Monte Carlo

• 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 simulation.
• 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 nature).
• 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.
  
  Canonical Ensemble 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.
  1. select an atom randomly and move it by a random displacement dx, dy, dz.
  2. Calculate the change in potential energy after displacement of the atom.
  3. If dV,O, accept the new configuration.
  4. If dV=0, then accept or reject this configuration based on the following:
     
     tij>x....accept.....................x and element of [0,1]
     
     tij<x....reject
• 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.
  Figure.