Gaussian probability distribution function

Under specific circumstances, alternative forms for ks j have been proposed like the parabolic or the truncated Gaussian probability distribution function for example [154]. 

Normalized one-dimensional Gaussian probability distribution function for occupying position x after random N steps from the origin (x = 0). 

Demonstrate that the Gaussian probability distribution function of a onedimensional random walk is normalized to unity ... 

The Gaussian probability distribution function, as discussed in Section 7.2.1, for a 3-dimensional random walk is... 

Applying the TABS model to the stress distribution function f(x), the probability of bond scission was calculated as a function of position along the chain, giving a Gaussian-like distribution function with a standard deviation a 6% for a perfectly extended chain. From the parabolic distribution of stress (Eq. 83), it was inferred that fH < fB near the chain extremities, and therefore, the polymer should remain coiled at its ends. When this fact is included into the calculations of f( [/) (Eq. 70), it was found that a is an increasing function of temperature whereas e( increases with chain flexibility [100],... 

One consequence of the positivity of a is that A A < (AU)0. If we repeat the same reasoning for the backwards transformation, in (2.9), we obtain A A > (AU)V These inequalities, known as the Gibbs-Bogoliubov bounds on free energy, hold not only for Gaussian distributions, but for any arbitrary probability distribution function. To derive these bounds, we consider two spatial probability distribution functions, F and G, on a space defined by N particles. First, we show that... 

Fig. 6.9. When one of the probability distribution functions f(W) and g(W) is a Gaussian, the other must also is a Gaussian with the same variance (a ). These two density functions peak at A A + and A A - f3a y, respectively. Their crossing point gives the free energy...

While radioactive decay is itself a random process, the Gaussian distribution function fails to account for probability relationships describing rates of radioactive decay Instead, appropriate statistical analysis of scintillation counting data relies on the use of the Poisson probability distribution function ... 

For the case of D-RADP-20 we have assumed a Gaussian probability distribution of transition temperatures, and therefore used an error function to fit the transition region. For D-RADP-25 the transition temperatures range from 138 K down to 118 K. In this temperature range, it is not possible to separate... 

In this respect xq is equal to a certain value k which replaces infinity as the upper integral limit in Eq. 2-2. So we can realize that k values depend on probability P. Again the GAUSSian (normal distribution) function is the simplest model function, because of its sole dependence on P which makes k = k(P). 

The probability distribution function for the fixed end-to-end distance R of macromolecule can be written down on either ground. In the simplest case, it is the Gaussian distribution... 

We will consider rubber elasticity in more detail later, but before you jump to that topic (if that s what you want to do next), it is important to consider probability distribution functions in a little more detail. The distribution of 1-dimensional drunken walks we obtained was shown earlier in Figure 8-31 and for large N the carve takes a Gaussian shape, by which we mean the probability distribution has the mathematical form given in Equations 8-6. 

Figure 3. (Left) The measured scaling exponents q v(q) (joined by dot-dashed straight lines) of the moments of the displacement Ax, as a function of the order q. The dashed line corresponds to 0.65 q while the dotted line corresponds to q 1.04. (Right) The normalized probability distribution function P(Ax(t)/a) versus Ai = Ax/a (a — exp(ln Ax(/ )) for the three times 0=500 (circles), + — 2 (diamonds) and / = 2/ (squares). The dashed line represents the Gaussian function.

The probability distribution function of the tube length L for a chain with N monomers is approximately Gaussian, with mean-square fluctuation of the order of the mean-square end-to-end distance of the chain. The tube length fluctuates in time, leading to stronger molar mass dependences of relaxation time, viscosity, and diffusion coefficient resembling experi-mental observations over some range of molar masses ... 

The name error function is chosen because of its frequent use in probability calculations involving the Gaussian probability distribution. Another form giving the same information is the normal probability integral ... 

Hi) Gaussian statistics. Chandler [39] has discussed a model for fluids in which the probability P(N,v) of observing N particles within a molecular size volume v is a Gaussian function of N. The moments of the probability distribution function are related to the -particle correlation functions g r r2,.. . , r, and... 

In the optimization process, the target function is the average Kullack-Leibler (KL) distance for all heavy atoms, which characterizes the difference between two Gaussian probability distributions defined by the theoretical and experimental ADPs [30,42,46], Given the eigenvalues (ofT, p e 1,2,3) and eigenvectors... 

Another approximation is obtained by observing that the two-point conditional probability distribution function G(x, Xq, t) of a Gaussian process exactly satisfies the following diffusion equation [297,301] ... 

Thus, as given by Eq. (1.42), the probability distribution function for the end-to-end vector R is Gaussian. The distribution has the unrealistic feature that R can be greater than the maximum extended length Nb of the chain. Although Eq. (1.42) is derived on the freely jointed chain model, it is actually valid for a long chain, where the central limit theorem is applicable, except for the highly extended states. 

Consider a strand of polymer chain between two cross-links. The vector R between the positions of the two cross-links changes with deformation. Any molecular theory on rubber elasticity is based on the probability distribution function for R. As seen in Chapter 1, if the number of segments N on the strand is large, the probability distribution (R, N) of the end-to-end vector R is a Gaussian function... 

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