Random Gaussian distribution

Figure 2. Statistics of current for the transmission through the Sinai billiard for T 0. The upper left panel shows the computed distribution for p = 2 together with the Porter-Thomas distribution P(p) (solid curve). In the inset in the same panel the computed wave function statistics f(p) for the real part of ip is compared with a random Gaussian distribution (solid curve). 

Suppose that we had a very large number of sine waves, all arranged so that they were in phase at time t = 0, with a random (Gaussian) distribution of frequencies. Eventually the waves will start to destructively interfere (cancel), and the amount of time the waves remain in phase depends on the width of the frequency distribution. For a distribution of frequencies with an uncertainty in frequency of 5% (Av = 0.05v0, where vq is the average frequency), it can be shown that the sum looks like Figure 5.13. 

The value of x is used as the average particle size of the powder, while r is a measure of the spread in the distribution of the particle size. In a random (Gaussian) distribution, about two thirds of the particles have sizes within the range of x s. 

Our computational cell is la rectangular parallelepiped, or sometimes a cube, with N in the range 672-2048. Our MD run is normally started by putting the ions at the lattice sites of a perfect bcc crystal, with a random Gaussian distribution of velocities. If the total kinetic energy of the ions is below a cer-... 

The random-bond heteropolymer is described by a Hamiltonian similar to (C2.5.A3) except that the short-range two-body tenn v.j is taken to be random with a Gaussian distribution. In this case a tliree-body tenn with a positive value of cu is needed to describe the collapsed phase. The Hamiltonian is... 

The two sources of stochasticity are conceptually and computationally quite distinct. In (A) we do not know the exact equations of motion and we solve instead phenomenological equations. There is no systematic way in which we can approach the exact equations of motion. For example, rarely in the Langevin approach the friction and the random force are extracted from a microscopic model. This makes it necessary to use a rather arbitrary selection of parameters, such as the amplitude of the random force or the friction coefficient. On the other hand, the equations in (B) are based on atomic information and it is the solution that is approximate. For ejcample, to compute a trajectory we make the ad-hoc assumption of a Gaussian distribution of numerical errors. In the present article we also argue that because of practical reasons it is not possible to ignore the numerical errors, even in approach (A). 

IlyperChem can either use initial velocilies gen eraled in a previous simulation or assign a Gaussian distribution of initial velocities derived from a random n iim her generator. Random numbers avoid introducing correlated motion at the beginn ing of a sim illation. ... 

These two methods generate random numbers in the normal distribution with zero me< and unit variance. A number (x) generated from this distribution can be related to i counterpart (x ) from another Gaussian distribution with mean (x ) and variance cr using... 

The Maxwell-Boltzmann velocity distribution function resembles the Gaussian distribution function because molecular and atomic velocities are randomly distributed about their mean. For a hypothetical particle constrained to move on the A -axis, or for the A -component of velocities of a real collection of particles moving freely in 3-space, the peak in the velocity distribution is at the mean, Vj. = 0. This leads to an apparent contradiction. As we know from the kinetic theor y of gases, at T > 0 all molecules are in motion. How can all particles be moving when the most probable velocity is = 0 ... 

If the data set is Puly nomial and the enor in y is random about known values of a , residuals will be distr ibuted about the regression line according to a normal or Gaussian distribution. If the dishibution is anything else, one of the initial hypotheses has failed. Either the enor dishibution is not random about the shaight line or y =f x) is not linear. 

The normal distribution of measurements (or the normal law of error) is the fundamental starting point for analysis of data. When a large number of measurements are made, the individual measurements are not all identical and equal to the accepted value /x, which is the mean of an infinite population or universe of data, but are scattered about /x, owing to random error. If the magnitude of any single measurement is the abscissa and the relative frequencies (i.e., the probability) of occurrence of different-sized measurements are the ordinate, the smooth curve drawn through the points (Fig. 2.10) is the normal or Gaussian distribution curve (also the error curve or probability curve). The term error curve arises when one considers the distribution of errors (x — /x) about the true value. 

In Langevin dynamics, we simulate the effect of a solvent by making two modifications to equation 15.1. First of all, we take account of random collisions between the solute and the solvent by adding a random force R. It is usual to assume that there is no correlation between this random force and the particle velocities and positions, and it is often taken to obey a Gaussian distribution with zero mean. 

The random force is taken from a Gaussian distribution with zero mean and variance... 

If a large number of replicate readings, at least 50, are taken of a continuous variable, e.g. a titrimetric end-point, the results attained will usually be distributed about the mean in a roughly symmetrical manner. The mathematical model that best satisfies such a distribution of random errors is called the Normal (or Gaussian) distribution. This is a bell-shaped curve that is symmetrical about the mean as shown in Fig. 4.1. 

We will now add random noise to each concentration value in Cl through C5. The noise will follow a gaussian distribution with a mean of 0 and a standard deviation of. 02 concentration units. This represents an average relative noise level of approximately 5% of the mean concentration values — a level typically encountered when working with industrial samples. Figure 15 contains multivariate plots of the noise-free and the noisy concentration values for Cl through C5. We will not make any use of the noise-free concentrations since we never have these when working with actual data. 

To better understand this, let s create a set of data that only contains random noise. Let s create 100 spectra of 10 wavelengths each. The absorbance value at each wavelength will be a random number selected from a gaussian distribution with a mean of 0 and a standard deviation of 1. In other words, our spectra will consist of pure, normally distributed noise. Figure SO contains plots of some of these spectra, It is difficult to draw a plot that shows each spectrum as a point in a 100-dimensional space, but we can plot the spectra in a 3-dimensional space using the absorbances at the first 3 wavelengths. That plot is shown in Figure 51. 

This result checks with our earlier calculation of the moments of the gaussian distribution, Eq. (3-66). The characteristic function of a gaussian random variable having an arbitrary mean and variance can be calculated either directly or else by means of the method outlined in the next paragraph. 

One consequence of Eq. (3-185) is the important result that the sum of a family of statistically independent, gaussianly distributed random variables is again gaussianly distributed. To show this, let... 

The central limit theorem thus states the remarkable fact that the distribution function of the normalized sum of identically distributed, statistically independent random variables approaches the gaussian distribution function as the number of summands approaches infinity—... 

For most systems in thermal equilibrium, it is sufficient to regard fB as random forces which follows a Gaussian distribution function with mean value = 0 and standard deviation = 2kBT 8(i — j) 8(t — t ) [44],... 

Typically in solution, a polymer molecule adopts a conformation in which segments are located away from the centre of the molecule in an approximately Gaussian distribution. It is perfectly possible for any given polymer molecule to adopt a very non-Gaussian conformation, for example an all-trans extended zig-zag. It is, however, not very likely. The Gaussian set of arrangements are known as random coil conformations. 

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