# A Guide to Monte Carlo Simulations in Statistical Physics by Kurt Binder, David P. Landau

By Kurt Binder, David P. Landau

Facing all points of Monte Carlo simulation of complicated actual structures encountered in condensed-matter physics and statistical mechanics, this e-book presents an creation to desktop simulations in physics. This variation now comprises fabric describing robust new algorithms that experience seemed because the earlier variation used to be released, and highlights contemporary technical advances and key purposes that those algorithms now make attainable. Updates additionally comprise a number of new sections and a bankruptcy at the use of Monte Carlo simulations of organic molecules. through the e-book there are lots of functions, examples, recipes, case reports, and workouts to aid the reader comprehend the cloth. it truly is perfect for graduate scholars and researchers, either in academia and undefined, who are looking to study thoughts that experience turn into a 3rd device of actual technology, complementing test and analytical idea.

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This separation of time scales (a phonon vibration time may be of the order of 10À13 seconds, the time between the moves of a vacancy can be 10 orders of magnitude slower) is due to the different length scales of these motions (vibrations take only one percent of a lattice spacing at low temperatures). Thus a simulation of the 40 2 Some necessary background dynamics of these hopping processes using the molecular dynamics method which numerically integrates Newton’s equations of motion, would suffer from a sampling of extremely rare events.

The question of non-ergodic behavior in the context of simulations is complex. For example, in the simulation of an Ising system which may have all spins up or all spins down, we may wish to keep the system from exploring all of phase space so that only positive values of the order parameter are observed. 1 Thermodynamics and statistical mechanics: a quick reminder 25 will be zero. A danger for simulations is that specialized algorithms may be unintentionally non-ergodic, thus yielding incorrect results.

However, as we shall see later, the usual form of Monte Carlo sampling, namely importance sampling Monte Carlo, leads to ‘dynamic’ correlations between subsequently generated observations fAi g. Then Eqn. 78) is replaced by (error)2 ¼ '2 ð1 þ 2(A =tÞ; n ð2:79Þ where t is the ‘time interval’ between subsequently generated states Ai , Aiþ1 and (A is the ‘correlation time’ (measured in the same units as t). 4 Markov chains and master equations The concept of Markov chains is so central to Monte Carlo simulations that we wish to present at least a brief discussion of the basic ideas.