Gaussian approximation, failings

The method will, however, fail badly if the Gaussian form is not a good approximation. For example, looking at the dynamics shown in Figure 2, a problem arises when a barrier causes the wavepacket to bifurcate. Under these circumstances it is necessary to use a superposition of functions. As will be seen later, this is always the case when non-adiabatic effects are present. 

In Fig. 14 the correlation function (q (t)q (O) is shown for the nonlinear potential in Eq. (3.85) at /3 = 10. This correlation function presents another nontrivial test of the various approximate methods because, classically, it can have no negative values while, quantum mechanically, it can be negative due to interference effects. Clearly, only the cumulant method can describe the latter effects. The classical result is extremely poor for this low-temperature correlation function. The CMD with semiclassical operators method also cannot give a correlation function with negative values in this case. This feature of the latter method arises because the correlation of the two operators at different times is ignored when the Gaussian averages are performed. Consequently, the semiclassical operator approximation underestimates the quantum real-time interference of the two operators and thus fails to... 

A third and major discrepancy, already referred to, is found at large deformations when the network chains fail to obey Gaussian statistics, even approximately. Considerable success is achieved in this case by using Eq. (1.5) in place of Eq. (1.1) for chain tensions in the network. 

The use of Stirling s approximation assumes a sufficiently large value of Njj. This assumption fails at high degrees of crosslinking and high extensions. The failure leads to non-Gaussian behavior. Note the similarity of the last term of this sum and that evaluated in the... 

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