Monte Carlo Simulation Methods - ideal gas. Calculating properties by integration.

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Presentation transcript:

Monte Carlo Simulation Methods - ideal gas

Calculating properties by integration

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). .   limit =lim N   (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. p mn - where p is the probability to accept the move  mn =  mn if  n >  m  mn =  mn. (  n /  m ) if  n <  m and if n=m 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(r N )/kT). If the Boltzmann factor is greater then the Random number the move is accepted.

Implementation rand(0,1)≤ exp(-∆V(r N )/kT) Random number generators Linear congruential method