Memory term

For small wavevectors the test particle density is a nearly conserved variable and will vary slowly in time. The correlation function in the memory term in the above equation involves evolution, where this slow mode is projected... 

The first term in (41) has been called the memory term since it contains the memory of the initial occupancy of the valence orbital. For slow atoms with large interactions, exp[—2P(oo)] will be small and the memory term insignificant. For fast atoms with weak interactions, exp[ —2P(oo)] will be close to unity, so the memory of the initial state is retained. 

Attention should be drawn to the fact that if the state with the swelling ratio Qc( u"1) of the nascent network is chosen as the reference state, the memory term h2 3 should be independent of the molecular weight of the elastic chains. This point is still somewhat controversial, and though experimental data support this statement in several cases35,36 there are other cases in which the opposite was observed14,22. ... 

X is the polymer solvent interaction parameter, which is a dimensionless quantity characterizing a polymer solvent system, x may generally depend on temperature, pressure, concentration, but only a little on the functionality/ h2>3 is the memory term, already discussed ... 

Recalling the definition of the memory term ft2 3. as given by Dusek and Prins3 ... 

Model networks can be considered as a new class of tridimensional polymeric material, with elastic chains of known and rather constant length, and branch points of almost constant functionality, with few defects, and of very satisfactory homogeneity. By combination of structural determinations and of swelling and deformation measurements it was shown that the existing theories of rubber elasticity are applicable, even though some ambiguity still exists as to what concerns the value and the meaning of the memory term. 

The integrals, Eqs. (B.3), exist quite generally by closing the contours C in the lower and upper complex planes, respectively, for extensions to the nonself adjoint case, see Figures 2.4, 2.5 and Appendix D. The retarded-advanced formulation including memory terms becomes... 

In general, couplings in Hp are sums of factorized terms, which can be used to express dissipative rates in terms of TCFs. The main features of dissipation rates can be discussed assuming that Hp contains a term with a s-region operator B(el) dependent on fast electronic variables, and a term B at) dependent on slow atomic electronic variables. The TCFs appearing in the memory term are C gat and Celectronic motions with... 

We remark that all the results of this section are obtained by using the B ark ai-Si I bey [30] fractional form of the Klein-Kramers equation for the evolution of the probability distribution function in phase space. In that equation, the fractional derivative, or memory term, acts only on the right-hand side—that is, on the diffusion or dissipative term. Thus, the form of the Liouville operator, or convective derivative, is preserved [cf. the right-hand side... 

The first tenn, iPCPpU describes the time evolution that would be observed if the system was uncoupled from the bath throughout the process (i.e. if QCP = PCQ = 0). The second term is the additional contribution to the time evolution of the system that results from its coupling to the bath. This contribution appears as a memory term that depends on the past history, / p(r), of the system. Consider the integrand in this term, written in the form ... 

The function Z t can be described as a memory term. Equations for multiple saturation and elution involving more than one memory term are also given by Hiester (HI). 

The memory term describes the changes of the network chain conformation in a solution at a concentration of the reacting system to the conformation in a dry state (reference state). 

It is now straightforward to integrate these equations of motion since the memory term has disappeared via the auxiliary variable z. Extensive use of this trick has been made for the GLE see for example Ref. 59. 

Let us now consider how, in accordance with the concept of model networks by the French authors, the functionahty of junctions, the molecular weight of the chains between junction points, the thermodynamic quality of a solvent, and the memory term will affect the swelling of model networks. 

Now, it is appropriate to define the memory term /j3 more precisely. Dusek and Prins relate the memory term to such concentration of polymeric segments at which no forces are exerted on crosshnks in other words, the chains are in the reference state ... 

The memory function M(k, pp" t) represents effects from the dynamics of collisional processes. Before embarking on the survey of the results of the generalized kinetic theory, let us see briefly how the basic equations of classical kinetic theories can be recovered by means of the memory-function equation (5.38). For instance, the Vlasov equation can be obtained by completely ignoring the memory term in Eq. (5.38) ... 

J, J j, and the memory term, (normalized with respect to gravity), should sum-... 

The first term describes the subsystem force field the second one is important if the medium surrounding the subsystem of interest is structured, such a situation is met in an enzyme s active site simulation (see below), otherwise, this term cancels out. The stochastic forces and memory term are subject to particular modeli-zations. 

The added mass and pressure gradient terms in Eq. (14) has the same form as that predicted by potential flow theory (see, e.g.. Ref. 142). Therefore these terms do not require modification. Finally, Odar [143] reported results for the memory term for rigid spheres at large Reynolds numbers. Thus one can develop an approximate equation of bubbles that should be reasonable for bubbles that are not highly deformed and which have Reynolds numbers that are O(IO ). Such an approach could, for example, be used to track small bubbles in turbulent channels or stirred tanks. 

As shown in Eq. 8, the potential splits into two parts upon stretching, one which restores the chain to its initial conformation at crosslinking, the memory term, and, a second, the lattice potential, which tends to move the chain along with the macroscopic sample. The lattice term in the potential deforms affinely, the chain, itself, tends to deform affinely if bi is much greater than bo. 

An important consequence of the memory-lattice model is that high moduli can be accommodated provided that the memory term in the potential function does not vanish. In terms of the model, a memory effect is present if is not equal to unity. This is evident in Eq(14d), low fluctuations corresponding to small Af lead to large values ofa A, the free energy of deformation, and corresponding large values of the retractive force. The coupling of the modulus to junction fluctuations does not appear in either the phantom network or fixed junction models. It arises here only because the minimum in the total potential is not exactly centered on the lattice. 

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