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 ebook offers an advent to laptop simulations in physics. This version now comprises fabric describing robust new algorithms that experience seemed because the past version used to be released, and highlights contemporary technical advances and key functions that those algorithms now make attainable. Updates additionally contain numerous new sections and a bankruptcy at the use of Monte Carlo simulations of organic molecules. through the ebook there are numerous functions, examples, recipes, case reports, and workouts to assist the reader comprehend the fabric. it truly is excellent for graduate scholars and researchers, either in academia and undefined, who are looking to examine recommendations that experience turn into a 3rd software of actual technology, complementing test and analytical concept.

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4 Phase diagrams Phase transitions occur as one of several different thermodynamic ﬁelds is varied. Thus, the loci of all points at which phase transitions occur form phase boundaries in a multidimensional space of thermodynamic ﬁelds. The classic example of a phase diagram is that of water, shown in pressure– temperature space in Fig. 1 Thermodynamics and statistical mechanics: a quick reminder 15 Fig. 3 (left) Schematic temperature dependence of the free energy and the internal energy for a system undergoing a ﬁrst order transition; (right) schematic temperature dependence of the free energy and the internal energy for a system undergoing a second order transition.

Since such algorithms are actually deterministic, the random number sequences which are thus produced are only ‘pseudo-random’ and do indeed have limitations which need to be understood. Thus, in the remainder of this book, when we refer to ‘random numbers’ it must be understood that we are really speaking of ‘pseudo-random’ numbers. These deterministic features are not always negative. For example, for testing a program it is often useful to compare the results with a previous run made using exactly the same random numbers.

As long as they do not play a major role, we can expect that Landau theory will yield correct predictions. Let us compare the ﬂuctuations in mðxÞ for a d-dimensional system over the ‘correlation volume’ $d with its mean value mo . e. for Landau theory to be valid the lattice dimensionality must be greater than or equal to the upper critical dimension du ¼ 4. In addition, below some lower critical dimensionality dl ﬂuctuations dominate completely and no transition occurs. In order to consider the tricritical point scenario depicted in Fig.