Long-time tails

The last issue we address concerns the existence of long-time tails in the discrete-time velocity correlation function. The diffusion coefficient can be written in terms of the velocity correlation function as... 

T. Ihle and D. M. Kroll, Stochastic rotation dynamics. II. Transport coefficients, numerics, and long-time tails, Phys. Rev. E 67, 066706 (2003). 

Beyond Tfl, whole molecules are moving and contributing to viscous flow [i.e., equation (44) describes the long-time tail of the stress relaxation curve or the onset of the flow regime]. 

At long times, when AH neutral-hydrogen formulation. But at early times, when AH may be a major fraction of nA, (121) can be much larger than the extrapolation of its long-time tail. 

Figure 24 Probability distributions for the waiting time for 10 dihedral transitions. Time is given in units of the average waiting time 10x. The distributions are peaked around 10 = 1 and are much broader than the Poisson distribution but approach it for high T. For low T, a high probability for short waiting times exists and a long time tail of the distribution develops.

We should point out that Eq. (42) indicates that the function G(s) can be obtained from the value of the friction kernel at t = 0. This is a consequence of the fact that the friction kernel is calculated in the clamping approximation. In any case, Eq. (42) allows for the calculation of G(s) without the numerical difficulties that plague the long-time tail of molecular dynamics simulations. 

As discussed by Kirkpatrick [10], this slow mode is important in the theories that include mode coupling effects. Such theories have been used to quantitatively understand the anomalous long-time tails of the stress-stress correlation function and the shear-dependent viscosity [3, 30, 34], observed in computer simulations. As mentioned earlier, a theory of glass transition has also been developed based on the softening of the heat mode. 

Thus we note that the memory kernel has a short-time and a long-time part. It is the long-time part which is not present in the viscoelastic model, becomes important in the supercooled-liquid-near-glass transition, and gives rise to the long-time tail of the dynamic structure factor. 

It has been discussed in the previous section that the long-time part in the memory function gives rise to the slow long-time tail in the dynamic structure factor. In the case of a hard-sphere system the short-time part is considered to be delta-correlated in time. In a Lennard-Jones system a Gaussian approximation is assumed for the short-time part. Near the glass transition the short-time part in a Lennard-Jones system can also be approximated by a delta correlation, since the time scale of decay of Tn(q, t) is very large compared to the Gaussian time scale. Thus the binary term can be written as... 

In determining the dynamic structure factor, the value of Do (7o) is needed to be specified. Here the decoupling is studied as a function of the change of density. A noticeable long-time tail in density relaxation appears only very near to the glass transition line. This makes the choice of 7o or p0 rather easy. It is found that, for the reduced temperature, T = 0.8, the reduced density is 0.91. 

The long-time tail in the solvent dynamic structure factor is responsible for the large value of the viscosity in supercooled liquid. The solute dynamics for smaller solutes are faster than that of the solvent. Thus, the solute dynamics is decoupled from this long-time tail of the solvent dynamic structure factor. 

The VACF calculated at p = 0.6 and at T = 0.7 is plotted in Fig. 17. The long-time tail present in the VACF is clearly demonstrated in the figure. As discussed before, it is the presence of this long-time t x tail in the VACF which gives rise to the divergence of the diffusion in the long time. It is found that the presence of the r1 tail in the VACF calculated at high density is not... 

The behavior of VACF and of D in one-dimensional systems are, therefore, of special interest. The transverse current mode of course does not exist here, and the decay of the longitudinal current mode (related to the dynamic structure factor by a trivial time differentiation) is sufficiently fast to preclude the existence of any "dangerous" long-time tail. Actually, Jepsen [181] was the first to derive die closed-form expression for the VACF and the diffusion coeffident for hard rods. His study showed that in the long time VACF decays as 1/f3, in contrast to the t d 2 dependence reported for the two and three dimensions. Lebowitz and Percus [182] studied the short-time behavior of VACF and made an exponential approximation for VACF [i.e, Cv(f) = e 2 ], for the short times. Haus and Raveche [183] carried out the extensive molecular dynamic simulations to study relaxation of an initially ordered array in one dimension. This study also investigated the 1/f3 behavior of VACF. However, none of the above studies provides a physical explanation of the 1/f3 dependence of VACF at long times, of the type that exists for two and three dimensions. 

Figure 21. The long-time tails of Cv(t) obtained from simulations plotted against tjtf at various densities at T = 1.0. t is the reduced time and tc is the time at which the long-time tail of C (r) started approaching zero. The different symbols from top to bottom represent the Cv(t) at reduced densities 0.3,0.4,0.6, and 0.82, respectively. The figure shows the dominance of 1 /r3 decay in C (f) at low and intermediate densities. This figure has been taken from Ref. 186.

It is interesting to note that despite the existence of long-time tails of C (f), we can still have a well-defined diffusion coefficient. We find that the decay of the longitudinal current mode is sufficiently fast to preclude... 

The low-frequency feature in the reduced spectral density corresponds to the long-time tail of the intermolecular response function, which is often denoted the intermediate response in the OKE literature (15,51). In most liquids, this portion of the response appears to be exponential over a significant time scale. Why this portion of the response is exponential and what information the time scale of this exponential holds is still poorly understood. For this reason, we have performed detailed temperature-dependent studies of the intermediate relaxation in six symmetric-top liquids acetonitrile, acetonitrile-d3, benzene, carbon disulfide, chloroform, and methyl iodide (56). 

The static experiments show that there is a complex between TBP and PNP at the dodecane/water interface. Even by simply mixing the two solutions it was clear that this interaction was time-dependent. However, the decay rate was sufficiently fast so that given the need to ensure uniform mixing in the bulk phases and the time required to accumulate a reasonable S/N, only the long time tail of the decay curve could be measured. No accurate estimate of the decay rate could be made in the Petri-dish. The solution was to construct a flow cell to measure the kinetics of the TBP and PNP interaction at the dodecane/water interface [48]. 

Out of the detailed mathematical aspects, some of them summarized in this section, there is a more general physical concept at the heart of the theory of error bounds. It is a fact that the memory function formalism provides in a natural manner a framework by which Ae short-time behavior, via the kernel of the integral equations, makes its effects felt in the long-time tail. The mathematical apparatus of continued fractions can adequately describe memory effects, and this explains the central role of this tool in the theory of relaxation. 

It is a remarkable point that the indices a and p seem to be free from the cluster size. Furthermore, it is worth noting that the intrinsic long-time tail of Plw(T) is the same as the universal scaling form of Eq. (10),... 

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