An Introduction to Monte Carlo Simulations in Statistical by K. P. N. Murthy

By K. P. N. Murthy

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At very high temperature, the spins behave as if they are independent of each other. Spinspin interactions are irrelevant compared to the thermal fluctuations. Entropy wins over energy completely. The macroscopic properties of the system are determined by entropic considerations only. As T decreases, the spin-spin interaction becomes more and more relevant and the correlation length diverges as T → Tc . If the system studied is finite, the correlation length, at best, can be of the order of the linear dimension of the system.

But there are fluctuations present all the time; fluctuations are a part and parcel of an equilibrium system. It is precisely because of these fluctuations that an equilibrium system manages to remain in equilibrium. g. the specific heat corresponds to energy fluctuations. 14 Hence when the trial microstate is of higher energy, we still accept it but with a probability less than unity; larger the energy increase, lower is the acceptance probability, since in equilibrium larger fluctuations are rarer.

Nothing happens to the dynamics for a very long time. But the Monte Carlo clock is ticking all the time. Can such a wastage of computer time be avoided? In other words, can we simulate a very slow dynamics by a fast algorithm? The answer is yes; in the year 1975, Bortz, Kalos and Lebowitz [57] proposed an event-driven algorithm that precisely does this. They called their algorithm the n-fold way. 15 What is the n-fold way? The n-fold way is an event - driven algorithm. We ensure that an event happens at every algorithmic time step.

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