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Explore Monte Carlo Simulation methods for calculating ideal gas properties and Metropolis Markov chain of events. Understand theoretical background, probability transitions, states equilibrium, and stochastic matrix. Practice accepting moves based on energies and Boltzmann factor. Learn to compare random numbers for move acceptance using the Linear congruential method. Enhance your understanding of gas simulations and state transitions in a computational framework.
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Monte Carlo Simulation Methods - ideal gas
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=limN (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 > 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(rN)/kT). If the Boltzmann factor is greater then the Random number the move is accepted.
Implementation rand(0,1)≤ exp(-∆V(rN)/kT) Random number generators Linear congruential method
