Schematic representation of ‘Microstates’. Integretion over a Markovian Web. G. C. Boulougouris, D. Frenkel, J. Chem. Theory Comput. 2005, 1,

Monte Carlo generate a Markovian chain between ‘Microstates’. Each Microstate is a point in a 3N dimensional phase-space defined By the positions of all “particle” of the system.

Monte Carlo generate a Markovian chain between ‘Microstates’.

Each point in a Markov-Chain depends only on the previous point whereas the overall probability of visiting each point is proportional to the imposed sampling probability (i.e Boltzmann statistics). Monte Carlo algorithms traditionally construct a Markov-Chain using two steps: Monte Carlo algorithms traditionally construct a Markov-Chain using two steps: 1.Starting from the current (‘old’) state (o) a trial move is attempted to a new state (n) according to a trial probability ( a ) 2.The trial state created in the former step is then accepted or rejected according to an acceptance rule that ensures “detailed balance” (or at least the less strict “balance”) between sampled states.

Monte Carlo generate a Markovian chain between ‘Microstates’.

Εvaluation of ensample averages.

The above equation is exact and in principle is sufficient to describe how the expectation value of a property can be evaluated by combining important sampling and integration over the local states of a Markovian Web (i.e for every state m sampled via important sampling an integration is performed over all n states for which ). Εvaluation of ensample averages. G. C. Boulougouris, D. Frenkel, J. Chem. Theory Comput. 2005, 1,

Performing the integration of microstates according to the underlining transition matrix The fact that the diagonal elements of our transition matrix are not defined in the same manner as the off- diagonal elements (since a transition from state m to each self is performed whenever a trail attempt to an other phase is rejected), indicates that the summation maybe broken in two terms.

Expression of the sum as a weighted average over the trail probability Using the same transition matrix to generates the important sampling and to perform the integration over neighboring microstates:

Same applications Enrichment of ensample averages by including all possible events in a trail move with more than one outcome. Combining more than one move e.g. a local and a non local move and enrich the ensemble average will all possible outcomes. Parallel Monte Carlo : Boulougouris G., D. Frenkel, J. Chem. Theory Comput. 2005, 1, Boulougouris G., D. Frenkel, J.Chem.Phys, accepted for publication (2005). Τ1Τ1 Τ2Τ2 Τ2Τ2 Τ1Τ1 Boulougouris G., D. Frenkel, J. Chem. Theory Comput. 2005, 1,