# reduciblity, arrhenius plots and the uroboros dragon, a reply to the preprint correlations[精选推荐pdf]

arXiv:nucl-ex/9709001v1 8 Sep 1997Reducibility, Arrhenius plots and the Uroboros Dragon1, a reply to the preprint “Correlations in Nuclear Arrhenius-Type Plots” by M.B. Tsang and P. Danielewicz.L.G. Moretto, L. Beaulieu, L. Phair, and G.J. WozniakNuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USAJune 5, 1997In previous publications [1–3], which have generated more attention than an- ticipated [4–9], we showed that the n-fragment multifragmentation probabili- ties at any given transverse energy Pn(Et), could be reduced by means of the binomial expressionPm n=m! n!(m − n)!pn(1 − p)m−n(1)to a single one fragment probability p(Et), where Et=PEisin2θi, the sum of the kinetic energies of all charged particles in an event weighted by the sine squared of their polar angle. This “reducibility” with Etwas taken as a strong indication that fragments were emitted nearly independently of one another with the same probability p(Et). We further noticed that, under the assump- tion that Etis proportional to the excitation energy E (and thus temperature T ∝√E t), the Arrhenius plots of log(1/p) versus 1/√E t, produced straight lines. We called this “thermal scaling” because it suggests the Boltzmann form for the elementary probability p ∝ exp(−B/T) orlog1 p=B T=B′√E t,(2)1The Uroboros dragon, or the snake eating its own tail, is an archetype that can be traced from the Egyptians through the Greeks and the Gnostic religions into the Alchemical writings of the Middle Ages and Renaissance all the way to modern times. A symbol of self reference (think of Escher’s hand drawing the hand that draws it.), it has been used to explain the universe. It has been applied by the authors of the preprint with a somewhat more modest intent.1with B representing the barrier for fragment emission.The present preprint attacks both of our claims. The first on the basis that ourbinomial fits could be matched in quality by a constrained Poissonian function. The second on the basis that the Arrhenius plots are allegedly shown to be the result of a strong autocorrelation.Since the second criticism is potentially more devastating than the first, we begin with it.As a preemptive strike, the authors raise concerns about the relevance of using√E tas a measure of the temperature. They cite as evidence to the contrary recent work exploring the impact parameter dependence of temperatures ex- tracted from excited state populations [10]. In that work a value of T=4.6 MeV is determined, largely independent of impact parameter for moderately central to very central collisions of Au+Au at 35 MeV/nucleon. We are not unduly concerned by this result since temperatures of 4-5 MeV are almost al- ways extracted from such studies, independent of bombarding energy [11,12], entrance channel mass asymmetry [11,12], as well as impact parameter [10,13]. Since such extracted temperatures seem to be insensitive to most features ofheavy-ion reactions, we would not be surprised to find that they are also in- sensitive to Et.Furthermore, temperatures extracted from the double isotope ratio method [14] do indeed show a broad range of temperatures for heavy ion reactions at intermediate energies [15].Coming to the heart of their criticism, the authors argue that the observed linearity of the Arrhenius plots is due to autocorrelation. However, a fa-tal flaw mars the derivation of the proposed explanation for linear Arrhe-nius plots [9] and undermines the conclusions of the preprint. This flaw is circular logic in the algebra, leading to a trivial identity that always holds, regardless of whether or not there exists an autocorrelation.Assuming with the authors that the binomial number of “throws” m is con- stant (for many experiments this is a good assumption) we consider the Ar- rhenius plot constructed with the average intermediate mass fragment (IMF) multiplicity hni = mp and Et. So we plot log1/hni versus 1/√Etand in all experimental cases we obtain a straight line. As we have observed [3], the au- thors also observe that Etis constructed both with the light charged particles and the IMFs. In the authors’ notationEt= (NC− hni)ELPt+ hniEIMFt,(3)where NCis the number of charged particles in an event and ELPtand EIMFtare2Fig. 1. Top panel: The transverse energy Etas a function of the average IMF multiplicity hni for the reaction Ar+Au, E/A=110 MeV. The line is a straight-linefit, Eq. (5). Bottom panel: hni as a function of Et. The fit is the exponential fit of Eq. (9) used in refs. [1–3].the average transverse energies of the light particles and IMFs, respectively. The use of hni to construct this independent variable Etcreates the danger of autocorrelation in the study of the dependence of hni on Et(as in the Arrhenius plot). This we also have discussed extensively [3].Now the authors [9] observe that hni and NCare globally very strongly cor-related. They fit this correlation with the linear formNC= a′+ b′hni(4)and finally they express Etas a function of the single variable hniEt= b?ab+ hni? (5)where a = a′ELPtand b = (b′− 1)ELPt+ EIMFt.A quicker but nearly equivalent procedure would have been to fit the equally3strongly correlated quantities Et, hni directly with the above form (see the top panel of Fig. 1) since NCand Etare very tightly correlated as well. Thevalues of a and b from a direct fit using Eq. (5) are 100 MeV and 220 MeV respectively. These values are nearly equal to those extracted by the authors (a=106 MeV, b=213 MeV) when starting with Eq. (4).A moment’s reflection shows that Eq. (5) contains the original experimental dependence of hni on Et(bottom panel of Fig. 1), and it does not differ in anysignificant way, other than in the linear fit, from the data used to construct the Arrhenius plots. It is just its inverse! Here is the start of the vicious circle.In fact, to fit the data they could have used with better results the Boltzmann- like functional form which gives linear Arrhenius plots as we have done in thebottom panel of Fig. 1. One can argue on which of the two is a better fit, but not on the fact that, for the two functions, one is approximately the inverse of the other.Armed with Eq. (5), the authors take an experimental Arrhenius plot which is linear, and they substitute Etwith the value calculated from Eq. (5). Since the linear form is a good approximation to Et, nothing much happens to the Arrhenius plot which of course remains linear. It simply means that the form of Eq. (5) is a good approximation to the data!Yet the authors make the surprising claim that the linearity in such a plot is the result of an autocorrelation because now one has hni both in the ordinate (as log1/hni) and in the abscissa (as 1/q b(a/b + hni)).In what follows we show that this technique for demonstrating autocorrelationsis so general and fundamentally flawed, that it will give the same result for all functions, even for those in which autocorrelation has been ruled out by construction. We call this method the Uroboros method, for reasons that will become clear below.As we noticed above, Eq. (5), either fitted directly to the data or indirectly through Eq. (4), gives an empirical approximation to the total dependence of Etupon hniEt= g(hni).(6)On the other hand, we have the total dependence of hni upon Etdirectly from the datahni = f (Et).(7)Clearly g (hni) is an approximation to the inverse of f (Et), g ≈ f−1. Now we4construct the Arrhenius plotF (hni) = F (f′(Et))(8)withhni = f′(Et) ∝ exp−K√E t! .(9)If the Arrhenius plot is linear it means the f′(Et) ≡ f. Hence,F (hni) = F (f (Et)) = F (f (g (hni))) = F? f? f−1(hni)?? = F (hni)(10)which is an identity, in the limit of g = f−1.The vicious circle of circular logic can be summarized as follows: given any function y = y(x), substitute in it x with its value x(y): y = y(x(y)). This gives always the identity y = y.This result proves, if there was any doubt, that any function is completely correlated with its own inverse, and it is a great source of straight lines, irre- spective of whether there is an autocorrelation or not.The flaw, in our specific case, lies with the fact that Eq. (6) gives the total empirical dependence of Eton hni irrespective of whether there is autocorre- lation or not and consequently it is the approximate inverse of Eq. (7) and of Eq. (9). Hence the Uroboros, the snake eating its tail.It is left for the diligent reader to test the universality of this method by applying the Uroboros to discover autocorrelations in the most venerable de- pendences in physics.It is clear from the previous considerations that the authors’ vicious circle does not prove anything, least of all autocorrelation.In order to appreciate the true role of autocorrelations, let as beforehni = f (Et)(11)andEt= Et(E,hni)(12)5where E is the excitation energy of the system. The dependence of hni on Etcan be best explored differentially∆hni =dhni dEt∆Et.(13)Now, the change in transverse energy can be written∆Et=∂Et ∂E?????n∆E +∂Et ∂ hni?????E∆hni(14)where the first term represents the change in Etdue to the change in exci- tation energy and the second term represents the trivial change in Etdue to autocorrelation. The partial derivative∂Et ∂hni???Eat constant E (not the total derivative) tells us about the autocorrelation.Substituting Eq. (14) in Eq. (13) we have∆hni1 −dhni dEt∂Et ∂ hni?????E! =dhni dEt∂Et ∂E?????hni∆E(15)where the complete left hand side is the increase in hni induced by ∆Ecorrected for the autocorrelation. Failure to appreciate the difference between ∂Et ∂hni???EanddEt dhnileads to disastrous circular reasoning.In fact, if we write∆Et=dEt dhni∆hni(16)and substitute in Eq. (13) we obtain as the authors did∆hni =dhni dEtdEt dhni∆hni = ∆hni,(17)or the Uroboros (again).Of course, the authors might argue that they know the first term of Eq. (14) to be close to zero because they believe the temperature changes at most by 1 MeV. Then indeed only the second term, or the autocorrelation would survive. This position is clearly untenable. In fact, were the authors correct, the derivative in Eq. (16) (which would now be the same as that in the second term of Eq. (14)) should have the value of approximately 20-40 MeV per6IMF, which is the measured mean transverse energy per IMF [16,17] for this particular system. Instead, it has the enormous value of 220 MeV per IMF! (See Fig. 1.) This alone should make the authors wonder and worry about constant temperature. In any case, as we pointed out above, their method will always give the same result, independent of the presence or abscence of an autocorrelation.For those who like to touch with their hands, here is a concrete example from an event generator code [3] which, by construction, generates complex fragment events according top = exp(−B/T)(18)withT =s E a,(19)where T is the temperature and E is the excitation energy of the system. In other words, the linear Arrhenius plot is contained in the events by con- struction. By using the independent variable E instead of Et, we are assured of avoiding the autocorrelation problem altogether.We construct the Arrhenius plot 1/hni versus 1/√E which turns out to be linear as expected (circles in the bottom panel of Fig. 2). Then we notice that E and hni can be plotted one against the other (circles in the top panel of Fig. 2) and we approximateE = a + bhni(20)by fitting the dependence with a straight line. Then we substitute E with the value given by Eq. (20) and note, as the authors, that the Arrhenius plot con- structed from this substitution (solid line in bottom panel of Fig. 2) remains linear over the range in Etwhere Eq. (20) well represents the simulation. Hopefully by now the game has become rather tedious, and nobody would argue that the result is just due to an autocorrelation.To verify the extent of autocorrelations in this model, we follow the same procedure as above but use Et, like in the data. Now, we expect some auto- correlation. The question is: how much? We obtain a linear Arrhenius plot, whose slope should coincide with the slope of the previous plot after scaling Et by 3/2. (The model assumes Et= 2/3E.) Indeed it does so to within betterthan 10 percent [3] (open squares of Fig. 2) This difference in slope is theextent of the autocorrelation problem. By redefining hni through an increase7Fig. 2. Top panel: The excitation energy E as a function of the average IMF mul- tiplicity from a binomial simulation [3] of the decay of a Xe nucleus (circles). The squares show the same correlation but using for E a scaled value of Et, E = 3/2Et. Bottom panel: The Arrhenius plot 1/hni versus 1/√E (circles) and 1/hni versus 1/p3/2Et(squares). The solid line represents the (erroneous) “self-correlation” ex- planation proposed by the authors (see text).of the element threshold (Zth) in the definition of IMF (IMF: Zth≤ Z ≤ 20), we can substantially reduce even this small autocorrelation. Yet the authors would conclude that in both cases, using E or Et, the linear Arrhenius plots are completely due to autocorrelation.An even better procedure, which we can apply directly to experimental data,is the following. We can redefine the IMFs to be the fragments of a single element, say Z=10. Now the contribution of the Z=10 fragments to Etis so small that it can be neglected and consequently autocorrelation is negligibleas well. Because of the restriction of the definition of an IMF (one single Z), the probability of emitting an IMF becomes very small, and in fact, we observe that the distribution of multiplicities becomes Poissonian. The Arrhenius plots of loghni versus 1/√Etremain linear and the linearity can be tested for each Z as in Fig. 3 without the worry of autocorrelation. The remarkable set of straight lines shown in Fig. 3, together with many other sets obtained for all the experiments we have analyzed, shows that thermal scaling is a real8Fig. 3. The average yield per event of the different indicated elements (symbols) as a function of 1/√Et. The lines are linear fits to the data.physical feature quite safe from the dreaded fangs of the Uroboros snake. Yet the procedure employed by the authors will always yield an autocorrelation, regardless of whether one is present or not.We now return to the first criticism, on which there is little to say.The authors claim that there is no a priori reason for choosing a binomialinstead of a constrained Poissonian, because they both fit the experimental multiplicity distributions equally well. We have no great argument with this. We actually like the Poissonian case and we would have chosen it if the data had not indicated a very high probability and a constraint (in size or tempo-ral) to boot [3]. A binomial satisfied both requirements. By requiring a fixed number of tosses m, it incorporates both a scale and a constraint. There are other ways to achieve the same goal. For instance, we could chop the Pois- sonian above a prescribed n or do something else, like the authors