Gaussian PDF

A Gaussian PDF in one dimension centred at n = a with variance cr is defined as... 

Gaussian PDFs are found for homogeneous inert scalar mixing in the presence of a uniform mean scalar gradient. However, for turbulent reacting flows, the composition PDF is usually far from Gaussian due to the non-linear effects of chemical reactions. 

In commercial CFD codes, the option of using a Gaussian PDF is often available. However, this choice has just one initial peak, and thus is unable to represent the initial conditions accurately. 

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... 

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. 

Turbulent mixing (i.e., the scalar flux) transports fluid elements in real space, but leaves the scalars unchanged in composition space. This implies that in the absence of molecular diffusion and chemistry the one-point composition PDF in homogeneous turbulence will remain unchanged for all time. Contrast this to the velocity field which quickly approaches a multi-variate Gaussian PDF due, mainly, to the fluctuating pressure term in (6.47). 

Choosing the drift coefficient to be linear in U and the diffusion matrix to be independent of U ensures that the Lagrangian velocity PDF will be Gaussian in homogeneous turbulence. Many other choices will yield a Gaussian PDF however, none have been studied to the same extent as the LGLM. 

H2 y is drawn from a Gaussian pdf with mean 0 and variance cr2. 

In conclusion, the ordinary central limit theorem (CLT) establishes that all the propagators sharing the same second moment (c2) yields in the time asymptotic limit the same gaussian pdf. What is the property that the propagators with a diverging second moment must have in common to produce the same pdf in the time asymptotic limit The answer to this question is equivalent to establishing the GCLT [45]. The answer to this important question rests on the anti-Fourier transform of the expression of Eq. (102), which turns out [46] to yield for v oc a distribution proportional to 1/ E,, with... 

It is straightforward to prove that this gaussian pdf is the solution of the following equation of motion ... 

Peak width thermal or static disorder Atomic disorder in the form of thermal and zero-point motion of atoms, and any static displacements of atoms away from ideal lattice sites, gives rise to a distribution of atom-atom distances. The PDF peaks are therefore broadened resulting in Gaussian shaped peaks. The width and shape of the PDF peaks contain information about the real atomic probability distribution. For example, a non-Gaussian PDF peak may suggest an anharmonic crystal potential. 

Fig. 8 The plot of spectral signal-to-artifact ratio of simulated spectrum/(ti, t2) = exp(—Ittii 300 Hz ti - 50 Hz ti - 27112 300 Hz tz - 50 Hz t2) in function of relative density of time domain points ( = p/Pn) comparing WP method and surface integration procedure (512 evolution time points of Gaussian PDF exp(-/ /2

It is a Gaussian PDF with zero mean so it decays as the radial distance to the origin increases in any direction. If there exist two or more models that fit the measurement equally well, using this radially decaying prior distribution helps trimming down the set of the optimal parameters to the one with the smallest 2-norm. 

PDF, the MH algorithm is carried out to simulate samples 0 2, 0 with the target PDF p( A kernel sampling density is constructed as a weighted sum of Gaussian PDFs centered among these samples to approximate p [8,240] ... 

This prior PDF is based on choosing a Gaussian PDF as a probability model for the eigen equation errors, where the prior covariance matrix Y,eq controls the size of these equation errors. The uncertainty in the equation errors for each mode are modeled as independent and identically distributed, so ... 

In the context of Bayesian system identification, the spread of the posterior PDE is a direct fundamental quantification of the remaining uncertainty associated with the modal parameters for a given assumed identification model and in the presence of the measured data. Since the posterior PDF is typically unimodal and it can be approximated by a joint Gaussian PDF, the uncertainty of the modal parameters can be quantified by the covariance matrix, which is called the posterior covariance matrix. The posterior covariance matrix is the inverse of the Hessian matrix of the NLLF, and it can be calculated for a given set of data. Clearly it depends on the particular set of data. 

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