Weak external force

In the linear-response regime, that is, under weak external forces F, these parameters x are considered as being independent of the strength of the forces. The dependence of an observable 6> on a force F is then the simple proportionality... 

The neighborhood of the Hopf bifurcation point is itself an important asymptotic regime where the description of the dynamics is greatly simplified, as we saw in Chap. 2. Confining ourselves again to systems of oscillators which are coupled (but not necessarily through diffusion), there seems to exist at least one more physical situation for which an equally simplified description of the dynamics is expected. This is when the mutual coupling of the oscillators is weak if necessary, weak external forces may be included. The aim of this chapter is to present a simple perturbation treatment appropriate for such circumstances. 

When using rubbers, non-linear mechanical properties are encountered in all situations of practical interest. Rubbers exhibit large deformations even under comparatively weak external forces and thus are mostly found outside the range of small strains. 

For a sufficiently weak external force F (r, t), the departure pt) = pt) - peq) from equilibrium is expected to be small. Then pt> peq>- Assuming t s to be the case, the nonequilibrium response function equilibrium response function... 

Once the Fock operators have been constructed from a set of MSOs, this matrix equation is linear in its unknowns. Its coefficients are dependent on time in a way determined by the forces driving the electrons. These forces are the nuclear Coulomb potentials in molecular collisions or dynamics, but they could also be weak external fields. 

The scaling prescription (59) embodies the assumption that the external force is so weak that it does not drive the TS trajectory out of the phase-space region in which the normal form expansion is valid. In the autonomous version of geometric TST, one generally assumes that this region is sufficiently large to make the normal form expansion a useful tool for the computation of the geometric objects. Once this assumption has been made, the additional condition imposed by Eq. (59) is only a weak constraint. 

The coils are mutually entangled, this fact is largely responsible for the special behaviour of polymers Between chain segments relatively weak interaction forces are present chain segments can move with respect to each other under the influence of relatively small external stresses. Consequently, the stiffness of polymers is rather low. 

Note that the distributions are cardinally different from those over smooth surfaces like in the previous cases. Again, two parts of the profiles have to be discussed. Over the top SCS s level z = h = 6 m, the wind velocity distributions grow monotonically in the case of a strong wind the temperature diminishes, as a rule. Few cases where the wind velocity diminishes over the SCS are characterized by a weak external wind so that the horizontal forced convection is perhaps comparative with the intense natural convective motion rising up from the heated and wetted air layer within SCS. 

Consider next a more general situation where the weak external perturbation is time-dependent, F = F(Z). We assume again that the force is weak so that, again, the system does not get far from the equilibrium state assumed in its absence. In this case, depending on the nature of this external force, two scenarios are usually encountered at long time. 

This theorem is also valid for many variational wavefunctions, e.g. for the Hartree-Fock one, if complete basis sets are used. As only the one-electron part of the Hamiltonian depends on the nuclear coordinates, H is a one-electron operator, and the evaluation of the Hellmann-Feynman forces is simple. Because of this simplicity, there have been a number of early suggestions to use the Hellmann-Feynman forces for the study of potential surfaces. These attempts met with little success, and the discussion below will show the reason for this. It is perhaps fair to say that the main value of the Hellmann-Feynman theorem for geometrical derivatives is in the insight it provides, and that numerical applications do not appear promising. For other types of perturbations, e.g. for weak external fields, the theorem is widely used, however. For a survey, see a recent book (Deb, 1981). 

The behavior of weakly nonhnear oscillators is discussed in the textbooks [30]. For a small external force F(t), the two-timing approximation holds. The motion of the crystal is almost sinusoidal and the consequences of the external force are captured by a slowly varying amphtude a(f) and a slowly varying phase 4>(f) ... 

The spinodal, binodal and critical point Equations derived on the basis of this theory will be discussed later. When the theory has been tested it has been found to describe the properties of polymer blends much better than the classical lattice theories 17 1B). It is more successful in interpreting the excess properties of mixtures with dispersion or weak attraction forces. In the case of mixtures with a strong specific interaction it suffers from the results of the random mixing assumption. The excess volumes observed by Shih and Flory, 9, for C6H6-PDMS mixtures are considerably different from those predicted by the theory and this cannot be resolved by reasonable alterations of any adjustable parameter. Hamada et al.20), however, have shown that the theory of Flory and his co-workers can be largely improved by using the number of external degrees of freedom for the mixture as ... 

As an example of application of the generalized Cahn-Hilliard equation, we consider the case when a droplet is set into slow motion due to either external forces or long-range interactions. We assume that the deviation from equihb-rium shape remains weak and can be treated as a small perturbation everywhere. The droplet mobility can be deduced then from integral conditions based on an equihbrium solution. This allows us to avoid solving dynantic equations exphcitiy and computing a perturbed shape. 

Once the glass transition temperature (7 ) has been exceeded, the intermolecular forces have become so weak that the influence of external forces can cause the macromolecules to slip apart from one another. The strength declines steeply, while the elongation leaps upward. In this temperature range, the plastic exists in a rubber-elastic or thermoelastic state. 

The study of very dilute solutions of simple solutes such as argon, methane, and the like is of interest for various reasons. First, these solutions reveal some anomalous properties in comparison with non-aqueous solutions and therefore present an attractive challenge to chemists, physicists, and biochemists. Second, aqueous solutions of a simple solute may be viewed as pure water subjected to a weak external field of force. Therefore, the study of such systems can contribute to our understanding of pure liquid water itself. Finally, and most... 

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