Monte Carlo Simulation Methods

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Monte Carlo Simulation Methods
- ideal gas
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Calculating properties by integration
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Theoretical background to Metropolis
Markov chain of events - the outcome of each
trial depends only on the preceding trial - each
trial belongs to a finite set of possible
outcomes ?mn - probability of moving from state m
to n ?(?1, ?2,. ?m, ?n,?N) - probability that
the system is in a particular
state ?(2) ?(1). ?
?(3) ?(2). ? ?(1). ?. ? ?limitlimN?? ?(1) ?N
- limiting (equilibrium) distribution ?mn -
probability to choose the two states m,n between
which the move is to be made (stochastic
matrix). ?mn ?mn. pmn - where p is the
probability to accept the move ?mn ?mn if ?n gt
?m ?mn ?mn. (?n/?m) if ?n lt ?m and if
nm In practice if the energy of the n state is
lower the move is accepted, if not a random
number between 0 and 1 is compared to the
Boltzmann factor exp(-?V(rN)/kT). If the
Boltzmann factor is greater then the Random
number the move is accepted.
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Implementation
rand(0,1) exp(-?V(rN)/kT)
Random number generators
Linear congruential method