Deviations from classical statistical theories

In the classical theory, it is assumed that the network is infinite, i.e. that no loose chain ends exist. Loose chain ends transfer the stress less efficiently than the other parts of the chain and it may be assumed that 

Loose chains which do not contribute to the elastic force reduce the number of load-carrying chain segments (Ne) to  

Other types of network defect also exist physical crosslinks and closed loops (Fig. 3.16). Physical crosslinks may be permanent with a locked-in conformation (Fig. 3.16(a)) of temporary by entanglement. The presence of the latter type leads to visco-elastic behaviour, i.e. to creep and stress relaxation. Intramolecular crosslinks decrease the interconnectivity and reduce the number of loadcarrying chains. 

The full derivation of the nominal stress (/)-strain relationship for a chain obeying non-Gaussian statistics is complicated. The final equation as derived by Kuhn is  

A chain obeying Gaussian statistics adapts to the following equation  

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Although mi completely satisfactory single theory of liquid helium has yet been formulated, one can say that most of the remarkable properties are qualitatively understood and are due 10 Ihe predominance nl quantum effects, including the dillerence in the statistics of the even and odd isotopes. Titus helium is the one example in nature of a quantum liquid, ail olher liquids showing only minor deviations from classical behavior. 

In the present work, we have used classical photon counting statistics in the weak laser field limit. In the case of strong laser intensity, quantum mechanical effects on the photon counting statistics are expected to be important. From theories developed to describe two-level atoms interacting with a photon field in the absence of environmental fluctuations, it is known that, for strong field cases, deviations from classical Poissonian statistics can become significant [68,99]. One of the well-known quantum mechanical... 

Two topical issues may be mentioned. The first is the definition of the potentials that are measured by different techniques, say by AFM, electrokinetically and externally imposed, and their relationships [11], The second is of a more theoretical nature and concerns modeling of the nondiffuse part of the double layer. The classical approach is through Stem theory [2], which in most cases is adequate, although it requires two additional parameters. A more recent development is in terms of ion correlations, essentially an advanced statistical theory whereby all coulombic ion-ion and ion-surface interaction pairs are counted and statistically summed [2]. This is a step forward over the smeared-out models of Gouy and Stern. The issue here is that cases must be found where deviations from Gouy theory cannot be interpreted on the basis of the Stem model... 

Booth" calculated the deviation of As from quadratidty by OnsagCT s model. Calculations of non-linear As-variations of higher order have also been poformed by Kielich in the Kirkwood-Frohlich semi-macroscopic approach taking into consideration statistical molecular correlations. Results such as these can be derived with the non-linear polarization (282). This treatment, however, is not directly applicable to the description of complete electric saturation, and we shall not develop it furtho- here. It appears preferable, for simplicity, to proceed within the framework of classical Langevin-Debye theory, which yidds results wdl adapted to numerical computations. ... 

In a very different context, in statistical mechanics theory of critical phenomena, corrections to classical exponents are calculated using a systematic series of mean field approximations. In this case, the deviation r from the mean-field value of a critical exponent is called coherent anomaly [173], Remember that ER(X) in Eqs. (115)—(118) corresponds to a bound state if XR < Xc and corresponds to a virtual state if XR >XC. Note that there is no other formal difference between bound and virtual states other than the sign in the logarithmic derivate of the wave function at r = R. Therefore there are no technical problems related with this fact. A relation between XR and Xc can be established for compact support potentials. In this case, using variational arguments, we obtain... 

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