New Observations on Fragment Multiplicities Wolfgang Bauer Michigan State University Work in collaboration with: Scott Pratt (MSU), Marko Kleine Berkenbusch (Chicago) Brandon Alleman (Hope College)

22 nd WWND - Wolfgang Bauer 2  Two (at least) thermodynamic phase transitions in nuclear matter: –“Liquid Gas” –Hadron gas  QGP / chiral restoration  Goal: Determine Order &Universality Class  Problems / Opportunities: –Finite size effects –Is there equilibrium? –Measurement of state variables ( , T, S, p, …) –Migration of nuclear system through phase diagram (expansion, collective flow)  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

22 nd WWND - Wolfgang Bauer 3 Width of Isotope Distribution, Sequential Decays  Predictions for width of isotope distribution are quite sensitive to isospin term in nuclear EoS  Complication: Sequential decay almost totally dominates experimentally observable fragment yields Pratt, Bauer, Morling, Underhill, PRC 63, (2001).

22 nd WWND - Wolfgang Bauer 4 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, W. Bauer, Int. J. Mod. Phys. E 7(2), 147 (1998)

22 nd WWND - Wolfgang Bauer 5 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)

22 nd WWND - Wolfgang Bauer 6 Buckyball Fragmentation 625 MeV Xe 35+ Cheng et al., PRA 54, 3182 (1996) Binding energy of C 60 : 420 eV

22 nd WWND - Wolfgang Bauer 7  Symmetric A+A collisions  Bubble and toroid formation  Imaginary sound velocity  Could also be a problem/opportunity for FAIR!CompressionCompression

22 nd WWND - Wolfgang Bauer 8 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.

22 nd WWND - Wolfgang Bauer 9 ISIS Data Analysis  Reaction: p,   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

22 nd WWND - Wolfgang Bauer 10 Comparison: Data & Theory  Very good agreement between theory and data –Filter very important –Sequential decay corrections huge 2 nd Moments Charge Yield Spectrum

22 nd WWND - Wolfgang Bauer 11 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, (2002)

22 nd WWND - Wolfgang Bauer 12 Detector Acceptance Filter Unfiltered Filtered

22 nd WWND - Wolfgang Bauer 13 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

22 nd WWND - Wolfgang Bauer 14 Essential: Sequential Decays

22 nd WWND - Wolfgang Bauer 15 The Competition … Work based on Fisher liquid drop model Same conclusion: Critical point is reached   J.B. Elliott et al., PRL 88, (2002)

22 nd WWND - Wolfgang Bauer 16 Freeze-Out Density  Percolation model only depends on breaking probability, which can be mapped into a temperature.  Q: How to map a 2-dimensional phase diagram?  A: Density related to fragment energy spectra WB, Alleman, Pratt nucl-th/

22 nd WWND - Wolfgang Bauer 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 System Size is the determining factor in the P(n) distributions Bauer, Pratt, PRC 59, 2695 (1999)

22 nd WWND - Wolfgang Bauer 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)

22 nd WWND - Wolfgang Bauer 19 English Word Frequency British language compound, 4124 texts, >100 million words

22 nd WWND - Wolfgang Bauer 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

22 nd WWND - Wolfgang Bauer 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, 4187 (1996)

22 nd WWND - Wolfgang Bauer 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, 3617 (1999)

22 nd WWND - Wolfgang Bauer 23 Zipf’s Law: First Attempt rank, r / Change System Size

22 nd WWND - Wolfgang Bauer 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:

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

22 nd WWND - Wolfgang Bauer 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! Bauer, Pratt, Alleman, Heavy Ion Physics, in print (2006)

22 nd WWND - Wolfgang Bauer 27 Zipf’s Law:  -dependence 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) Resulting distributions: Zipf Mandelbrot

22 nd WWND - Wolfgang Bauer 28Zipf-MandelbrotZipf-Mandelbrot  Limiting distributions for cluster size vs. rank  Exponent WB, Alleman, Pratt nucl-th/

22 nd WWND - Wolfgang Bauer 29SummarySummary  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 Research funded by US National Science Foundation Grant PHY

22 nd WWND - Wolfgang Bauer 30 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”

22 nd WWND - Wolfgang Bauer 31 Computer Hard Drive  Genome like a computer hard drive.  Memory is like chromosomes.  Files analogous to genes.  To delete a file, or gene, delete entry point (= start codon).

22 nd WWND - Wolfgang Bauer 32 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

22 nd WWND - Wolfgang Bauer 33SimulationSimulation  Random numbers are generated to determine where cuts are made.  Here length is 300 and number of pieces is 30.

22 nd WWND - Wolfgang Bauer 34 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.

22 nd WWND - Wolfgang Bauer 35 Power Law – Percolation Theory  Assumes pieces fall according to a power law.  Average length of piece N is:

22 nd WWND - Wolfgang Bauer 36 Data from Human Chromosomes 1, 2, 7, 10, 17, and Y. Plotted against Exponential and Power Law models Gene Data Alleman, Pratt, Bauer 2005

22 nd WWND - Wolfgang Bauer 37 Influence of Sequential Decays Critical fluctuations Blurring due to sequential decays