Abstract
Since the beginning of 1950s the Monte Carlo method (MC) has served as one of the major numerical tools of computer modeling of physical processes. Its specific feature is based on statistical modeling as opposed to deterministic calculations of finite difference type. In performing MC calculations one literally tries to counterfeit random quantities distributed according to the known laws of physics, or — especially in statistical physics — to counterfeit processes leading to known physical behavior, e.g. to the settlement of thermodynamic equilibrium. Such counterfeits are known by various names, such as “imitation”, “simulation”, “modeling”, and also “numerical experiment.” To make the terminology more precise, the words “statistical” and “computational” are often added. This refinement is related to the fact that statistics requires a lot of observations, attempts or trials, and one cannot do without modern computers.
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Terminologically one should not confuse the mathematical expectation with AVRG(X) (as it historically appears in physical theories since the nineteenth century). The former is a theoretically exact quantity (as if we were be able to calculate the integral (7.1)) while the latter is an estimate (as if we were able to observe the coordinates rN M times).
However, MC calculations can be facilitated by other means. While the speed of computer operations has almost reached its physical limit, rapid progress in the design of chips for computer memory does not seem to be slowing down. The accessible amount of computer memory doubles every year, and can reach
The corresponding Fortran codes are available via the Computer Physics Communications Program Library. Randomness is guaranteed for up to 1018 numbers produced by each generator. Recently the same authors [36] have studied a “more random” 128-bit sequence 20 = 1, 2n+1 = (Q, A) mod 2128, qn+i = Qn+i /2.0128 and selected more than 2000 multipliers for it.
Note that if AT = T2 — T1 is large, simulation lengths for T1 and T2 will differ considerably and statistical errors will be uncorrelated. In this case, however, physical quantities corresponding to these temperatures will be significantly different, and therefore the absence of correlations of statistical errors does not play a role.
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© 2001 Springer-Verlag Berlin Heidelberg
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Kalikmanov, V.I. (2001). Monte Carlo methods. In: Statistical Physics of Fluids. Texts and Monographs in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04536-7_7
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DOI: https://doi.org/10.1007/978-3-662-04536-7_7
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