Random number generation properties

The energies may be random within some fixed range. Random-number generators use this property intentionally. 

In this method a random number generator is used to move and rotate molecules in a random fashion. If the system is held under specified conditions of temperature, volume and number of molecules, the probability of a particular arrangement of molecules is proportional to exp(-U/kT), where U is the total intermolecular energy of the assembly of molecules and k is the Boltzmann constant. Thus, within the MC scheme the movement of individual molecules is accepted or rejected in accordance with a probability determined by the Boltzmann distribution law. After the generation of a long sequence of moves, the results are averaged to give the equilibrium properties of the model system. 

Often we are only interested in the equilibrium structure of a set of molecules. When the system is at high densities or includes a large number of conformations, other methods become computationally unfeasible, and one often uses Monte Carlo (MC) techniques. Monte Carlo methods use random number generators to integrate systems with very high degrees of freedom. These integrals are then used in theories of statistical mechanics to evaluate thermodynamic properties of materials. ... 

It is a property of two-way arrays that, if a matrix X is generated of size (/ x J) with I x J random numbers (assuming a perfect random number generator), then X has less than full rank with probability zero. This has consequences for measured data sets in chemistry if a matrix X is the result of a measurement it will always be of full rank, because measurement noise cannot be avoided in experiments (apart from problems with, e.g., rounded off noise and limited AD conversion which can lower the rank of X). 

It is only the first class of applications to which this chapter is devoted, because these computations require the highest quality of random numbers. The ability to do a multidimensional integral relies on properties of uniformity of n-tuples of random numbers and/or the equivalent property that random numbers be uncorrelated. The quality aspect in the other uses is normally less important simply because the models are usually not all that precisely specified. The largest uncertainties are typically due more to approximations arising in the formulation of the model than those caused by lack of randomness in the random number generator. 

Another example is from the numerical study of phase transitions. Renormalization theory has proved accurate for the basic scaling properties of simple transitions. The attention of the research community is now shifting to corrections to scaling, and to more complex models. Very long simulations (also of the MCMC type) are done to investigate this effect, and it has been discovered that the random number generator can influence the results [3-6]. As computers become more powerful, and Monte Carlo methods become more commonly used and more central to scientific progress, the quality of the random number sequence becomes more important. 

This chapter is structured as follows. First we discuss the desired properties that random number generators should have. Next we discuss several... 

In this section we discuss some of the desired properties of good random number generators. We shall then explain specific implications of these for parallel random number generation. 

The molecular dynamics approach is a simulation method which essentially solves the equations of motion for a collection of particles within a periodic boundary. This allows for specific transport properties such as the diffusion coefficient to be calculated for example, the effect of nonstoichiometry on ionic conductivity in Na P"Al203 has been investigated using a molecular dynamics approach. " An alternative simulation involves a Monte Carlo approach and relies on statistical sampling by random number generation, yielding... 

Property 1 indicates tliat tlie pdf of a discrete random variable generates probability by substitution. Properties 2 and 3 restrict the values of f(x) to nonnegative real numbers whose sum is 1. An example of a discrete probability distribution function (approaching a normal distribution - to be discussed in the next chapter) is provided in Figure 19.8.1. 

---