1. Crossing the Coexistence Line of the Ising Model at Fixed Magnetization L. Phair, J. B. Elliott, L. G. Moretto

2. Fisher Droplet Model (FDM) FDM developed to describe formation of drops in macroscopic fluids FDM allows to approximate a real gas by an ideal gas of monomers, dimers, trimers,... ”A-mers” (clusters) The FDM provides a general formula for the concentration of clusters n A (T) of size A in a vapor at temperature T Cluster concentration n A (T ) + ideal gas law PV = T vapor densityvapor pressure 

3. Motivation: nuclear phase diagram for a droplet? What happens when you build a phase diagram with “vapor” in coexistence with a (small) droplet? T c ? critical exponents?

4. Magnetic transition Isomorphous with liquid- vapor transition Hamiltonian for s-sites and B-external field Ising model (or lattice gas)

5. Finite size effects in Ising … seek ye first the droplet and its righteousness, and all … things shall be added unto you… ? A0A0 finite lattice or finite drop? Grand-canonical Canonical (Lattice Gas)

6. Lowering of the isobaric transition temperature with decreasing droplet size Clapeyron Equation for a finite drop Clapeyron equation Integrated Correct for surface

7. Example of vapor with drop The density has the same “correction” or expectation as the pressure Challenge: Can we describe p and  in terms of their bulk behavior?

8. Clue from the multiplicity distributions Empirical observation: Ising multiplicity distributions are Poisson – – Meaning: Each fragment behaves grand canonically – independent of each other. – – As if each fragments’ component were an independent ideal gas in equilibrium with each other and with the drop (which must produce them). – – This is Fisher’s model but for a finite drop rather than the infinite bulk liquid

9. Clue from Clapeyron Rayleigh corrected the molar enthalpy using a surface correction for the droplet Extend this idea, you really want the “separation energy” Leads naturally to a liquid drop expression A0A0 A0-AA0-A A EiEi EfEf

10. Finite size effects: Complement Infinite liquidFinite drop Generalization: instead of E S (A 0, A) use E LD (A 0, A) which includes Coulomb, symmetry, etc.(tomorrow’s talk by L.G. Moretto) Specifically, for the Fisher expression: Fit the yields and infer T c (NOTE: this is the finite size correction)

11. Fisher fits with complement 2d lattice of side L=40, fixed occupation  =0.05, ground state drop A 0 =80 T c = 2.26 +- 0.02 to be compared with the theoretical value of 2.269 Can we declare victory?

12. Going from the drop to the bulk We can successfully infer the bulk vapor density based on our knowledge of the drop.

13. From Complement to Clapeyron In the limit of large A 0 >>A Take the leading term (A=1)

14. Summary Understand the finite size effects in the Ising model at fixed magnetization in terms of a droplet (rather than the lattice size) –Natural and physical explanation in terms of a liquid drop model (surface effects) –Natural nuclear physics viewpoint, but novel for the Ising community Obvious application to fragmentation data (use the liquid drop model to account for the full separation energy “cost” in Fisher)

15. Complement for Coulomb NO  Data lead to T c for bulk nuclear matter

16. (Negative) Heat Capacities in Finite Systems Inspiration from Ising –To avoid pitfalls, look out for the ground state

17. Coulomb’s Quandary Coulomb and the drop 1)Drop self energy 2)Drop-vapor interaction energy 3)Vapor self energy Solutions: 1)Easy 2)Take the vapor at infinity!! 3)Diverges for an infinite amount of vapor!!

18. Generalization to nuclei: heat capacity via binding energy No negative heat capacities above A≈60 At constant pressure p,

19. The problem of the drop-vapor interaction energy If each cluster is bound to the droplet (Q<0), may be OK. If at least one cluster seriously unbound (|Q|>>T), then trouble. –Entropy problem. –For a dilute phase at infinity, this spells disaster! At infinity,  E is very negative  S is very positive  F can never become 0.

20. Vapor self energy If Drop-vapor interaction energy is solved, then just take a small sample of vapor so that E Coul(self) /A << T However: with Coulomb, it is already difficult to define phases, not to mention phase transitions! Worse yet for finite systems Use a box? Results will depend on size (and shape!) of box God-given box is the only way out!

21. We need a “box” Artificial box is a bad idea Natural box is the perfect idea –Saddle points, corrected for Coulomb (easy!), give the “perfect” system. Only surface binds the fragments. Transition state theory saddle points are in equilibrium with the “compound” system. For this system we can study the coexistence –Fisher comes naturally

22. A box for each cluster Saddle points: Transition state theory guarantees in equilibrium with S s s Coulomb and all Isolate Coulomb from  F and divide away the Boltzmann factor s

23. Solution: remove Coulomb This is the normal situation for a short range Van der Waals interaction Conclusion: from emission rates (with Coulomb) we can obtain equilibrium concentrations (and phase diagrams without Coulomb – just like in the nuclear matter problem)

24. d=2 Ising fixed magnetization (density) calculations M = 0.9,  = 0.05 M = 0.6,  = 0.20, inside coexistence region outside coexistence region inside coexistence region, T > T c

25. Inside coexistence region: – –yields scale via Fisher & complement – –complement is liquid drop A max (T): Surface tension  =2 Surface energy coefficient: – –small clusters square- like: S c 0 =4  – –large clusters circular: L c 0 =2  Cluster yields from all L, M,  values collapse onto coexistence line Fisher scaling points to T c d=2 Ising fixed magnetization M (d=2 lattice gas fixed average density  ) T = 0 T>0 Liquid drop VacuumVapor L L A0A0 A max

26. Inside coexistence region: – –yields scale via Fisher & complement – –complement is liquid drop A max (T): d=3 Ising fixed magnetization M (d=3 lattice gas fixed average density  ) T = 0 T>0 Liquid drop VacuumVapor L L A0A0 A max Cluster yields collapse onto coexistence line Fisher scaling points to T c c 0 (A  +(A max (T)-A)  -A max (T)  )  /T Fit: 1≤A ≤ 10, A max (T=0)=100 n A (T)/q 0 (A(A max (T)-A)  A max (T)) - 

27. Complement for excited nuclei Complement in energy –bulk, surface, Coulomb (self & interaction), symmetry, rotational Complement in surface entropy –  F surface modified by  No entropy contribution from Coulomb (self & interaction), symmetry, rotational –  F non-surface =  E, not modified by  A0-AA0-AA A0A0

28. Complement for excited nuclei Fisher scaling collapses data onto coexistence line Gives bulk T c =18.6±0.7 MeV p c ≈ 0.36 MeV/fm 3 Clausius-Clapyron fit:  E ≈ 15.2 MeV Fisher + ideal gas:  c ≈ 0.45  0 Full curve via Guggenheim Fit parameters: L(E * ), T c, q 0, D secondary Fixed parameters: , , liquid-drop coefficients

29. Conclusions Nuclear dropletsIsing lattices Surface is simplest correction for finite size effects (Rayleigh and Clapeyron) Complement accounts for finite size scaling of droplet For ground state droplets with A 0 <<L d, finite size effects due to lattice size are minimal. Surface is simplest correction for finite size effects(Rayleigh and Clapeyron) Complement accounts for finite size scaling of droplet In Coulomb endowed systems, only by looking at transition state and removing Coulomb can one speak of traditional phase transitions Bulk critical point extracted when complement taken into account.