Transcript Power Point

Integretion over a Markovian Web.
G. C. Boulougouris, D. Frenkel, J. Chem. Theory Comput. 2005, 1, 389-393
Schematic representation
of ‘Microstates’.
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’.
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:
acco  n
1  acco  n
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.
 An   
A 
n

n
n
An   m mn
n

n
n
m

 n    m mn
n
 m mn
m
m

 A  
n
n
mn
m

n
m
m
 m mn

 A  
n
m
mn
n

m
m
n
 m mn

Εvaluation of ensample averages.

 A  
n
m
mn
n

m
m

 m mn
  A 
m
m
n
n
  
m
n
m

mn

mn
n
mn
1
n


m  An mn 

m
n


 An  n 
  An mn  m
 m
n
m
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 ).
G. C. Boulougouris, D. Frenkel, J. Chem. Theory Comput. 2005, 1, 389-393
Performing the integration of microstates according to the underlining transition matrix
am n
 mn  amn accmn
for m  n
 mm  1    mn
nm
 An   
A
n
n

mn
m
The fact that the diagonal elements of our transition
matrix are not defined in the same manner as the offdiagonal 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.
A
nm
n
mn
 Am (1    mn )
nm
m
  An am  n accm  n  Am (1   am  n accm  n )  m 
nm
nm

nm
  An am  n accm  n  Am (  am  n   am  n accm  n )  m 
nm
nm
nm
  am  n  An accm  n  Am (1  accm  n )   m
nm
amn  1
Expression of the sum as a weighted average over the trail probability
am n
Using the same transition matrix to generates the important sampling
and to perform the integration over neighboring microstates:
 An   
A
n
n

nm

mn
m

nm
am  n  An accm  n  Am (1  accm  n ) 
m
am  n  An accm  n  Am (1  accm  n ) 

nm
 An   

am  n
 An accm  n  Am (1  accm  n ) 

m
amn , n  m 
m
Same applications
Enrichment of ensample averages by including all possible events in a trail
move with more than one outcome.
Boulougouris G., D. Frenkel, J.Chem.Phys, accepted for publication (2005).
Combining more than one move e.g. a local and a non local move and enrich
the ensemble average will all possible outcomes.
Boulougouris G., D. Frenkel, J. Chem. Theory Comput. 2005, 1, 389-393
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Parallel Monte Carlo :
Boulougouris G., D. Frenkel, J. Chem. Theory Comput. 2005, 1, 389-393
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