1. Free energies and phase transitions

2. Condition for phase coexistence in a one-component system:

3. The Gibbs “Ensemble”

4. NVT Ensemble Fluid

5. NVT Ensemble G L L G

6. Gibbs Ensemble G L Equilibrium!

8. Distribute n 1 particles over two volumes Change the volume V 1 Displace the particles

9. Distribute n 1 particles over two volumes:

10. Integrate volume V 1

11. Displace the particles in box 1 and box2

12. Probability distribution

13. Particle displacement Volume change Particle exchange

14. Acceptance rules Detailed Balance:

15. Displacement of a particle in box 1

17. Acceptance rules Adding a particle to box 2

18. Moving a particle from box 1 to box 2

28. Analyzing the results (1) Well below T c Approaching T c

29. Analyzing the results (2) Well below T c Approaching T c

30. Analyzing the results (3)

31. Condition for phase coexistence in a one-component system:

32. With normal Monte Carlo simulations, we cannot compute “thermal” quantities, such as S, F and G, because they depend on the total volume of accessible phase space. For example: and

33. F cannot be computed with importance sampling Generate configuration using MC: with

34. Solutions: 1.“normal” thermodynamic integration 2.“artificial” thermodynamic integration 3.“particle-insertion” method

35. How are free energies measured experimentally? P

36. Then take the limit V 0  . Not so convenient because of divergences. Better:  0, as V 0  

37. This approach works if we can integrate from a known reference state - Ideal gas (“T=  ”), Harmonic crystal (“T=0”), Otherwise: use “artificial” thermodynamic integration (Kirkwood) Suppose we know F(N,V,T) for a system with a simple potential energy function U 0 : F 0 (N,V,T). We wish to know F 1 (N,V,T) for a system with a potential energy function U 1.

38. Consider a system with a mixed potential energy function (1- )U 0 + U 1 : F (N,V,T). hence

39. Or: And therefore Examples of application:

40. F( ) 01 F(0) F(1) The second derivative is ALWAYS negative: Therefore: Good test of simulation results…

41. 1.Any atomic or molecular crystal. Reference state: Einstein crystal 2.Nematic Liquid crystal: Start from isotropic phase. Switch on “magnetic field” and integrate around the I-N critical point isotropicnematic density field

42. Chemical Potentials

43. Particle insertion method to compute chemical potentials But N is not a continuous variable. Therefore

44. Does that help? Yes: rewrite s is a scaled coordinate: 0  s<1 r = L s (is box size)

45. Now write then

46. And therefore but

47. So, finally, we get: Interpretation: 1.Evaluate  U for a random insertion of a molecule in a system containing N molecule. 2.Compute 3.Repeat M times and compute the average “Boltzmann factor” 4.Then

49. Lennard-Jones fluid

50. Other ensembles: NPT NPT: Gibbs free energyNVT: Helmholtz free energy NVT: The volume fluctuates! NPT:

51. Hard spheres Probability to insert a test particle!

52. Particle insertion method to compute chemical potentials But N is not a continuous variable. Therefore

53. ACCEPTANCE OF RANDOM INSERTION DEPENDS ON SIZE

55. Particle insertion continued…. therefore But also

56. As before: With s a scaled coordinate: 0  s<1 r = L s (is box size)

57. Now write

58. And therefore

59. Interpretation: 1.Evaluate  U for a random REMOVAL of a molecule in a system containing N+1 molecule. 2.Compute 3.Repeat M times and compute the average “Boltzmann factor” 4.Then DON’T EVEN THINK OF IT!!!

60. What is wrong? is not bounded. The average that we compute can be dominated by INFINITE contributions from points that are NEVER sampled. What to do? Consider:

61. And also consider the distribution p 0 and p 1 are related:

62. Simulate system 0: compute f 0 Simulate system 1: compute f 1 Fit f 0 and f 1 to two polynomials that only differ by a constant.

63. Chemical potential System 1: N, V,T, USystem 0: N-1, V,T, U + 1 ideal gas System 0: test particle energySystem 1: real particle energy

64. Does it work for hard spheres? consider  U=0

65. Problems with Widom method: Low insertion probability yields poor statistics. For instance: Trial insertions that consist of a sequence of intermediate steps. Examples: changing polymer conformations, moving groups of atoms, …

66. What is the problem with polymer simulations? Illlustration: Inserting particles in a dense liquid Trial moves that lead to “hard-core” overlaps tend to be rejected.

67. ANALOGY: Finding a seat in a crowded restaurant. Can you seat one person, please…

68. Next: consider the random insertion of a chain molecule (polymer). Waiter! Can you seat 100 persons… together please! Random insertions of polymers in dense liquids usually fail completely…

69. (Partial) Solution: Biased insertion. Berend Smit’s lecture tomorrow…

70. Simulations of soft matter are time consuming: 1.Because of the large number of degrees of freedom (solution: coarse graining) 2.Because the dynamics is intrinsically slow. Examples: 1.Polymer dynamics 2.Hydrodynamics 3.Activated dynamics

71. Slow dynamics implies slow equilibration. This is particularly serious for glassy systems. 6 hours 8 hours Mountain hikes..

72. after a bit of global warming…

76. .. 20 minutes

77. Sampling the valleys.. Combine this….. …with this

78. Parallel Tempering COMBINE Low-temperature and high- temperature runs in a SINGLE Parallel simulation

79. In practice: System 1 at temperature T 1 System 2 at temperature T 2 Boltzmann factor Total Boltzmann factor

80. SWAP move System 1 at temperature T 2 System 2 at temperature T 1 Boltzmann factor Total Boltzmann factor

81. Ratio

82. Systems may swap temperature if their combined Boltzmann factor allows it.

83. number of MC cycles Window 0 10000 20000 300 200 100 NOTES: 1.One can run MANY systems in parallel 2.The control parameter need not be temperature

84. Application: computation of a critical point INSIDE the glassy phase of “sticky spheres”: GLASSY