1. Nuclear Multifragmentation and Zipf’s Law Wolfgang Bauer Michigan State University Work in collaboration with: Scott Pratt (MSU), Marko Kleine Berkenbusch (Chicago) Brandon Alleman (Hope College) Work in collaboration with: Scott Pratt (MSU), Marko Kleine Berkenbusch (Chicago) Brandon Alleman (Hope College)

2. 2  Two (at least) thermodynamic phase transitions in nuclear matter: –“Liquid Gas” –Hadron gas  QGP / chiral restoration  Problems / Opportunities: –Finite size effects –Is there equilibrium? –Measurement of state variables ( , T, S, p, …) –Migration of nuclear system through phase diagram  Structural Phase Transitions (deformation, spin, pairing, …) –have similar problems & questions –lack macroscopic equivalent Nuclear Matter Phase Diagram Source: NUCLEAR SCIENCE, A Teacher’s Guide to the Nuclear Science Wall Chart, Figure 9-2 (expansion, collective flow)

3. 3HistoryHistory

4. 4 Influence of Sequential Decays Critical fluctuations Blurring due to sequential decays

5. 5 Width of Isotope Distribution, Sequential Decays  Predictions for width of isotope distribution are sensitive to isospin term in nuclear EoS  Complication: Sequential decay almost totally dominates experimentally observable fragment yields Pratt, WB, Morling, Underhill, PRC 63, 034608 (2001).

6. 6 Isospin: RIA Reaction Physics rp-process r-process  Exploration of the drip lines below charge Z~40 via projectile fragmentation reactions  Determination of the isospin degree of freedom in the nuclear equation of state  Astrophysical relevance  Review: B.A. Li, C.M. Ko, WB, Int. J. Mod. Phys. E 7(2), 147 (1998)

7. 7 Cross-Disciplinary Comparison  Left: Nuclear Fragmentation  Right: Buckyball Fragmentation  Histograms: Percolation Models  Similarities: –U - shape (b-integration) –Power-law for imf’s (1.3 vs. 2.6) –Binding energy effects provide fine structure Data: Bujak et al., PRC 32, 620 (1985) LeBrun et al., PRL 72, 3965 (1994) Calc.: W.B., PRC 38, 1297 (1988) Cheng et al., PRA 54, 3182 (1996)

8. 8 Buckyball Fragmentation 625 MeV Xe 35+ Cheng et al., PRA 54, 3182 (1996) Binding energy of C 60 : 420 eV

9. 9 ISiS BNL Experiment  10.8 GeV p or  + Au  Indiana Silicon Strip Array  Experiment performed at AGS accelerator of Brookhaven National Laboratory  Vic Viola et al.

10. 10 ISIS Data Analysis  Reaction: p,  +Au @AGS  Very good statistics (~10 6 complete events)  Philosophy: Don’t deal with energy deposition models, but take this information from experiment!  Detector acceptance effects crucial –filtered calculations, instead of corrected data  Parameter-free calculations Marko Kleine Berkenbusch Collaboration w. Viola group Residue Sizes Residue Excitation Energies

11. 11 Comparison: Data & Theory  Very good agreement between theory and data –Filter very important –Sequential decay corrections huge 2 nd Moments Charge Yield Spectrum

12. 12 Scaling Analysis  Idea (Elliott et al.): If data follow scaling function with f(0) = 1 (think “exponential”), then we can use scaling plot to see if data cross the point [0,1] -> critical events  Idea works for theory  Note: –Critical events present, p>p c –Critical value of p c was corrected for finite size of system M. Kleine Berkenbusch et al., PRL 88, 0022701 (2002)

13. 13 Effects of Detector Acceptance Filter Unfiltered Filtered

14. 14 Scaling of ISIS Data  Most important: critical region and explosive events probed in experiment  Possibility to narrow window of critical parameters –  : vertical dispersion –  : horizontal dispersion –T c : horizontal shift   2 Analysis to find critical exponents and temperature

15. 15 Essential for Scaling of Data: Correction for Sequential Decays

16. 16 The Competition … Work based on Fisher liquid drop model Same conclusion: Critical point is reached   J.B. Elliott et al., PRL 88, 042701 (2002)

17. 17 IMF Probability Distributions Moby Dick Moby Dick: IMF: word with ≥ 10 characters Nuclear Physics Nuclear Physics: IMF: fragment with 20 ≥ Z ≥ 3 System Size is the determining factor in the P(n) distributions System Size is the determining factor in the P(n) distributions

18. 18 Zipf’s Law  Back to Linguistics  Count number of words in a book (in English) and order the words by their frequency of appearance  Find that the most frequent word appears twice as often as next most popular word, three times as often as 3rd most popular, and so on.  Astonishing observation! G. K. Zipf, Human Behavior and the Principle of Least Effort (Addisson-Wesley, Cambridge, MA, 1949)

19. 19 English Word Frequency British language compound, >4000 texts

20. 20 DJIA-1st Digit  1st digit of DJIA is not uniformly distributed from 1 through 9!  Consequence of exponential rise (~6.9% annual average  Also psychological effects visible 1st digit # of occurrences

21. 21 Zipf’s Law in Percolation  Sort clusters according to size at critical point  Largest cluster is n times bigger than n th largest cluster M. Watanabe, PRE 53 (‘96)

22. 22 Zipf’s Law in Fragmentation  Calculation with Lattice Gas Model  Fit largest fragments to A n = c n -  At critical T: crosses 1  New way to detect criticality (?) Y.G. Ma, PRL 83 (‘99)

23. 23 Zipf’s Law: First Attempt rank, r /

24. 24 Zipf’s Law: Probabilities (1)  Probability that cluster of size A is the largest one = probability that at least one cluster of size A is present times probability that there are 0 clusters of size > A  N(A) = average yield of size A : N(A) = aA -   N(>A) = average yield of size A : ( V = event size)  Normalization constant a from condition:

25. 25 Zipf’s Law: Probabilities (2)  Use Poisson statistics for individual probabilities:  Put it all together:  Average size of biggest cluster (Exact expression!)

26. 26 Zipf’s Law: Probabilities (3)  Probability for given A to be 2nd biggest cluster:  Average size of 2nd biggest cluster:  And so on … (recursion relations!)

27. 27 Zipf’s Law:  -dependence 2.00 2.18 2.33 2.50 2.70 3.00 5.00 Expectation if Zipf’s Law was exact Verdict: Zipf’s Law does not work for multifragmentation, even at the critical point! (but it’s close) W.B., Pratt (2005) Resulting distributions: Zipf Mandelbrot

28. 28 Human Genome  1-d partitioning problem of gene length distribution on DNA  Human DNA consist of 3G base pairs on 46 chromosomes, grouped into codons of length 3 base pairs –Introns form genes –Interspersed by exons; “junk DNA”

29. 29 Computer Hard Drive  Genome like a computer hard drive.  Memory is like chromosomes.  A files analogous to genes.  To delete a file, or gene, delete beginning.

30. 30 Recursive Method Number of ways a length A string can split into m pieces with no piece larger than i. Probability the l th longest piece has length i

31. 31SimulationSimulation  Random numbers are generated to determine where cuts are made.  Here length is 300 and number of pieces is 30.

32. 32 Assumption: Relaxed Total Size  The number of pieces falls exponentially.  From this assumption the average piece size is obtained.  Also, the average size of the longest piece.

33. 33 Power Law – Percolation Theory  Assumes pieces fall according to a power law.  Average length of piece N is:

34. 34 Data from Chromosomes 1, 2, 7, 10, 17, and Y. Plotted against Exponential and Power Law models in Green. Gene Data Alleman, Pratt, WB 2005

35. 35SummarySummary  Scaling analysis (properly corrected for decays and feeding) is useful to extract critical point parameters.  “Zipf’s Law” does not work as advertised, but analysis along these lines can dig up useful information on critical exponent , finite size scaling, self-organized criticality  Gene length distribution as a 1d partitioning problem is interesting and not solved Research funded by US National Science Foundation Grant PHY-0245009