How to overcome free energy
barriers in grandcanonical
simulations
Peter Virnau, M. Müller, B. Mognetti, L. Yelash,
K. Binder
Leipzig, 11/2007
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)
Grandcanonical Simulations
P(n)
liquid
gas
particle number n
Problem: Regions of small probability (free energy barriers)
algorithms -> applications
Free energy barriers
P n exp (-H(n)/kT)
modify Hamiltonian:
weight function:
P(n)
H´(n) = H(n) + kT w(n)
ideal: w(n)=ln P(n)
good estimate for w(n)
-> “flat” histogram
particle number n
algorithms -> applications
How to generate a good w(n) ?
1. Histogram-reweighting
(extrapolate data from
-> limited range, tedious
P(n)
previous simulation)
2. Adjust w(n) during
simulation
i.e., Wang-Landau
(-> violates detailed balance)
algorithms -> applications
particle number n
Idea: - simulate windows
successively
- generate w(n) for the
next window by
extrapolation
P(n) P(1) P(2)
P(n)

...
P(0) P(0) P(1) P(n - 1)
P(n)
Successive umbrella sampling
Adv: - works everywhere
- efficient
P. Virnau and M.Müller, J. Chem. Phys. (2004)
algorithms -> applications
particle number n
Successive umbrella sampling
Analysis: no violation of detailed balance
-> errors are controlled and can be calculated:
P(n)
P(1)
D
 n D
 n  D0
P(0)
P(0)
Errors are independent of window size
single window: t  O(n2) window size one: t  n O(1) = O(n)???
D0

n
t
n
t  n O(n) = O(n2)
algorithms -> applications
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 -> applications
Alkanes
Mapping 3CH2=1 bead
Potential
Lennard-Jones + FENE
algorithms -> applications
Carbon dioxide
Mapping: CO2= 1 LJ bead
+ quadrupole moment
algorithms -> applications
Benzene
Mapping: C6H6= 1 LJ bead
+ quadrupole moment
algorithms -> applications
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
Critical points
Binder cumulants
U4 = <M4>/<M2>2
M = r-<r>
algorithms -> applications
Free energy
Relation btw. P(n) und F:
DF/kT
F = -kT ln(Zcan)
= -kT lnP(n) + const.
Interface tension: g  DF/2L2
particle number n
algorithms -> applications