Peter%20Virnau,%20M.%20M

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How to overcome free energy barriers in
grandcanonical simulations
Peter Virnau, M. Müller, B. Mognetti, L. Yelash,
K. Binder
Leipzig, 11/2007
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Overview Credits

• Grandcanonical Simulations
• (How to determine critical points?)
• How to overcome free energy barriers?
  Successive Umbrella Sampling
• (How to calculate interface tension?)
• Applications
• Systematic approach to model and coarse-grain
  small molecules (Noble gases, Alkanes, CO2,
  benzene)

G.M. Torrie and J.P. Valleau, J. Comput. Phys.
23, 187 (1977) B. Berg and T. Neuhaus, Phys. Rev.
Letters 68, 9 (1992) F. Wang and D.Landau, Phys.
Rev. Letters 86, 2050 (2001)
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Grandcanonical Simulations
Problem Regions of small probability (free
energy barriers)
algorithms -gt applications
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Free energy barriers
(-H(n)/kT) modify Hamiltonian H(n)
H(n) kT w(n) weight function ideal w(n)ln
P(n) good estimate for w(n) -gt flat histogram
algorithms -gt applications
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How to generate a good w(n) ?
1. Histogram-reweighting (extrapolate data from
previous simulation) -gt limited range,
tedious 2. Adjust w(n) during
simulation i.e., Wang-Landau (-gt violates
detailed balance)
algorithms -gt applications
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Successive umbrella sampling
Idea - simulate windows successively
- generate w(n) for the next window
by extrapolation Adv - works
everywhere - efficient
P. Virnau and M.Müller, J. Chem. Phys. (2004)
algorithms -gt applications
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Successive umbrella sampling
Analysis no violation of detailed balance
-gt errors are controlled and can be
calculated Errors are independent of
window size single window t O(n2) window
size one t n O(1) O(n)???
D 0 ? t n t n
O(n) O(n2)
algorithms -gt applications
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Noble gases

• Potential Lennard-Jones
• Strategy
• Equate critical points of simulation and
  experiment
• e and s
• How well is the rest of the diagram described?

algorithms -gt applications
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Alkanes
Mapping 3CH21 bead
Potential Lennard-Jones FENE
algorithms -gt applications
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Carbon dioxide
Mapping CO2 1 LJ bead quadrupole moment
algorithms -gt applications
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Benzene
Mapping C6H6 1 LJ bead quadrupole moment
algorithms -gt applications
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Take home messages

• How to overcome free energy barriers?
• Successive umbrella sampling
• Modeling
• Equate critical points of simulation and
  experiments to obtain simulation parameters
• ? very good agreement with phase change data

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Critical points
Binder cumulants U4 ltM4gt/ltM2gt2 M r-ltrgt
algorithms -gt applications
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Free energy
Relation btw. P(n) und F F -kT ln(Zcan)
-kT lnP(n) const.
Interface tension g DF/2L2
algorithms -gt applications