Short-time expansion

For the numerical implementation of the QCL equation, the Liouville operator S is decomposed into a zero-order part Sq which is easy to evaluate and a nonadiabatic transition part if whose evaluation is difficult. The splitting suggests that we (a) employ a short-time expansion of the full exponential by use of a first-order Trotter formula... 

Short-time expansion. The Taylor series expansion of the correlation function is a useful representation at small times,... 

Fig. 23. Velocity autocorrelation functions from the Stockmayer simulation of CO, the Gaussian memory based on / and /, and the short time expansion...

As discussed in Section IX, there are several approximate schemes to stitch together the short- and long-time dynamic behavior. While the initial decay is determined by the short-time expansion, the long-time decay is determined... 

One choice could be to extend the short-time expansion to fourth or even sixth order in time and then remove the terms of such orders from the density and the current terms. Note that the transverse current contribution starts at the sixth order while the density term starts at the fourth order. Such a scheme can certainly be more successful, although it does reduce the simplicity of the present scheme. 

In this section we elaborate on the strong-coupling regime, as defined by the condition Jo/ojc > 1- In the SCL the main contribution to the time integral in Eq. (456) arises from short times. Hence a short-time expansion of J> t) may already give reasonable results and it allows, additionally, to find an analytical expression for P(E). At t -C coj1 we find,... 

In this equation g(t) represents the retarded effect of the frictional force, and /(f) is an external force including the random force from the solvent molecules. We see, in contrast to the simple Langevin equation with a constant friction coefficient, that the friction force at a given time t depends on all previous velocities along the trajectory. The friction force is no longer local in time and does not depend on the current velocity alone. The time-dependent friction coefficient is therefore also referred to as a memory kernel . A short-time expansion of the velocity correlation function based on the GLE gives (fcfiT/M)( 1 — (g/M)t2/(2r) + ), where r is the decay time of g(t), and it therefore does not have a discontinuous first derivative at t = 0. The discussion of the properties of the GLE is most easily accomplished by using so-called linear response theory, which forms the theoretical basis for the equation and is a powerful method that allows us to determine non-equilibrium transport coefficients from equilibrium properties of the systems. A discussion of this is, however, beyond the scope of this book. 

Problem 6.4. Show that the short time expansion of the position autocorrelation function is... 

The expressions, as written, explicitly contain the expression ( ), which has to be obtained by integration of the velocity V(R). Since we are using a short-time expansion anyway, this exact expression may be replaced by the approximation... 

The two lowest cumulants are then k, = /li, and Kj = /itj Mi- When only these are different from zero, the function F(Gaussian distribution. This is in fact a distribution frequently encountered in impulsive collision processes where the duration of collisions is short compared with the periods of internal motions. Then one can make a short-time expansion of ln[F(/)/F(0)j and keep only two terms in the cumulant expansion. 

The second case occurs when, in addition to neglecting the velocities of the slow variables, the remaining TCF can be obtained from short-time expansions as described in the section on cumulants, or from asymptotic expansions as shown for rotational motions in ref. [16]. This can lead to diagonal expressions in the X variables of the form... 

Slow rotational motion short-time expansions... 

Since the rotational periods of most polyatomics are usually much longer than vibrational periods and longer than the collision times, the correlation due to molecular rotation can be evaluated by means of the short-time expansions discussed in section 2.2. For this purpose, the overall TCF is written... 

Fig. 6.21 Comparison of the simulation data (solid line) and various fitting formulae at r = 0.19. The short time expansion of the Rouse model and the Kohlrausch law are represented by a dashed line with long dashes and by a dashed-dotted line, respectively. The dashed line with the short heavy dashes corresponds to the short time expansion of the scaling function (eq. [6.29]) whereas the dotted line refers to its long time part. The model is the same as in Fig. 6.20.

If the temperature is reduced to 7 = 0.18 (see Fig. 6.22) only the initial part of the first step of the correlator can adequately be fitted by the short time expansion of eq. (6.29), whereas for longer times (f) decays much more strongly than predicted by the idealized MCT. Hence, the idealized theory overestimates the freezing tendency of the polymer melt in this simulation. This result is, however, theoretically expected if one takes the above-mentioned hopping processes into account. In this extended version of the MCT it turns out that the asymptotic expansions of the idealized theory are only applicable to the initial decay of the correlator, whereas more compli-... 

In a second class, a density expansion is used to obtain a series approximation for G (e) that is accurate at short times and small ncent rat ions. One approach is to construct a Fade approximant for G (e) from this series that causes G (t) to decay to zero at long times. A second approach is to construct cumulant approximants for the series by inverting the Laplace transform of the density exMnsion and re-expressing the series as a short-time expansion for fn[G (t)]. 

At this junction it is common to discuss various short time expansions and/or long time situations, i.e. to consider partitions of relevant time scales. We will principally mention two interdependent scales, i.e. a global relaxation time - ei and a local collision time Tc. For instance, during Tc the amplitude

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