Comment on "Negative heat capacities and first order phase transitions in nuclei" by L.G. Moretto et al
F. Gulminelli
In a recent paper L.G. Moretto et al \cite{Moretto} claim that the negative heat capacities presented in our previously published paper \cite{PRL2000} are "artifacts" coming from the use of periodic boundary conditions in the Lattice-Gas calculations. We stress in this comment that this claim is wrong: in ref. \cite{PRL2000} we did {not} use periodic boundary conditions and anyhow the boundary conditions are irrelevant for the statistical ensemble used in \cite{PRL2000}. The second claim of \cite{Moretto} is that, because of the Coulomb repulsion, systems "with $A>60$ should present no anomalous negative heat capacities". We show that this conclusion is contradicted by exact Lattice-gas simulations including Coulomb forces which present negative heat capacities even for $A>200$.
moretto.tex
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\begin{document}
%
\title{Comment on "Negative heat capacities 
and first order phase transitions in
nuclei" by L.G. Moretto et al  }
%
%
\author{ F. Gulminelli} 
\affiliation{LPC Caen (IN2P3 - CNRS / EnsiCaen et Universit\'{e}), F-14050
Caen C\'{e}dex, France }

\author{Ph.Chomaz}
\affiliation{GANIL ( DSM - CEA / IN2P3 - CNRS),
B.P.5027, F-14021 Caen C\'edex, France}
   
            

\begin{abstract}
In a recent paper L.G. Moretto et al \cite{Moretto}
claim that the negative heat capacities presented in our 
previously published paper \cite{PRL2000} are
"artifacts" coming from the use of periodic boundary 
conditions in the Lattice-Gas calculations. We stress 
in this comment that this claim is wrong: in ref. \cite{PRL2000}
we did {\it not} use periodic boundary conditions and anyhow the boundary 
conditions are irrelevant for the statistical ensemble used in 
\cite{PRL2000}.
The second claim of \cite{Moretto} is 
that, because of the Coulomb repulsion, systems 
"with $A>60$ should present no anomalous
negative heat capacities". We show that this conclusion  
is contradicted by exact Lattice-gas simulations including
Coulomb forces which present negative heat capacities
even for $A>200$. 
\end{abstract}
\maketitle   
%In fact, the partition  
%studied in ref \cite{Moretto},
%a single nucleus in equilibrium with a gas of monomers,
%is too schematic to infer general conclusions 
%in the multifragmentation regime associated with negative 
%heat capacities. 
 
Let us start with the discussion about the boundary conditions. 
It is clearly stated in  ref. \cite{PRL2000} 
 %(third paragraph of page 3588) 
that"$L^3$ [the lattice size] is 
large enough (typically greater than $20^3$ lattice 
sites) so that the boundary conditions do not affect 
the calculations with a constraining $\lambda$." This
means that the results of ref.  
\cite{PRL2000} do not depend upon the conditions used
at the boundary, and in fact the calculations were made
{\it without} periodic boundary conditions. At that time
we checked the independence of the boundaries 
comparing $N=8000$ and $N=27000$ lattices.
%To give an example, 
%Let us consider the case 
%presented in figure 4 of \cite{PRL2000}. These results are 
%independent of the boundary conditions as we checked at that time
%comparing $N=8000$ and $N=27000$ lattices.
To see a sizeable effect of the boundary, we have
decreased the size of the lattice.  
A fast calculation of $50000$ events  
for $216$ particles in a $N=5832$ lattice
at an energy $E=0.4\epsilon $ 
confined by a Lagrange multiplier $\log \lambda=-8 $ , 
gives a temperature $T=(0.687 \pm 0.004)\epsilon$, 
a heat capacity $C = -16.3 \pm 0.2$, 
and a kinetic energy fluctuation $\sigma_k^2/T^2 =1.64 \pm 0.03$.
If periodic boundary conditions are imposed, the temperature
becomes $T=(0.682 \pm 0.004)\epsilon$, 
the heat capacity $C = -16.9 \pm 0.4 $ 
and the fluctuation $\sigma_k^2/T^2 = 1.63 \pm 0.07$. 
The temperature decrease in the phase transition region is $\Delta T=(7.6 \pm
0.4)10^{-3} \epsilon$ ($\Delta T=(6.7 \pm 0.6)10^{-3} \epsilon$) with
(without) periodic boundary conditions.
This  shows that
%, even for lattices smaller then the ones used in 
%reference \cite{PRL2000}, 
the boundary conditions do not affect the thermodynamics.
%extracted negative heat capacities and temperature back bending. 
%As stated in ref.\cite{PRL2000},
%to get a full independence of the boundary, slightly bigger lattices 
%($L\geq 20$) have to be considered for a system of this size.
%Therefore, the criticism of ref. \cite{PRL2000} in ref. 
%\cite{Moretto} is clearly wrong: the negative heat capacities
%presented in ref.   \cite{PRL2000} have not been obtained with periodic
%boundary conditions and in fact do not depend on
%the boundary conditions at all.
%In this  discussion it is important to remember that 
%boundary conditions %define a volume, and so they 
%matter when a constant volume (isochore) ensemble 
%is discussed. 
Indeed, in ref. \cite{PRL2000} we have analyzed 
an ensemble of particles 
%constrained by a radial Lagrange potential
in the vacuum for 
which only the average spatial extension of the system
is defined (isobar ensemble). Then, provided that the lattice in which the system 
has still to be discretized
for technical reasons is big enough, 
boundary conditions are irrelevant since the particles never explore 
the outer region. 

In a previous paper \cite{PRL1999} 
we have considered a canonical 
Lattice Gas model in a box of constant volume (isochore 
ensemble), and there we have used periodic boundary conditions.  
In these calculations negative compressibilities 
%and "chemical" susceptibilities 
are reported but the heat capacity is always positive.
In fact, %ref. \cite{PRL1999} deals with a
in the canonical ensemble the heat 
capacity is proportional to the energy variance 
and can never be negative \cite{Shro}. 

Concerning the isochore microcanonical ensemble,
%a huge literature can be found ( for nuclear 
%physics applications see for example
% ref. \cite{Richert}). 
the published results \cite{Richert} show that
%, for the liquid-gas phase transition occurring 
%in small systems %(with size typical of nuclear systems)
the heat capacity 
at constant volume $C_V$ is always positive.
%It should be noticed that 
%This is also what one should expect 
%from the behavior of $C_V$ at the thermodynamic limit \cite{Moretto1}. 
Therefore, contrary to what is claimed in ref. \cite{Moretto}, 
published results dealing with Lattice Gas calculations 
with typical nuclear size systems and periodic boundary conditions 
\cite{Richert,IMFM} do not show any negative heat capacity.  

We may incidentally note that recently 
a debate has started 
in the statistical physics community after the observation of a negative 
heat capacity branch for very large systems in 
the the constant-magnetization microcanonical  
Ising model \cite{Pliemling} (equivalent to the 
microcanonical isochore Lattice Gas model). 
%It has been recently shown \cite{IMFM} that this anomaly in $C_V$ is indeed 
%ue to the phenomenon analogous to the one 
%iscussed in the Moretto et al paper \cite{Moretto} : 
%ontributions of the surface to the equilibrium between a macroscopic 
%drop of liquid and its vapor. 
In these cases however, the densities involved are 
so low %and the sizes so large 
that once again boundary conditions are irrelevant. 
Moreover, the sizes
discussed are orders of magnitude larger than the nuclear ones.

Finally we would like to stress that we have recently demonstrated
that negative heat capacities in finite systems 
are the origin of first order phase 
transitions with a finite latent heat: 
%in infinite systems:  
to present an energy discontinuity %in 
%the caloric curve 
at the thermodynamic limit, 
finite systems {\it must} present a negative microcanonical 
heat capacity if the number 
of particles is large enough \cite{zeroes}. 

%In conclusion, there is no room in published results 
%of Ising based models 
%for a possible artifact due to periodic 
%boundary conditions as stated in ref. \cite{Moretto}.
\begin{figure}[tbp]
%\begin{center}
%\vspace{-1.cm}
\includegraphics[width=.9\linewidth]{fig1.eps}
%\vspace{-1.cm}  
\caption{ Caloric curve for the Lattice Gas hamiltonian
augmented with the Coulomb interaction in the microcanonical isobar ensemble
($\log \lambda = -7, A=207, Z=82$) from ref.\protect\cite{vero}. The coupling
between neighboring sites is fixed to $\epsilon = 5.5 MeV$.   }
%\end{center}
\end{figure} 


Let us now comment about the effect of Coulomb.  
The conclusion of ref.\cite{Moretto} is 
based on a simple model of a unique cluster in 
equilibrium with a gas. However, 
%if we look at the typical partitions 
%in the negative heat capacity region 
%of systems with typical nuclear sizes,
%we can easily see that 
this configuration is %by far 
not  the most probable 
one both in statistical models \cite{smm,gross} and in nuclear physics 
experiments at high excitation energy  \cite{data}. 
In particular, the negative heat capacity 
region is characterized by a large ($m \geq 3$) 
multiplicity of fragments \cite{palluto}. 
Therefore any conclusion drawn considering only the specific channel of
multiplicity 1 is not valid. 
If we would follow the simple model of ref. \cite{Moretto} 
and consider partitions with $m=3-5$ fragments of similar mass
%as experimentally measured 
\cite{data}, 
this would lead to a figure analogous to 
figure 5 of ref. \cite{Moretto} but with an abscissa approximately 
multiplied by  $m$ (or more, since all the light particles have to be included),
because the inter-fragment coulomb interaction  
can be neglected if the volume is large. 
One would then find $A \approx 200$ as %with such a multifragmentation pattern 
the limit of negative heat capacities. 
%Moreover, the mass and charge 
%which should be discussed are the total one including not only the 
%largest fragment but all the particles and clusters. Depending upon 
%the available volume this may push to even higher masses the limit of 
%negative heat capacities.  

Of course fragments do not have the same size and 
to get a quantitative result %estimation of the limiting source mass
one needs to compute the relative weight of all the possible 
channels; this is indeed what is done in nuclear statistical models
\cite{gross,smm,raduta} which show that the intermediate state between 
the compound nucleus and the vaporized system is %nothing but
multifragmentation: the presence of many drops in equilibrium.
%This shows that 
%The model of ref. \cite{Moretto} is 
%may be relevant on a mesoscopic
%or macroscopic scale but is simply
%not physical for microscopic systems %in this energy regime 
%and cannot be used to infer strong conclusions.
% as done in ref. \cite{Moretto}.
 
In the case of the Lattice Gas model the different partitions 
and their relative
weight can be calculated without any approximation.  
%The upper part of Figure 1 shows a typical 
%fragment size distribution in the the negative heat capacity region
%of the uncharged Lattice Gas model %in the isobar ensemble 
%\cite{europhys}. 
%(no Coulomb force has been included in the
%calculation presented in this reference).  
In the negative heat capacity energy regime there are in average 
three fragments of size greater than 4 \cite{PRL2000}.
The sudden opening of this multifragment channel\cite{gross} 
causes a convex intruder in the entropy that is responsible 
of the negative heat capacity. %(here again results are obviously 
%independent of the boundary conditions).   
The effect of the Coulomb interaction in the Lattice Gas model 
can be appreciated from  
oFigure 1 which shows a caloric curve at constant $\lambda=\beta p$
for a system of $A=207$ particles and charge $Z=82$ \cite{vero}.
%The effect of Coulomb is a trivial global shift in total energy plus a decrease
%in the transition temperature as expected \cite{bonche}. 
A clear backbending is visible. 
%contrary to the 
%claim of ref.\cite{Moretto}
%that the heat capacity of systems with $Z>26$ should always be positive. 
In the charged Lattice-gas model, 
the energy interval corresponding to the backbending
%amplitude of the backbending 
%as well as the value of the latent heat 
decreases with increasing charge \cite{vero}, 
showing that the idea  that Coulomb
tends to suppress the negative heat capacity 
is qualitatively correct.
However, when all the available channels are 
correctly weighted, %as in this exact
%microscopic model, 
the quantitative effect is very different
then the one reported in ref. \cite{Moretto}.
%A systematic study of the charged lattice-gas model is in progress,
%and we have already seen that
%, for standard values of the system size, 
%the backbending seems to vanish only for a charges bigger 
%than $Z=92$ \cite{vero}. 
Only a slight effect of the Coulomb interaction has
also been reported in the isochore framework \cite{carmona}.

Obviously the result of Figure 1 is model dependent.
Different macroscopic models show a higher \cite{raduta} or lower \cite{smm}
sensitivity to the Coulomb depending on the detailed implementation 
of the Coulomb and nuclear interactions.
On the other hand molecular dynamics (which can be solved exactly as the Lattice
Gas model) leads to caloric curves which are almost independent of Coulomb
\cite{claudio}. Whether this can be interpreted as an evidence of metastable
long time tails, the relaxation time of long range interactions being
excessively long, is a subject of debate \cite{tsallis}.   
In any case these examples and the results of Figure 1 show that 
a definitive understanding of the effect of a non saturating long range
interaction on a first order phase transition cannot certainly be achieved
through the oversimplified model of ref. \cite{Moretto}.
  

 
 
\begin{thebibliography}{99}

\bibitem{Moretto} L.G. Moretto et al, 
Phys. Rev. C66 (2002) 041601.

\bibitem{PRL2000} P.Chomaz, V. Duflot and F.Gulminelli, 
Phys. Rev. Lett. 85(2000)3587.

\bibitem{PRL1999} F.Gulminelli and P.Chomaz, 
Phys. Rev. Lett. 82(1999)1402.

\bibitem{Shro} E. Schroedinger, 
"Statistical Thermodynamics", Cambridge, 1952

\bibitem{Richert} X. Campi and H. Krivine, Nucl. Phys. A 620 (1997);  
J. M. Carmona et al., Phys. Rev.C 61 (2000) 037304.  

%\bibitem{Moretto1} L. G. Moretto et al., Phys. Rev. Lett. 76 (1996) 2822.

\bibitem{IMFM} F. Gulminelli et al., cond-mat/0302177, submitted to
Phys. Rev. E. 

\bibitem{Pliemling} M. Pleimling and A. Hueller, 
J. Stat. Phys. 104 (2001) 971; K.Binder, Physica A 319 (2002) 99.
 
\bibitem{zeroes} Ph. Chomaz and F. Gulminelli, cond-mat/0210456 and 
Physica A, in press.

\bibitem{gross} D. H. E. Gross,  "Microcanonical
thermodynamics: phase transitions in finite systems", 
Lecture notes in Physics vol. 66, World Scientific (2001).


\bibitem{data} M.Bruno et al., Phys. Lett. B 292 (1992) 251;
	       G. Bizard et al., Phys. Lett. B 302 (1993) 162;
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	       M. F. Rivet et al., Phys. Lett. B 430 (1998) 217;
               L. Beaulieu et al., Phys. Rev. Lett. 84 (2000) 5971.

\bibitem{palluto} M. D'Agostino et al., Nucl. Phys. A699 (2002) 795.

\bibitem{smm} J.~P.~Bon\-dorf et al., Phys.~Rep. 257 (1995) 133; 
J.~P.~Bondorf, A.~S.~Botvina, I.~N.~Mishustin, Phys. Rev. 
58C (1998) R27.

\bibitem{raduta} A. H. Raduta, A. R. Raduta, Nucl. Phys. A703 (2002) 876.

%\bibitem{europhys} F. Gulminelli, Ph. Chomaz, V. Duflot, 
                     %     Europhys. Lett. 50 (2000) 434.
		     %V. Duflot, Ph. Chomaz and F. Gulminelli, Phys. Lett. 
		     %B476 (2000) 279.

\bibitem{vero} V.Duflot-Flandrois, PhD. Thesis (2001), Universit\'e de Caen,
preprint GANIL T/01/05. 

\bibitem{carmona} J. M. Carmona et al., Eur. Phys. Journ. A 11 (2001) 87;
J. Pan and S. Das Gupta, Phys. Rev. C 57 (1998) 1839.
 
\bibitem{claudio} C.Dorso and M. Ison, private communication and to be
published.

\bibitem{tsallis} V. Latora, A. Rapisarda, C. Tsallis, Phys. Rev. E 64 (2001)
056134; B. J. C. Cabral and C. Tsallis, 
Phys. Rev. E 66 (2002) 065101.

\end{thebibliography}

\end{document}

nucl-th0304058.extracted_bib
\begin{thebibliography}{99}

\bibitem{Moretto} L.G. Moretto et al, 
Phys. Rev. C66 (2002) 041601.

\bibitem{PRL2000} P.Chomaz, V. Duflot and F.Gulminelli, 
Phys. Rev. Lett. 85(2000)3587.

\bibitem{PRL1999} F.Gulminelli and P.Chomaz, 
Phys. Rev. Lett. 82(1999)1402.

\bibitem{Shro} E. Schroedinger, 
"Statistical Thermodynamics", Cambridge, 1952

\bibitem{Richert} X. Campi and H. Krivine, Nucl. Phys. A 620 (1997);  
J. M. Carmona et al., Phys. Rev.C 61 (2000) 037304.  

%\bibitem{Moretto1} L. G. Moretto et al., Phys. Rev. Lett. 76 (1996) 2822.

\bibitem{IMFM} F. Gulminelli et al., cond-mat/0302177, submitted to
Phys. Rev. E. 

\bibitem{Pliemling} M. Pleimling and A. Hueller, 
J. Stat. Phys. 104 (2001) 971; K.Binder, Physica A 319 (2002) 99.
 
\bibitem{zeroes} Ph. Chomaz and F. Gulminelli, cond-mat/0210456 and 
Physica A, in press.

\bibitem{gross} D. H. E. Gross,  "Microcanonical
thermodynamics: phase transitions in finite systems", 
Lecture notes in Physics vol. 66, World Scientific (2001).


\bibitem{data} M.Bruno et al., Phys. Lett. B 292 (1992) 251;
	       G. Bizard et al., Phys. Lett. B 302 (1993) 162;
	       A. Schuttauf et al., Nucl. Phys. A 607 (1996) 457;
	       M. F. Rivet et al., Phys. Lett. B 430 (1998) 217;
               L. Beaulieu et al., Phys. Rev. Lett. 84 (2000) 5971.

\bibitem{palluto} M. D'Agostino et al., Nucl. Phys. A699 (2002) 795.

\bibitem{smm} J.~P.~Bon\-dorf et al., Phys.~Rep. 257 (1995) 133; 
J.~P.~Bondorf, A.~S.~Botvina, I.~N.~Mishustin, Phys. Rev. 
58C (1998) R27.

\bibitem{raduta} A. H. Raduta, A. R. Raduta, Nucl. Phys. A703 (2002) 876.

%\bibitem{europhys} F. Gulminelli, Ph. Chomaz, V. Duflot, 
                     %     Europhys. Lett. 50 (2000) 434.
		     %V. Duflot, Ph. Chomaz and F. Gulminelli, Phys. Lett. 
		     %B476 (2000) 279.

\bibitem{vero} V.Duflot-Flandrois, PhD. Thesis (2001), Universit\'e de Caen,
preprint GANIL T/01/05. 

\bibitem{carmona} J. M. Carmona et al., Eur. Phys. Journ. A 11 (2001) 87;
J. Pan and S. Das Gupta, Phys. Rev. C 57 (1998) 1839.
 
\bibitem{claudio} C.Dorso and M. Ison, private communication and to be
published.

\bibitem{tsallis} V. Latora, A. Rapisarda, C. Tsallis, Phys. Rev. E 64 (2001)
056134; B. J. C. Cabral and C. Tsallis, 
Phys. Rev. E 66 (2002) 065101.

\end{thebibliography}