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2. Quantitative increase brings about qualitative change - Hegel The whole is greater than the sum of its parts It is a fundamental idea in condensed matter physics

3. A plausible answer is that there is no such a clear-cut critical size eg. thermodynamic limit The present talk aims to deliver an alternative definite answer

4. N, V → ∞ at fixed N/V Analytic partition function generates analytic function only Well known statement : Phase transition arises only in this limit

5. Easy to observe discontinuous phase transitions Avogadro Number is still finite ! ~ 10^26 << ∞ Density of H 2 O

6.  Our primary interest lies on a system with definite particle number N  We wish to study the precise dependence on N

7.  Canonical partition function : key quantity Z N (¯, V) = Tr(e - ¯H ) where ¯ = (k B T ) - 1

8. P = (1/¯)  V ln Z N S = k B (1{¯  ¯ ) ln Z N E = {  ¯ ln Z N C V = (k B /N) ¯ 2  2 ¯ ln Z N

9.  Realistic constraint  Helps to realize phase transition  Finding the inverse : P(T,V) → V(T,P)

10. Water in a box At constant V : heating up If you open it : easy to boil

11.  Temperature derivative :  T  P =  T { (  T P/  V P)  V  Specific heat : C P =  T  P (E +PV )/N = C V { (  V S) 2 /(k B ¯N  V P)  Denominator  V P may vanish !

12.   V P possesses indefinite sign  V P = ¯  (  V E Ã {  V E Ã  ) 2  {  2 V E Ã    V P  0 Stable  V P  0 Unstable

13. To generate instability & singularity : Canonical Ensemble + Constant Pressure constraint

14.  Many identical particles... distinguishable : (1/2)^N identical : 1 / (N+1) have nontrivial statistics

15.  Grand canonical partition function Z = ¦ Ã (1 { η e - ¯E Ã ) - 1 = exp(  k=1 ¸ k η k /k) where ¸ k :=  Ã e - k¯E Ã

16. Partition sum : (1+1+1, 1+2, 3) for N=3 Hardy-Ramanujan formula ~ exp (N 1/2 )

17. det(  N ) →  : Distinguishable Determinant have 2 N terms

18.  η Z =  η exp(  k ¸ k η k /k) = Z  k ¸ k η k - 1 N 2 calculation : Our key formula for Numerical Computation

19. Identical bosonic particles collapse to the ground state at low T

20.  Ideal Bose gas in cubic box : Dirichlet where  Functions of dimensionless q : Z N, ¯  ¯ = q ln q  q, V  V = ({2/d) q ln q  q

21. ¿ V := k B ( 2 m/¼ 2 ~ 2 ) T(V / N) 2/d ¿ P := k B ( 2 m/¼ 2 ~ 2 ) d/(d+2) TP - 2/(d+2) Á := {(1/N) ¯V 2  V P v P := ( 2 m/¼ 2 ~ 2 ) d/(d+2) (V / N)P d/(d+2) e P := k B ( 2 m/¼ 2 ~ 2 ) d/(d+2) TP - 2/(d+2)

22. ¿ V := k B ( 2 m/¼ 2 ~ 2 ) T(V / N) 2/d ¿ P := k B ( 2 m/¼ 2 ~ 2 ) d/(d+2) TP - 2/(d+2) Á := {(1/N) ¯V 2  V P v P := ( 2 m/¼ 2 ~ 2 ) d/(d+2) (V / N)P d/(d+2) e P := k B ( 2 m/¼ 2 ~ 2 ) d/(d+2) TP - 2/(d+2)

23. ¿ V = {N - 2/d /ln q ¿ P = {1/(ln q [(2/d) q  q ln Z N ] 2/(d+2) ) Á = {(1/N)[(({2/d) q ln q  q ) 2 {({2/d) q ln q  q ] ln Z N v P = (¿ V /¿ P ) - 2/d e P = {(1/N)({2/d) 2/(d+2) ( 2 m/¼ 2 ~ 2 ) d/(d+2)  (q ln q  q ln Z N ) (2d+2)/(d+2)

24. As : Condensate

25. As : Classical ideal gas

26.  d = 3  Fortran 90 program based on Recurrence relation  Extended Precision : 35 significant digits  Used Supercomputer system with 480 nodes & 5.6TFlop/s

27. N = 1, 10, 100, 10000 Constant volume curves

28. Constant pressure curves

29. Supercooling & Superheating points : 1 atm & Helium-4 mass : 1.686, 1.689 K ( cf. 2.17K, 4.22K )

30. Ideal Bose gas under constant pressure for N ≥ 7616 1 st order phase transition : supercooling/heating, BEC, triple valued Cp, discontinuities

31.  Dimension & Geometry dependence : Cubic box in 3D → 7616   V P = 0 is only way for finite system to realize singularity

32.  Low energy strong coupling limit of YM matrix model H ~ tr { ½ (¦ I ) 2 + ¼ g 2 [X I,X J ] 2 + fermionic }

33. ¸ k = #(q k ) 3 (s - k +1+s k ), s = exp({¯!)

34. Separation of drop and V expansion

35.  Z = ¦ à (1 + η e - ¯E à ) = exp(  k=1 ¸ k η k /k) with ¸ k = ({1) k - 1  à e - k¯E à  Expand ¦ à (1 + η e - ¯E à ) directly with à = ( n x, n y, n z, spin ) Danke schön !