Displacements, logarithmic divergence

The weak logarithmic divergence with the sample size L is known as the Landau-Peierls instability. As a result, for sufficiently large L the fluctuations become of the order of the layer spacing, which means that the layer structure would be wiped out. However, for samples in the miUimolar range and typical values of the elastic moduli K 10 N and B 10 N/m, the layer displacement amplitude a = (ifi) does not exceed 0.5-0.7 run. For a typical smectic period d ra 3 nm this gives relative displacements ajd k, 0.2 the smectic layers are still well defined. Nevertheless, the displacements are large compared to those of a typical 3D crystal for which ... 

In many cases considered in later chapters, >(r, t) contains a logarithmic divergence. This point is noted below, in the present section, in a general context. It is therefore necessary, and in fact convenient, to differentiate the displacement equation in (2.8.7) with respect to x. On doing so, we can present the Kolosov-Muskhelishivili equations in an alternative form which is more convenient for practical applications. Let... 

Brownian fluctuations, inertia, nonhydrodynamic interactions, etc.) to lead to exponential divergence of particle trajectories, and (2) a lack of predictability after a dimensionless time increment (called the predictability horizon by Lighthill) that is of the order of the natural logarithm of the ratio of the characteristic displacement of the deterministic mean flow relative to the RMS displacement associated with the disturbance to the system. This weak, logarithmic dependence of the predictability horizon on the magnitude of the disturbance effects means that extremely small disturbances will lead to irreversibility after a very modest period of time. 

The lack of long-range translational order in the 2D harmonic lattice is reflected in the mean-square displacement of a particle from its lattice site, which diverges logarithmically with increasing system size [47],... 

Even though the mean-square displacement diverges logarithmically, the nearest-neighbor separation has small fluctuations, and has a well-defined and finite average value in the thermodynamic limit. This is why the harmonic approximation is expected to apply to the low temperature 2D solid, even though the mean-square displacement diverges. 

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