1. Monte Carlo methods (II) Simulating different ensembles

2. E1 E0
Accept with probability
exp[-(E2-E1)/kBT]
Accept E1

3. Configuration Xo, energy Eo
Perturb Xo: X1 = Xo + DX
Compute the new energy (E1)
E1<Eo ? N
Draw Y from U(0,1) Y
Compute W=exp[-(E1-Eo)/kT]
A:=A+A(Xo)
W>Y? Y
Xo=X1, Eo=E1 N

4. periodic bondary contidtions
Infinite systems:
periodic bondary contidtions
Minimum image convention (a particle is not supposed to interact with its image).
Spherical cut-off satisfying minimum-image convention

5. New positions in periodix boxlx

6. Space-filling polyhedra that can serve as periodic boxes
Qian. Strahs, Schlick, J. Comput. Chem., 2001, 15,

7. Example: hexagonal prism/elongated dodecahedron
Qian. Strahs, Schlick, J. Comput. Chem., 2001, 15,

8. Example: solute molecules in non-cubic boxes
Qian. Strahs, Schlick, J. Comput. Chem., 2001, 15,

9. Cut-off on short-range interactions
Simple truncation
Truncation and shift

10. Truncation correction (LJ potential)

11. Characteristic function
Ensemble types
Type
Parameters
Characteristic function
Microcanonical
N, V, E
ln W
Canonical
N, V, T
ln Q
Isothermal-isobaric
N, p, T
ln D
Grand canonical
m, V, T
ln X

12. Microcanonical ensemble
N, V, E defined
Canonical ensemble
N, V, T defined

13. Isothermic-isobaric ensemble Grand canonical ensemble
N ,T, p defined
Grand canonical ensemble
m , T, V defined

14. NVE Monte Carlo simulations
V(x1)
Ed1
accept
V(x1)
Ed0
V(x0)
Ed0
reject
V(x1)
Ed1
accept
M. Creutz, Phys. Rev. Lett., 1983, 50,

15. NPT Monte Carlo sampling
Scaled variables

16. Acceptance criterion
For coordinate change with keeping the box dimensions – as in canonical MC.
For change of box dimensions keeping the scaled coordinates constant
It should be noted that even though the scaled coordinates remain constant under this move, the actual coordinates don’t.
Therefore U(sN,Vold)<>U(sN,Vnew)

17. Reference algorithms for MC/MD simulations (Fortran 77)
M.P. Allen, D.J. Tildesley, „Computer Simulations of Liquids” , Oxford Science Publications, Clardenon Press, Oxford, F11: Monte Carlo simulations of Lennard-Jones fluid.

18. mVT Monte Carlo simulations
Applicable, e.g., in studying adsorption phenomena when equilibration with the reservoir of gas/liquid would take years of computation.

19. Acceptance criterion
For coordinate change with keeping the number of molecules constant – as in canonical MC.
For insertion/deletion of a molecule
The larger the molecule, the less is the probability of accepting insertion.

20. Ergodicity

21. Computing averages with Metropolis Monte Carlo
It should be noted that all MC steps are considered, including those which resulted in the rejection of a new configuration. Therefore, if a configuration has a very low energy, it will be counted multiple times.

22. Importance of proper counting
Analytical (solid lines) and simulated (symbols) equation of state of LJ fluid. Units are atomic units corresponding to scaled coordinates. Open squares: only new accepted configurations counted. Solid squares: all configurations (old and new) counted after a move.

23. Detailed balance (Einstein’s theorem)
old
new

24. Importance of detailed balance
Analytical (solid lines) and simulated (symbols) equation of state of LJ fluid. Units are atomic units corresponding to scaled coordinates. Open squares: detailed balance not satisfied. Solid squares: detailed balance satisfied.

25. MC Simulations of chain molecules: moves
endmove
spike
crankshaft

26. More moves

27. Configurational-bias Monte Carlo
w=2/3*1/3
w=2/3
Rosenbluth and Rosenbluth, J. Chem. Phys., 1955, 23,