Long-wavelength modes

In the phase transition model (4) long wavelength modes are coupled to short wavelengths modes by nonlinear couplings (21). It is not known whether this model can be solved exactly except for numerical methods. 

Here N is the normalization constant and the coefficients Ai,A2 and B will be determined by Schrodinger equation. We are interested in the reduced density matrix for the long wavelength mode , which is obtained by integrating the density matrix To(i, 2, t) over the short wavelength mode 2 = reduced density matrix for the long wavelength mode can be written in the form... 

Therefore, we conclude that the long wavelength mode neither decoheres nor is classically correlated before the phase transition. However, after the phase transitions, the unstable long wavelength mode becomes classical, gaining both quantum decoherence and classical correlation. Thus an order parameter appears from long wavelength modes (S.P. Kim et.al., 2000 2002 2001). 

The above property of the chemical potential offers a convenient device for discussing the relaxational dynamics of near-equilibrium systems. By near-equilibrium one usually has in mind a system where each part, has reached equilibrium locally, but long wavelength modes have not been completely relaxed. This allows one to define a. local chemical potential /t(x) whose spatial variation serves as the driving force towards global equilibrium. 

The last term on the right-hand side of Eq. (8.8.10) simplifies considerably if we confine our attention only to low k values (which correspond to long wavelength modes)... 

The trivial solution p(x) = 0 is unstable, i.e., the population grows, if k/, = rF 0)-D nk/L) > 0 for somefc. Since the growth rates k/, decrease monoton-ically with k, the dominant mode is the long-wavelength mode = 1. It determines... 

In this way, we regard as a function of t, t, and s. The reason why we should assume this slow space dependence is clear from the discussion in the previous section. To put it briefly, we are treating the circumstance where the long-wavelength modes inside the circles in Fig. 2.2 are generally present with nonvanishing amplitudes, while the shorter-wavelength modes outside the circles are absent. 

J. Noolandi (Xerox Research Center of Canada) What is the importance of the long-wavelength modes in your model, and could you clarify the relationship of your model to magnetic spin models ... 

The reduced mobility of the chains can also be seen in the relaxation of the long-wavelength modes of the long chain. In the reptation model, the relaxation time of a Rouse mode p, with N/p > Ne is enlarged by a factor of NjNe, giving ... 

The distribution of retardation times is also modified by the presence of the network. Long wavelength modes are damped by the network and the distribution of chain lengths between crosslinks should lead to a softened cutoff at the long relaxation time end. This complexity suggests that actual equilibrium data should be taken for very long times, because full equilibrium may require extensive patience. Even if the network were perfect with identical numbers of monomers between crossUnks and no chain ends or loops, the physical location of the crossUnks would reflect the Gaussian statistics and the response would still be complex. 

The experimentalist does not ordinarily measure a conformational state time correlation function. Usually a technique is sensitive to some first or second order tensor, an orientational quantity such as a dipole moment along or perpendicular to the chain, the direction of a C-H bond, a polarizability, or a transition moment of a chromophore. It is even more difficult to analyze these objects orientational time correlation functions. Furthermore they may exhibit both rapid relaxation due to conformational transitions and slow relaxation due to coupling to long wavelength modes. We have suggested fitting the short time part to the simplest function which accounts for single and pair transitions, ... 

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