Numbers random generators

A source of random numbers is required by any Monte Carlo experiment. It is certainly possible, in principle, to produce numbers that are random in that they are the result of some random physical process such as radioactive decay, but such techniques are almost never used today. Instead one uses a mathematical relation that produces a sequence of numbers that will pass a specified battery of statistical tests. The numbers are not random in that their sequence is determined by the generator, but various statistical tests cannot distinguish them from random numbers. To be more specific we want a sequence of numbers / = 1,2,3. that are uniform in the interval (0,1) and that are not seriously correlated. A possible sequence of statistical tests would examine uniformity of in the unit interval, of 2i 2i+i in the unit square, of 3h 3i+u 31+2 in the unit cube, and so on until correlation behavior of a sufficient order (for the experiment in question) has been considered. 

Although these generators are popular there are many pitfalls in their use. For instance, the generator 

This generator performs well in a pairs test but drastically fails a triples test. (This can be circumvented by replacing 65,539 by 54,891.) 

Linear congruential RNGs are based on the recursion for integer random numbers r, 

Monte Carlo and chain growth methods for molecular simulations 

The period can be increased, if two or more previously generated random numbers are used to generate a new one. One such class of RNGs is formed by the lagged Fibonacci generators 

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As an alternative to the random selection of particles it is possible to move the atom sequentially (this requires one fewer call to the random number generator per iteration) Alternatively, several atoms can be moved at once if an appropriate value for the maximun displacement is chosen then this may enable phase space to he covered more efficiently. 

Marsaglia G, A Zaman and W W Tsang 1990. Towards a Universal Random Number Generate Statistics and Probability Letters 8 35-39. 

Sharp W E and C Bays 1992. A Review of Portable Random Number Generators. Computers at Geosciences 18 79-87. 

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

HyperChem can either use initial velocities generated in a previous simulation or assign a Gaussian distribution of initial velocities derived from a random number generator. Random numbers avoid introducing correlated motion at the beginning of a simulation. 

If Restart is not checked then the velocities are randomly assigned in a way that leads to a Maxwell-Boltzmann distribution of velocities. That is, a random number generator assigns velocities according to a Gaussian probability distribution. The velocities are then scaled so that the total kinetic energy is exactly 12 kT where T is the specified starting temperature. After a short period of simulation the velocities evolve into a Maxwell-Boltzmann distribution. 

Table 10.1 shows the seleetion of parents for mating from the initial population. If a random number generator is used to generate numbers between 0.0 and 1.0, then the eumulative probability values in Table 10.1 is used as follows ... 

The random number generator produeed the following values 0.326, 0.412, 0.862 and 0.067. Henee Parent 1 was seleeted twiee. Parents 2 and 4 onee and Parent 3 not at all. The seleeted parents were randomly mated with random ehoiee of erossover points. The fitness of the first generation of offsprings is shown in Table 10.2. 

The next spin of the random number generator produeed values 0.814, 0.236, 0.481 and 0.712, giving the roulette wheel hits shown in Table 10.2. 

This is a technique developed during World War II for simulating stochastic physical processes, specifically, neutron transport in atomic bomb design. Its name comes from its resemblance to gambling. Each of the random variables in a relationship is represented by a distribution (Section 2.5). A random number generator picks a number from the distribution with a probability proportional to the pdf. After physical weighting the random numbers for each of the stochastic variables, the relationship is calculated to find the value of the independent variable (top event if a fault tree) for this particular combination of dependent variables (e.g.. components). 

Monte Carlo simulation uses computer programs called random number generators. A random number may be defined as a nmnber selected from tlie interval (0, 1) in such a way tliat tlie probabilities that the number comes from any two subintervals of equal lengtli are equal. For example, the probability tliat tlie number is in tlie subinter al (0.1, 0.3) is the same as the probability tliat tlie nmnber is in tlie subinterval (0.5, 0.7). Random numbers thus defined are observations on a random variable X having a uniform distribution on tlie interval (0, 1). Tliis means tliat tlie pdf of X is specified by... 

Suppose that using Monte Carlo simulation witli 10 simulated values of Ta and 10 simulated values of Tb, it is desired to estimate an average value of Ts. First, 20 random numbers are generated. Tliese are shown in columns 1 and 4 of Table 20.6.2. Regard each of the random numbers generated as the value of tlie cdf of a standard nonnal variable Z. Let Zi be tlie simulated value of Z corresponding to 0.10, tlie first random number in colunm 1. Then, since 0.10 is tlie value of tlie cdf for Z = Zi,... 

The obvious lesson to be taken away from this amusing example is that how well a net learns the desired associations depends almost entirely on how well the database of facts is defined. Just as Monte Carlo simulations in statistical mechanics may fall short of intended results if they are forced to rely upon poorly coded random number generators, so do backpropagating nets typically fail to ac hieve expected re.sults if the facts they are trained on are statistically corrupt. 

Determine the copolymer composition for a styrene-acrylonitrile copolymer made at the azeotrope (62 mol% styrene). Assume = 1000. One approach is to use the Gaussian approximation to the binomial distribution. Another is to synthesize 100,000 or so molecules using a random number generator and to sort them by composition. 

The program makes use of the random number generator in ISIM, RAND, to cause random fluctuations in the feed temperature, To. Eliminating RAND will shorten the run time. 

Returns a pseudo-random number in Y in the range 0 to 1. X is the seed for the random number generator Natural logarithm X must be positive <... 

Fig. 4. A schematic two-dimensional illustration of the idea for an information theory model of hydrophobic hydration. Direct insertion of a solute of substantial size (the larger circle) will be impractical. For smaller solutes (the smaller circles) the situation is tractable a successful insertion is found, for example, in the upper panel on the right. For either the small or the large solute, statistical information can be collected that leads to reasonable but approximate models of the hydration free energy, Eq. (7). An important issue is that the solvent configurations (here, the point sets) are supplied by simulation or X-ray or neutron scattering experiments. Therefore, solvent structural assumptions can be avoided to some degree. The point set for the upper panel is obtained by pseudo-random-number generation so the correct inference would be of a Poisson distribution of points and = kTpv where v is the van der Waals volume of the solute. Quasi-random series were used for the bottom panel so those inferences should be different. See Pratt et al. (1999).

The random number generator in all simulations was RAN3 (Press et al., 1992). Computations were made in FORTRAN 90. To reduce cancellation errors, theoretical values of r for PAR were computed in double precision. 

Figure 3.3a-c shows graphs of correlation coefficient r versus p2 f°r different pj in WEG, FAN, and PAR. The curves are limiting values of r evaluated as described in Appendix 3B. The symbols are the average r s from 1000 simulations of 500 random correlated coordinates for different pj and p2. The agreement is excellent, except for r < 0.1. The small deviation probably is caused by minor imperfections in the random number generator. 

Hamiltonian representative of state a between simulation cells i and j. rand[0 l] represents a uniform random number generated from 0.0 < rand[0 l] < 1.0. If the random exchange for a given pair of A states is rejected, the simulation cells are swapped back. Nsampie steps are being performed again, until the next exchange. 

The algorithm also incorporates tacticity control for vinyl chains. The random number generator is used to choose between d- and l-versions of the transformation matrix. A single parameter controls the relative probability of d- and l-residues. The poly(p-fluorostyrene) results presented here are for atactic (stereochemically irregular) chains. 

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