0$. The systems in $\CC_a,\, a>0$ should be thought as ``thermostats'' acting on the system $\CC_0$ across the separating walls via their mutual pair interactions. %\input fig \eqfig{210pt}{90pt}{ \ins{80pt}{60pt}{$\V X_0,\V X_1,\ldots,\V X_n$} \ins{43pt}{27pt}{$\st\ddot{\V X}_{0i}=-\dpr_i U_0(\V X_0)-\sum_{a} \dpr_i U_a(\V X_0,\V X_i)+\V F_i$} \ins{43pt}{10pt}{$\st\ddot{\V X}_{ai}=-\dpr_i U_a(\V X_a)- \dpr_i U_a(\V X_0,\V X_i)-\a_a \dot{\V X}_{ai}$} }{fig}{} \0{Fig.1: \nota Schematic illustration of the geometry. The equations of motion are written here assuming unit mass for the particles.} \* The thermostats {\it temperature} $T_a$ is defined to be proportional to the kinetic energy via the Boltzmann's constant $k_B$. Setting $K_a\=\fra12\sum_a\dot{\V X}_{a}^2\defi \fra32 k_B T_a\,N_a$, it is supposed constant and kept such by the action of suitable (phenomenological) forces on the $i$-th particle in $\CC_a$ of the form $-\a_a \dot{\bf X}_{ai}$. The $\a_a$ can be taken $$\a_a = \fra{W_a-\dot U_a}{3 N_a k_B T_a}\Eq(e2.1)$$ % where where the work performed by the system on the thermostats particles $-\sum \dot{\V X}_a\cdot\Dpr_{\V X_a}U_a(\V X_0,\V X_a)$ can be called $W_a=\dot Q_a$ = heat given to the thermostat $\CC_a$ by the system in $\CC_0$. The external forces $\V F_i$ are assumed to be purely positional. Given the above dynamical model (for heat transport) remark that it is {\it reversible} and time reversal is just the usual velocity reversal (because the thermostat forces are {\it even} under global velocities change). Furthermore the divergence of the total phase space volume can be immediately computed and turns out to be $\s(\V X)=\s_0(\V X)+\dot U(\V X)$ with $$\s_0(\V X)=\sum_{a=1}^n \fra{\dot Q_a}{k_B T_a}\,\fra{3N_a-1}{3 N_a}= \sum_{a=1}^n \fra{\dot Q_a}{k_B T_a}\Eq(e2.2)$$ % where $U(\V X)=\sum_{a=1}^n \fra{\dot U_a}{k_B T_a}\,\fra{3N_a-1}{3 N_a}$, and $O(N_a^{-1})$ has been neglected in the last equality in \equ(e2.2). When computing time averages the ``extra term'' $\dot U$ will not contribute because being a time derivative its average will be $\fra1T(U(S_T\V X)-U(\V X))$ and therefore will give a vanishing contribution for large $T$ and the average of $\sum_{a=1}^n \fra{\dot Q_a}{k_B T_a}=\s_0$ will be the average of $\s$. If the interaction $U$ is bounded also the fluctuations of the averages of $\s$ and $\s_0$ will coincide. In the cases in which the interactions are not bounded (\eg Lennard-Jones repulsive cores) care has to be exercised in the fluctuations analysis: the picture does not change except in a rather well understood, trivial, way and this will not be discussed here, \cite{CV03,BGGZ05}. Note that $\s_0$ is a ``boundary term'', in the sense that it depends on the forces through the boundaries and the forces are supposed short range. Thus the question arises whether such kind of thermostats and short range interactions can lead to stationary states: this is not obvious but the ``efficiency'' of such thermostats has been investigated in molecular dynamics simulations, \cite{AES01,GG06}, leading (not surpringly) to the result that the thermostat mechanism in Fig.1 can lead to stationary states (even in presence of additional positional forces stirring the particles in $\CC_0$). Since the thermotats are regarded in equilibrium the above expression shows that $\s(\V X)$ can be ``legitimately'' called the entropy increase of the reservoirs: so the mechanical notion of phase space contraction acquires a clear physical meaning: and this is a no small achievement of a long series of works based on simulations of molecular dynamics, \cite{EM90,Ru99}. \* \0{\bf 3.} {\it Comments} \numsec=3\numfor=1\* \0(1) Other studies of fluctuations have been proposed: they are rather different and apply to systems which are not stationary. The object of study are initial data {\it sampled within an equilibrium distribution} of a Hamiltonian system and subsequently evolved with the equations of motion of a dissipative time reversible system. Then the phase space contraction averaged over a time $\t$, $a\defi\fra1\t\ig_0^\t\s(S_t x)dt$, will be such that the probability $P_0(a)$ {\it with respect to the initial equilibrium distribution} for $a$ to have a given value is such that $$\fra{P_0(a)}{P_0(-a)}=e^{a \t}.\Eq(e3.1)$$ % This is an exact identity, immediately following from the definitions. It involves {\it no error terms}, unlike the ``similar'' \equ(e1.6) that can be written also as $\fra{P(p)}{P(-p)}=e^{p\s_+\t+O(1)}$, with $P$ the {\it probability with respect to the stationary distribution}, which is {\it singular} with respect to the equilibrium distributions if $\s_+>0$. It has been claimed that, being valid for all times, it implies the fluctuation relation, \equ(e1.6), for stationary states (at least when the stationary state exists). This would imply a simple, direct and {\it assumptionless} derivation of the fluctuation theorem in \equ(e1.7) and should hold in spite of the fact that in \cite{GC95} an assumption about the chaotic nature of the motions is needed to derive it, together with a rather detailed understanding of the nature of chaotic systems. However a derivation of the fluctuation theorem \equ(e1.6) from \equ(e3.1) involves considering \equ(e3.1) {\it after} the limit $\t\to\io$ has been performed: a rather unclear procedure (note that the {\it r.h.s.} depends on $\t$). Leaving aside the logical consitency problems it should be kept in mind that in the stationary state, at least in the interesting cases in which $\s_+>0$ and there is dissipation, the statistics of motion will be controlled by a distribution that has nothing to do with the initial equilibrium distribution in which the averages in \equ(e3.1) are considered. Therefore the claim is incorrect and it is no surprise that some kind of chaos has to be present to obtain the fluctuation relation \equ(e1.6). In fact one can give examples of simple systems in which \equ(e3.1) holds for all times, the system evolves towards a stationary state and nevertheless the \equ(e1.6) {\it does not hold}, \cite{CG99}. The confusion has crept into the literature and even affected experiments: this can only be explained by a certain lack of attention to the literature due to the urge to find an easy way of testing the large fluctuations in real systems (fluids, granular materials, or even biological systems). Another aspect of \equ(e3.1) is that it involves $a$ rather than $p=a/\s_+$. This is clearly a matter of convention: however care has to be exercised because the fluctuation relation \equ(e1.6) is valid for $|p|

} is the same as \eqnum except that \ref{