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Unformatted text preview: John von Neumann Institute for Computing Introduction to Monte Carlo Methods Daan Frenkel published in Computational Soft Matter: From Synthetic Polymers to Proteins, Lecture Notes, Norbert Attig, Kurt Binder, Helmut Grubmuller, Kurt Kremer (Eds.), John von Neumann Institute for Computing, Julich, NIC Series, Vol. 23 , ISBN 3000126414, pp. 2960, 2004. c 2004 by John von Neumann Institute for Computing Permission to make digital or hard copies of portions of this work for personal or classroom use is granted provided that the copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise requires prior specific permission by the publisher mentioned above. http://www.fzjuelich.de/nicseries/volume23 Introduction to Monte Carlo Methods Daan Frenkel FOM Institute for Atomic and Molecular Physics, Kruislaan 407, 1098 SJ Amsterdam, The Netherlands Email: frenkel@amolf.nl These give an introduction to Monte Carlo simulations. After a general introduction of the approach and practical implementation, special attention is paid to the used of biased sampling methods in the context of polymer simulations 1 Introduction The Monte Carlo techniques that are described in this chapter can be used to compute the equilibrium properties of classical manybody systems. In this context, the word clas sical means that the core motion of the constituent particles obeys the laws of classical mechanics. This is an excellent approximation for a wide range of materials. Only when we consider the translational or rotational motion of light atoms or molecules (He, H 2 , D 2 ,) or vibrational motion with a frequency such that h > k B T , should we worry about quantum effects. These lecture notes provide a somewhat selective introduction to the Monte Carlo (MC) method. The selection reflects my own personal bias. It is largely (but not completely) based on the more complete description given in ref. 1 Before embarking on a description of the MC method, I should first briefly explain the role of computer simulations in general. This topic is best discussed by considering the situation that prevailed before the advent of electronic computers. At that time, there was only one way to predict the outcome of an experiment, namely by making use of a theory that provided an approximate description of the system under consideration. The reason why an approximate theory was almost always used is that there are very few model sys tems for which the equilibrium properties can be computed exactly (examples are the ideal gas, the harmonic crystal and a number of lattice models, such as the twodimensional Ising model for ferromagnets), and even fewer model systems for which the transport proper ties can be computed exactly. As a result, most properties of real materials were predicted on the basis of approximate theories (examples are the van der Waals equation for dense gases, the DebyeHuckel theory for electrolytes, or the Boltzmann equation to describe the...
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 Spring '10
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