Identical ideal individual hypothesis

Jiao-Kai Chen

School of Physics and Information Engineering,
Shanxi Normal University, Linfen 041004, P. R. China
E-Mail<EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

The identical ideal individual hypothesis is proposed. According to this
hypothesis, the identical ideal individuals should be classified into two
classes: the bosonic individuals and the fermionic individuals. The bosonic
individuals can occupy the same behavior state while the fermionic individuals
can not be in the same behavior state. We propose that human beings and many
species of animals are fermionic, which can not occupy the same behavior state
according to the Pauli exclusion principle. An unified theoretical explanation
is given for the natures of two important and seemingly irrelated phenomena in
psychology: the existence of the personal space and the behavior
differentiation under high population density condition.

Keywords: Behavior coordinate system, identical ideal individual, fermionic
individual, bosonic individual, personal space, behavior differentiation

##  1 Introduction

The application of concepts and principles of quantum theory in physics to
psychology which we call quantum psychology here for simplicity is promising
and is attracting more and more attention (Aerts, 2009; Aerts & Aerts, 1995;
Aerts, Gabora & Sozzo, 2013; Atmanspacher, Römer & Walach, 2002; Bagarello,
2019; Bagarello, Basieva & Khrennikov, 2018; Bagarello, Basieva, Pothos &
Khrennikov, 2018; Bordley, 1998; Bruza, Wang & Busemeyer, 2015; Busemeyer &
Bruza, 2012; Busemeyer, Wang, Khrennikov & Basieva, 2014; Chen, 2019; Haven &
Khrennikov, 2013; Khrennivov, 1999; Melkikh, 2019; Pothos & Busemeyer, 2009;
Rossi, 1994; Triffet & Green, 1996; Valle, 1989). Quantum psychology has been
used to investigate many puzzling phenomena and problems accumulated in
psychological science which can not be solved by classical theories (Busemeyer
& Bruza, 2012) . Based on the ideal individual model, behavior coordinate
system and quantum probability (Chen, 2019) , we assume that the behavior
state of an ideal individual should be represented by a behavior state
function and then propose the identical ideal individual hypothesis.

The existence of the personal space (Hall, 1966; Katz, 1937; Sommer, 1959;
Stern, 1938; Uexkull, 1937) and the behavior differentiation under high
population density condition (Calhoun, 1962; Evans, 1979; Marsden, 1972)are
two important and confusing issues in psychology. The natures of these two
phenomena remain mysterious and irrelated in old theories. We suggest that
these two phenomena are quantum effects. By applying the identical ideal
individual hypothesis, we offer a possible theoretical explanation in an
unified approach for the natures of these two seemingly irrelated phenomena.

In the remainder of this paper we first introduce the concepts of the behavior
coordinate system and the behavior state function. Second, we propose the
identical ideal individual hypothesis. Third, we discuss the consequences of
the identical ideal individual hypothesis. We offer a possible theoretical
explanation in an unified approach for the natures of the existence of the
personal space and the behavior differentiation under high population density
condition. Fourth, the conclusions are given.

##  2 Preliminaries

###  2.1 Behavior coordinate system

As position of a point particle is the basic variable in Newtonian mechanics,
behavior or behavior position of an individual is the basic variable in
psychology. Behavior refers to the observable actions of an ideal individual,
e.g. speech, body movement and emotional expression. (In this work, the case
of human beings is discussed, other species can be investigated similarly.)
Except for the intrinsic quantities such as age and gender, quantities such as
motivation, emotion and personality are supposed to be functions of behavior
and the derivative of behavior with respect to time. In other words, these
quantities can be revealed by behavior and its derivative (Tolman, 1948) .

An ideal individual is an idealization of humans or other species of animals
(Chen, 2019) . For an ideal individual, every possible behavior can occur with
equal probability. These possible behaviors are elements of the set whose
members are the behaviors of humans and animals that have occurred in the
past, are occurring at present and will occur in the future. Different
behaviors show different properties and make distinction among different
species and different individuals.

Let the behavior coordinate system be a coordinate system that specifies each
behavior point uniquely in a behavior space by a set of numerical behavior
coordinates (Hock, 2015; Rosenhan, 1973) . The reference lines are
$q_{1}$-axis, $q_{2}$-axis, $q_{3}$-axis and so on. To describe the behaviors
of human beings, we can assume that $q_{1}$-axis is the speech-axis,
$q_{2}$-axis is the body-movement-axis and $q_{3}$-axis is the emotional-
expression-axis. For simplicity, we assume the behavior coordinate system is a
Cartesian coordinate system, and the behavior coordinate space is a
n-dimensional Euclidean space $\mathbb{R}^{n}$. Although the behavior
coordinate system is far from being well established now, it is useful for
solving many problems and it can give some enlightening results.

###  2.2 Behavior state function

The behavior state of an ideal individual is assumed to be represented by a
behavior state function ${\Psi({\bf x},{\bf q},t)}$ (Chen, 2019) which
represents one behavior pattern. ${\Psi({\bf x},{\bf q},t)}$ is a vector in
the Hilbert space, where $t$ is time, ${\bf x}$ denotes the spatial space
vector ${\bf x}=(x,y,z)$. The spatial position in psychology is some different
from that in physics because an organism has volume and cannot be regarded as
a particle in psychology although it can be regarded as a particle in physics
which is defined to be an object of insignificant size. ${\bf q}$ denotes the
behavior space vector ${\bf q}=(q_{1},q_{2},q_{3},\cdots)$. $q_{1}$ is the
speech coordinate, $q_{2}$ is the body-movement coordinate, $q_{3}$ is the
emotional-expression coordinate, and $\cdots$ denotes other coordinates.

Adopting the Born hypothesis which gives the probability interpretation of the
wave function in quantum mechanics, the behavior state function ${\Psi({\bf
x},{\bf q},t)}$ is assumed to be a probability amplitude, such that
${\left|\Psi({\bf x},{\bf q},t)\right|^{2}}$ gives the probability of finding
an ideal individual at a behavior point ${\bf q}$ at time $t$ and spatial
point ${\bf x}$. There is the normalization condition

$\int{\left|\Psi({\bf x},{\bf q},t)\right|^{2}}{\rm d}{\bf q}{\rm d}{\bf
x}=1.$ (1)

The integration is over whole behavior space and spatial space. In many cases,
the spatial coordinates have small or even no effects on an individual’s
behavior and then the spatial coordinates can be neglected. In some cases, the
spatial space is taken as part of the environment. In consequence, the
behavior coordinates are highlighted in these cases and the spatial
coordinates can be integrated out,

${\psi({\bf q},t)}=c\int{{\Psi({\bf x},{\bf q},t)}}{{\rm d}{\bf
x}},\quad\int\psi^{\ast}({\bf q},t)\psi({\bf q},t){\rm d}{\bf q}=1,$ (2)

where $c$ is a normalization factor. We assume the superposition principle
holds for $\Psi$ and $\psi$,

$\Psi=\sum_{i=1}^{n}a_{i}\Psi_{i},\quad\psi=\sum_{i=1}^{n}b_{i}{\psi_{i}}.$
(3)

In many references (Busemeyer & Bruza, 2012; Haven & Khrennikov, 2013) , the
behavior state function ${\psi({\bf q},t)}$ has been used actually to discuss
problems although the behavior variable ${\bf q}$ is not given explicitly.

##  3 Identical ideal individuals

Human beings and animals are macroscopic objects whose mechanical motion in
spatial space is described not by quantum mechanics but by Newtonian
mechanics. The considered quantum effects are not on mechanical motion but on
behaviors. The ideal individuals can be regarded as being identical
approximately under some conditions and then some quantum effects will become
prominent. For simplicity, let ${\bf x}_{1}={\bf x}_{2}=\cdots={\bf x}$,
$t_{1}=t_{2}=\cdots=t$, where ${\bf x}_{i}$ is the spatial position of the
$i$-th individual at time $t_{i}$.

Two ideal individuals are identical if there should be no experiment that
detects any intrinsic difference between them. The identical ideal individuals
are those individuals that have the same age, gender, etc. and behave in the
same manner under equal conditions. Suppose there is a system of two
indistinguishable individuals $1$ and $2$. Because of the identity of the
ideal individuals, the states of the system obtained from each other by merely
interchanging the two individuals must be completely equivalent. This means
that, as a result of this interchange, the state function of the system can
change only by an unimportant phase factor. This operation is carried out by
the operator ${\hat{P}_{12}}$

${\hat{P}_{12}}\Psi({\bf x},{\bf q}_{1},t;{\bf x},{\bf q}_{2},t)=\Psi({\bf
x},{\bf q}_{2},t;{\bf x},{\bf q}_{1},t)={\rm e}^{{\rm i}\lambda}\Psi({\bf
x},{\bf q}_{1},t;{\bf x},{\bf q}_{2},t),$ (4)

where $\Psi({\bf x},{\bf q}_{1},t;{\bf x},{\bf q}_{2},t)$ is the behavior
state function, $\lambda$ is some real constant. ${\hat{P}_{12}}$ is the
permutation operator,

${\hat{P}_{12}}={\hat{P}_{12}}^{-1},\quad{\hat{P}_{12}}{\hat{P}_{12}}=I,\quad{\hat{P}_{12}}^{{\dagger}}={\hat{P}_{12}},$
(5)

where $I$ is the identity operator. A second exchange of these two individuals
recreates the original state. Hence,

${\hat{P}_{12}}^{2}\Psi({\bf x},{\bf q}_{1},t;{\bf x},{\bf q}_{2},t)={\rm
e}^{2{\rm i}\lambda}\Psi({\bf x},{\bf q}_{1},t;{\bf x},{\bf
q}_{2},t)=\Psi({\bf x},{\bf q}_{1},t;{\bf x},{\bf q}_{2},t),$ (6)

yielding

${\rm e}^{{\rm i}\lambda}=\pm 1.$ (7)

Therefore either

${\hat{P}_{12}}{\Psi_{s}}={\Psi_{s}}\;\;{\rm
or}\;\;{\hat{P}_{12}}{\Psi_{a}}=-{\Psi_{a}}\;\;{\rm holds},$ (8)

where

${\Psi_{s}}=\frac{1}{\sqrt{2}}(1+{\hat{P}_{12}})\Psi({\bf x},{\bf
q}_{1},t;{\bf x},{\bf
q}_{2},t),\;\;{\Psi_{a}}=\frac{1}{\sqrt{2}}(1-{\hat{P}_{12}})\Psi({\bf x},{\bf
q}_{1},t;{\bf x},{\bf q}_{2},t).$ (9)

We call the behavior state function ${\Psi_{s}}$ with the eigenvalue $+1$
symmetric and ${\Psi_{a}}$ with the eigenvalue $-1$ antisymmetric with respect
to the exchange of two indistinguishable ideal individuals.

Whether individuals are described by a symmetric or an antisymmetric behavior
state function will depend on their nature. The identical ideal individuals
should be classified into two classes: the bosonic individuals and the
fermionic individuals. The bosonic individuals can be described by a symmetric
behavior state function ${\Psi_{s}}$ and they can occupy the same behavior
state. The fermionic individuals are described by an antisymmetric behavior
state function ${\Psi_{a}}$ and they can not be in the same behavior state. If
the fermionic individuals occupy the same behavior state, we can obtain
${\Psi_{a}}=0$ from Eq. (9), which we call “zero state” or “dead state”. This
result is known as the Pauli exclusion principle which is sometimes expressed
by stating that no two fermionic individuals can occupy the same behavior
state.

##  4 Discussions

If the difference between the spatial positions and some intrinsic properties
of individuals are indistinguishable approximately or can be neglected to some
extent as the ideal individuals are congregated with high population density
111The population density is the number of organims per unit volume, $n=N/V$,
where $N$ is the number of organisms, $V$ is the spatial volume occupied by
organisms. The population density is high if $n{\geq}n_{c}$, where $n_{c}$ is
the critical value of population dentisy. $n_{c}=1/V_{ps}$, where $V_{ps}$ is
the personal space of this species of organisms. Under high population density
condition, the organisms can be regarded as being indistinguishable by their
spatial positions in quantum psychology. , these ideal individuals are assumed
to be identical.

If the identical ideal individuals are bosonic, they can occupy the same
behavior state and a large fraction of them can occupy the lowest behavior
state under some conditions as a Bose-Einstein condensate. It can be expected
that the bosonic individuals will congregate not only in behavior space but
also in spatial space under some conditions. Evidently, the bosonic
individuals who have the same behavior state and are assembled with high
population density are not in the advantageous position in competition and
survival of species according to Darwin’s theory of nature selection because
they as a species are vulnerable in suddenly appeared harsh environments. We
conjecture that the species whose members are bosonic will be small in number
if there exist this kind of species in nature and their living environment
will be very different from the environment where the fermionic species live.

If the identical ideal individuals are fermionic, they can not occupy the same
behavior state according to the Pauli exclusion principle. The fermionic
individuals taking the same behavior state or behavior pattern are repulsive.
Consequently, we can conjecture that the fermionic individuals need their
personal spatial space to be distinguished by spatial positions and then they
are not identical again so that they can behave not only in different patterns
but also in the same pattern. This discussion offers a possible theoretical
explanation for the existence of the personal space (Sommer, 1959) which is a
quantum effect in spatial space. Simultaneously, human beings and many species
are expected to be fermonic. According to the above discussions, we also
suggest that not only human beings but also other species of organisms have
their personal spaces if their behavior states can be described by the
behavior state function.

The fermionic individuals occupying the same behavior state will be in “zero
state”, otherwise, they will be forced by the Pauli exclusion principle to be
in different behavior states or to form different behavior patterns. This
discussion gives a possible theoretical explanation for the behavior
differentiation under the high population density condition (Calhoun, 1962) .
The behavior differentiation is a quantum effect in behavior space. Let the
behavior state be represented by the behavior state function $\Psi_{nm}$,
where $n$ and $m$ are quantum numbers. For degenerate states, there are many
possible values of $m$ for a given value of $n$. Therefore, some rats can
behave in the same behavior pattern with different $m$ which does not violate
the Pauli exclusion principle. It offers a possible explanation for the
quantities of rats exhibiting the same behavior pattern under high population
density condition (Calhoun, 1962) .

The existence of the personal space and the behavior differentiation under
high population density condition seem irrelated and the natures of them has
been remaining mysterious ever before. Now we tend toward believing that they
are quantum phenomena arising from the identity of the ideal individuals. The
unified explanations for the natures of these two phenomena can be regarded as
an evidence that psychology can be described by quantum theory. More prominent
quantum effects are expected to be observed in the behaviors of simple
organisms if their behaviors can be described by quantum theory. Experiments
on simple organisms such as amoeba, paramecium and bacteria under high
population density condition are expected.

##  5 Conclusions

Based on the behavior coordinate system, the ideal individual model and
quantum probability, we assume that the behavior state of an ideal individual
should be represented by a behavior state function. Then we propose the
identical ideal individual hypothesis. According to this hypothesis, the
identical ideal individuals should be classified into two classes: the bosonic
individuals and the fermionic individuals. The bosonic individuals can occupy
the same behavior state while the fermionic individuals can not be in the same
behavior state. We conjecture that the bosonic species will be small in number
or be very different from the fermionic species if they exist in nature. We
propose that human beings and many species of animals are fermionic, which can
not occupy the same behavior state according to the Pauli exclusion principle.

The existence of the personal space and the behavior differentiation under
high population density condition are explained theoretically in an unified
approach by using the identical ideal individuals hypothesis. The existence of
the personal space is a quantum effect in spatial space caused by the identity
of the ideal individuals while the behavior differentiation under high
population density condition is a quantum effect in behavior space.

## References

  * (1)
  * Aerts (2009) Aerts, D. (2009). Quantum structure in cognition. J. Math. Psychology 53, 314-348.
  * Aerts & Aerts (1995) Aerts, D. & Aerts, S. (1995). Applications of quantum statistics in psychological studies of decision processes. Found. Sci. 1, 85-97.
  * Aerts, Gabora & Sozzo (2013) Aerts, D., Gabora, L. & Sozzo, S. (2013). Concepts and Their Dynamics: A Quantum-Theoretic Modeling of Human Thought. Topics in Cognitive Sciences 5, 737-772.
  * Atmanspacher, Römer & Walach (2002) Atmanspacher, H., Römer, H. & Walach, H. (2002). Weak quantum theory: complementarity and entanglement in physics and beyond. Found. Phys. 32(3), 379-406.
  * Bagarello (2019) Bagarello, F. (2019). Quantum Concepts in the Social, Ecological and Biological Sciences. New York: Cambridge University Press.
  * Bagarello, Basieva & Khrennikov (2018) Bagarello, F., Basieva, I. & Khrennikov, A. (2018). Quantum field inspired model of decision making: Asymptotic stabilization of belief state via interaction with surrounding mental environment. J. Math. Psychology 82, 159-168.
  * Bagarello, Basieva, Pothos & Khrennikov (2018) Bagarello, F., Basieva, I., Pothos, E. M. & Khrennikov, A. (2018). Quantum like modeling of decision making: Quantifying uncertainty with the aid of Heisenberg CRobertson inequality. J. Math. Psychology 84 49-56.
  * Bordley (1998) Bordley, R. F. (1998). Quantum mechanical and human violations of compound probability principles: toward a generalized Heisenberg uncertainty principle. Operations Research 46(6), 923-926.
  * Bruza, Wang & Busemeyer (2015) Bruza, P. D., Wang, Z. & Busemeyer, J. R. (2015). Quantum cognition: a new theorectical approach to psychology. Trends in Cognitive Sciences 19(7), 383-393.
  * Busemeyer & Bruza (2012) Busemeyer, J. R. & Bruza, P. D. (2012). Quantum Models of Cognition and Decision. Cambridge: Cambridge University Press.
  * Busemeyer, Wang, Khrennikov & Basieva (2014) Busemeyer, J. R., Wang, Z., Khrennikov, A. & Basieva, I. (2014). Applying quantum principles to psychology. Phys. Scr. T163, 014007.
  * Calhoun (1962) Calhoun, J. B. (1962). Population density and social pathology. Scientific American 206(3), 139-148.
  * Chen (2019) Chen, J. K. (2019). A novel quantum theory of psychology. arXiv: 1903.02633V2 [q-bio.NC].
  * Evans (1979) Evans, G. W. (1979). Behavioral and psychological consequences of crowding in humans. Journal of Applied Social Psychology 9, 27-46.
  * Hall (1966) Hall, E. T. (1966). The Hidden Dimension. New York: Doubleday.
  * Haven & Khrennikov (2013) Haven, E. & Khrennikov, A. (2013). Quantum Social Science. Cambridge: Cambridge University Press.
  * Hock (2015) Hock, R. R. (2015). Forty Studies That Changed Psychology: Explorations into the History of Psychological Research 7ed. Boston: Pearson Education Limited.
  * Katz (1937) Katz, D. (1937). Animals and Men. New York: Longmans, Green.
  * Khrennivov (1999) Khrennivov, A. (1999). Classical and Quantum Mechanics on Information Spaces with Applications to Cognitive, Psychological, Social, and Anomalous Phenomena. Foundations of Physics 29(7), 1065-1098.
  * Marsden (1972) Marsden, H. M. (1972). Crowing and Animal behavior. In J. E. Wohlhill & D. H. Carson (Eds.), Environment and the Social Sciences. Washington, DC: American Pshyological Association.
  * Melkikh (2019) Melkikh, A. V. (2019). Thinking as a quantum phenomenon. BioSystems 176, 32-40.
  * Pothos & Busemeyer (2009) Pothos, E. M. & Busemeyer, J. R. (2009). A quantum probability explanation for violations of ‘rational’ decision theory. Proc. R. Soc. B 276, 2171-2178.
  * Rosenhan (1973) Rosenhan, D. L. (1973). On being sane in insane places. Science 179, 250-258.
  * Rossi (1994) Rossi, E. L. (1994). A quantum psychology for the future? Psychological Perspectives 30(1), 6-11.
  * Sommer (1959) Sommer, R. (1959). Studeis in Personal Space. Sociometry 22(3), 247-260.
  * Stern (1938) Stern, W. (1938). General Psychology, translated by H. D. Spoerl. New York: Macmillan.
  * Tolman (1948) Tolman, E. C. (1948). Cognitive maps in rats and men. Psychological Review 55, 189-208.
  * Triffet & Green (1996) Triffet, T. & Green, H. S. (1996). Consciousness: Computing the Uncomputable. Mathl. Comput. Modeling 24(3), 37-56.
  * Uexkull (1937) Uexkull, J. Von. (1937). A Stroll through the Worlds of Animals and Men, in Instinctive Behavior, Claire Schiller ed. New York: International University Press.
  * Valle (1989) Valle, R. S. (1989). Relativistic Quantum Psychology. In: R. S. Valle, R. von Eckartsberg (eds) Metaphors of Consciousness. New York: Plenum Press.


Dynamical ensembles equivalence

in fluid mechanics∗ ∗<EMAIL_ADDRESS>#96-???, and chao-
dyn@xyz.lanl.gov,#960????

G. Gallavotti

Fisica, Università di Roma La Sapienza, P.le Moro 2, 00185, Roma, Italia.

Abstract: Dissipative Euler and Navier Stokes equations are discussed with the
aim of proposing several experiments apt to test the equivalence of dynamical
ensembles and the chaotic hypothesis.

PACS: 47.52.+j, 05.45.+b, 05.70.Ln, 47.70.-n

Keywords: Navier Stokes, Chaotic hypothesis, Caos, Nonequilibrium ensembles

§1 Reversible dissipation in Euler and Navier Stokes equations. In [Ga6] one
finds an analysis leading to a conjecture on the equivalence between the
irreversible NS equation and a reversible equation that was called GNS
(”gaussian NS equation”).

The ideas discussed in [Ga6] are, however, much more general and it is worth
pointing out some other applications, as well as several possible tests that
seem under reach of present day experimental (numerical and ”real”)
techniques.

I shall focus on fluid mechanics problems considering a fluid that:

(1) is enclosed in a periodic box $\Omega$ with side $L$, possibly with a few
disks (”obstacles”) removed so that no infinite straight path can be found in
$\Omega$ that avoids the obstacles,

(2) is incompressible with density $\rho$.

I shall consider four distinct evolution equations for this fluid, all of
dissipative nature.

$\eqalign{\dot{{\,\underline{u}\,}}+u_{\kern-3.0pt\lower 6.6pt\hbox
to3.1298pt{$\widetilde{}$\hfill}}\kern 2.0pt\,\cdot\partial_{\kern-3.0pt\lower
6.6pt\hbox to3.1298pt{$\widetilde{}$\hfill}}\kern
2.0pt\,\,{\,\underline{u}\,}=-{1\over\rho}{\,\underline{\partial}\,}p+{\,\underline{g}\,}+\nu\Delta{\,\underline{u}\,},\qquad{\,\underline{\partial}\,}\cdot{\,\underline{u}\,}=0&\qquad{\rm
NS}\cr\dot{{\,\underline{u}\,}}+u_{\kern-3.0pt\lower 6.6pt\hbox
to3.1298pt{$\widetilde{}$\hfill}}\kern 2.0pt\,\cdot\partial_{\kern-3.0pt\lower
6.6pt\hbox to3.1298pt{$\widetilde{}$\hfill}}\kern
2.0pt\,\,{\,\underline{u}\,}=-{1\over\rho}{\,\underline{\partial}\,}p+{\,\underline{g}\,}+\beta\Delta{\,\underline{u}\,},\qquad{\,\underline{\partial}\,}\cdot{\,\underline{u}\,}=0&\qquad{\rm
GNS}\cr\dot{{\,\underline{u}\,}}+u_{\kern-3.0pt\lower 6.6pt\hbox
to3.1298pt{$\widetilde{}$\hfill}}\kern 2.0pt\,\cdot\partial_{\kern-3.0pt\lower
6.6pt\hbox to3.1298pt{$\widetilde{}$\hfill}}\kern
2.0pt\,\,{\,\underline{u}\,}=-{1\over\rho}{\,\underline{\partial}\,}p+{\,\underline{g}\,}-\chi{\,\underline{u}\,},\qquad{\,\underline{\partial}\,}\cdot{\,\underline{u}\,}=0&\qquad{\rm
ED}\cr\dot{{\,\underline{u}\,}}+u_{\kern-3.0pt\lower 6.6pt\hbox
to3.1298pt{$\widetilde{}$\hfill}}\kern 2.0pt\,\cdot\partial_{\kern-3.0pt\lower
6.6pt\hbox to3.1298pt{$\widetilde{}$\hfill}}\kern
2.0pt\,\,{\,\underline{u}\,}=-{1\over\rho}{\,\underline{\partial}\,}p+{\,\underline{g}\,}-\alpha{\,\underline{u}\,},\qquad{\,\underline{\partial}\,}\cdot{\,\underline{u}\,}=0&\qquad{\rm
GED}\cr}$ $None$

In the case $\Omega$ contains obstacles a ”no friction” boundary condition
will be imposed on $\partial\Omega$, i.e.
${\,\underline{u}\,}\cdot{\,\underline{n}\,}=0$ if ${\,\underline{n}\,}$ is
the normal to $\partial\Omega$. The first equation is the well known Navier
Stokes equation with $\nu$ being the viscosity.

The second equation, introduced in [Ga6] and called GNS, has a multiplier
$\beta$ defined so that the total vorticity $\eta
L^{3}=\rho\int{\,\underline{\omega}\,}^{2}\,dx$, with
${\,\underline{\omega}\,}=\partial\wedge{\,\underline{u}\,}$ being the
vorticity, is a constant of motion; this means that:

$\beta({\,\underline{u}\,})={\int\big{(}{\,\underline{\partial}\,}\wedge{\,\underline{g}\,}\cdot{\,\underline{\omega}\,}+{\,\underline{\omega}\,}\cdot\,(\omega_{\kern-3.0pt\lower
6.6pt\hbox to3.1298pt{$\widetilde{}$\hfill}}\kern
2.0pt\,\cdot\partial_{\kern-3.0pt\lower 6.6pt\hbox
to3.1298pt{$\widetilde{}$\hfill}}\kern
2.0pt\,{\,\underline{u}\,})\big{)}\,d{\,\underline{x}\,}\over\int{\,\underline{\omega}\,}^{2}\,d{\,\underline{x}\,}}$
$None$

The third equation will be called the Euler dissipative equation, ED: it
represents a non viscous ideal fluid moving in a ”sticky background”: $\chi$
is a ”sticky” viscosity. The model is, as far as I know, not a good model for
any physical situation, but it is interesting to consider it mainly for
comparison purposes.

The fourth equation will be called GED equation, gaussian dissipative Euler
equation and here $\alpha$ is a multiplier defined so that the total (kinetic)
energy $\varepsilon L^{3}={\rho\over 2}\int{\,\underline{u}\,}^{2}\,dx$ is a
constant of motion in spite of the action of the force ${\,\underline{g}\,}$;
this means that $\alpha$ is given by:

$\alpha({\,\underline{u}\,})={\int{\,\underline{g}\,}\cdot{\,\underline{u}\,}\over\int{\,\underline{u}\,}^{2}}$
$None$

A similar equation, with a constraint on the energy contained in each
“momentum shell” to be constant was considered in [ZJ], which is the first
paper in which the idea of a reversible Navier Stokes equation is advanced and
studied. The energy content of each “momentum shell” was fixed to be the value
predicted by Kolmogorov theory, [LL].

Note that both the GED and the GNS equations have a symmetry in
${\,\underline{u}\,}$, so that they are reversible in the sense that, if
$V_{t}$ is the flow describing the equation solution (so that $t\to
V_{t}{\,\underline{u}\,}={\,\underline{u}\,}(t)$ is the solution with initial
data ${\,\underline{u}\,}$), then the transformation
$i:{\,\underline{u}\,}\to-{\,\underline{u}\,}$ anti-commutes with the time
evolution $V_{t}$:

$i\,V_{t}\,=\,V_{-t}\,i$ $None$

We shall avoid (as it is, unfortunately, always the case in the current
literature) considering the problem of proving the global existence of
solution to the equations (1.1) (the problem is in fact open, see [Ga1]) and
we shall consider the truncated equations with momentum cut off $K$.

The truncation will be performed on a suitable orthonormal basis for the
divergenceless fields in $\Omega$: given the boundary conditions we consider
it natural to use the basis generated by the minimax principle applied to the
Dirichlet quadratic form $\int_{\Omega}(\partial_{\kern-3.0pt\lower 6.6pt\hbox
to3.1298pt{$\widetilde{}$\hfill}}\kern
2.0pt\,\,{\,\underline{u}\,})^{2}d{\,\underline{x}\,}$ defined on the space of
the $C^{\infty}(\Omega)$ divergenceless fields ${\,\underline{u}\,}$ with
${\,\underline{u}\,}\cdot{\,\underline{n}\,}=0$ on $\partial\Omega$. The basis
fields ${\,\underline{u}\,}_{j}$ will verify:
$\Delta{\,\underline{u}\,}_{j}=-E_{j}{\,\underline{u}\,}_{j}+{\,\underline{\partial}\,}_{j}\mu$,
for a suitable multiplier $\mu_{j}$, with ${\,\underline{u}\,}_{j},\mu_{j}\in
C^{\infty}$ and $E_{j}$ are eigenvalues).

For instance in the case of no obstacles let the
${\,\underline{u}\,}=\sum_{{\,\underline{k}\,}\neq{\,\underline{0}\,}}{\,\underline{\gamma}\,}_{{\,\underline{k}\,}}e^{i{\,\underline{k}\,}\cdot{\,\underline{x}\,}}$
be the velocity field represented in Fourier series with
${\,\underline{\gamma}\,}_{{\,\underline{k}\,}}={\overline{\,\underline{\gamma}\,}_{-{\,\underline{k}\,}}}$
and ${\,\underline{k}\,}\cdot{\,\underline{\gamma}\,}_{{\,\underline{k}\,}}=0$
(incompressibility condition); here ${\,\underline{k}\,}$ has components that
are integer multiples of the ”lowest momentum” $k_{0}={2\pi\over L}$. Then
consider the equation:

$\dot{{\,\underline{\gamma}\,}}_{{\,\underline{k}\,}}=-\vartheta({{\,\underline{k}\,}}){\,\underline{\gamma}\,}_{{\,\underline{k}\,}}-i\sum_{{\,\underline{k}\,}_{1}+{\,\underline{k}\,}_{2}={\,\underline{k}\,}}({\,\underline{\gamma}\,}_{{\,\underline{k}\,}_{1}}\cdot{\,\underline{k}\,}_{2})\,\Pi_{{\,\underline{k}\,}}{\,\underline{\gamma}\,}_{{\,\underline{k}\,}_{2}}+{\,\underline{g}\,}_{{\,\underline{k}\,}}$
$None$

where the ${\,\underline{k}\,}$’s take only the values
$0<|{\,\underline{k}\,}|<K$ for some momentum cut–off $K>0$ and
$\Pi_{{\,\underline{k}\,}}$ is the projection on the plane orthogonal to
${\,\underline{k}\,}$. This is an equation that defines a ”truncation on the
momentum sphere with radius $K$ of the equations (1.1)” if:

$\eqalign{\vartheta({{\,\underline{k}\,}})=-\nu{{\,\underline{k}\,}}^{2}&{\rm
NS\
case}\cr\vartheta({{\,\underline{k}\,}})=-\beta{{\,\underline{k}\,}}^{2}&{\rm
GNS\ case}\cr\vartheta({{\,\underline{k}\,}})=-\chi&{\rm ED\
case}\cr\vartheta({{\,\underline{k}\,}})=-\alpha&{\rm GED\ case}\cr}$ $None$

For simplicity we may suppose, in this no obstacles cases, that the mode
${\,\underline{k}\,}={\,\underline{0}\,}$ is absent, i.e.
${\,\underline{\gamma}\,}_{{\,\underline{0}\,}}={\,\underline{0}\,}$: this can
be done if, as we suppose, the external force ${\,\underline{g}\,}$ does not
have a zero mode component (i.e. if it has zero average).

In order that the resulting cut–off equations be physically acceptable, and
supposing that
${\,\underline{g}\,}_{{\,\underline{k}\,}}\neq{\,\underline{0}\,}$ only for
$|{{\,\underline{k}\,}}|\sim k_{0}$, one shall have to fix $K$ large. For
instance in the NS case it should be much larger than the Kolmogorov scale
$K=(\eta\nu^{-2})^{1/4}$, where $\nu\eta$ is the average dissipation of the
solutions to (1.6) with $K=+\infty$ (determined on the basis of heuristic
dimensional considerations by
$\eta\sim{|{\,\underline{g}\,}|^{2}L^{2}\nu^{-2}}$): see [LL].

We shall use the same cut off for the other equations with
${{\,\underline{k}\,}}$ replaced by the basis label $j$ and
$|{{\,\underline{k}\,}}|$ replaced by $\sqrt{E_{j}}$, which is certainly a
natural choice for the GNS equation.

For the ED and GNE equations the choice of $K$ should be made by developing a
theory analogous to Kolmogorov’s theory. We only attempt a preliminary
analysis in §6 as the latter equations have no physical interpretation (of
which I am aware) and they are used here only for the purpose of illustrating
some interesting mechanisms and theories. Below we always refer to the
truncated equations, unless otherwise stated.

It is easy, in the no obstacles cases, to express the coefficients
$\alpha,\beta$ for the cut off equations:

$\eqalignno{&\alpha={\sum_{0<{{\,\underline{k}\,}}|<K}{\overline{\,\underline{g}\,}}_{{\,\underline{k}\,}}\cdot{\,\underline{\gamma}\,}_{{\,\underline{k}\,}}\over\sum_{0<{{\,\underline{k}\,}}|<K}{\,\underline{\gamma}\,}_{{\,\underline{k}\,}}^{2}}\qquad\beta=\beta_{i}+\beta_{e}&(1.7)\cr&\beta_{e}={\sum_{{\,\underline{k}\,}\neq{\,\underline{0}\,}}{{\,\underline{k}\,}}^{2}{\,\underline{g}\,}_{{\,\underline{k}\,}}\cdot{\overline{\,\underline{\gamma}\,}}_{{\,\underline{k}\,}}\over\sum_{{\,\underline{k}\,}}{\,\underline{k}\,}^{2}|{\,\underline{\gamma}\,}_{{\,\underline{k}\,}}|^{2}}\cr&\beta_{i}={-i\sum_{{\,\underline{k}\,}_{1}+{\,\underline{k}\,}_{2}+{\,\underline{k}\,}_{3}={\,\underline{0}\,}}{\,\underline{k}\,}_{3}^{2}\,({\,\underline{\gamma}\,}_{{\,\underline{k}\,}_{1}}\cdot{\,\underline{k}\,}_{2})\,({\,\underline{\gamma}\,}_{{\,\underline{k}\,}_{2}}\cdot{\,\underline{\gamma}\,}_{{\,\underline{k}\,}_{3}})\over\sum_{{\,\underline{k}\,}}{\,\underline{k}\,}^{2}|{\,\underline{\gamma}\,}_{{\,\underline{k}\,}}|^{2}}\cr}$

where the ${\,\underline{k}\,}$’s take only the values
$0<|{\,\underline{k}\,}|<K$ for some momentum cut–off $K>0$ and
$\Pi_{{\,\underline{k}\,}}$ is the orthogonal projection on the plane
perpendicular to ${\,\underline{k}\,}$.

The cases in which the region $\Omega$ contains obstacles is very similar but
we cannot write simple expressions for the basis fields and therefore the
equations, although formally very similar to (1.5), (1.7) cannot be written
very explicitly.

The solutions of the equations (1.5), or to the corresponding ones in the
obastacles cases, will be denoted
$V^{\nu,ns}_{t}{\,\underline{u}\,},V^{\eta,gns}_{t}{\,\underline{u}\,},V^{\chi,ed}_{t}{\,\underline{u}\,},V^{\varepsilon,ged}_{t}{\,\underline{u}\,}$
when the initial datum is ${\,\underline{u}\,}$. Or in general:

$V^{\xi}_{t}{\,\underline{u}\,},\quad\xi=(\nu,ns),\,(\eta,gns),\,(\chi,ed),\,(\varepsilon,ged)$
$None$

Keeping the forcing ${\,\underline{g}\,}$ constant we shall admit that for
each equation, i.e. for each choice of $\xi$, there is a unique stationary
distribution $\mu_{\xi}$ describing the statistics of all initial data
${\,\underline{u}\,}$ that are randomly chosen with a ”Liouville
distribution”, i.e. (in the no obstacle cases, to fix the ideas) with a
distribution $\mu_{0}(d{\,\underline{\gamma}\,})$ proportional to the volume
measure
$\delta_{\xi}\,\prod_{|{\,\underline{k}\,}|<K}d{\,\underline{\gamma}\,}_{{\,\underline{k}\,}}$,
where the delta function is present only in the case of the reversible
equations and fixes the constants of motion to the value prescribed by the
first label in $\xi$.

This means that given ”any observable” $F$ on the phase space ${\cal F}$ (of
the velocity fields with momentum cut–off $K$) it is:

$\lim_{T\to\infty}{1\over
T}\int_{0}^{T}F(V^{\xi}_{t}{\,\underline{\gamma}\,})dt=\int_{\cal
F}F({\,\underline{\gamma}\,}^{\prime})\mu_{\xi}(d{\,\underline{\gamma}\,}^{\prime}){\buildrel
def\over{=}}\langle{F}\rangle_{\xi}$ $None$

for all choices of ${\,\underline{\gamma}\,}$ except a set of zero Liouville
measure. The distribution $\mu_{ns}$ will be called the SRB distribution for
the Eq. (1.5),(1.6) (see the ”zero-th law” in [UF],[GC2]).

A particular role will be plaid by the averages
$\langle{\varepsilon}\rangle_{\xi},\langle{\eta}\rangle_{\xi}$ as well as by
the averages of $\langle{\alpha}\rangle_{\xi},\langle{\beta}\rangle_{\xi}$ and
of the entropy production rate $\sigma({\,\underline{\gamma}\,})$ that is
defined by the divergence of the r.h.s. of the cut off equations.

Consider explicitly only the no obstacles case: if $D_{K}$ is the number of
modes ${{\,\underline{k}\,}}$ with $0<|{{\,\underline{k}\,}}|<K$ then the
number of (independent) components of
$\\{{\,\underline{\gamma}\,}_{{\,\underline{k}\,}}\\}$ is $2D_{K}$ and, see
(1.5), setting
$2{\overline{D}}_{K}=\sum_{|{\,\underline{k}\,}|<K}2{\,\underline{k}\,}^{2}$
(which in the case with obstacles become
$2{\overline{D}}_{K}=\sum_{\sqrt{E_{j}}<K}\sqrt{E_{j}})$), one finds that
$\sigma$ is given by:

$\eqalign{\sigma=&2{\overline{D}}_{K}\nu\qquad\kern
42.67912pt\xi=(\nu,ns)\cr\sigma=&2{\overline{D}}_{K}\beta-{\overline{\beta}}_{e}-{\overline{\beta}}_{i}\qquad\xi=(\eta,gns)\cr\sigma=&2D_{K}\chi\qquad\kern
42.67912pt\xi=(\chi,ed)\cr\sigma=&2D_{K}\alpha-\alpha\kern
42.67912pt\xi=(\varepsilon,ged)\cr}$ $None$

where ${\overline{\beta}}_{i},{\overline{\beta}}_{e}$ are suitably defined,
e.g. in the no obstacles cases:

${\overline{\beta}}_{e}={\sum_{{\,\underline{k}\,}}{\,\underline{k}\,}^{4}{\overline{\,\underline{g}\,}}_{{\,\underline{k}\,}}\cdot{\,\underline{\gamma}\,}_{{\,\underline{k}\,}}\over\sum_{{\,\underline{k}\,}}{\,\underline{k}\,}^{2}|{\,\underline{\gamma}\,}_{{{\,\underline{k}\,}}}|^{2}}-2{\big{(}\sum_{{\,\underline{k}\,}}{\,\underline{k}\,}^{2}{\overline{\,\underline{g}\,}}_{{\,\underline{k}\,}}\cdot{\,\underline{\gamma}\,}_{{\,\underline{k}\,}}\big{)}\big{(}\sum_{{\,\underline{k}\,}}{\,\underline{k}\,}^{4}{\,\underline{\gamma}\,}_{{\,\underline{k}\,}}^{2}\big{)}\over\big{(}\sum_{{\,\underline{k}\,}}{\,\underline{k}\,}^{2}|{\,\underline{\gamma}\,}_{{{\,\underline{k}\,}}}|^{2}\big{)}^{2}}$
$None$

so that $\sigma\simeq 2{\overline{D}}_{K}\beta$ for $\xi=(\eta,gns)$ and
$\sigma\simeq 2D_{K}\alpha$ for $\xi=(\varepsilon,ged)$.

The equivalence of dynamical ensembles conjecture, [Ga6], is the following:1

1 … è tanto nuova e, nella prima apprensione, remota dal verisimile, che
quando non si avesse modo di dilucidarla e renderla più chiara che’l Sole
meglio sarebbe il tacerla che’l pronunziarla; però, già che me la son lasciata
scappare di bocca…, [G], p. 231.

Conjecture NS: The statistics $\mu_{\nu,ns},\mu_{\eta,gns}$ of the NS
equations and of the GNS equations respectively are equivalent in the limit of
large Reynolds number provided the parameters $\eta$ and $\nu$ are so related
that $\langle{\sigma}\rangle_{\nu,ns}=\langle{\sigma}\rangle_{\eta,gns}$.

Here equivalent means that the ratios of the averages of the same observables
with respect to the two distributions approaches $1$ as $R\to\infty$. The
Reynolds number is defined here to be $R=\varepsilon^{1/3}L^{4/3}\nu^{-1}$.

A corresponding conjecture can be formulated for the ED and GED equations:

Conjecture ED: The statistics $\mu_{\chi,ed},\mu_{\varepsilon,ged}$ of the ED
equations and of the GED equations respectively are equivalent in the limit of
large Reynolds number provided the parameters $\varepsilon$ and $\chi$ are so
related that
$\langle{\sigma}\rangle_{\chi,ed}=\langle{\sigma}\rangle_{\varepsilon,ged}$.

The above stated conjectures are closely analogous to the familiar statements
on the equivalence of thermodynamic ensembles, with the thermodynamic limit
replaced by the limit $R\to\infty$ of infinite Reynolds number. They can be
substantially weakened for the purposes of possible applications.

The idea of non equilibrium ensembles and their possible equivalence is not
really new: the recent literature contains many hints in such direction. The
clearest is perhaps [ZJ]. See also [Ga2] (§4) and [MR].

On heuristic grounds, the conjectures would be justified if one did accept
that the entropy creation rate reaches its average on a time scale that is
fast compared to the hydrodynamical scales. The coefficients
$\alpha\simeq(2D_{K})^{-1}\sigma$, and
$\beta\simeq(2{\overline{D}}_{K})^{-1}\sigma$, see (1.10), would be confused
with their time averages $\langle{\alpha}\rangle_{\varepsilon,gne}$ or
$\langle{\beta}\rangle_{\eta,gns}$ and identified with the viscosity constants
$\nu$ or $\chi$.

In this way the GNS and the NS equations would be equivalently good: both
being the macroscopic manifestation of two equivalent microscopic dissipation
mechanisms: one explicitly specified by the Gaussian constraint of constant
dissipation and the other with dissipation unspecified a priori  but
phenomenologically modeled by a constant viscosity. Likewise one can view the
GED and the ED equations as macroscopically equivalent: one with constant
energy and the other with constant sticky viscosity $\chi$.

The interest of the above conjectures is that the same physical system in
which irreversible dissipation occurs (the NS or ED equations) can be
described equivalently by a reversibly dissipative system (the GNS or GED
equations).

For reversible systems a general principle, the chaotic hypothesis, can be
reasonably assumed to hold and to imply consequences that seem to be non
trivial, see [GC1], [GC2], [Ga3], [Ga4],[Ga5].

The next section is devoted to a quick discussion of some of the established
consequences of the principle and §3%§6 will deal with heuristic ideas and
with describing a possible scenario for the phenomenology of the equations
(1.1). The scenario will be developed without any pretension of rigor and it
will present what will appear as the simplest among many other possibilities.
It leads (implicitly) to several possible experimental tests of the chaotic
hypothesis and of the other ideas involved in its development: the tests can
also be viewed, independently, just as interesting experiments proposals.

In the following we shall always consider the NS equations and the GNS
equations with parameters fixed so that $\mu_{\nu,ns}$ and $\mu_{\eta,gns}$
are equivalent by the conjecture NS, and likewise we shall always consider the
ED and GED equations with parameters fixed so that $\mu_{\chi,ed}$ and
$\mu_{\varepsilon,ged}$ are equivalent by the conjecture ED.

§2 The fluctuation theorems.

In reference [GC1],[GC2] the chaotic hypothesis was presented as a
reformulation of an older principle due to Ruelle, [R1]. It gave us the
possibility of some quantitative parameterless ”predictions”, in various
cases, see also [Ga3], [Ga4], [Ga5]. The hypothesis is:

Chaotic hypothesis: A chaotic many particle system or fluid in a stationary
state can be regarded, for the purpose of computing macroscopic properties, as
a smooth dynamical system with a transitive Axiom A global attractor. In
reversible systems it can be regarded, for the same purposes, as a smooth
transitive Anosov system.

The main result of [GC1] is the fluctuation theorem that gives a property of
the variable $p=p({\,\underline{\gamma}\,})$ defined in terms of the
contraction rate $\sigma_{0}$ of the attractor surface elements by:

${1\over\tau}\int_{-\tau/2}^{\tau/2}\sigma_{0}(V_{t}{\,\underline{\gamma}\,})\,dt=\langle{\sigma_{0}}\rangle_{+}p$
$None$

which can be regarded as a random variable with the distribution
$\pi_{\tau}(p)dp$ that it inherits from $\mu_{\eta,gns}$ or
$\mu_{\varepsilon,ged}$; here $\langle{\sigma_{0}}\rangle_{+}$ is the average
of $\sigma_{0}$ with respect to the distribution $\mu_{\eta,gns}$ or
$\mu_{\varepsilon,ged}$.

Note that $\sigma_{0}$ should not be confused with $\sigma$, (1.8): thinking
of the attractor as a smooth surface $\sigma_{0}$ is the contraction rate of
its surface elements, which is different from the contraction rate $\sigma$ of
the phase space volume elements, see §4%§6.

If the conjectures of §1 are accepted $\langle{\sigma_{0}}\rangle_{+}$ is also
the $\mu_{\nu,ns}$ or $\mu_{\chi,ed}$ average of $\sigma_{0}$ or at least
tends to it as $R\to\infty$.

If $\langle{\sigma_{0}}\rangle_{+}>0$, see [R2] for a discussion of the
conditions for this inequality (”Ruelle’s H-theorem”), and if
$\zeta(p)=\lim_{\tau\to\infty}{1\over\tau}\log\pi_{\tau}(p)$ then the
fluctuation theorem of [GC1] gives the following large deviation relation, see
also [Ga3], [Ge], for the equations GNS and GED:

${\zeta(p)-\zeta(-p)\over\langle{\sigma_{0}}\rangle_{+}p}=1,\qquad{\rm for\
all\ }p$ $None$

which, in the case of nonequilibrium statistical mechanics, has been
interpreted as an extension of the fluctuation dissipation theorem to large
forcing fields, [Ga5]. Here ”for all” $p$ means for all possible values of $p$
(which is in general a bounded quantity).

The fluctuation theorem (2.2) says that the distribution of $p$ is
multifractal, not surprisingly since $\zeta(p)$ can be regarded as a kind of
generalized sum of Lyapunov exponents in the sense of [FP], [BJPV], and the
odd part of $\zeta(p)$ is linear.

A more general fluctuation theorem concerns the joint distribution of the
variable $p$ and of any other variable $q=q({\,\underline{\gamma}\,})$ that is
similarly defined in terms of an observable $Q$ which is odd under the time
reversal operation that is defined on the attractor, i.e. :

${1\over\tau}\int_{-\tau/2}^{\tau/2}dt\,Q(V^{\xi}{\,\underline{\gamma}\,})=\langle{Q}\rangle_{+}\,q$
$None$

If $\pi_{\tau}(p,q)$ denotes the joint probability density of the observables
$p,q$ and if $\zeta(p,q)=\lim_{\tau\to\infty}{1\over\tau}\log\pi_{\tau}(p,q)$
then it follows from the chaotic hypothesis that the distributions for $p,q$,
with respect to the statistics $\mu_{\xi}$, $\xi=(\beta,gns)$ or
$\xi=(\alpha,ged)$, verify:

${\zeta(p,q)-\zeta(-p,-q)\over\langle{\sigma_{0}}\rangle_{+}p}=1,\qquad{\rm
for\ all\ }p,q$ $None$

which, in the case of nonequilibrium statistical mechanics, has been
interpreted as an extension of the Onsager’s reciprocity to large forcing
fields, [Ga5].

In the case of the equations (1.1), second and fourth, the above relations are
applicable when the motions are chaotic, i.e. have at least one positive
Lyapunov exponent (which should happen as soon as $R$ is large enough,
excepted possibly very special cases, see [Ma]); and it gives an interesting
parameterless prediction if the contraction rate $\sigma_{0}$ can be related
to the contraction rate $\sigma$ of the GNS equations.

If the conjecture in §1 holds in the strong sense of complete asymptotic
equivalence between the “ensembles” $\mu_{\eta,gns}$ and $\mu_{\nu,ns}$ (or
$\mu_{\varepsilon,ged}$ and $\mu_{\chi,ed}$) then (2.2) should also hold for
other models of the viscous stationary states, like the one given by the
classical NS equation in particular: note the quantities $\sigma$ in (1.10)
are still fluctuating variables even when the evolution considered is given by
the NS or ED equation. A check of this property might be under reach of
present day experimental techniques.

A check of (2.2) or (2.4) might be more accessible in the case of fluid
systems because they often show chaotic motions with relatively few degrees of
freedom so that the large fluctuations, that must occur in order to make
possible testing the fluctuation theorems, are more likely occur and be
observable.

But this also means that the attractors have a very small size, compared to
that of phase space: hence a problem in the interpretation of (2.2) is the
fact that the quantity $\sigma_{0}$ that appears in the theorem does coincide
with the easily determined (see (1.10)) contraction of volume in the phase
space only if the attractor is dense in phase space.

This is a property that can be expected in the case of nonequilibrium
statistical mechanics under small or moderately strong forcing, see [BGG] for
a discussion of this point, but it cannot be expected at large forcing or in
fluid systems. In the latter case one should use the contraction rate of the
volume elements on the attractor, see [BGG], [BG]. In the GNS systems it is
likely that the property never really holds.

This might render (2.2) quite useless as it is usually unrealistic to hope to
determine the attractor equation with accuracy sufficient to compute its area
contraction per unit time (assuming, as the chaotic hypothesis implies, that
the attractor can be regarded as a smooth surface).

Nevertheless in nonequilibrium statistical mechanics further analysis is
possible, based on an important symmetry of the Lyapunov exponents spectrum,
and one can relate easily the area contraction on the surface defined by the
attractor and the total phase space volume contraction. Hence one can try to
push further the analysis of [BGG], [BG] to see if one can say something also
in the case of the GNS equations.

§3 A scenario for experimental checks of the fluctuation theorem. The Lyapunov
spectrum of ED and GED equations. Pairing rule and Axiom C.

What follows is a very heuristic analysis aimed at giving an argument for the
explicit form that the fluctuation theorem (2.2) will take in the case of the
GED equations: and therefore by the equivalence conjecture also for the ED
equations. No pretention of mathematical rigor is present and the idea is to
illustrate the simplest possible scenario that I consider possible. The
interest is (apart from the subjective feeling of a certain beauty) that the
discussion suggests experiments and checks that have intrinsic interest and
that do not seem to have yet been considered in the literature.

We consider first the case of (1.1) in a domain $\Omega$ with obstacles: in
spite of the appearances this is an easier case because in this case we can
imagine forcing the system with a locally conservative force which is not
globally conservative, like a field roughly parallel to one axis and tangent
to the obstacles (one can imagine a uniformly charged fluid under an
electromotive constant electric field).

Note that in order to have a non trivial forcing the forcing field must be non
globally conservative: otherwise its effect would be just that of altering the
pressure.

The Euler equations can, in general, be regarded as hamiltonian equations for
a system whose configurations are the diffeomorphisms of the box $\Omega$ (in
our case a torus with, possibly, a few holes) containing the fluid: they are
not directly in hamiltonian form in the same sense as the (closely analogous)
Euler equations for a rigid body with a fixed point are not immediately
hamiltonian (e.g. they involve half the number of actual equations of motion).

In this way the GNS or GED equations can be regarded as hamiltonian equations
(approximately so, because the cut–off $K$ destroys this property) modified by
the action of a non conservative force ${\,\underline{g}\,}$ and by the
gaussian constraint that the total vorticity or the total energy are
constants.

Of course we exploit the ”slip” (i.e. no friction) boundary conditions in
order to be able to conclude the hamiltonian nature of the Euler equations.

The phase space will then consist of a space larger than the above ${\cal F}$,
see (1.9): its points $({\,\underline{u}\,},{\,\underline{\delta}\,})$ will be
(cut–off) velocity fields and (cut–off) displacement fields describing the
positions of the fluid particles with respect to a reference configuration. We
call this the “full phase space” of the equations (1.1).

The equations for the displacements will be in all models (1.1):

$\dot{{\,\underline{\delta}\,}}={\,\underline{u}\,}({\,\underline{\delta}\,},t),\qquad{\,\underline{\delta}\,}({\,\underline{x}\,},0)={\,\underline{\delta}\,}_{0}({\,\underline{x}\,})$
$None$

which, once ${\,\underline{u}\,}$ is known from (1.3), (1.2), permit us to
compute the physical fluid flow and the positions
${\,\underline{\delta}\,}({\,\underline{x}\,},t)$ of the fluid particles that
at time $0$ were at the points
${\,\underline{\delta}\,}_{0}({\,\underline{x}\,})$, away from the reference
configuration position ${\,\underline{x}\,}\in\Omega$.

In the case in which the ${\,\underline{u}\,}$ verify truncated equations also
(3.1) have to be truncated, for instance by replacing each
$e^{i{{\,\underline{k}\,}}\cdot{\,\underline{\delta}\,}({\,\underline{x}\,})}$
in ${\,\underline{u}\,}({\,\underline{\delta}\,},t)$ by its truncated Fourier
expansion.

The system motions (describing velocity and displacement fields) can be
regarded as motions with $2D_{K}$ degrees of freedom where, for instance in
the no obstacles case, $D_{K}$ is the number of non zero modes
${\,\underline{k}\,}$ with $0<|{\,\underline{k}\,}|<K$ (because each
${\,\underline{\gamma}\,}_{{\,\underline{k}\,}}$ has two complex components
but
${\,\underline{\gamma}\,}_{-{\,\underline{k}\,}}={\overline{\,\underline{\gamma}\,}}_{{\,\underline{k}\,}}$).
This means that $4D_{K}$ coordinates are necessary to describe the motion.

Hence there are $4D_{K}$ Lyapunov exponents, $2D_{K}$ from the velocity
equations (1.2) and $2D_{K}$ from the displacements equations (3.1).

In view of the equivalence conjectures we study the equations GNS and GED when
convenient and the NS or ED when convenient.

Out of the $4D_{K}$ exponents one has to extract, in the GNS or GED cases, one
exponent that is trivially $0$ because of the conservation of the dissipation
rate and one exponent that is trivially zero and corresponds to the vector
field given by the r.h.s. of the GNS equation. Furthermore in the GNS or GED
cases two more vanishing Lyapunov exponents are associated with other
constants of motion.1

1 In the case of the GNS equations the helicity
$\int{\,\underline{\omega}\,}\cdot{\,\underline{u}\,}\,dx$ is a constant of
the motion and such is $\int{\,\underline{\delta}\,}^{2}dx$ for the
displacement equations.

The other $2N=4D_{K}-4$ exponents, or in the ED, NS cases all the $2N=4D_{K}$
exponents, can be ordered in two groups the first containing the first $N$
exponents in decreasing order and the second the remaining $N$ ones in
increasing order.

The exponents of the first group are denoted $\lambda^{+}_{j}$, $j=1,\ldots,N$
and the ones in the second group are denoted $\lambda^{-}_{j}$,
$j=1,\ldots,N$. We call the two exponents $(\lambda^{+}_{j},\lambda^{-}_{j})$
a pair.

We consider first in detail the ED and GED equations. In the above context it
seems reasonable that in the full phase space of the GED and ED equations a
pairing rule holds:

${\lambda^{+}_{j}+\lambda^{-}_{j}\over 2}=const$ $None$

at least when the forcing is locally conservative as we suppose from now on
unless otherwise stated. The value of the constant will be called the ”pairing
level” or ”pairing constant”, which must be ${1\over
2}\langle{\sigma}\rangle_{+}$, see (1.10).

The pairing rule, in fact, formally holds in the present ED case. One can
easily adapt the proof in [Dr]: this is discussed in the Appendix A1.

The rule then holds also for GED as a consequence of the conjecture ED. A
direct proof can probably be made along the lines of the work [DM]. In fact
the constraint imposed by the definition of the multiplier $\alpha$, (1.7), is
a constraint of the type called isokinetic in [DM] and their proof seems to
apply ”without change”, although I did not check the details (the appendix A1
should give the background for such an analysis).

In the cases in which (3.2) has been proved, [DM], [Dr], it holds also in a
far stronger sense: the local Lyapunov exponents,2

2 i.e. the logarithms of the eigenvalues of $\sqrt{J_{t}^{T}J_{t}}$, if
$J_{t}$ is the local jacobian matrix of the evolution operator $V_{t}$, other
than those relative to the directions of the flow or of the imposed
constraints or of other constants of motion of which the Lyapunov exponents
are the averages, are paired as in (3.2) to a constant that is $j$ independent
but, of course, is dependent on the point in phase space. We call this the
strong pairing rule. See the final comments.

Note that the Lyapunov exponents of the full system can also be easily divided
into velocity exponents, i.e. the ones of the GED or ED equations, and the
displacement exponents, i.e. the others (which cannot be measured from the GED
or ED evolution alone but require also (3.1)). In fact if we denote
symbolically by $(x,y)$ the pair
$({\,\underline{u}\,},{\,\underline{\delta}\,})$ then the jacobian matrix of
the equations is described by a matrix having the form $\pmatrix{A&0\cr
B&C\cr}$ where $A,B,C$ are operators.

For a further classification of the exponents we shall think that the Lyapunov
exponents are divided into three classes that we call viscous, inertial and
slow. The following scenario will be again summarized and enriched in the
figure in §4.

(1) The slow exponents (”slow pairs”) consist of $M$ pairs of exponents the
largest of which is $\leq 0$ and it is a velocity exponent corresponding to
slow motions of the velocity field, while the other (necessarily $<0$)
exponent of the pair is a displacement exponent and corresponds to a fast
approach to the stationary state of some of the displacement variables.

(2) The viscous exponents (”viscous pairs”) consist of $V$ negative velocity
exponents describing the fast approach to the stationary state of the viscous
degrees of freedom of the velocity field: their paired positive exponents are
displacement exponents associated with chaotic motions of the displacement
variables.

(3) The remaining $2P=N-M-V$ pairs (”inertial pairs”) have one $>0$ and one
$<0$ Lyapunov exponents: $P$ of the pairs are pairs of velocity exponents and
the $P$ other pairs are displacement exponents. The $P$ pairs of velocity
exponents are the only pairs of exponents of the equations for the velocity
field that contain one positive and one negative element: they describe the
gross characteristics of the chaotic motion on the attractor. It follows that
the three types of exponents can in principle be uniquely identified among the
$N$ exponents of the velocity field equations, see also below.

The existence of a certain number denoted $P$ above of pairs of exponents, for
the velocity field evolution, that are pairs of exponents of opposite sign
does not follow simply from the fact that we are collecting togheter pairs
containing a $>0$ exponent. In principle the $>0$ exponents of the velocity
field could be paired with negative displacement exponents. We think that it
is natural that the $>0$ Lyapunov exponents for the velocity field are paired
with $<0$ exponents of the velocity field because we associate such pairs with
the motions on the attractor. Since the GED equations are reversible it
follows from [BG] that if the motions are also supposed to verify a geometric
property called in [BG] Axiom C property (a simple extension of the paradigm
of turbulent behavior, see [R1], that is the Axiom A property) then there must
be an equal number $P$ of positive and negative exponents for the restriction
of the GED equations to their attractor. It seems therefore natural to think
that they form $P$ pairs.

The equality of the number of $>0$ and $<0$ exponents for the motion on the
attractor for the velocity fields is due to the existence, in reversible Axiom
C systems, of a local time reversal map $i^{*}$ that transforms the attractor
into itself anti-commuting with the time evolution, even when (and this is the
rule in fluid dynamics) the attractor itself is not time reversal invariant:
see [BG]. We proceed under the assumption that the Axiom C property is
verified: for a complete discussion of the property we must unfortunately
refer to [BG].

In Axiom C systems the time reversal symmetry ”cannot be lost”: when it is
spontaneously broken (because the attractor is not time reversal invariant) it
is replaced by a ”weaker” symmetry, good enough to make ”effectively
reversible” all the motions on the attractor, a relation similar to the one in
fundamental Physics between $T$ and $TCP$ (the latter being the ”real” time
reversal as the first is not a symmetry of the world we see).

(4) The other $P$ pairs should consist of displacement exponents exhibiting a
rather symmetric behavior with respect to that of the GED exponents. Below we
are going to suggest that very similar properties hold for the NS and GNS
equations: in that case this further appealing symmetry seems compatible with
(and in fact it was suggested by) the data on the velocity Lyapunov spectrum
for models (”GOY shell models”) whose behavior is ”believed” to be related to
NS equations: see [BJPV], figure in p. 71, taking into account that the
pairing level in such data is very small because the viscosity is very small.

In the above scenario the existence of the other $P$ pairs of displacement
exponents is assumed in order to make the total count of the number of
exponents correct and is not based on evidence of any other kind. The
displacement exponents do not seem to have been considered in the literature,
at least not in conjunction with the velocity field exponents (not
surprisingly in view of the difficulty of the measurements).

Thus if we measure the Lyapunov exponents for the GED equations alone we
expect to find $P$ pairs of opposite sign exponents paired at the value
${1\over 2N}\langle{\sigma}\rangle_{+}$ for some $P\leq N$.

It seems reasonable that the $P$ pairs of displacement exponents coincide with
the $P$ pairs of inertial exponents for the velocity field equations: but this
is not really necessary in order that an unambiguous identification of the
three type of exponents be possible. They are already identified by the above
properties.

However if the $P$ pairs of velocity inertial exponents and the $P$ pairs of
displacement inertial exponents do coincide we see that, by the pairing rule,
the knowledge of the Lyapunov spectrum for the velocity equations implies that
all the displacement exponents are knwon as well: no need to compute them.

§4 Predictions of the fluctuation theorem for ED and GED.

With the scenario developed in §3 we reconsider the fluctuation theorem and
note that it is easy to check, by evaluating the divergence of the r.h.s. of
the equations (1.2), (3.1), that the volume in the full ($2N$ dimensional)
phase space contracts at the same rate $\sigma$ at which the volume in
velocity space does. Fluid incompressibility, and absence of displacement
variables in the equations for the velocity field, imply this property.

Furthermore if the strong pairing rule is assumed the total volume contraction
in the full phase space, including the displacement variables, will be
$\sigma({\,\underline{\gamma}\,})$ and it will be related to the contraction
$\sigma_{0}({\,\underline{\gamma}\,})$ of the area on the attractor surface by
$\sigma_{0}({\,\underline{\gamma}\,})={2P\over
2N}\sigma({\,\underline{\gamma}\,})$, see [BGG] where the same mechanism was
first exploited. This gives proportionality between the ”apparent” contraction
rate $\sigma$ and the ”true” contraction rate $\sigma_{0}$ on the attractor
for the GNS equations.

As discussed at the end of §2 the fluctuation theorem holds for the
fluctuations of $\sigma_{0}$ so that the fluctuations of $\sigma$ will verify
(2.2) but with a r.h.s in which $1$ is replaced by ${P\over N}$ where $P$ is
the number of pairs of Lyapunov exponents for the GED equations with one
positive element.

If the number of degrees of freedom is increased by increasing $K$ one should
expect, therefore, that the constant ${P\over N}\langle{\sigma}\rangle_{+}$
approaches $P\langle{\alpha}\rangle_{+}$ because the number of ”true
exponents” (i.e. inertial exponents) should not change as soon as $K$ is so
large that the motion is well described by the truncated equations: in fact it
is believed that the attractor dimension does not depend on the truncation
scale $K$ (as long as it is large enough).

Since the conjecture of §2 implies $\langle{\alpha}\rangle_{+}=\chi$, the
constant should approach $P\chi$, at least if $R$ (i.e. the Reynolds number)
is large. The fluctuation theorem will thus take the form:

${\zeta(p)-\zeta(-p)\over p}=P\,\chi$ $None$

if the variable $p$ is defined as in (2.1) but with the measurable $\sigma$
replacing the a priori difficult to measure $\sigma_{0}$, and $\zeta(p)$ is
the limit of $\tau^{-1}\log\pi_{\tau}(p)$ with $\pi_{\tau}(p)$ being the
$\mu_{\varepsilon,ged}$ distribution of $p$. The number $P$ is accessible by
measurements performed only on the GED equations and not involving the
displacement variables (being the number of positive Lyapunov exponents of the
GED equations).

The above analysis is somewhat conjectural but experiments, at least numerical
ones, are possible to check the picture; e.g. one could attempt at:

(1) checking the just derived slope $P\chi$ or

(2) checking the following picture, representing the above classification of
the exponents:

$\lambda$ $\lambda$ $\lambda$ $x={j\over M}$ $x={j\over P}$ $x={j\over V}$
${\scriptscriptstyle pairing\atop level}$ (4.2)

The continuous line in the first graph gives the value ($\leq 0$) of the
$j$–th (among $M$) slow Lyapunov exponent (as a function of ${j\over M}$) of
the GED equations; the dashed line is the graph of the paired exponents (of
the displacement equations) and the intermediate line is the pairing constant.
The exponents are defined only for $x={j\over M}$ but the graphs give,
instead, continuous (or dashed) lines for visual aid.

The second graph gives the values of the $j$-th (among $P$) pair of inertial
exponents of the GED equation (one positive and one negative per pair): here
too we use the continuous curves even though the number of such exponents will
ususally be much smaller than the total number and therefore a discrete
representation would be more appropriate.

The negative curve in the third graph is the graph of the $j$-th viscous
exponent (out of $V$) of the GED equation; the corresponding positive curve
(dashed) is the curve of the companion exponents which correspond to
displacement exponents. A fourth graph giving the other $P$ displacement
exponents (in pairs of one negative and one positive) would be qualitatively
equal to the second graph (with the curves dashed for consistence of
notation).

The graphs are not experimental data: they are just sketches illustrating the
”simplest” picture that I considered reasonably possible. They should be taken
as a conjecture, and they suggest performing experimental evaluation of the
exponents for a check of the ideas of the present paper.

What do we imagine to happen when the equations are changed by enlarging the
cut off (in the velocity as well as in the displacement variables) or by
changing the forcing? Suppose that the cut off is already so large that adding
one extra mode does not really affect the qualitative and quantitative
features of the motion. Then adding one mode, i.e. increasing the total
dimension of the system by $2$, should add one pair of viscous exponents at
the end of the spectrum respectively equal to $0$ and $-\chi$, as drawn in the
third graph in the figure. While changing the forcing should, from time to
time as the forcing changes, change the category of some exponents. Namely the
simplest picture would be that one of the vanishing slow GED exponents
”becomes” positive and one of the viscous ”becomes” inertial and paired with
it; a symmetric evolution should take place with the displacement exponents.
Or vice-versa. The attractor changes dimension by $2$ units, see [BG],[BGG],
at each of such events.

In order that the latter picture be possible one needs that at a transition
the viscous spectrum bottom consists of a pair of a $>0$ displacement exponent
and a negative viscous exponent coinciding with the inertial exponents top
pair; and at the same should happen for the bottom pair of the inertial and
the top pair of the slow spectra.

The case of periodic boundary conditions does not fit in the above analysis of
the pairing rule because on the torus there is no way of forcing the system
with a locally conservative but globally non conservative force field with $0$
average. Nevertheless some kind of pairing might still occurr under simple non
conservative forcing acting only on some large scale modes, see §5.

It has been pointed out to me by F. Bonetto that consistency of the above
picture requires that the sum of the displacement exponents be exactly $0$:
the two of us have indeed been able to verify that this property is formally
exactly verified in the ED equations. And this led to a correction of the
graphs drawn in the figure above that I had originally drawn without taking
such property into account. We shall come back on this point in a future
study. Note also that the fact that the sum of the displacement exponents
vanishes provides a natural test that the truncation that one is using is
actually large enough for having reached cut-off independence of the
asymptotic properties ot the motions: this happens at the cut off value where
the sum of the displacements exponents vanishes: further addition of modes
only makes longer the flat part in the third graph of the figure above.

§5 The NS and GNS equations. Extension of the pairing rule.

We turn to the NS and GNS equations, whose interest is far greater than the
just studied ED or GED equations.

One is tempted to say that the scenario should be the same. However the
pairing rule analysis, which is essential for the physical interpretation of
the results, is no longer naively possible, not even at a heuristic level.

A pairing rule, first pointed out in special non constant friction cases in
[EM], p. 281, has been proved only in the case of systems subject to special
gaussian constraints, see [DM], but it has apparently a much wider validity,
see [EM], [ECM1], [SEM], [DPH] and it is likely to hold also in the cases GNS
and NS, at least in some sense.

Also the argument in [Dr] implies the existence of pairing in systems that are
obtained from hamiltonian systems by adding to them an irreversible constant
friction term proportional to the momenta in a system of canonical
coordinates. And the argument in [DM] is restricted to ”isokinetic”
constraints precisely because they are reversible constraints that are
obtained by adding to a hamiltonian system a suitable force proportional to
the canonical momenta. Since this is an essential feature for the validity of
(3.2) the latter becomes doubtful in the cases (that include NS and GNS
equations) in which the friction or thermostat forces are proportional to the
canonical momenta via a matrix $C$ which is not the identity (it is the
laplacian in the case of the NS or GNS equations).3

3 I owe to F. Bonetto the clarification of this point.

In such cases one could envisage that (3.2) is replaced, in the GNS equations
case, by a relation like:

${\lambda_{j}^{+}+\lambda_{j}^{-}\over 2}=\langle{\beta}\rangle\,c_{j}$ $None$

where $\langle{\beta}\rangle$ is the $\mu_{\eta,gns}$ average (in the case of
the NS equations one would write $\nu$ instead of $\langle{\beta}\rangle$);
and $c_{j}$ is some suitable function of $j$, that might be related to the
spectrum of the matrix $C$. However attempting at establishing such a
connection would lead to too many too detailed assumptions at this stage and
one would like not to rely on them. And from the proofs in [Dr], [DM] it seems
unlikely that a pairing rule can hold in a strong form, i.e. that (5.1) holds
for the local exponents if $\langle{\beta}\rangle$ is replaced by $\beta$.

We therefore define $c_{j}$ by the (5.1) without linking $c_{j}$ to the matrix
$C$. However we shall suppose that (5.1) holds in a ”almost local” form in the
sense that on a rapid time scale (5.1) becomes true also for the local
exponents. This means that, up to an error that tends to zero very quickly
with the time $\tau$, the logarithms of the eigenvalues of the matrix
$(J_{\tau}^{T}(x)J_{\tau}(x))^{1/2\tau}$, with $J_{\tau}(x)$ being the
jacobian matrix for the evolution operator $V_{\tau}$ at $x$, divided by
$\tau$ verify ${1\over
2}(\lambda^{+}_{j}+\lambda^{-}_{j})=c_{j}\beta_{\tau}(x)$ with
$\beta_{\tau}(x)$ denoting the average
${1\over\tau}\int_{-\tau/2}^{\tau/2}\beta(V_{t}x)\,dt$.

This property, together with the Axiom C assumption, suffices to extend, in a
suitable form, the validity of the predictions (i.e. conjectures) discussed in
this section for the ED and GED equations to the case of the NS and GNS
equations as follows.

We first remark that the really relevant feature of the pairing rule, as far
as the applications in [BGG] and above are concerned, is not the constancy of
the pairing but, rather, the fact that some kind of pairing takes place on a
fast enough time scale. secondly we assume that this is actually the case for
the GNS equations. These are the main remark and the main assumption on which
we base the analysis of the fluctuation theorem predictions in the NS and GNS
equations.

If a relation (5.1) holds the constants $c_{j}$ will have to add up to the sum
of the Lyapunov exponents that can be derived as the average value of
$\sigma$: this means that
$\langle{\sigma}\rangle_{+}=2{\overline{D}}_{K}\langle{\beta}\rangle_{\eta,gns}=(\sum_{j=1}^{2D_{K}}c_{j})\langle{\beta}\rangle_{\eta,gns}$,
up to terms neglegible as the Reynolds number tends to $\infty$, see (1.10)
and comments preceding it.

Furthermore let $I$ be the set of $P$ inertial pairs (i.e. of pairs of
Lyapunov exponents $\lambda^{+}_{j},\lambda^{-}_{j}$ with one positive
element) and suppose that the (5.1) becomes valid on a sufficiently fast time
scale, then the values of $\langle{\sigma}\rangle_{+}$ and
$\langle{\sigma_{0}}\rangle_{+}$ would have ratio (see [BGG]) $(\sum_{j\in
I}c_{j})/(\sum_{j}c_{j})$ so that:

$\langle{\sigma_{0}}\rangle={\sum_{j\in
I}c_{j}\over\sum_{j}c_{j}}\langle{\sigma}\rangle_{+}=(\sum_{j\in
I}c_{j})\,\langle{\beta}\rangle_{\eta,gns}=(\sum_{j\in I}c_{j})\nu\,{\buildrel
def\over{=}}\,{\overline{P}}\,\nu$ $None$

having used the conjecture NS of §1 equating
$\langle{\beta}\rangle_{\eta,gns}$ to $\nu$.

Then if a local time reversal exists on the attractor (i.e. if the geometric
Axiom C is assumed as well, [BG], for the dynamics generated by the GNS
equations) the fluctuations of the observable $\sigma$ will have a ”free
energy” (or a ”generalized sum of Lyapunov exponents” to adhere to the
terminology in [FP], [BJPV]) $\zeta(p)$, in the sense of (4.1), with an odd
part $p\,{\overline{P}}\,\nu$, with ${\overline{P}}$ defined in (5.2). This is
a property whose validity can be conceivably tested in, real or numerical,
moderately turbulent systems. At least the linearity in $p$ of
$\zeta(p)-\zeta(-p)$ should be observable.

Note also that, in all cases, the pairing rule is trivially valid in the case
of no forcing: in fact the equivalence criterion in the conjecture in §2
requires that in absence of forcing one has to take $\eta=0$ or
$\varepsilon=0$: i.e. the stationary state is, in that case, the trivial (non
chaotic) flow
${\,\underline{u}\,}={\,\underline{0}\,},{\,\underline{\delta}\,}=const$.

The assumption that the forcing be locally conservative has not been used and
disappears togheter with the constancy of the pairing: the above more general
pairing hypothesis (see (5.1) and the comment following it) is more ”flexible”
and does not require the special hypothesis of local conservativity of the the
forcing.

§6 Relation between the NS and ED equations. The barometric formula.

Finally we discuss another main point of our analysis.

In reference [Ga6] the argument leading to the conjecture NS above can be
interpreted as saying that NS and ED are also in some sense equivalent.

The argument is based on the constancy of the dissipation rate $\varepsilon$
in a stationary flow at high Reynolds number and on the microscopic
reversibility. In some sense the GNS equations emerge as even more natural
than the NS equations.

A criticism can be raised, however. In fact one can argue that the energy is
also constant in a stationary state and one could develop the argument in
[Ga6] to imply that the GED equations are also a good model for a fluid
motion.

Since clearly one should not expect NS and ED to be equivalent this looks at
first as an unsoluble logical contradiction. Which can furthermore be
conceivably easily checked to occurr.

However on further thought the contradiction can be resolved and one should
think that all what has been deduced is that there should be a relation
between the stationary states of ED (or GED equivalently) and of NS (or GNS).
The relation to which I think is the kind of relation that one also finds in
equilibrium statistical mechanics in gases in a strongly varying external
field of intensity $g$, like the gravity field.

Locally a gas in a field looks just like a homogeneous gas in equilibrium, but
globally over a length scale $H$ over which the external potential really
changes ($\beta mgH\sim 1$, if $\beta$ is the inverse temperature and $m$ the
particles mass) one will see that pressure and the density are not constants
and one gets the barometric formula, see [MP].

Likewise we can expect that the stationary states of ED (or equivalently of
GED) are also “locally” equivalent to stationary states for NS (or GNS): in
the sense that if we only look at observables depending on field components
${\,\underline{u}\,}_{{\,\underline{k}\,}}$ with modes ${{\,\underline{k}\,}}$
on a certain scale $|{{\,\underline{k}\,}}|\sim\kappa$ whose size depends on
the dissipation then we should see essentially no difference. The precise
relation that determines $\kappa$ will be called barometric formula: it should
be easy to determine the formula on the basis of dimensional considerations.
Locality is here to be interpreted in momentum space rather than in coordinate
space.

The determination of the barometric formula amounts essentially at a
development of the analogous of the Kolmogorov theory for the ED equations.

We now attempt at a development of such theory, in the no obstacles case for
simplicity, on the basis of a few assumptions that we did not subject to as
much criticism as they certainly deserve: a criticism that will be undertaken
in a separate publication. We follow closely the ideas (and imitate the
assumptions) of the exposition of Kolmogorov’s theory in [LL]. We set
$\rho=1$.

It seems reasonable to suppose that in the ED case the stationary distribution
equipartitions the energy among the modes, i.e.
$\langle{|{\,\underline{\gamma}\,}_{{\,\underline{k}\,}}|^{2}}\rangle=\gamma^{2}$
for all ${\,\underline{k}\,}$ in the ”inertial range”
$L^{-1}\ll|{{\,\underline{k}\,}}|\ll k_{\chi}$ where $k_{\chi}$ is the
”Kolmogorov scale”, to be determined below. Hence
$\gamma^{2}(k_{\chi}L)^{3}=\varepsilon$.

Then a velocity variation characteristic of the momentum scale $\kappa$ is
$v_{\kappa}^{2}=\langle{(\sum_{|{{\,\underline{k}\,}}|\in[\kappa/2,\kappa]}{\,\underline{\gamma}\,}_{{\,\underline{k}\,}})^{2}}\rangle$
and, assuming statistical independence of the distribution of the various
${\,\underline{\gamma}\,}_{{\,\underline{k}\,}}$, we get
$v_{\kappa}^{2}=(\kappa L)^{3}\gamma^{2}$ up to a constant factor.

The scale $k_{\chi}$ can be determined because it has to be a momentum scale
that only depends on $\varepsilon$ and $\chi$ so that
$k_{\chi}=({\varepsilon\over\chi^{2}})^{1/2}$ up to a constant factor: this is
a momentum scale analogous to the Kolmogorov scale $k_{\nu}$ of the NS
equations, which is $k_{\nu}=({\eta\over\nu^{2}})^{1/4}$.

For purposes of comparison we note that the quantity called $\varepsilon$ in
the Kolmogorov’s theory (”K41-theory”), see [LL], corresponds to $\eta\nu$ of
the present paper.

In this case the energy distribution (i.e. the amount $K(k)dk$ of energy per
unit volume and between $k$ and $k+dk$) is $K(k)={3\varepsilon\over
4\pi}{k^{2}\over k_{\chi}^{3}}$, for $k<k_{\chi}$: very different from the
Kolmogorov’s $k^{-5/3}$ law.

In the K-41 theory a key role is plaid by the quantity $v_{\kappa}^{3}\kappa$
which is identical to $\eta\nu$ for all $k_{0}\ll\kappa\ll k_{\nu}$. Therefore
we compute the value of $v_{\kappa}^{3}\kappa$ in our case and we find:

${v_{\kappa}^{3}\kappa\over\varepsilon\chi}={((\kappa
L)^{3}\gamma^{2})^{3/2}\kappa\over\varepsilon\chi}={\big{(}(k_{\chi}L)^{3}\gamma^{2}\big{)}^{3/2}k_{\chi}\over\varepsilon\chi}({\kappa\over
k_{\chi}})^{11/2}={\varepsilon^{3/2}k_{\chi}\over\varepsilon\chi}\big{(}{\kappa\over
k_{\chi}}\big{)}^{11/2}$ $None$

and we see that the quantity $v_{\kappa}^{3}\kappa$ does depend on $\kappa$ in
the ED case. Given $\kappa$ the SRB statistics for the ED equations driven
with a total energy $\varepsilon$ gives to this quantity the same value that
it has in the SRB statistics for the NS equation driven with a total vorticity
$\eta$ if:

${\varepsilon\chi\over\eta\nu}=\big{(}{\kappa\over k_{\chi}}\big{)}^{-{11\over
2}}$ $None$

provided (of course) $\kappa$ is greater than the Kolmogorov scales
$k_{\nu},k_{\chi}$.

The “barometric formula” is then the statement of equivalence between NS and
ED on scale $\kappa$, i.e. if one only looks at field properties depending on
${\,\underline{\gamma}\,}_{{\,\underline{k}\,}}$ for ${1\over
2}\kappa<|{{\,\underline{k}\,}}|<\kappa$, if (6.2) holds and $\kappa\gg
k_{\nu},k_{\chi}$.

If we look at a different scale $\kappa^{\prime}=2^{n}\kappa$ for some (large)
$n$ then we can expect equivalence between ED (or GED) and NS (or GNS) but the
pairs $\varepsilon,\eta$ should now be such that (6.2) holds on the new scale:
the analogy with the usual barometric formula for the Boltzmann Gibbs
distribution in the gravity field justifies the name given to (6.2). We see
that $\eta\nu$ plays the role of the gravity, $\varepsilon\nu$ plays the role
of the chemical potential and $\kappa/k_{\chi}$ plays the role of the height.

The above analysis seems to be fully consistent with the numerical results in
[ZJ] who first proposed, in a different context, a picture very close to the
one developed here.

It is clear that this point of view has several consequences: for instance in
particular it tells us that that the shape of the pairing curve in (5.1)
cannot be arbitrary (i.e. $\langle{\beta}\rangle
c_{j}\sim\nu{{\,\underline{k}\,}}_{j}^{2}$ if the modes are ordered in
increasing order). This is a point on which I hope to return in a later
analysis.

Also: the equivalence between NS and ED on a given momentum scale makes more
interesting the ideas in [Ga4] and a test of the Onsager reciprocity derived
in the latter paper seems now quite feasible and seems also to have
conseqiences for the real NS equations.

A further remark is that although (6.2) depends on the validity of the K-41
theory and of the corresponding theory for ED equations the barometric formula
can be developed independently of such theories: hence any modification of the
K-41 theory (and of the corresponding theory for ED) will lead to a barometric
formula, with a relation between $\varepsilon,\eta,\kappa$, possibly more
complicated than (6.2).

Acknowledgements: I am grateful to L. Biferale, G. Paladin and A. Vulpiani for
the information about the GOY model spectral symmetry and to G. Gentile, C.
Liverani, V. Mastropietro and particularly to F. Bonetto for many clarifying
discussions and suggestions: his criticism has been very important for
correcting slips and improving various points. This work is part of the
research program of the European Network on: ”Stability and Universality in
Classical Mechanics”, # ERBCHRXCT940460 and it has been partially supported
also by MRS grants #40%,60%, and INFN.

Appendix A1: The hamiltonian formalism for Euler equations and Dressler’s
theorem for ED.

To check the applicability of the results on pairing of [Dr] to the ED
equations we must check that the equations can be written, in canonical
coordinates for some hamiltonian function $H$, in the form:

$\eqalign{\dot{{\,\underline{q}\,}}=&{{\,\underline{\partial}\,}}_{{\,\underline{p}\,}}H\cr\dot{{\,\underline{p}\,}}=&-{{\,\underline{\partial}\,}}_{{\,\underline{q}\,}}H+{\,\underline{F}\,}-\chi{\,\underline{p}}\cr}$
$None$

where ${\,\underline{F}\,}$ is such that
$\partial_{q_{j}}F_{i}=\partial_{i}F_{j}$ without being necessarily
${\,\underline{F}\,}=-{{\,\underline{\partial}\,}}V$ for some globally defined
$V$ (the latter would be a trivial case). The labels for the components of
${\,\underline{q}\,}$ are ${\,\underline{x}\,},i$ with
${\,\underline{x}\,}\in\Omega$ and $i=1,2,3$. The partial derivatives are,
correspondingly, functional derivatives; we shall ignore this because a
”formally proper” analysis is easy and leads to the same results. By ”formal”
we do not mean rigorous, but only rigorous if the functions we consider have
suitably strong smoothness properties: a fully rigorous treatment is of course
impossible for want of reasonable existence, uniqueness and regularity
theorems for the Euler equations or the Navier Stkes equations in $3$
dimensions.

Consider first the Euler equations. They can be derived from the lagrangian:

${\cal
L}_{0}(\dot{{\,\underline{\delta}\,}},{\,\underline{\delta}\,})={\rho\over
2}\int\dot{{\,\underline{\delta}\,}}^{2}d{\,\underline{x}\,}$ $None$

($\rho=$ density) defined on the space ${\cal D}$ of the diffeomorphisms
${\,\underline{x}\,}\to{\,\underline{\delta}\,}({\,\underline{x}\,})$ of the
box $\Omega$, by imposing the ideal constraint:

$\det
J{\;\equiv\;}\det{\partial{\,\underline{\delta}\,}\over\partial{\,\underline{x}\,}}({\,\underline{x}\,})={{\,\underline{\partial}\,}}\delta_{1}\wedge{{\,\underline{\partial}\,}}\delta_{2}\cdot{{\,\underline{\partial}\,}}\delta_{3}{\;\equiv\;}1$
$None$

In fact, if $Q$ is a Lagrange multiplier, the stationarity condition for:

${\cal L}(\dot{{\,\underline{\delta}\,}},{\,\underline{\delta}\,})={\rho\over
2}\int\dot{{\,\underline{\delta}\,}}^{2}d{\,\underline{x}\,}+\int
Q({\,\underline{x}\,})\det
J({\,\underline{\delta}\,})({\,\underline{x}\,})\,d{\,\underline{x}\,}$ $None$

leads to:

$\rho\dot{{\,\underline{\delta}\,}}=-(\det
J)\,J^{-1}{{\,\underline{\partial}\,}}Q=-\det
J\,{\partial{\,\underline{x}\,}\over\partial\delta_{\kern-3.0pt\lower
6.6pt\hbox to3.1298pt{$\widetilde{}$\hfill}}\kern
2.0pt\,}\cdot\partial_{\kern-3.0pt\lower 6.6pt\hbox
to3.1298pt{$\widetilde{}$\hfill}}\kern 2.0pt\,Q$ $None$

so that setting
${\,\underline{u}\,}({\,\underline{\delta}\,}({\,\underline{x}\,}))=\dot{{\,\underline{\delta}\,}}({\,\underline{x}\,})$,
$p({\,\underline{\delta}\,})=Q({\,\underline{x}\,})$ if
${\,\underline{\delta}\,}={\,\underline{\delta}\,}({\,\underline{x}\,})$, we
see that:

${d{\,\underline{u}\,}\over dt}=-{1\over\rho}{{\,\underline{\partial}\,}}p$
$None$

which are the Euler equations. And the multiplier $Q({\,\underline{x}\,})$ can
be computed as:

$Q({\,\underline{x}\,})=p({\,\underline{\delta}\,})=-\big{[}\Delta^{-1}({{\,\underline{\partial}\,}}u_{\kern-3.0pt\lower
6.6pt\hbox to3.1298pt{$\widetilde{}$\hfill}}\kern
2.0pt\,\cdot\partial_{\kern-3.0pt\lower 6.6pt\hbox
to3.1298pt{$\widetilde{}$\hfill}}\kern
2.0pt\,\,{\,\underline{u}\,})\big{]}_{{\,\underline{\delta}\,}}$ $None$

where the functions in square brackets are regarded as functions of the
variable ${\,\underline{\delta}\,}$ and the differential operators also
differentiate over such variable. After the computation the variable
${\,\underline{\delta}\,}$ has to be set equal to
${\,\underline{\delta}\,}({\,\underline{x}\,})$.

Therefore by using the lagrangian:

${\cal
L}_{i}(\dot{{\,\underline{\delta}\,}},{\,\underline{\delta}\,})=\int\big{(}{\rho\dot{{\,\underline{\delta}\,}}({\,\underline{x}\,})^{2}\over
2}-\big{[}\Delta^{-1}({{\,\underline{\partial}\,}}u_{\kern-3.0pt\lower
6.6pt\hbox to3.1298pt{$\widetilde{}$\hfill}}\kern
2.0pt\,\cdot\partial_{\kern-3.0pt\lower 6.6pt\hbox
to3.1298pt{$\widetilde{}$\hfill}}\kern
2.0pt\,\,{\,\underline{u}\,})\big{]}_{{\,\underline{\delta}\,}({\,\underline{x}\,})}(\det
J({\,\underline{\delta}\,})|_{{\,\underline{x}\,}}-1)\big{)}\,d{\,\underline{x}\,}$
$None$

we generate lagrangian equations for which the ”surface” $\Sigma$ of the
incompressible diffeomorphisms in the space ${\cal D}$ is invariant: these are
the diffeomorphisms
${\,\underline{x}\,}\to{\,\underline{\delta}\,}({\,\underline{x}\,})$ such
that
$J({\,\underline{\delta}\,})={{\,\underline{\partial}\,}}\delta_{1}\wedge{{\,\underline{\partial}\,}}\delta_{2}\cdot{{\,\underline{\partial}\,}}\delta_{3}{\;\equiv\;}1$
at every point ${\,\underline{x}\,}\in\Omega$.

Then $\Sigma$ is invariant in the sense that the solution to the lagrangian
equations with initial data ”on $\Sigma$”, i.e. such that
${\,\underline{\delta}\,}\in\Sigma$ and
${{\,\underline{\partial}\,}}\cdot\dot{{\,\underline{\delta}\,}}({\,\underline{x}\,})=0$,
evolve remaining ”on $\Sigma$”.

The hamiltonian for the lagrangian (A1.8) is obtained by computing the
canonical momentum ${\,\underline{p}\,}({\,\underline{x}\,})$ and the
hamiltonian as:

$\eqalign{{\,\underline{p}\,}({\,\underline{x}\,})=&{\delta{\cal
L}_{i}\over\delta\,{\,\underline{\delta}\,}({\,\underline{x}\,})}=\rho\dot{{\,\underline{\delta}\,}}({\,\underline{x}\,})+\ldots\cr
H({\,\underline{p}\,},{\,\underline{q}\,})=&{1\over
2}\big{(}G({\,\underline{q}\,}){\,\underline{p}\,},{\,\underline{p}\,})\cr}$
$None$

where $G({\,\underline{q}\,})$ is a suitable quadratic form that can be read
directly from (A1.8) (but it has a somewhat involved expression of no interest
for us), and the $\ldots$ (that can also be read from (A1.8)) are terms that
vanish if ${\,\underline{\delta}\,}\in\Sigma$ and
${{\,\underline{\partial}\,}}\cdot\dot{{\,\underline{\delta}\,}}=0$, i.e. they
vanish on the incompressible motions.

The above is well known and shows that the Euler flow can be interpreted as a
gedodesic flow on the surface $\Sigma$ of the incompressible diffeomorphisms
of the box $\Omega$ enclosing the fluid, see appendix 2 in [A].

Modifying the Euler equations by the addition of a force
${\,\underline{f}\,}({\,\underline{x}\,})$ such that locally
${\,\underline{f}\,}({\,\underline{x}\,})=-{{\,\underline{\partial}\,}}\Phi({\,\underline{x}\,})$
means modifying the equations into:

${d{\,\underline{u}\,}\over
dt}=-{{\,\underline{\partial}\,}}p-{{\,\underline{\partial}\,}}_{{\,\underline{x}\,}}\Phi$
$None$

which can be derived from a lagrangian:

${\cal
L}^{\Phi}_{i}(\dot{{\,\underline{\delta}\,}},{\,\underline{\delta}\,}))={\cal
L}_{i}(\dot{{\,\underline{\delta}\,}},{\,\underline{\delta}\,}))-\int\Phi({\,\underline{\delta}\,}({\,\underline{x}\,}))\,d{\,\underline{x}\,}$
$None$

which leads to the equations:

$\dot{{\,\underline{u}\,}}({\,\underline{\delta}\,}({\,\underline{x}\,}))=-{1\over\rho}{{\,\underline{\partial}\,}}_{{\,\underline{\delta}\,}}p({\,\underline{\delta}\,}({\,\underline{x}\,}))+{{\,\underline{\partial}\,}}_{{\,\underline{\delta}\,}}\Phi({\,\underline{\delta}\,}({\,\underline{x}\,}))$
$None$

Hence the ED equations have the form:

$\eqalign{\dot{{\,\underline{q}\,}}=&\partial_{{\,\underline{p}\,}}H\cr\dot{{\,\underline{p}\,}}=&-\partial_{{\,\underline{q}\,}}H-{\,\underline{F}\,}-\chi{\,\underline{p}}\cr}$
$None$

at least as far as the motions which have an incompessible initial datum are
concerned. This is true because the ED equations which have an incompressible
initial datum evolve it by keeping it incompressible.

The Lyapunov exponents of the equation (A1.13) verify the pairing rule by the
analysis in [Dr]. However the pairing takes place in the full phase space of
the diffeomorphisms of $\Omega$, including the incompressble ones.

It is not difficult to see, by using that the constraint to stay on the
surface $\Sigma$ is holonomous, that one can find canonical coordinates
${\,\underline{\pi}\,},{\,\underline{\kappa}\,}$,
${\,\underline{\pi}\,}^{\perp},{\,\underline{\kappa}\,}^{\perp}$ describing
the motions on $\Sigma$ or, respectively, transversally to it. And the
equations for ${\,\underline{\pi}\,}^{\perp},{\,\underline{\kappa}\,}^{\perp}$
are, near $\Sigma$ and for ${\,\underline{\pi}\,}^{\perp}$ small,
$\dot{{\,\underline{\pi}\,}}^{\perp}=-\chi{\,\underline{\pi}\,}^{\perp}$ and
$\dot{{\,\underline{\kappa}\,}}^{\perp}={\,\underline{\pi}\,}^{\perp}$ so that
the corresponding Lyapunov exponents are trivially paired in pairs $0,-\chi$
with pairing sum $-{\chi\over 2}$.

Since we have seen above that all the exponents are paired at the level
${\chi\over 2}$ this means that all the physically intereting exponents
(relative to the incompressible motions, i.e. relative to the
${\,\underline{\pi}\,},{\,\underline{\kappa}\,}$ coordinates) are also paired
at the same level, as claimed in §4.

References.

[A]  Arnold, V.: Méthodes Mathématiques de la Mécanique classique, Ed. MIR,
Moscow, 1974.

[BG]  Bonetto, F., Gallavotti, G.: Reversibility, coarse graining and the
chaoticity principle, preprint in mp$\\_$arc@ math.utexas.edu, #96-50.

[BGG]  Bonetto, F., Gallavotti, G., Garrido, P.: Chaotic principle: an
experimental test, in mp$\\_$arc@ math. utexas. edu # 154, 1996.

[BJPV] Bohr, T., Jensen M.H., Paladin, G., Vulpiani, A.: Dynamical systems
approach to turbulence, Cambridge Nonlinear series, 1996.

[DPH]  Dellago, C., Posch, H., Hoover, W.: Lyapunov instability in system of
hard disks in equilibrium and non-equilibrium steady states, preprint,
Institut für Experimentalphysik, Universität Wien, Austria, August 1995.

[Dr]  Dressler, U.: Symmetry property of the Lyapunov exponents of a class of
dissipative dynamical systems with viscous damping, Physical Review, 38A,
2103–2109, 1988.

[DM]  Dettman, C.P., Morriss, G.P.: Proof of conjugate pairing for an
isokinetic thermostat, N.S. Wales U., Sydney 2052, preprint, 1995.

[ECM1]  Evans, D.J.,Cohen, E.G.D., Morriss, G.P.: Viscosity of a simple fluid
from its maximal Lyapunov exponents, Physical Review, 42A, 5990–5997, 1990.

[EM]  Evans, D.J., Morriss, G.P.: Statistical mechanics of nonequilibrium
liquids, Academic Press, 1990.

[FP]  Frisch, U., Parisi, G.: in Turbulence and predictability, p. 84–87, ed.
by M. Ghil, R. Benzi, G. Parisi, North Holland, 1984.

[G]  Galilei, G.: ”Discorsi intorno a due scienze nuove”, ”Opere”, Salani,
Firenze, 1964.

[Ga1]  Gallavotti, G.: Some rigorous results about 3D Navier Stokes, Les
Houches 1992 NATO-ASI meeting on “Turbulence in spatially extended systems”,
ed. R. Benzi, C. Basdevant, S. Ciliberto; p. 45–81; Nova Science, NY, 1993.

[Ga2]  Gallavotti, G.: Ergodicity, ensembles, irreversibility in Boltzmann and
beyond, Journal of Statistical Physics, 78, 1571–1589, 1995. See also Topics
in chaotic dynamics, Lectures at the Granada school, ed. Garrido–Marro,
Lecture Notes in Physics, 448, 1995. And, for mathematical details, Reversible
Anosov diffeomorphisms and large deviations., Mathematical Physics Electronic
Journal, 1, (1), 1995, (http:// www. ma. utexas. edu/ MPEJ/ mpej.htlm).

[Ga3]  Gallavotti, G.:Chaotic hypothesis: Onsager reciprocity and
fluctuation–dissipation theorem, in<EMAIL_ADDRESS>#95-288, 1995.
In print in Journal of Statistical Physics, 1996.

[Ga4]  Gallavotti, G.: Chaotic principle: some applications to developed
turbulence, archived in<EMAIL_ADDRESS>#95-232, and chao-dyn@
xyz.lanl.gov, #9505013.

[Ga5]  Gallavotti, G.: Extension of Onsager’s reciprocity to large fields and
the chaotic hypothesis, mp$\\_$ arc 96-109; or chao-dyn 9603003.

[Ga6]  Gallavotti, G.: Equivalence of dynamical ensembles and Navier Stokes
equations, mp$\\_$ arc 96-131; or chao-dyn 9604006.

[Ge]  Gentile, G.: Large deviation rule for Anosov flows, preprint
http://chimera. roma1. infn. it, 1996.

[GC1]  Gallavotti, G., Cohen, E.G.D.: Dynamical ensembles in nonequilibrium
statistical mechanics, Physical Review Letters, 74, 2694–2697, 1995.

[GC2]  Gallavotti, G., Cohen, E.G.D.: Dynamical ensembles in stationary
states, in print in Journal of Statistical Physics, 1995.

[LL]  Landau, L., Lifshitz, V.: Mécanique des fluides, MIR, Moscow, 1966.

[Ma]  Marchioro, C.: An example of absence of turbulence for any Reynolds
numberb, Communications in mathematical Physics, 105, 99–106, 1986. And An
example of absence of turbulence for any Reynolds number: $II$, Communications
in mathematical Physics, 108, 647–651, 1987.

[MP]  Marchioro, C., Presutti, E.: Thermodynamics of particle systems in the
presence of external macroscopic fields: $I$ classical case, Communications in
Mathematical Physics, 27, 146– 154, 1972.

[MR]  Morriss, G.P., Rondoni, L.: Equivalence of ”nonequilibrium” ensembles
for simple maps, preprint, U.of N.S. Wales, Sidney, 1996

[R1]  Ruelle, D.: Chaotic motions and strange attractors, Lezioni Lincee,
notes by S. Isola, Accademia Nazionale dei Lincei, Cambridge University Press,
1989; see also: Ruelle, D.: Measures describing a turbulent flow, Annals of
the New York Academy of Sciences, 357, 1–9, 1980.

[R2]  Ruelle, D.: Positivity of entropy production in non equilibrium
statistical mechanics, IHES preprint, 1995.

[SEM]  Sarman, S., Evans, D.J., Morriss, G.P.: Conjugate pairing rule and
thermal transport coefficients, Physical Review, 42A, 2233–2242, 1992.

[UF]  Uhlenbeck, G.E., Ford, G.W.: Lectures in Statistical Mechanics, American
Mathematical society, Providence, R.I., pp. 5,16,30, 1963.

[ZJ]  Zhen-Su She, Jackson, E.: Constrained Euler system for Navier–Stokes
turbulence, Physical Review Letters, 70, 1255–1258, 1993.

Internet access: All the Author’s quoted preprints can be found and freely
downloaded (latest postscript version including corrections of misprints and
errors) at:

http://chimera.roma1.infn.it

in the Mathematical Physics Preprints page. e-mail address of author:
<EMAIL_ADDRESS>