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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)

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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

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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, 034608 (2001).

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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)

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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)

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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

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

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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.

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22 nd WWND - Wolfgang Bauer 9 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

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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

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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, 022701 (2002)

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

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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

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22 nd WWND - Wolfgang Bauer 14 Essential: Sequential Decays

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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, 042701 (2002)

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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/0512101

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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)

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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)

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

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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

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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)

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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)

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

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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:

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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!)

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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)

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22 nd WWND - Wolfgang Bauer 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) Resulting distributions: Zipf Mandelbrot

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22 nd WWND - Wolfgang Bauer 28Zipf-MandelbrotZipf-Mandelbrot Limiting distributions for cluster size vs. rank Exponent WB, Alleman, Pratt nucl-th/0511007

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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-0245009

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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”

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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).

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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

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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.

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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.

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

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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

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

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