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in a recent paper @xcite , we pointed out that microcanonical calculations in the lattice gas model ( lgm ) with constant energy are no harder to implement than canonical calculations with constant temperature .
we will call the first mlgm , and the second , clgm . for practical cases at hand @xmath1 or 200 ) , the calculations use monte - carlo simulations with metropolis algorithm .
we found that in lgm , as used in nuclear disintegration problems , there is no `` backbending '' in the caloric curve for systems as small as @xmath2kr whether in microcanonical or canonical treatments . by `` backbending ''
one means an `` s '' shape when energy is plotted along the y - axis and temperature along the x - axis . since microcanonical treatments seem to lead to backbending for small systems ( 100 particles is small enough ) in other models @xcite , our findings need some clarification .
motivated by this , we present here results for a very small system , 11 particles in @xmath0 lattice . here
we can avoid monte - carlo samplings and do exact ( though it still requires some numerical work which is easy ) computations .
the results are quite interesting and not only explain our previous findings but also shed light on several connections between microcanonical and canonical calculations .
as our objective is solely pedagogical , we assume there is just one kind of particles ( nucleons ) .
we take the number of particles to be 11 .
the lattice space is @xmath0 .
this then implies a freeze - out density 0.41@xmath3 which is somewhat higher than the freeze - out density used in lattice gas model calculations @xcite .
the nearest neighbour bonds are attractive : @xmath4mev to get the nuclear matter binding energy correct .
the nuclear lattice gas model which is denoted here by lgm is an extension of the standard textbook lattice gas model as discussed , for example , in @xcite .
we denote the standard lattice gas model by slgm .
the difference is simple : in slgm , the nucleons are frozen in their lattice sites . in lgm , dictated by the physics of the nuclear problem ,
they are given momenta . in clgm
, these momenta are generated using a maxwell - boltzmann distribution . in mlgm , they are taken from a uniform distribution within a sphere in momentum space .
the addition of kinetic energy , however , changes the caloric curve in an interesting and profound way
. we will find it useful to discuss the caloric curves in both slgm and lgm .
chronologically , it is easier to discuss slgm first , then point out how lgm modifies the results . in both the models
the key quantities are @xmath5= the number of configurations with @xmath6 nearest neighbour bonds for the case of 11 particles in @xmath0 lattice sites . once these are known both canonical and microcanonical calculations are readily done .
the degeneracy factors are given in the small table .
they can be obtained with little effort in this simple case .    [
cols="^,^,^,^ " , ]     _ table i : degenracy factors @xmath7 with @xmath6 nearest neighbour bonds . _
instead of writing @xmath7 we will find it convenient to write @xmath8 as a function of @xmath9 where @xmath9 is the excitation energy . the degeneracy factor @xmath10 as a function of @xmath11
is plotted in fig .
the distribution is discrete but in fig.1 we show it as a continuous distribution and label the y - axis by @xmath12 .
if one wants to define a temperature , the standard practice in the microcanonical model is to compute @xmath13 ( see @xcite ) .
an inspection of fig.1 shows that as a function of excitation energy the temperature will rise first , approach @xmath14 , will then switch towards @xmath15 and as the excitation energy will further increase the temperature will approach 0 from the negative side .
this happens because in slgm there is an upper bound to energy .
this is of course well - known for spin 1/2 systems in a magnetic field if the kinetic energy of the spin system is suppressed @xcite . in nuclear shell model , for example , this will happen if one restricts oneself to limited shell model orbitals .
this is well - known to practitioners @xcite .    the caloric curve in microcanonical slgm is shown in fig .
in plotting this curve we used degeneracies of successive discrete points in the excitation energy and divided by @xmath16 to get the temperature .
notice that in the positive side of the temperature there is no anomalous behaviour .
if we plot @xmath9 along the y - axis and @xmath17 along the x - axis , there is a `` giant '' size backending at about half the excitation energy available to the system .
but this is merely a reflection of the fact that the excitation energy available to the system is finite .
this will drastically change in the nuclear lgm where availability of kinetic energy will remove the upper limit .
for canonical calculation , we pick a positive temperature : to get the caloric curve we compute @xmath18 . subtracting out the ground state energy we obtain the plot in fig .
the same procedure can be used for negative temperature .
both are used in fig .
the similarity between caloric curves calculated in the microcanonical and canonical models is obvious although there are quantitative differences .
from slgm we now turn to nuclear lgm which serves as a model for nuclear disaasembly .
this was the case presented in @xcite .
the excitation energy can come from two sources now : kinetic and potential .
consequently , we compute @xmath19 where @xmath20 is discrete and taken from the table and @xmath21 is taken to be the integral @xmath22 @xmath23 in our chosen case is 11 .
now there is no upper limit to @xmath9 . in fig .
3 we have plotted @xmath19 . the most important difference from fig . 1
is that the negative temperature zone has completely disappeared .
thus the difference in the caloric curves obtained from slgm and lgm will be profound .
there are two ways one can calculate the temperature in the microcanonical model .
one is the standard formula : @xmath24 where @xmath25 the other intuitive approach would be to make the following ansatz .
although we are talking of one system only , formally eq .
( 5.2 ) is similar to that of two systems characterised by @xmath8 and @xmath26 which share energy with each other but are insulated from the rest of the universe so that the total energy @xmath9 does not change .
if the systems characterised by @xmath8 and @xmath27 are large then the sum above would be dominated by the largest term in the sum which is obtained when the temperature of each subsystem is the same , i.e. , @xmath28 .
we now use @xmath29 .
this leads to @xmath30 .
this @xmath31 and the standard definition of @xmath17 agree quite well as can be seen in fig.4 .
notice also there is no backbending in the microcanonical caloric curve .
if one wants to use the microcanonical nuclear lgm for practical calculations with nucleon numbers about 100 or higher and also wants to obtain a value for temperature , getting the temperature from kinetic energy is the only easy option .    in fig.4
we have also shown the caloric curve in nuclear lgm in the canonical model .
this agrees with the microcanonical calculation quite well .
in the particular example ( 11 particles in @xmath0 boxes in the nuclear lgm ) , one has exact expressions for microcanonical density of states .
one can also compute numerically the canonical partition function . in nuclear physics one
often has numerical values for canonical or grand canonical partition functions .
the direct expression for the microcanonical density of state is usually intractable and in order to obtain a value one uses the saddle - point approximation @xcite .
we can use the nuclear lgm to test the accuracy of the saddle - point approximation since here both the microcanonical density of state and the canonical partition function are directly calculable .
the microcanonical density of states and the canonical partition function are related by @xmath32 .
the inverse transformation is @xmath33 .
the saddle - point approximation for this integral leads to @xmath34}{\sqrt{2\pi ( < e^2>-<e>^2)}}\end{aligned}\ ] ] where the value of @xmath35 is so chosen that at this value @xmath36 .
the saddle - point approximation for the density of states is also compared to the exact density of state in fig.3 . except for low excitation energies
, the saddle - point approximation is seen to be excellent .
we performed an exact microcanonical calculation of the caloric curve of 11 particles in a @xmath0 lattice .
the caloric curve does not have a backbending which means there was no negative specific heat in this model for 11 particles .
we then conclude that the phenomenon of backbending can be quite model dependent .
this work was supported in part by the natural sciences and engineering council of canada and by _ le fonds pour la formation de chercheurs et laide  la recherche du qubec_. we acknowledge communications with professor dieter gross .
for pedagogical reasons we compute the caloric curve for 11 particles in a @xmath0 lattice . 
 monte - carlo simulation can be avoided and exact results are obtained . 
 there is no back - bending in the caloric curve and negative specific heat does not appear . 
 we point out that the introduction of kinetic energy in the nuclear lattice gas model modifies the results of the standard lattice gas model in a profound way .