Monte Carlo methods (II) Simulating different ensembles
-
Upload
bonnie-todd -
Category
Documents
-
view
215 -
download
0
description
Transcript of Monte Carlo methods (II) Simulating different ensembles
Monte Carlo methods (II) Simulating different ensembles E1 E0
Accept with probability exp[-(E2-E1)/kBT] Accept E1 Configuration
Xo, energy Eo
Perturb Xo: X1 = Xo + DX Compute the new energy (E1) E1Y? Y Xo=X1,
Eo=E1 N 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 New positions
in periodix box
lx Space-filling polyhedra that can serve as periodic boxes
Qian. Strahs, Schlick, J. Comput. Chem., 2001, 15, Example:
hexagonal prism/elongated dodecahedron
Qian. Strahs, Schlick, J. Comput. Chem., 2001, 15, Example: solute
molecules in non-cubic boxes
Qian. Strahs, Schlick, J. Comput. Chem., 2001, 15, Cut-off on
short-range interactions
Simple truncation Truncation and shift Truncation correction (LJ
potential) 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
Microcanonical ensemble
N, V, E defined Canonical ensemble N, V, T defined
Isothermic-isobaric ensemble Grand canonical ensemble
N ,T, p defined Grand canonical ensemble m , T, V defined 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, NPT Monte Carlo sampling
Scaled variables 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 dont. Therefore
U(sN,Vold)U(sN,Vnew) 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. mVT Monte Carlo
simulations
Applicable, e.g., in studying adsorption phenomena when
equilibration with the reservoir of gas/liquid would take years of
computation. 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. Ergodicity 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. 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. Solidsquares: all configurations (old and new) counted
after a move. Detailed balance (Einsteins theorem)
old new 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.
Solidsquares: detailed balance satisfied. MC Simulations of chain
molecules: moves
endmove spike crankshaft More moves Configurational-bias Monte
Carlo
w=2/3*1/3 w=2/3 Rosenbluth and Rosenbluth, J. Chem. Phys., 1955,
23,