Gaussian joint

In fully developed homogeneous turbulence,7 the one-point joint velocity PDF is nearly Gaussian (Pope 2000). A Gaussian joint PDF is uniquely defined by a vector of expected values (j, and a covariance matrix C ... 

Having found the independent linear combinations of bond vectors [see Eqs. (2.1.17), (2.1.27), and (2.1.28)], through the central-limit theorem it is easy to construct the Gaussian joint distribution and the associated quadratic potential. Adopting for simplicity the periodic-chain transform, we have... 

Fig. 9.5 Gaussian joint normal distribution of the variables Zj and Zg with the means and covariances of those of the distribution in fig. 9.8. (From [1].)...

The function 0(Vj) in eq. (16) represents a N(0,1) standard normal PDF, 0(V, V2,V3 R ) a 3-d1mens1onal Gaussian joint distribution with zero mean and unit standard deviations and 4>(.) the standard cumulative normal probability. All components of the correlation matrix [pjj]... 

Where 0 denotes the standard deviation of the RV W <, 02(vj,Vj,p jj) the two-dimensional Gaussian Joint distribution with zero mean, unit standard deviation and correlation coefficient p )j, and... 

It can be shown that the right-hand side of Eq. (3-208) is the -dimensional characteristic function of a -dimensional distribution function, and that the -dimensional distribution function of afn, , s n approaches this distribution function. Under suitable additional hypothesis, it can also be shown that the joint probability density function of s , , sjn approaches the joint probability density function whose characteristic function is given by the right-hand side of Eq. (3-208). To preserve the analogy with the one-dimensional case, this distribution (density) function is called the -dimensional, zero mean gaussian distribution (density) function. The explicit form of this density function can be obtained by taking the i-dimensional Fourier transform of e HsA, with the result.45... 

We conclude this section by deriving an important property of jointly gaussian random variables namely, the fact that a necessary and sufficient condition for a group of jointly gaussian random variables 9i>- >< to be statistically independent is that E[cpjCpk] = j k. Stated in other words, linearly independent (uncorrelated),46 gaussian random variables are statistically independent. This statement is not necessarily true for non-gaussian random variables. 

Fig. 77.—Gaussian density distribution of the chain displacement vec tors for chain molecules consisting of 10 freely jointed segments, each of length 1 = 2.5 A. The end-to-end length r is in Angstrom units and W(x, y, z) is expressed in A". ...

Equations (10) and (11) are characteristic of the Gaussian distribution, Eq. (8), irrespective of the relationship of 0 to chain dimensions in any given instance. In the particular case of the freely jointed chain assumes the value given by Eq. (6). Substituting Eq. (6) inEqs. (10) and (11) yields... 

Derivation of the Gaussian Distribution for a Random Chain in One Dimension.—We derive here the probability that the vector connecting the ends of a chain comprising n freely jointed bonds has a component x along an arbitrary direction chosen as the x-axis. As has been pointed out in the text of this chapter, the problem can be reduced to the calculation of the probability of a displacement of x in a random walk of n steps in one dimension, each step consisting of a displacement equal in magnitude to the root-mean-square projection l/y/Z of a bond on the a -axis. Then... 

For poly(methylene), an exclusion distance (hard sphere diameter) of 2.00 A was used to prevent overlap of methylene residues. The calculation reproduced the accepted theoretical and experimental characteristic ratios (mean square unperturbed end-to-end distance relative to that for a freely jointed gaussian chain with the same number of segments) of 5.9. This wps for zero angular bias and a trans/gauche energy separation of 2.09 kJ mol". ... 

For Gaussian random variables, an extensive theory exists relating the joint, marginal, and conditional velocity PDFs (Pope 2000). For example, if the one-point joint velocity PDF is Gaussian, then it can be shown that the following properties hold ... 

The scalar fields appearing in Figs. 3.7 to 3.9 were taken from the same DNS database as the velocities shown in Figs. 2.1 to 2.3. The one-point joint velocity, composition PDF found from any of these examples will be nearly Gaussian, even though the temporal and/or spatial variations are distinctly different in each case.11 Due to the mean scalar... 

A simple functional form that can be used to approximate the joint PDF of the Ns composition variables is the joint Gaussian PDF ... 

Note that, by definition, = 0. Moreover, starting from (5.336), it is easily shown that (0in0 > = I and (0deP0dep) = 0- It then follows from the assumed joint Gaussian PDF for 0 that 0dep = 0, and that the joint PDF of 0m is given by... 

Comparing (5.377) with (3.105) on p. 85 in the high-Reynolds-number limit (and with e = 0), it can be seen that (5.378) is a spurious dissipation term.149 This model artifact results from the presumed form of the joint composition PDF. Indeed, in a transported PDF description of inhomogeneous scalar mixing, the scalar PDF relaxes to a continuous (Gaussian) form. Although this relaxation process cannot be represented exactly by a finite number of delta functions, Gs and M1 1 can be chosen to eliminate the spurious dissipation term in the mixture-fraction-variance transport equation.150... 

Homogeneous, linear Fokker-Planck equations are known to admit a multi-variate Gaussian PDF as a solution.33 Thus, this closure scheme ensures that a joint Gaussian velocity PDF will result for statistically stationary, homogeneous turbulent flow. 

While the form of this term is the same as the viscous-dissipation term in the conditional acceleration, the modeling approach is very different. Indeed, while the velocity field in a homogeneous turbulent flow is well described by a multi-variate Gaussian process, the scalar fields are very often bounded and, hence, non-Gaussian. Moreover, joint scalar... 

The scalars must remain bounded (e.g., the mixture fraction must lie between 0 and 1), but as the variances decrease to zero the shape of the joint PDF about ) should be asymptotically Gaussian. 

Generate a mixing model that predicts the correct joint scalar PDF shape for a given scalar covariance matrix, including the asymptotic collapse to a Gaussian form. 

The functional form of H(< ) will control the shape of the joint composition PDF. For example, if H = I the PDF will evolve to a Gaussian form. 

The most common choice is for the components of Z to be uncorrelated standardized Gaussian random variables. For this case, ez z) = z = diag(szj,. .., szNs), i.e., the conditional joint scalar dissipation rate matrix is constant and diagonal. 

In other words, if some of the components of are linearly dependent, then so should an equal number of components of Z. As an example, G could be diagonal so that

would be independent of Z andhave the same correlation structure as ++/>). The joint dissipation rate matrix c/ could be found using the LSR model (Fox 1999). 

The search for the form of W of vulcanized rubbers was initiated by polymer physicists. In 1934, Guth and Mark2 and Kuhn3) considered an idealized single chain which consists of a number of links jointed linearly and freely, and derived the probability P that the end-to-end distance of the chain assumes a given value. The resulting probability function of Gaussian type was then substituted into the Boltzmann equation for entropy s, which reads,... 

Since the joint probability density of the complete set of positions and momenta is Gaussian, the distribution of any subset must also be Gaussian. But the characteristic function corresponding to a Gaussian density takes a simple form, and in the present case is... 

This shows that for a Gaussian distribution the covariance matrix defined in (3.8) is equal to A-1. It follows that a Gaussian distribution is fully determined by the averages of the variables and their covariance matrix. In particular, if the variables are uncorrelated, A-1 is diagonal and hence also A, so that the variables are also independent. Thus, provided that it is known that the joint distribution is Gaussian, uncorrelated implies independent (compare the Exercise in 3). This independence can always be achieved by a linear, and even by an orthogonal, transformation of the variables. 

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