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Chaos, SRB distribution, Entropy, Nonequilibrium 
Thermodynamics, Fluids, Navier-Stokes, Thermostats, Fluctuation 
theorem, Chaotic hypothesis
---------------0607300913169
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Pacchetti stilistici%%%%%%%%%%%%%%
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%%%%%%\usepackage{fancyhdr}\pagestyle{fancy}{}\fancyhead{}\fancyfoot{}
\begin{document}
%\BOZZA

\ausilio
\fiat


\centerline{\titolone Microscopic chaos and}

\centerline{\titolone macroscopic entropy in fluids}
\*
\centerline{\bf Giovanni Gallavotti}

\centerline{Fisica and I.N.F.N. Roma 1}
\centerline{
30 July 2006 ;
%\today
}

\*\*

\0{\bf Abstract: \it In nonequilibrium thermodynamics ma\-cro\-scopic
entropy creation plays an important role. Here we study the
relationship it bears with the phase space contraction,
which has been recently proposed as an apparently alternative 
quantity.}
\*

\0{\it 1. Phase space contraction and entropy creation}
\numsec=1\numfor=1\*

Studying stationary states of mechanical systems in interaction with
thermostats the latter are modeled by systems of particles subject to
anholonomic constraints. The equations of motion take the form
$\dot x=f_{\V E}(x)$ where $x$ is a point in phase space and $\V E$
are parameters controlling the size of the acting nonconservative forces. 

It has appeared natural to define {\it entropy creation rate} the
divergence $\s(x)\defi-\sum_j \dpr_{x_j} f_{\V E}(x)$.  There are
other natural definitions of entropy creation rate and here we study
their relation with the above phase space divergence. The aim is to
find such a relationship for a system that can be considered to be
described as a continuum following macroscopic equations.

%\input fig
\eqfig{110pt}{90pt}{}{fig}{Fig1}

\0{\nota Fig.1 Reservoirs occupy finite regions outside $\CC_0$,
\eg sectors $\CC_i\subset R^3$, $i=1,2\ldots$. Their particles are
constrained to have a {\it total} kinetic energy $K_i$ constant, by
suitable forces, so that the reservoirs ``temperatures'' $T_i$ are
well defined.} 
\*

To be concrete and in rather ample generality we imagine a system
$\CC_0$ of particles enclosed in a container, also called $\CC_0$,
with elastic boundary conditions surrounded by a few thermostats which
consist of particles interacting with the system via short range
interactions, through a portion $\dpr_i{\CC_0}$ of the surface of ${\CC_0}$, and
subject to the constraint that the total kinetic energy of the $N_i$
particles in the $i$-th thermostat is $K_i=\fra12 \dot{\V
X}_i^2=\fra32 N_i k_B T_i$.  A symbolic illustration is in Fig.1.

Assuming unit mass,
the equations of motion will be

$$\eqalignno{
&\ddot{\V X}_0=-\dpr_{\V X_0}\Big( U_0(\V X_0)+\sum_{j>0}
W_{0,j}(\V X_{0},\V X_j)\Big)+\V E(\V X_0),\cr
&\ddot{\V X}_i=-\dpr_{\V X_i}\Big( U_i(\V X_i)+
W_{0,i}(\V X_{i},\V X_j)\Big)-\a_i \dot{\V X}_i&\eq(e1)  \cr
}$$
%
with $\a_i$ such that $K_i$ is a constant. Here $W_{0,i}$ is the
interaction potential between particles in $\CC_i$ and in $\CC_0$,
while $U_0,U_i$ are the internal energies of the particles in
$\CC_0,\CC_i$ respectively. We imagine that the energies $W_{i,j},U_j$
are due to {\it smooth} translation invariant pair
potentials; repulsion from the boundaries of the containers will be
elastic reflection. It is assumed, in Eq.\equ(e1) that there is no
direct interaction between different thermostats: their particles
interact directly only with the ones in $\CC_0$.  Here $\V E({\V X}_0)$
denotes possibly present external positional forces stirring the
particles in $\CC_0$.

Since the work per unit time that particles outside the thermostat
$\CC_i$ (hence in $\CC_0$) exercise on the particles in it, is
$-\dpr_{{\V X}_{i}}W_{0,i}(\V X_{0},{\V
X}_i)\cdot\dot{\V X}_i$ and it can be interpreted as the ``amount of
heat $Q_i$ entering'' the thermostat $\CC_i$, energy
conservation yields

$$\fra{d}{dt} \big(\fra{1}2\dot{\V X}_i^{2}+ U_i)\=\dot U_i=-
\a_i \dot{\V X_i}^2
+Q_i\Eq(e2)$$
%
and the contraints on the thermostats kinetic energies give
$\a_i\=\fra{Q_i-\dot U_i}{3N_i k_B T_i}$.

Set $x=(\V X_i,\dot{\V X}_i)_{i=0,..}$ and write \equ(e1) as $\dot x=
f_{\V E}(x)$. The divergence $\s(x)=-\sum_j\dpr_{x_j}f_{\V E,j}(x)$ of
the equations of motion in phase space is readily computed from
\equ(e1), see also \cite{Ga06}, and is $\s(x)={\sum_{i>0}}
\fra{Q_i}{k_B T_i} +\dot R$ or $\s(x)=\e(x)+\dot R$ with

$$\e(x)={\sum_{i>0}} \fra{Q_i}{k_B T_i}\Eq(e3)$$
%
where $R=\sum_{i>0} (1-\fra1{3 N_i}) U_{i}$
and $\fra{Q_i}{k_B T_i}$ should really be
$(1-\fra1{3 N_i}) \fra{Q_i}{k_B T_i} $: this simplification is made
just to simplify the formulae as, in any event, we are interested in
cases in which $N_0,N_i\gg1$. 
%The $\fra{1}2\dot{\V X}_i^{2}$
%does not appear in $R$ since it would not contribute to $\dot R$.
\*

\0{\it Remark:} (i) The $\e(x)\defi{\sum_{i>0}} \fra{Q_i}{k_B T_i}$
can be called naturally the {\it entropy creation rate} and, therefore, 
Eq.\equ(e2) have a physical meaning: {\it entropy is created at the
boundary of the system}. Creation {\it
really} takes place where the walls get in contact with the
thermostats, where the temperatures $T_i$ are defined.  
\\
(ii) Note that if particles in $\CC_0$ were {\it also} subject to an
isokinetic constraint $\fra{1}2(\dot{\V X}_0)^2=\fra32 N_0 k_B T_0$  
phase space contraction would simply be changed by the addition of
$\fra {Q_0}{k_B T_0}$ with $Q_0$ being the work done per unit time by
the thermostats in $\CC_i$, $i>0$, on particles in $\CC_0$; also
$R$ will contain an extra term proportional to $\dot U_0$.
\\
(iii) The divergence $\s(x)$ is {\it different} from 
the entropy creation rate $\e(x)$. Their difference is  a
``total time derivative'', therefore the time averages
$a\defi \fra1\t\ig_{-\fra\t2}^{\fra{\t}2} \s(S_t x)\,dt$
and $a_0\defi \fra1\t\ig_{-\fra\t2}^{\fra{\t}2} \e(S_t x)\,dt$
are related by 

$$
a=a_0+\fra1\t\big(R(S_{\fra\t2}x)-R(S_{-\fra\t2}x)\big)
\Eq(e4)$$
%
which means that the observables $a$ and $a_0$ will have the {\it
same} distribution with respect to any stationary distribution in the
limit $\t\to\io$ if $R$ is a bounded function (as in our case). More
general and ``singular'' interaction potentials could be considered to
reach essentially equivalent conclusions, \cite{BGGZ05}.  
\\
(iv) Note that also phase space contraction of a system in contact with
isokinetic thermostats has a precise physical meaning as it {\it
equals minus the sum of the dimensionless free energy creation rates}
$-\fra{\dot U_i}{k_B T_i}+\fra{Q_i}{k_B T_i}$ of the thermostats.

\* 
\0{\it 2. Macroscopic fluids}
\*

The above analysis shows that the two notions of entropy creation
rate $\e(x)$ in Eq.\equ(e3) and of phase space contraction $\s(x)$ are
{\it related but different}. They have the same stationary average, as
they differ by a total derivative $\dot R$. This implies
that not only the averages of $\s$ and $\e$ are equal but also that
the fluctuations of the finite time averages, \ie of $a$ and $a_0$ in
Eq\equ(e4), are the same: so that properties known for the
fluctuations of $\s$ imply corresponding properties for the
fluctuations of the physically meaningful entropy creation rate
$\e$. This is relevant because, in the literature, several results
have been derived concerning the fluctuations of the time averages of
the phase space contraction, see \cite{Ga02}.

Therefore it is of some interest to see what the above mechanical
notion of entropy creation rate becomes in a system which can be
considered as a continuum in a stationary state and in local
equilibrium. In fact for such a system an independent definition of
entropy creation is classical, \cite{DGM84}. We check that the two
notions coincide.

Consider, in $\CC_0$, a system of particles which
can be regarded as a continuum in a stationary state and in contact
with fixed walls on which, at each boundary point $\Bx\in\dpr \CC_0$,
temperature is prescribed at a value $T(\Bx)$ because the surface
element $d s_\Bx$ is in contact with a thermostat (as idealized in
Fig.1). Then the entropy creation rate according to Eq.\equ(e3) will
be

$$\e=\ig_{\dpr\CC_0}\fra{Q(\Bx)}{k_B T(\Bx)} 
ds_{\Bx},\Eq(e5)$$
%
where $Q(\Bx)$ is the amount of work per unit time and unit surface
that the fluid performs on the thermostat in contact with the
surface element $ds_\Bx$, while phase space contraction $\s$
will differ from this by
$-\fra{d}{dt}\ig_{\dpr\CC_0}\fra{U_{ext}(\Bx)}{k_B T(\Bx)}\,ds_\Bx$
where $U_{ext}(\Bx)$ is the internal potential energy of the same
thermostat.

There are no complete derivations of the Navier Stokes equations,
(NS), from molecular models: however all attempts (which achieve the
result under reasonable extra assumptions) deal with limiting regimes
implying restrictions on initial data involving a length scaling of
$O(\d^{-1})$, a time scaling of $O(\d^{-2})$, (hence) a velocity
scaling $O(\d)$ and become exact in the limit as $\d\to0$.

Here we shall assume that the NS equations can be also obtained from a
molecular model under a suitable scaling of space and time variables.
We shall therefore consider microscopic initial data with Maxwellian
velocity distribution and with position distributions with average
fields (of density $\r(\V x)$, of kinetic energy, \ie temperature
$T(\V x)$, and velocity $\V u(\V x)$) consistent with initial values
corresponding to a continuum. And we shall suppose that they evolve so
that average velocity, density, kinetic energy satisfy NS with good
approximation, and exactly in the limit in which some scaling
parameter $\d\to0$.

Physically $\d$ is a parameter measuring ``how far from a continuum
the microscopic structure is''; it can be identified with the ratio
between the molecular free path and the length scale of the variation
of the macroscopic velocity and temperature fields.

Therefore in the limit $\d\to0$ each volume element will contain an
infinite number of particles and fluctuations will be
suppressed; however the {\it average} entropy creation will be defined
and, by Eq.\equ(e3), be

$$\media{\e}=-\ig_{\dpr\CC_0}\k \fra{\V n(\Bx)\cdot\V\dpr\, T(\Bx)}{k_B T(\Bx)} 
ds_{\Bx}\Eq(e6)$$
%
where $\k$ is the thermal conductivity and the
average is intended over a time scale long compared to the microscopic
time evolution but macroscopically short. 

Suppression of fluctuations will not mean that the averages defining
$\V u(\V x), T(\V x)$ over such time scales will not continue to vary
even in the stationary state. However {\it global quantities}, \eg
$\ig_{\CC_0} \V u(\V x)\,d\V x$ or $\ig_{\CC_0} \log T(\V x)\,d\V x$,
will have such averages varying over a longer time scale.

Eq.\equ(e6) is the expression corresponding to Eq.\equ(e3) derived
from molecular dynamics and it must be compared, for compatibility,
with the familiar expressions for the entropy creation rate in systems
described by macrosopic continua equations, \cite{DGM84}.

We consider here a viscous and thermally conducting
fluid with density $\r$ and in local equilibrium.  Thus the
local equilibrium entropy density $s$ depends on temperature and density
$s=s(T,\r)$. Then, if $\VV\t'$ is the stress tensor
$\t'_{ij}=(\dpr_i u_j+\dpr_j u_i)$ in terms of the velocity field $\V
u$, $\h$ is the dynamical viscosity
and $U(\V x)$ is the internal energy density, the NS equation 
are, \cite[p.6,18]{Ga02},

$$\eqalignno{
(1)\kern0.3truecm&\dpr_t\r+\V\dpr\cdot(\r\V u)=0\cr
(2)\kern0.3truecm&\dpr_t\V u+\W u\cdot\W\dpr\, \V u=-{1\over\r}\V\dpr\, p
   +\fra{\h}\r \D\V u+\V g&\eq(e7)\cr
(3)\kern0.3truecm&\dpr_t U+\V \dpr\cdot(\V u U)=\h\,\VV\t'\, 
\V\dpr \,\W u+\k\D T-p\,\V\dpr\cdot\V  u\cr
(4)\kern0.3truecm&T\,(\dpr_t s+\V \dpr\cdot(\V u s))\,=\,\h\,
   \VV\t'\, \V\dpr \,\W u+\k\D T\cr
}$$
%
The conditions at the boundary of the fluid container ${\CC_0}$ will
be time independent, $T=T(\Bx), \V n(\Bx)\cdot \V u(\Bx)=0$ with $\V
n$ = outer normal (elastic boundary), or $\V u=\V0$ (no slip
boundary).  Here $\V g$ is a (nonconservative) external force
generating the fluid motion and $p$ is the physical pressure.

As mentioned, Eq.\equ(e7) are macroscopic equations that can be valid
only in some limiting regime. Given a system of particles with short
range pair interactions let $\d$ be a dimensionless scaling parameter;
then a typical conjecture is: for suitably restricted and close to
local equilibrium initial data (see \cite[p.21]{Ga02} for examples)
{\it on time scales of $O(\d^{-2})$ and space scales $O(\d^{-1})$ the
evolution follows the incompressible NS equation}, \cite[p.30]{Ga02}.

The classical entropy creation rate in nonequilibrium
thermodynamics of an {\it incompressible fluid} is

$$k_B \media{\e}=\ig_{\CC_0}\Big(\k\, \big(\fra{\V\dpr T}{T}\big)^2
+\h\, \fra1T{\VV\t'\,\V\dpr \W u}\Big)\,d\V x.\Eq(e8)$$
%
By integration by parts and use of the first and
fourth of \equ(e7), $k_B \media{\e}_\m$ becomes, if $S\defi \ig_{\CC_0}
s\,d\V x$,

$$
\eqalignno{
&\ig_{\CC_0}\Big(-\k\,\V\dpr T\,\cdot\,\V\dpr T^{-1}
+\h \,\fra1T{\VV\t'\,\V\dpr \W u}\Big)\,d\V x=\cr
&=
-\ig_{\dpr {\CC_0}} \k\, \fra{\V n\cdot\V\dpr T}T \,ds_\Bx+
\ig_{\CC_0}\fra{(\k\D T+\h\, \VV\t'\V\dpr\W u)}T d\V x=\cr
\noalign{\vglue.05mm}
&=
-\ig_{\dpr {\CC_0}} \k \,\fra{\V n\cdot\V\dpr T}T\,ds_\Bx+
\dot S+\ig_{\CC_0} 
\V u\cdot\V\dpr s\,d\V x=            &\eq(e9)
\cr
&=
-\ig_{\dpr {\CC_0}}\k\, \fra{ \V\dpr T\cdot\V
  n}T\,ds_\Bx+\dot S\cr}
$$
%
\ie it {\it still leads to} the expression Eq.\equ(e6), ``local on the
boundary'' or ``localized at the contact between system and
thermostats'', since $\V u\cdot\V n\=0$ by the boundary conditions,
{\it plus} the time derivative of the total ``thermodynamic entropy''
of the fluid.
\*

\0{\it Remarks:} (i) An identical analysis can be performed for
{\it Rayleigh's convection model}, widely used to test ideas on
turbulence since \cite{Lo63}: the result is the same because the extra
term that would appear in Eq.\equ(e9), see \cite[p. 47]{Ga02}, would be
proportional to $\ig_{\CC_0} u_z d\V x$ which vanishes because the
motion has no net momentum in the $z$ direction.
\\
(ii) It should be noted that in the limit $\d\to0$, \ie when the NS
equations are expected to become rigorously exact, the 
Eq.\equ(e8) simplifies: only the first term in \rhs remains
because the velocity $\V u$ scales as $O(\d)$, \cite[p.26]{Ga02}.


\* 
\0{\it 3. Incompressible continua viewpoint}
\*

The above analysis leads to a further natural question: whether the
phase space contraction and the entropy creation rate can be computed
if we imagine, as it is tempting to do, the small macroscopic volume
elements of an incompressible fluid {\it in local equilibrium and
observed on a short but macroscopic time scale} as a collection of
small thermostats in contact with reservoirs consisting in the
neighboring volume elements: would the results be consistent?

However the volume elements $E=d\V x$ are not separated by walls,
hence they can exchange particles, and they also move.  In order to be
able to treat volume elements as systems on their own we imagine that
their size is $\x$ with $\x$ macroscopically small but microscopically
large: certainly such length scale $\x$ is $\ll L=\big(\fra1T\fra{\dpr
T}{\dpr x}\big)^{-1}$ where $L$ is the macroscopic scale of the
container ${\CC_0}$.

Furthermore we have to assume that molecules diffuse in a
characteristic evolution time, over a distance $\ll \x$. The diffusion
coefficient is $D= O(\fra{k_B T}{m r^2 \r v})$ with $v$ the average
speed, $v=O(v_{sound})$, $m$ the mass of the molecules, $r$ their
radius and $\r$ the numerical density, and a characteristic time scale
is $\th=\fra{m D}{k_B T}$. The distance traveled by diffusion in
the latter time scale is $(D\th)^{\fra12}$ ($\sim 10^{-2}$cm in
air at normal conditions).  In stationary turbulence $\x$ has also to
be small compared to the Kolmogorov scale.

Finally we assume that the local quantities, velocity field and
temperature field,  $\V u(\V x),T(\V x)$, evolve on a time scale much
slower than the microscopic time scale $\th$ and can be considered
constant on that time scale. Then the expression Eq.\equ(e8) should be
regarded as an average over a long microscopic time but over
a short macroscopic time.

If the conditions that allow us to consider a volume element in a
fluid as a thermostated system in a stationary state in contact with
thermostats made of the neighboring elements, \ie that the quantity
$\d$ introduced after \equ(e7) is small and the diffusion across the
elements boundaries is not so important to make the identity of the
volume elements ill defined (\ie $\d\ll\x\ll L$) we can apply
the analysis leading from Eq.\equ(e1) to Eq.\equ(e3) and conclude that
{\it up to a total derivative $\dot R$} the phase space contraction of
the total system, \ie fluid plus thermostats, is

$$\e(x)=\sum_E \sum_{E'} \fra{Q_{E,E'}}{k_BT_{E'}}\Eq(e10)$$
%
where $Q_{E,E'}$ is the amount of work that the particles in a given
volume element $E$ perform over the neighboring elements $E'$, see
Eq.\equ(e3).

We assume that the local average kinetic energies can be regarded as
non fluctuating (\ie we assume to be close to thermal equilibrium):
this is an assumption that appears often (if tacitly) in
nonequilibrium thermodynamics and it will be used here to deduce that
the average values of products of temperatures and velocities equal
the products of the averages.

If the average heat current is $-\k\, \V\dpr T$ and the element $E$ is
imagined with the bases orthogonal to the gradient of $T$, the average
contribution to $Q_{E,E'}$ for $E'$ adjacent to the upper base of $E$
is $-\k \,\V\dpr T\cdot \V n\, \x^2$ ($\x^2$ = area of the base) and it is
opposite to the contribution from the lower base; therefore the
quantity $\sum_{E'} \fra{Q_{E,E'}}{T_{E'}}$ has average 

$$ -\k\, \V\dpr T\cdot \V n \x^2\,(\fra1{T_+}-\fra1{T_-})=-\k
\V\dpr T \cdot \V\dpr T^{-1}\x^3,\Eq(e11)$$
%
if $T_\pm$ are the temperatures at the two bases; therefore summing
over $E$: $k_B\media{\e}_\m=\ig_{\CC_0}
\V\dpr(\fra1{T})\cdot(-\k\,\V\dpr T) \,d {\V x}$ which can be written
in the more familiar form $\k\,\ig_{\CC_0} \big(\fra{\V\dpr T}
T\big)^2d\V x$. Then if $\dpr_t T=0,\D T=0$ and $\V u=\V 0$ Eq.\equ(e8), hence
Eq.\equ(e6), follows by partial integration.

More generally in presence of time dependence and non vanishing
velocity field there will be an extra amount of energy
transfered to elements adjacent to $E$ and due to diffusion across the
bases: it can be evaluated in the same way as above to be $\h
\VV\t'\cdot \V n\,(\W u_+-\W u_-)\x^2$ if $\VV\t'$ is the stress
tensor and it changes $\media{\e}_\m$ to Eq.\equ(e8), hence to
Eq.\equ(e6) plus $\dot S$ (see Eq.\equ(e9)) if account is taken of the
fourth of Eq\equ(e7).

Finally it is interesting to remark that not only the entropy creation
rate but also the phase space contraction can be computed along the
above lines. Regarding each volume element $E$ as a thermostated
system in a stationary state with a fixed temperature, the phase space
contraction is given by $\ig_{\CC_0}\big(\e(\V x)-\fra{\dot U(\V x)}{k_B
T(\V x)}\big)\,d\V x-\ig_{\dpr\CC_0 }\fra{\dot U_{ext}(\V x)}{k_B T(\V x)}
\,ds_{\V x}$ where $U(\V x)$ denotes the energy density and
$U_{ext}(\V x)$ denotes the potential energy of the thermostats (the
kinetic energies do not appear because in this approximation they are
supposed to be constant as each volume element is regarded to have a
constant kinetic energy, \ie a well defined temperature). 

Energy conservation, $\dot U(\V x)= \VV\t'\cdot\V\dpr \W u+\k\D T$,
see (3) in Eq.\equ(e7), and partial integration of the contribution
$\k\big(\fra{\V\dpr T}T\big)^2$ to $\e(\V x)$, see Eq.\equ(e8), in the
integral $\ig_{\CC_0}\big(\e(\V x)-\fra{\dot U(\V x)}{k_B T(\V
x)}\big)\,d\V x$ leave us with a boundary term {\it which is just
Eq.\equ(e6) minus} $\ig_{\dpr \CC_0}\fra{\dot U_{ext}(\V x)}{k_B T(\V
x)}ds_{\V x}$ as it could be guessed from the expression for $\s$
preceding Eq.\equ(e3). Therefore even by regarding the fluid volume
elements as thermostated systems in stationary nonequilibrium leads to
the expected general relation between entropy creation $\e$ and phase
space divergence $\s$ discussed in Sec.1, namely (if $x$ denotes the
fields determining the state of the fluid)

$$\s(x)=\e(x)-\ig_{\dpr \CC_0}\fra{\dot U_{ext}(\V x)}{k_B T(\V
x)}ds_{\V x}\Eq(e12).$$

Analysis of compressible fluids is unfortunately more difficult
and should also be attempted: the first difficulty will be, of course, that
it is not clear under which scaling the compressible NS equations
should hold as a reasonable approximation.

\*
\0{\bf Acknowledgements: \rm
I am indebted to A.Giuliani, F.Zamponi and, in particular, to F.Bonetto
for important suggestions.}

%\*\0\revtex
%\vfill\eject

%\nota 
%\bibliography{0Bibcaos} 
%\bibliographystyle{apsrev}
\bibliographystyle{unsrt}
%\input chfluidi.bbl
\begin{thebibliography}{1}

\bibitem{Ga06}
G.~Gallavotti.
\newblock Irreversibility time scale.
\newblock {\em Chaos}, 16:023130 (+7), 2006.

\bibitem{BGGZ05}
F.~Bonetto, G.~Gallavotti, A.~Giuliani, and F.~Zam\-poni.
\newblock Chaotic {H}ypothesis, {F}luctuation {T}heorem and {S}ingularities.
\newblock {\em Journal of Statistical Physics}, 123:39--54, 2006.

\bibitem{DGM84}
S.~de~Groot and P.~Mazur.
\newblock {\em Non equilibrium thermodynamics}.
\newblock Dover, Mineola, NY, 1984.

\bibitem{Ga02}
G.~Gallavotti.
\newblock {\em Foundations of Fluid Dynamics}.
\newblock (second printing) Springer Verlag, Berlin, 2005.

\bibitem{Lo63}
E.~Lorenz.
\newblock Deterministic non periodic flow.
\newblock {\em Journal of Atmospheric Science}, 20:130--141, 1963.

\end{thebibliography}

\0e-mail: {\tt giovanni.gallavotti@roma1.infn.it}
\\
web: {\tt http://ipparco.roma1.infn.it}
\end{document}

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