Long-Time Behavior

2 Long-Time Behavior In a short time scale, 5i(k,T) exhibits a complicated pattern, reflecting complex motions of different parts of the polymer chain. Over a long time, however, the motion is simplified. It is dominated by the translation of the chain as a whole in the solution for any conformation. We can prove the dominance of the center-of-mass motion as follows. 

the displacement ri,(0) - ry(r) between bead i at time 0 and bead j at time t on chain 1 consists of three parts  

The first and third terms are equal to by definition. Only the second term grows with time because of diffusion of the chain as a whole. After a long time, the 

The above discussion applies only to a long time. In a short time. In i(r) may exhibit non-k behavior. In Section 3.4, we will learn in more details how Si(k,T) depends on k and rfor a bead-spring model. 

Case 1. Single steady state (Sj 0 and S2 0) There is very little cross talk between the two species. Hence, both species coexist at all t and k = kuk22 - ku)/ ku + kji - 2 12). The system is robust in that it has only one realizable steady-state composition (x, = J and x = 1- J where = 2/[ i + S2]), whatever the feed composition. 

Cases II. Single steady state Case Ila. 5i 0 and S2 0, but not both zero Case Ilb. 5i 0 and S2 0, but not both zero) There is some cross talk between a very reactive species and a refiactoiy species. The latter dominates the system s long-time behavior, which is robust in that = k22 and X2 = 1 in Case Ila, and = ku and Xj = 1 in Case lib. 

as the interaction intensifies, steady-state multiplicity and stability come into play. In the case of uniformly coupled bimolecular reactions kij = kjkj), the interaction is not strong enough to induce steady-state multiplicity the system behaves similarly to Case II above. 

Numerical calculations demonstrate that a mixture willi as few as three reactants can exhibit a rich variety of behaviors. The results also point to the importance of varying the feed composition in kinetics studies and that longtime rate data can be used for estimating kinetic constants. 

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While none of the 256 possible radius r = 1 binary valued CA are believed to be capable of universal computation, the rule whose long time behavior has proven to be the most difficult to understand fully is rule R22 ... 

The time evolution of the discrete-valued CA rule, F —> F, is thus converted into a two-dimensional continuous-valued discrete-time map, 3 xt,yt) —> (a y+i, /y+i). This continuous form clearly facilitates comparisons between the long-time behaviors of CA and their two-dimensional discrete mapping counter-... 

Figure 13a shows the contribution of translational diffusion. The translational diffusion only describes the experimental data for the smaller momentum transfer Q = 0.037 A. Figure 13b presents S(Q,t), including the first mode. Obviously, the long-time behavior of the structure factor is now already adequately represented, whereas for shorter times the chain apparently relaxes much faster than calculated. 

The macroscopic long-time behavior of dense polymer liquids exhibits drastic changes if permanent cross-links are introduced in the system [75-77], Due to the presence of junctions the flow properties are suppressed and the viscoelastic liquid is transformed into a viscoelastic solid. This is contrary to the short-time behavior, which appears very similar in non-cross-linked and crosslinked polymer systems. 

The long-time behavior (Q(Q)t) > 1 of the coherent dynamic structure factors for both relaxations shows the same time dependence as the corresponding incoherent ones... 

The diverging longest relaxation time, Eq. 1-6, sets the upper limit of the integral. The solid (gel) contribution is represented by Ge. The crossover to any specific short-time behavior for A < A0 is neglected here, since we are mostly concerned with the long-time behavior. 

The cross-over to the glass at short times (or to other short-time behavior) is neglected here, which is justified as long as we only try to predict the long-time behavior, which is most affected by the solidification process. 

Derksen (2006a) continued along this line of approach and—by means of a clever strategy—mimicked the long-time behavior of solids suspension in an unbaffled tall stirred tank equipped with four hydrofoil impellers (Lightnin A310). The time span covered by his LES amounted to some 20,000 impeller revolutions (some 20 min). Running a LES for a Reynolds number of 1.6 x 10 over the entire time span is not an option, and for that reason a particular flow... 

Thus, as noted by Yeung andPope (1993), since the molecular diffusivities do not appear on the right-hand side, molecular differential diffusion affects the coherency only indirectly, i.e., through inter-scale transfer processes which propagate incoherency from small scales to large scales. The choice of the model for the scalar transfer spectra thus completely determines the long-time behavior of pap in the absence of mean scalar gradients. 

To summarize, the results presented for five representative examples of nonadiabatic dynamics demonstrate the ability of the MFT method to account for a qualitative description of the dynamics in case of processes involving two electronic states. The origin of the problems to describe the correct long-time relaxation dynamics as well as multi-state processes will be discussed in more detail in Section VI. Despite these problems, it is surprising how this simplest MQC method can describe complex nonadiabatic dynamics. Other related approximate methods such as the quantum-mechanical TDSCF approximation have been found to completely fail to account for the long-time behavior of the electronic dynamics (see Fig. 10). This is because the standard Hartree ansatz in the TDSCF approach neglects all correlations between the dynamical DoF, whereas the ensemble average performed in the MFT treatment accounts for the static correlation of the problem. 

It has been reported (4-6) that elastomers undergo very longterm relaxation processes in stress relaxation and creep experiments. The long time behavior of shear modulus can be represented by (18)... 

TAD has been demonstrated to be very effective for studying the long-time behavior of defects produced in collision cascades [25,26]. An MD/TAD procedure has also been applied to the simulation of thin-film growth of Ag... 

You have now learned about how to use DFT calculations to compute the rates of individual activated processes. This information is extremely useful, but it is still not enough to fully describe many interesting physical problems. In many situations, a system will evolve over time via many individual hops between local minima. For example, creation of catalytic clusters of metal atoms on metal oxide surfaces involves the hopping of multiple individual metal atoms on a surface. These clusters often nucleate at defects on the oxide surface, a process that is the net outcome from both hopping of atoms on the defect-free areas of the surface and in the neighborhood of defects. A characteristic of this problem is that it is the long time behavior of atoms as they move on a complicated energy surface defined by many different local minima that is of interest. 

With this simplification of the two dimensional step flow problem, we can study the long time behavior of the step train well beyond the initial onset of instability. We start with an array of 40 steps with small perturbations from an initial uniform configuration. We discretize the y coordinate so that each step has 2000 segments. Periodic boundary conditions are used in x and y direction. The time evolution problem of Eqs. (15) using (16) is converted into a set of difference equations. We control the time step so... 

It can be shown that for the totally absorbing boundary condition, and for R r, the long-time behavior of the pair survival probability W(l) is described by... 

Despite the differences in the long-time behavior (due to the lower cutoff in the 1 // case), these two examples allow us to generalize to any dephasing spectrum with a monotonically decreasing system-bath coupling strength as a function of frequency. The optimal modulation for such spectra will be an energy-constrained chirped modulation, with modifications due to other spectral characteristics, for example, cutoffs. 

The cut off parameter a is of order unity. Its value is somewhat arbitrary, reflecting the inability of this continuous spectrum to represent the long time behavior of the Rouse model precisely. Thus, with Eq. (4.33) and Eqs. (3.24) and (3.25) ... 

Fig. 5.3. The dipole autocorrelation function, long-time behavior (schematic). Xd and xc are times of the order of the mean duration of a collision and the mean time between collisions, respectively.

Long-time behavior of correlation functions. The dipoles induced in successive collisions are correlated as Fig. 3.4 on p. 70 suggests. As a consequence, the dipole autocorrelation function has a negative tail of a duration comparable to the mean time between collisions, Fig. 5.3. Furthermore, the area under the negative tail is of similar order of magnitude as the area under the positive (or intracollisional) part of C(r). If the neg-... 

With initial concentrations of A = 0.5, B = 0.2, and C = 0.3 numerically solve the transient problem to predict A(r), B(t), and C(t). Based on the solution, explain in physical terms the short-time behavior and the long-time behavior. Explain the observed behavior in terms of stiffness. 

Solve the nominal problem for an initial condition of A = 0.9 and B = 0.1. Explain the short-time and long-time behavior of the solution in the context of stiffness. 

Planck equation. In this way the actual equations of motion need be solved only during At, which can be done by some perturbation theory. The Fokker-Planck equation then serves to find the long-time behavior. This separation between short-time behavior and long-time behavior is made possible by the Markov assumption. 

The creep of a viscoelastic body or the stress relaxation of an elasacoviscous one is employed in the evaluation of T] and G. In such studies, the long-time behavior of a material at low temperatures resembles the short-time response at high temperatures. A means of superimposing data over a wide range of temperatures has resulted which permits the mechanical behavior of viscoelastic materials to be expressed as a master curve over a reduced time scale covering as much as twenty decades (powers of ten). 

Computer experiments on condensed media simulate finite systems and moreover use periodic boundary conditions. The effect of these boundary conditions on the spectrum of different correlation functions is difficult to assess. Before the long-time behavior of covariance functions can be studied on a computer, there are a number of fundamental questions of this kind that must be answered. 

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