-Generate a set of states in phase space that are sampled from the complete set in
accordance with the probability density 
, but not necessarily
for microcanonical ensemble (i.e. not using true equations of motion)
-Invent a means of generating from one state point 
another point
(need not have a physical interpretation)
-This prescription should satisfy the following conditions
1. probability density should not change as system evolves
2. any reasonable starting distribution 
should tend to this stationary
solution as the simulation proceeds
3. ergodicity should hold (although it can't be proven, there should be a reasonabe argument)
where here 
runs over succession of 
states generated by this prescription
-Must ensure adequate sampling
How do we generate a sequence of random states so that by the end of
the simulation each state has occurred with the appropriate probability?
A. Markov chain
a sequence of trials that satisfies 2 conditions:
1. the outcome of each trial belongs to a finite set of outcomes
called the state space
2. the outcome of each trial depends only on the outcome of the trial that immediately precedes it
The 2 states 
and 
are linked by a transition
probability 
, which is the
probability of going from state m to state n.
B. Metropolis Method
The matrix 
is symmetric ( 
) and stochastic.
Since this solution only involves the ratio 
it is independent of the partition function 
.
There is a lot of flexibility in choosing . One definition is as follows. To construct
a state n from a state m, one atom (i)
is chosen at random and displaced from
its position 
with equal probability to any point 
inside a cube
R of side 
centered at .
On the computer there are a large but finite
number 
of new positions for the atom i, so:
Thus, at the beginning of a MC move an atom is picked at random and given a uniform
random displacement along each of the coordinate directions. The maximum displacement
is 
and governs the size of the region R. The appropriate element of the transition
element depends on the relative probabilities of the initial state m and the final state n.
(Note that the factor 
is automatically included in making the move.)
1. if 
(downhill in energy) then 
and Eq. (M1) applies - accept new configuration
2. if 
(uphill in energy) then 
and Eq. (M2) applies - accept move with
probability 
.
This ratio can be expressed as 
. To accept a move with
this probability, a random number is generated uniformly between 0 and 1. If it is
less than then the move is accepted; otherwise it is rejected. Thus,
over the course of the run the net result is that energy changes are accepted with a probability . If the uphill move is rejected, the system remains in state m and this old configuration is recounted as a new state in the chain.
We can summarize this procedure by saying we accept any move (uphill or downhill) with probability min(1, ).
The maximum allowed displacement
governs the size of the trial move
-If it is too small then a large fraction of moves are accepted but phase space
is explored slowly (i.e. consecutive states are highly correlated).
-If it is too large then nearly all the trial moves are rejected and again there is little
movement through phase space.
-Typically
is adjusted so that about half the trial moves are rejected.
Sometimes the selection of atoms to move is done sequentially (i.e., in
order of atom index) rather than randomly. This is equally valid.
The length of an MC simulation is measured in cycles (N trial moves
whether selected sequentially or randomly). The computer time in a MC
cycle is comparable to that in a MD time step.
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