- Monte Carlo methods most frequently used in computational
statistical mechanics. Especially useful is simulation of liquids.
- Monte Carlo methods are probabilistic, rather than deterministic,
procedures; atoms are moved more or less randomly during the course of the
- Over the course of some several ,million attempted steps, a large
number of energetically accessible configurations of the system are
explored. This collection of energetically accessible configurations, or
states, is called an ensemble. Thermodynamic and other properties of the
system can be computer as ensemble averages during the simulation.
Depending on the property of interest, several hundred thousand to several
million steps may be necessary to compute the property accurately (some
properties ' converge' more rapidly than others). For example, average
internal energy may converge relatively quickly, but heat capacity may
require a much larger ensemble sample to compute reliability.
- It is desirable to use a large random step size and attain high step
acceptance ratio so as to sample a large number of energetically accessible
( and distinctly different) states in a limited number of steps. Usually,
the step size is adjusted so that approximately 50% of the attempted steps
are accepted. Use of a large random step size in biomolecule simulations
generally yields poor step acceptance ratios. Use of small step sizes can
improve the acceptance ratio to efficient levels(~50%), but severely
restricts the sampling of configuration space, leading to slow convergence
of calculated properties.
- For large biomolecules with many internal degrees of freedom, MC
methods are generally less efficient than molecular dynamics methods for
the calculation of thermodynamic properties, and MC simulations do not
provide direct dynamic information about the system (more static in
- As with MD, reliability of the simulation is limited by the quality of
interactions as represented by the potential function. No amount of
improvement in the sampling technique (MC) or algorithms constructed
(MD)will compensate for inaccurate or inappropriate potential functions.
Consider a canonical ensemble with fixed N,V,T. Suppose we are interested
in the equilibrium value of some thermodynamic quantity A, whre A=A(q3N),
with Q the statistical mechanics:
An example would be if A was the potential energy function and the
configurational part of the internal E of the system.
- Molecular systems cannot typically be solved with straight MC since
most random numbers that come up generate configurations that are
unrealistic (unrealistic molecul;ar interactions). If E is large, the
exponent in the above expression will be very small and this particular
value will not contribute significantly to the average property. For a
system of many water molecules in a box, most of the random configurations
will have unreasonable contacts==>high==>slow convergence!
- Metropolis Method
Uses Boltzmann type probabilities to increase the convergence
procedure. Based on a Markov chain.
- For N configurations, a transition probability matrix if formed:
where, pij and tij are given by
and does not depend on where you come from (2nd order Markov Chain)
- We sample according to this probability of transition.
- select an
atom randomly and move it by a random displacement dx, dy, dz.
- Calculate the change in potential energy after displacement of the atom.
- If dV,O, accept the new configuration.
- If dV=0, then accept or
reject this configuration based on the following:
tij>x....accept.....................x and element of [0,1]
- For simulations involving liquid water, all accessible configurations
within kT of the minimum are going to be acceptable. This selection is
related to the Boltzmann probability.
- Moving water molecules around is much easier than solute since the
H-bond well depth is ~5 kcal/mol. For a solutemolecule, however, movement
of an atom involves bond well depths~1000kcal/mole or so. So, moving an
atom a very small amount will still produce a significant change in energy
and so the whole procedure will take way too long to converge.
- This is 'harmonic" MC since we ignore that individual normal modes are
actually tied to each other, and you can't really talk about movement of a
particular internal coordinate.
- Desirable to observe only small differences between configurations (few
koalas). The overall motion of the system may be significant, but motion
between steps must be small. If a single step produces an E change too
large, it is likely to be rejected and if too few are accepted, convergence
is slow. On the other hand, if you accept too many, you may as well go
through the rigorous procedure.