ become labels of a fluid
element located at the point $q_{\xi}$ with velocity $\dot{q}_{\xi}$.

More formally $({{\bf q}},{\bf\dot{{\bf q}}})\in Dif({\mathcal{T}}^{d})\times
Lin({\mathcal{T}}^{d}){\buildrel def\over{=}}{\mathcal{F}}$ where
$Dif({\mathcal{T}}^{d})$ is the space of the ${\mathcal{C}}^{\infty}$
diffeomorphisms of ${\mathcal{T}}^{d}$ and $Lin({\mathcal{T}}^{d})$ the space
of the ${\mathcal{C}}^{\infty}$ vector fields with $0$ average: for each
$({{\bf q}},{\bf\dot{{\bf q}}})$ the vector $\dot{q}_{\xi}\in R^{d}$ is
considered applied to the point $q_{\xi}$. and $({\rm div}{\bf\dot{{\bf
q}}})_{\xi}=\sum_{i=1}^{d}{\partial}_{q_{\xi}}\dot{q}_{\xi}$.

Actually we concentrate on the subspace of $({{\bf q}},{\bf\dot{{\bf q}}})\in
SDif({\mathcal{T}}^{d})\times(SLin({\mathcal{T}}^{d})){\buildrel
def\over{=}}S{\mathcal{F}}\subset{\mathcal{F}}$ where the evolution of an
incompressible fluid takes place: $SDif({\mathcal{T}}^{d})$ being the volume
preserving diffeomphisms and $SLin({\mathcal{T}}^{d})$ the $0$-divergence
vector fields, i.e. for each such pair $({{\bf q}},{\bf\dot{{\bf q}}})$ it is
$({\rm div}{\bf\dot{{\bf q}}})_{\xi}=0$.

If the positions $q_{\xi}$ are moved the variation of $\dot{q}_{\xi}$ is
proportional to $\frac{{\partial}\dot{q}_{\xi}}{{\partial}{q_{\eta}}}$; the
Lagrangian is:

${\mathcal{L}}({\bf\dot{{\bf q}}},{{\bf
q}})=\int_{{\mathcal{T}}^{d}}\frac{1}{2}\Big{(}\dot{q}_{\xi}^{2}-Q({\bf\dot{{\bf
q}}},{{\bf q}})_{\xi}\Big{)}d\xi$ (A.1)

where $Q$ is the quadratic form on ${\mathcal{F}}$:

$-\frac{1}{4\pi}\int_{{\mathcal{T}}^{d}}d\gamma\sum_{r,s=1}^{d}\Big{(}\frac{1}{{}^{\prime\prime}|q_{\xi}-q_{\gamma}|^{\prime\prime}}\Big{)}\,\frac{{\partial}\dot{q}_{\gamma,r}}{{\partial}q_{\gamma,s}}\,\frac{{\partial}\dot{q}_{\gamma,s}}{{\partial}q_{\gamma,r}}$
(A.2)

and ${-(4\pi)^{-1}}/{{}^{\prime\prime}|x-y|^{\prime\prime}}$ symbolizes the
Green’s function for the Laplacian on ${\mathcal{T}}^{3}$: 191919i.e. formally
the summation over images $y+2\pi{\bf n},\,{\bf n}\in Z^{d}$; which makes
sense if the kernel is applied to a smooth function with $0$ average. so that
$\Delta_{q_{\xi}}Q_{\xi}=({{\mbox{\boldmath$\partial$}}_{\kern-3.0pt\lower
7.0pt\hbox{$\widetilde{}$}}\kern 3.0pt}{{\bf
p}})_{\xi}({\mbox{\boldmath$\partial$}}{{{\bf p}}_{\kern-3.0pt\lower
7.0pt\hbox{$\widetilde{}$}}\kern 3.0pt})_{\xi}$. Hence ${\mathcal{L}}$ is a
metric over ${\mathcal{F}}$ and the flow it generates runs over its geodesics.

Addition of $Q$ corresponds to a force which has the property that it keeps
data in $S{\mathcal{F}}$ inside $S{\mathcal{F}}$ as long as they evolve
smoothly in time: this is checked in the following.

The Lagrangian ${\mathcal{L}}$ leads to the canonical variables via:

$\displaystyle{p_{\xi,i}{\buildrel def\over{=}}}$ | $\displaystyle{{}\dot{q}_{\xi,i}-\frac{1}{4\pi}\int_{{\mathcal{T}}^{d}}d\lambda\int_{{\mathcal{T}}^{d}}d\gamma\frac{1}{\\{^{\prime\prime}|q_{\lambda}-q_{\gamma}|^{\prime\prime}\\}}}$
---|---
|
$\displaystyle{{}\cdot\delta_{i,r}\frac{{\partial}\delta(q_{\xi}-q_{\gamma})}{{\partial}q_{\gamma,s}}\frac{{\partial}\dot{q}_{\gamma,s}}{{\partial}q_{\gamma,r}}{\buildrel
def\over{=}}\dot{q}_{\xi,i}+A({{\bf q}},{\bf\dot{q}})_{\xi,i}}$
(A.3)

where in the last equality (defining $A$) $p_{\xi,i}=\dot{q}_{\xi,i}$ and
$A=0$ hold if ${{\bf q}}\in SDif({\mathcal{T}}^{d})$ and ${\bf\dot{{\bf
q}}}\in SLin({\mathcal{T}}^{d})$, i.e. if
$\sum_{s=1}^{d}\frac{{\partial}\dot{q}_{\gamma,s}}{{\partial}q_{\gamma,s}}=0$:
so that the diffeomorphism ${{\bf q}}$ is incompressible, and also $\dot{{\bf
q}}$ is divergenceless, ${\rm div}{\bf\dot{{\bf q}}}=0$, and the double
integral vanishes because $\int
d\lambda\\{^{\prime\prime}|q_{\lambda}-q_{\gamma}|^{\prime\prime}\\}^{-1}$ is
a constant and integration by parts over $\gamma$ becomes possible as
$d\gamma=dq_{\gamma}$ if ${{\bf q}}\in SDif({\mathcal{T}}^{d})$.

Defining the linear operator $A^{-1}({{\bf q}},{{\bf p}})$, obtained by
inverting ${{\bf p}}={\bf\dot{{\bf q}}}+A({{\bf q}},{\bf\dot{q}})$, the
equation for ${\bf\dot{{\bf p}}}$ is readily obtained, at least if $({{\bf
p}},{{\bf q}})\in S{\mathcal{F}}$:

$\dot{q}_{\xi}=p_{\xi},\qquad\dot{p}_{\xi}={\partial}_{q_{\xi}}{\mathcal{L}}({\bf\dot{q}},{{\bf
q}})={\partial}_{q_{\xi}}\frac{1}{2}Q({\bf\dot{{\bf q}}},{{\bf q}})$ (A.4)

and the Hamiltonian:

$H({{\bf p}},{{\bf q}}){\buildrel def\over{=}}\int_{{\mathcal{T}}^{d}}({{\bf
p}}_{\xi}\cdot\dot{{\bf q}}_{\xi}-{\mathcal{L}}({{\bf p}}-A^{-1}({{\bf
q}},{{\bf p}}),{{\bf q}})_{\xi})d\xi$ (A.5)

yields canonical equations, which for data in $S{\mathcal{F}}$ are:

$\displaystyle{{}\dot{q}_{\xi}=p_{\xi},\quad\dot{p}_{\xi}=-{\partial}_{q_{\xi}}P_{\xi}({{\bf
q}},{{\bf p}})}$
---
$\displaystyle{{}P_{\xi}{\buildrel
def\over{=}}\int_{{\mathcal{T}}^{d}}d\gamma\sum_{r,s=1}^{d}\Big{(}\frac{-(4\pi)^{-1}}{{}^{\prime\prime}|q_{\xi}-q_{\gamma}|^{\prime\prime}}\Big{)}\,\frac{{\partial}p_{\gamma,r}}{{\partial}q_{\gamma,s}}\,\frac{{\partial}p_{\gamma,s}}{{\partial}q_{\gamma,r}}}$
(A.6)

which hold only if $({{\bf p}},{{\bf q}})\in S{\mathcal{F}}$, while in
${\mathcal{F}}$ the equations would be more involved (but uninteresting for
the present purposes) although still Hamiltonian.

The equations can be written for data in $S{\mathcal{F}}$ setting
$x=q_{\xi},u(x)=p_{\xi},P(x)=P_{\xi}$. Then
$\frac{dp_{\xi,i}}{dt}={\partial}_{t}p_{\xi,i}+\sum_{j}p_{\xi,j}{\partial}_{q_{\xi,j}}p_{\xi,i}$
and:

$\displaystyle{{}{{\mbox{\boldmath$\partial$}}\cdot u}(x)=0,\
P(x)=-\kern-2.84526pt\sum_{r,s=1}^{d}\kern-2.84526pt\Delta^{-1}({\partial}_{s}u_{r}(x){\partial}_{r}u_{s}(x))}$
---
$\displaystyle{{}{\partial}_{t}q_{\xi}=u(q_{\xi}),\quad{\partial}_{t}u(x)=-{u_{\kern-3.0pt\lower
7.0pt\hbox{$\widetilde{}$}}\kern
3.0pt}(x)\cdot{{\mbox{\boldmath$\partial$}}_{\kern-3.0pt\lower
7.0pt\hbox{$\widetilde{}$}}\kern 3.0pt}u(x)-{\mbox{\boldmath$\partial$}}P(x)}$
$\displaystyle{{}\kern 108.12054pt+{{\bf f}}(x)+\nu\Delta u(x)}$
(A.7)

where the terms in the third line are added in the case there is forcing and
NS viscosity: the $\nu\Delta u$ will be replaced in the next Appendix C by
Ekman’s viscosity $-\nu u$ and the equation thus modified will be used to
exhibit a special symmetry arising in this case.

Therefore the Eq.(A.7) coincide with the Navier Stokes equations and their
solutions will remain in $S{\mathcal{F}}$ s long as they remain smooth: for
data not in $S{\mathcal{F}}$ only solutions local in time can be envisaged and
the equations would be more involved.202020Representing a constrained motion
as a special case of unconstrained motion subject suitable extra forces
follows a familiar prototype. A point mass constrained on a circle of radius
$R$, centered at the origin $O\in R^{3}$, can be seen as a point subject to a
centripetal force evolving under the Lagrangian
${\mathcal{L}}=\frac{1}{2}{{\bf\dot{{\bf q}}}^{2}}-\frac{(|{{\bf
q}}|-R)}{R}{{\bf\dot{{\bf q}}}^{2}}$. This leads to ${{\bf p}}={\bf\dot{{\bf
q}}}\theta$ with $\theta=(1-2\frac{|{{\bf q}}|-R}{R})$ and, for the
Hamiltonian, ${\mathcal{H}}=\frac{1}{2}\frac{{{\bf p}}^{2}}{\theta}$, to the
equations ${\bf\dot{{\bf q}}}={{\bf p}}\theta^{-1},\ {\bf\dot{{\bf
p}}}=-\frac{{{\bf p}}^{2}}{\theta^{2}R}\frac{{{\bf q}}}{|{{\bf q}}|}$. Thus it
appears that the phase space $R^{6}$ is analogous to ${\mathcal{F}}$, the maps
of the circle $\xi\to s$, mapping the arc $\xi$ 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 long as it is defined (i.e. as long
as $|{{\bf q}}|\neq 0,\infty$), which for data initially on the circle and
initial velocity tangent to it is a uniform rotation. On such motions the
Hamiltonian value is $\frac{1}{2}{{\bf p}}^{2}$ as $\theta=1$, alike the
corresponding vanishing of $Q$ on $S{\mathcal{F}}$ in Eq.(1.3),(A.5).

##  Appendix B Euler´s equation Jacobian

The Jacobian is obtained by taking suitable functional derivatives of the
transport term and the pressure term, before applying the projection operator
$\cal P$ in Eq.(2.4). The contribution of the transport term is (before
applying $\cal P$):

$\frac{{\partial}\dot{u}_{i}(x)}{{\partial}u_{j}(y)}=-\delta(x-y){\partial}_{x_{j}}u_{i}(x)-u_{k}(x)\delta_{i,j}{\partial}_{x_{k}}\delta(x-y)$
(B.1)

where the second term is an antisymmetric operator in
$L_{2}({\mathcal{T}}^{d})\times R^{d}$. The contribution from the pressure
term is

$\displaystyle{\frac{{\partial}\ {\partial}_{x_{i}}P(x)}{{\partial}u_{j}(y)}=}$ | $\displaystyle{{}-2{\partial}_{x_{i}}\int_{{\mathcal{T}}^{d}}dz\,\Delta^{-1}(x-z)}$
---|---
|
$\displaystyle{{}\cdot{\partial}_{z_{k}}\delta(z-y){\partial}_{z_{j}}u_{k}(z)}$
$\displaystyle{=}$ | $\displaystyle{{}2({\partial}_{x_{i}}{\partial}_{y_{k}}\Delta^{-1}(x-y)){\partial}_{y_{j}}u_{k}(y)}$
(B.2)

and in both Eq.(B.1),(B.2) summation over $k$ is intended.

The latter operator does not contribute to the Jacobian because acting on a
divergenceless field yields $0$; therefore the symmetric part of the Jacobian
is the multiplication operator, in $L_{2}({\mathcal{T}}^{d})\times R^{d}$, by:

$W_{i,j}(x)=-\frac{1}{2}({\partial}_{x_{j}}u_{i}(x)+{\partial}_{x_{j}}u_{i}(x))$
(B.3)

followed by the orthogonal projection $\cal P$ on the subspace of the
divergenceless fields $SLin({\mathcal{T}}^{d})\subset
L_{2}({\mathcal{T}}^{d})\times R^{d}$, Sec.I.

##  Appendix C The pairing symmetry in $\bf E^{*}$

Symmetry on the Lyapunov spectrum for the fluid equations is not really
surprising, at least not in the case of the equations obtained from Euler´s
equations in Lagrangian form when forcing and viscosity of the form $-\nu{{\bf
u}}$ (Ekman´s viscosity) are added. The ultimate reason is that 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 $H({{\bf
p}},{{\bf q}})\in R^{2n}$ be a $n$-degrees of freedom Hamiltonian and to its
equations a friction $-\nu{{\bf p}}$ is introduced turning the equations of
motion into ${\bf\dot{\bf p}}=-{\partial}_{{\bf q}}H({\bf p},{\bf q})-\nu{\bf
p},\,{\bf\dot{{\bf q}}}={\partial}_{{\bf p}}H({\bf p},{\bf q})$.

The Jacobian matrix $J_{\nu}$ (i.e. the linearization of the flow in phase
space $({\bf p},{\bf q})\in R^{2n}$) is a symplectic matrix up to the
contribution from the friction force: $J_{\nu}=J_{0}$ \+ the contribution from
the friction force.

The latter is a diagonal matrix with the first $\frac{n}{2}$ elements $=-\nu$
and the rest $=0$: hence it is the sum of a diagonal matrix with the first
half entries equal to $-\frac{\nu}{2}$ and the others equal to
$+\frac{1}{2}\nu$ plus a second diagonal matrix with all the $2n$ elements
$=-\frac{\nu}{2}$.

Hence the Jacobian matrix of the equations with friction is recognized to be
the sum of the symplectic Jacobian matrix of the Hamiltonian $H({\bf p},{\bf
q})+\frac{\nu}{2}{\bf p}\cdot{\bf q}$ and of the constant diagonal matrix
$-\frac{\nu}{2}$: namely $J_{\nu}=J_{0}-\frac{\nu}{2}$.

Therefore the eigenvalues will be equal to $-\frac{\nu}{2}$ plus those of a
symplectic matrix: such eigenvalues arise in pairs of opposite sign. The same
pairing will remain true for products of Jacobians $J_{\nu}({{\bf u}}_{i})$
and therefore the Lyapunov exponents will arise in pairs with sum exactly
$=-\nu$.

In Sec.I the equations $\bf E^{*}$ in ${\mathcal{F}}$ defined by Eq.(1.3) are
geodesic flows (Appendix A) which evolve in ${\mathcal{F}}$ leaving invariant
the manifold $S{\mathcal{F}}\subset{\mathcal{F}}$. The Jacobian is therefore
symplectic even if restricted to $S{\mathcal{F}}$.

Hence if viscosity and forcing are added, with force depending only on
$q_{\xi}$ and friction expressed as $-\nu p_{\xi}$ (Ekman´s viscosity),
obtaining dissipative equations that can be called $\bf IE^{*}$ (i.e.
Lagrangian Euler’s with Ekman’s viscosity), the pairing to $-\nu$ is valid by
above argument on the geodesic flow.

The argument in [9] can also be applied to $\bf RE^{*}$, i.e. to the equation
obtained by adding to $\bf E^{*}$ the forcing and a reversible viscosity
$-\alpha(p)p_{\xi}$ with, if $x=q_{\xi},u(x)=p_{\xi}$, and $\alpha(u)$ given
by item (1) in Eq.(3.1).

The $\bf RE^{*}$ equation, in 2D and 3D, implies energy conservation:
therefore there is a $0$ Lyapunov exponent that can be added to the $0$
exponent associated with the flow direction.

Discarding the latter $0$ exponents the important new idea added in [9] to the
above summarized work [10] will imply that the Lyapunov exponents of the
Lagrangian motion of $\bf RE^{*}$ on $S{\mathcal{F}}$ are paired to the
average of $\alpha({{\bf u}})$ (if they exist, i.e. in 2D).212121In [9] the
force is supposed conservative: in $\bf RE^{*}$ the force is solenoidal
(divergenceless). Yet this does not affect the result because what really
matters is that the Jacobian of the equation with $0$ viscosity is a
symplectic matrix; and this is implied by the Hamiltonian nature of the
evolution. Note that in the above preliminary analysis of the method in [10]
the pairing would also follow, and for the same reason, if a Lorentz or
Coriolis type of force were added, like ${\mbox{\boldmath$\varphi$}}({{\bf
q}})\wedge\dot{{\bf q}}$ with ${\mbox{\boldmath$\varphi$}}({{\bf q}})$ an
incompressible field, i.e. ${\mbox{\boldmath$\varphi$}}({{\bf
q}})={\mbox{\boldmath$\partial$}}\wedge{\bf A}({{\bf q}})$.

The analysis does not apply to the $2{\mathcal{N}}-2$ exponents of the
truncated equations: because it is unclear whether the argument in [9] applies
to them.

So in 2D the Lyapunov exponents of the equations $\bf IE^{*},RE^{*}$ will
verify a pairing symmetry by reflection at the level $-\nu$ or to the average
of $\alpha$ respectively. But this neither applies to the Navier Stokes
equations, because the viscosity in not a force proportional to ${{\bf u}}$,
nor to $\bf IE^{*}$ in 3D because there is no global existence uniqueness
result on the flow $S_{t}$. About the regularized $\bf E^{*},IE^{*},RE^{*}$
equations pairing might just be approximate.

See [15, Sec.6] for heuristic considerations about possible extensions of the
above remarks to the NS equations considered as equations for the velocity
field only. A particular intriguing question is: is there a relation between
the Lyapunov spectrum of $\bf E^{*}$, Eq.(1.2), and the corresponding spectrum
of the simple Euler equation $\bf E$, Eq.(1.1) ?

For instance is it possible that the $\bf E^{*}$ (Lagrangian Euler equations,
Eq.(1.2)) has a Lyapunov spectrum which is just the one of $\bf E$ (the Euler
equations $\bf E$, Eq.(1.1)) with each exponent counted twice, [19] ? If so
the pairing in the simple Euler equations $\bf E$ with forcing and Ekman
viscosity added would hold at the level $-\frac{1}{2}\nu$ and in the
corresponding reversible equations at the level of the average of
$-\frac{1}{2}\alpha({{\bf u}})$.

##  Appendix D The Axiom C.

To describe the main features of the Axiom C, [6], consider first the simpler
case of a reversible diffeomorphism $S$, i.e. such that there is a
diffeomorphism, $I$ such that $IS=S^{-1}I,I^{2}=1$. Imagine that the
attracting set ${\mathcal{A}}$ differs from its time reversal image
$I{\mathcal{A}}=\cal R$ and that CH holds.

The tangent space at a generic point $z$ is supposed to be smoothly decomposed
as $T_{u}(z)\oplus T_{s}(z)\oplus T_{m}(z)$. If $z\in{\mathcal{A}}$ or
$z\in\cal R$ then $T_{s}(z),T_{u}(z)$ coincide with the tangent, at $z$, to
the stable manifold of $S$ on ${\mathcal{A}}$ or $\cal R$ respectively;
furthermore for each ball $U_{\delta}(x)\subset{\mathcal{A}}$, of radius
$\delta$, consider the manifolds $W_{i}(x)\cap U_{\delta}(x),i=u,s$, and
assume that they can be continued into smooth manifolds $W_{+},W_{-}$
everywhere tangent $T_{s}\oplus T_{m}$ and $T_{u}\oplus T_{m}$ and which
intersect $\cal R$ in a single point $\widetilde{x}=Px$ if $\delta$ is small
enough: thus defining $P$ as a map between ${\mathcal{A}},\cal R$.

Finally, as the labels s,u suggest, the vectors in $T_{s},T_{u}$ uniformly
contract exponentially as time tends to $+\infty$ or $-\infty$ respectively,
while vectors in $T_{m}$ contract exponentially as $t\to\pm\infty$ (i.e. in
both directions, being ´squeezed´ on ${\mathcal{A}}$ and $\cal R$).

The case of a flow $S_{t}$ can be described similarly by imagining that
$T_{m}$ contains also the neutral direction $\frac{d}{dt}S_{t}{{\bf u}}$ and
contracts transversally to it. In this context Axiom C adds to the Axiom B,
[42], the assumption of the existence of a repeller $\cal R$ intersected by
the manifolds emerging from ${\mathcal{A}}$.

The latter property permits to establish the map $P$, thus allowing to define
the composition $\widetilde{I}=PI$, acting as a time reversal on
${\mathcal{A}}$ and $\cal R$, because the invariance of the manifolds implies
that on ${\mathcal{A}}\cup\cal R$ it is $PS_{t}=S_{t}P$: so that
$\widetilde{I}S_{t}=S_{-t}\widetilde{I}$ on ${\mathcal{A}}\cup\cal R$ (note
that $\widetilde{I}$ is not defined outside ${\mathcal{A}}\cup\cal R$). See
[6] for more details.

Acknowledgements: I am indebted to M. Cencini for pointing out an error on the
first version of the paper.

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