We introduce for canonical fragmention models an exact method for computing
expectation values which exclude the largest cluster. This method allows for
the computation of the reduced multiplicity and other quantities of interest
introduced by Campi, and a comparison shows that the percolation model and a
recent canonical model differ mostly only in small respects in these ensemble
averages.Comment: 7 pages, revtex 3.0, 2 figs. available on reques

Chase, K. C.

Mekjian, A. Z.

English

arXiv

We introduce for canonical fragmention models an exact method for computing expectation values which exclude the largest cluster. This method allows for the computation of the reduced multiplicity and other quantities of interest introduced by Campi, and a comparison shows that the percolation model and a recent canonical model differ mostly only in small respects in these ensemble averages

Chase, K C

Mekjian, A Z

CERN Document Server

RU9552
Exact methods for Campi plots
K. C. Chase and A. Z. Mekjian
Department of Physics, Rutgers University
Piscataway, New Jersey 08855
(August 30, 1995)
Abstract
We introduce for canonical fragmentation models an exact method for com-
puting expectation values which exclude the largest cluster. This method
allows for the computation of the reduced multiplicity and other quantities of
interest introduced by Campi, and a comparison shows that the percolation
model and a recent canonical model dier mostly only in small respects in
these ensemble averages.
25.70.Np, 21.65+f, 05.70.Ce, 05.70.Jk
Typeset using REVTEX
1
Campi and Krivine [1,2] introduced a method for distinguishing fragmentation models
from one another. By comparing various expectation values in which the largest cluster
is excluded but the particle number and fragment multiplicity are held xed, they showed
that the percolation model has a distinctly dierent behavior than many competing nuclear
fragmentation models.
In this paper, we analyze another statistical weight we have been using recently and show
that it shares many of the same properties as percolation theory, a point already apparent
from a consideration of critical exponents [3]. The method used in this paper is unusual in
that it is an exact computational method. Monte Carlo sampling is avoided by exploiting
some properties of the partition functions, and as such there are no statistical errors, an
immense improvement over earlier methods.
We begin by assuming that each fragmentation outcome happens with a probability
proportional to the Gibbs weight [4{7]
W (~n) =
Y
k1
xnkk
nk!
; (1)
where nk is the number of fragments of size (or charge) k, and xk is a parameter associated
with k sized fragments. We then dene the microcanonical partition function as
Z
(m)
A (~x) =
X
m(A)
W (~n) ; (2)
where m(A) is the set of partitions of A nucleons into m fragments, i.e.
P
k knk = A,P
k nk = m. These partition functions satisfy the identity
@Z
(m)
A
@xk
= Z
(m−1)
A−k (~x) ; (3)
which allows for the computation of the partition functions recursively from
P
khnki = m,
since
hnki = xk
@
@xk
lnZ(m)A = xk
Z
(m−1)
A−k
Z
(m)
A
: (4)
Campi dened reduced moments as moments in which the largest cluster is excluded
from the measure, i.e.
2
Ms(~n) 
X
k
ksnk − k
s
max (5)
where kmax is the size of the largest cluster. This suggested the denition of the reduced
variance γ2 [1,2] for a fragmentation event should be
γ2(~n) =
M2M0
M21
= (m− 1)
P
k k
2nk − k2max
(A− kmax)2
; (6)
Its expectation value can be computed by breaking events into classes specied by kmax, and
summing the expectation values over those classes with the appropriate weight, i.e.
hγ2i = (m− 1)
X
kmax
Pr(kmax)
P
k k
2hnki(kmax)− k2max
(A− kmax)2
; (7)
where Pr(kmax) is the probability of kmax being the largest cluster size and hnki(kmax) is the
expectation value of nk when kmax is xed.
To compute expectation values in which the largest cluster size is xed we need to
compute the partition function for such ensembles. Clearly this partition function is given
by all the terms in the microcanonical partition function which have xkmax as the highest xk
in the term. Consider
Z
(m)
A (kmax)  Z
(m)
A (x1; : : : ; xkmax; 0; : : : ; 0)
−Z(m)A (x1; : : : ; xkmax−1; 0; : : : ; 0) : (8)
We see that Z(m)A (kmax) is the partition function for ensembles with xed maximum cluster
size kmax, since the rst term collects all terms with xkmax or lower, and the second term
eliminates those terms which don’t have an xkmax. From this result we can determine Pr(kmax)
and hnki(kmax), which are given by
Pr(kmax) =
Z
(m)
A (kmax)
Z
(m)
A (x1; : : : ; xA)
(9)
hnki(kmax) =
8>>>><>>>>:
0 k > kmax
Z
(m−1)
A−k (x1;:::;xkmax ;0;:::;0)
Z
(m)
A (kmax)
k = kmax
Z
(m−1)
A−k (kmax)
Z
(m)
A (kmax)
k < kmax
(10)
3
This method is quite general and can be applied to other models. For example, equiparti-
tioning models, which have weights given by
W (~n) =
Y
k1
xnkk (11)
can also be analyzed by this method with some minor modications. For example, xk = 1
is the model used by Sobotka and Moretto [8].
With these identities there is sucient information to compute hγ2i and other reduced
moments for any xk. We use xk = x=k for a variety of reasons discussed elsewhere [9].
Campi and Krivine [2] following Mekjian [4] considered this model with  = 1:0 and showed
that its reduced variance and other related expectation values had a distinctly dierent
behavior than the percolation model. Plotting the expected reduced variance vs. (m−1)=A,
hγ2i(m) has a single peak. The location, height and width of this peak for the two models
(and other models they considered) are completely dierent, suggesting the usefulness of this
plot in distinguishing fragmentation models. The choice  = 0 was considered by Gross,
et. al. [10{12], and a dierent model was analyzed by Pan and Das Gupta [13].
Since that time our interest has turned to the choice  = 2:5 because of similarities with
percolation theory and Bose condensation. Namely, the sudden appearance of an innite
cluster in the innite A limit and the presence of condensation phenomena. As such, we
have recomputed the Campi plots for this model and have discovered that they duplicate
the percolation model results in many respects. Figure 1(a) shows the results. The height
and location of the peak of the reduced variance hγ2i are the same in both models. The only
signicant dierence is the width of the peak which is larger in the Gibbs model than in the
percolation model. Plots of hkmaxi(m) vs. (m− 1)=A given in Fig. 1(b) are also very similar
for both models, and the scaling behavior of the position, width and height with changing
A also agree.
Another plot suggested by Campi [14] is to divide the event space by the maximum
cluster size of the event kmax and the reduced second moment M2 and plot the probability
of the canonical model being at any particular point on the graph. This can be also be done
4
exactly for Gibbs models in a way completely analogous to the way given above. Dene
the partition function ZA(m2; ~x) as the sum of the Gibbs weight Eq. (1) over all partition
vectors ~n which satisfy
P
k knk = A,
P
k k
2nk = m2. This can be computed by the following
recursion,
ZA(m2; ~x) =
1
A
X
k
kxkZA−k(m2 − k
2; ~x) (12)
with Z0(m2; ~x) = m2;0. If we dene Z as in Eq. (8), we nd again the partition function
conditioned on kmax being xed, which is proportional to the probability of having an event
with both m2 and kmax. Figure 2 plots this probability prole as a contour plot, which
reveals the events are centered on a particular region in this phase space. The slopes of the
edges of this region are related to the critical exponents according to Campi [14].
Clearly there are dierences between percolation theory and a Gibbs model, but the dif-
ferences are not as large as originally suggested by early computations. Indeed the reduced
variance might not reliably distingush percolation from a simple Gibbs model. A dierent
method is needed to distingush these models. However, the idea of excluding the largest
cluster from the ensemble averages is a standard procedure in percolation theory [15], and
this new technique for doing that analytically in the Gibbs models shows a particular advan-
tage of these models over percolation models, which we hope will encourage further interest
in them.
This work supported in part by the National Science Foundation Grant No. NSFPHY
92-12016.
5
REFERENCES
[1] X. Campi, Phys. Lett. B208, 351 (1988); Nucl. Phys. A495 259c (1989).
[2] X. Campi and H. Krivine, Nucl. Phys. A545 161c (1992); Z. Phys. A 344, 81 (1992).
[3] K. C. Chase and A. Z. Mekjian, Rutgers University Preprint RU9542.
[4] A. Z. Mekjian, Phys. Rev. Lett. 64, 2125 (1990); Phys. Rev. C 41, 2103 (1990).
[5] A. Z. Mekjian and S. J. Lee, Phys. Rev. A 44, 6294 (1991).
[6] S. J. Lee and A. Z. Mekjian, Phys. Lett. A149, 7 (1990); Phys. Rev. C 45, 365 (1992);
45, 1284 (1992).
[7] A. J. Cole, P. J. Lindstrom, H. J. Crawford and B. G. Harvey, Phys. Rev. C 39, 891
(1989).
[8] L. G. Sobotka and L. G. Moretto, Phys. Rev. C 31, 668 (1985).
[9] K. C. Chase and A. Z. Mekjian, Phys. Rev. C 49, 2164 (1994).
[10] D. H. E. Gross, A. Ecker and A. R. DeAngelis, Ann. Physik (8) 1, 340 (1992);
[11] , D. H. E. Gross, A. Ecker and A. R. DeAngelis, Nucl. Phys. A545, 187c (1992).
[12] H. R. Jaqaman, A. R. DeAngelis, A. Ecker and D. H. E. Gross, Nucl. Phys. A541, 492
(1992).
[13] J. Pan and S. Das Gupta, Phys. Lett. 344B, 29 (1995); Phys. Rev. C 51, 1384 (1995).
[14] X. Campi, J. Phys. A 19, L917 (1986).
[15] D. Stauer and A. Aharony, Introduction to percolation theory, (Taylor and Francis,
London, 1992).
6
FIGURES
FIG. 1. Expected reduced multiplicity hγ2i(m) (a) and largest cluster size kmax=A (b) vs.
(m− 1)=A for  = 1:0; 2:5 and the percolation model at A = 125.
FIG. 2. Campi probability contour plot for Z = 79,  = 2:5 at the critical point x = xc.
The axes are logarithmic, with the largest cluster size on the y-axis, and the ratio of the reduced
moments M2=M1 on the x-axis. The central rings are higher in probability than the outer rings.
7


Exact methods for Campi plots

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