Non-Equilibrium Correlation Functions

The non-equilibrium moments of the normal co-ordinates for an entangled system are defined by expression (4.17) with accuracy up to first-order terms in the velocity gradients. It is written down once more with the label of normal co-ordinates 

The functions Mu(s) and / (s) are defined in previous sections. To describe the most slow relaxation, we use expressions (4.32) and (4.33) and find that 

This expression demonstrates that the relaxation time t , defined by relation (4.27), is the relaxation time of the mean square normal co-ordinate, or 

At a constant velocity gradient, expression (4.43) takes the form 

There are no major difficulties in calculating the mean square normal coordinate when more general formulae (4.28) and (4.29) for the functions Mu(s) and /ij/(s) are used. In this case three sets (branches) of relaxation times 

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Here the subscript 1 indicates the tracer ion. The function A/ is related to X by the first-order non-equilibrium correlation functions. 

This contrasts with relation (5.16), which led to a non-physical conservation law for J. Eqs. (5.28) and Eq. (5.30) make it possible to calculate in the high-temperature limit the relaxation of both rotational energy and momentum, avoiding any difficulties peculiar to EFA. In the next section we will find their equilibrium correlation functions and determine corresponding correlation times. 

Figure 16-12. Left normalized non-equilibrium response function for the electron energy gap in SCA at different densities and 450 K. Right equilibrium spatial correlations between the center of the first excited state rj and the nitrogen site of ammonia for the supercritical states. Solid and dashed lines correspond to adiabatic trajectories with forces taken from the ground and first excited electronic states, respectively. Adapted from Ref. [28]...

The essential characteristic of the equilibrium correlations is that they originate in a system starting from non-correlated states. We recall also that the correct form of the equilibrium correlations can be obtained if one admits that for long times the velocity distribution function takes a Maxwellian form. 

In general, the scalar Taylor microscale will be a function of the Schmidt number. However, for fully developed turbulent flows,18 l.,p L and /, Sc 1/2Xg. Thus, a model for non-equilibrium scalar mixing could be formulated in terms of a dynamic model for Xassociated with working in terms of the scalar spatial correlation function, a simpler approach is to work with the scalar energy spectrum defined next. 

For LiH and LiD, 244-term non-BO wave functions were variationaUy optimized. The initial guess for the LiH non-BO wave function was built by multiplying a 244-term BO wave function expanded in a basis of explicitly correlated functions by Gaussians for the H nucleus centered at and around (in all three dimensions) a point separated from the origin by the equilibrium distance of 3.015 bohr along the direction of the electric field. Thus the centers... 

The indices k in the Ihs above denote a pair of basis operators, coupled by the element Rk. - The indices n and /i denote individual interactions (dipole-dipole, anisotropic shielding etc) the double sum over /x and /x indicates the possible occurrence of interference terms between different interactions [9]. The spectral density functions are in turn related to the time-correlation functions (TCFs), the fundamental quantities in non-equilibrium statistical mechanics. The time-correlation functions depend on the strength of the interactions involved and on their modulation by stochastic processes. The TCFs provide the fundamental link between the spin relaxation and molecular dynamics in condensed matter. In many common cases, the TCFs and the spectral density functions can, to a good approximation, be... 

The point is that this approach ignores the distinctive feature of the bi-molecular process - its non-equilibrium character. The fundamental result known in the theory of non-equilibrium systems [2, 3] is that they tend to become self-organised to a degree which could be characterised by the joint correlation functions, Xv(r, t) and Y(r, t). The idea to use n t)r as a small parameter were right, unless there are no other distinctive parameters of the same dimension as tq. 

Joint distribution of BB and AB pairs is shown in Fig. 6.44. The distribution of similar mobile particles B at long times in the asymmetric case practically is the same as in the symmetric case (when X = Xb). The behaviour of Xb (r, t) is determined by the Coulomb repulsion of B s for which the non-equilibrium screening effect does not take place. In its turn, some deviation for the joint dissimilar functions Y(r, t) seen in Fig. 6.44 for the symmetric and asymmetric cases is a direct consequence of different screening effects in the latter case the effective recombination radius increases in time which results in an increase of the Y(r,t) gradient at r = ro at long times this correlation function itself strives for the Heaviside step-like form. 

The non-equilibrium particle distribution is clearly observed through the joint correlation functions plotted in Fig. 6.47. Note that under the linear approximation [74] the correlation function for the dissimilar defects Y (r, t) increases monotonically with r from zero to the asymptotic value of unity Y(r —y oo,t) = 1. In contrast, curve 1 in Fig. 6.47 (f = 101) demonstrates a maximum which could be interpreted as an enriched concentration of dissimilar pairs, AB, near the boundary of the recombination sphere, r tq. With increasing time this maximum disappears and Y(r, t) assumes the usual smoothed-step form. The calculations show that such a maximum in Y(r, t) takes place within a wide range of the initial defect concentrations and for a random initial distribution of both similar and dissimilar particles used in our calculations X (r, 0) = Y(r > 1,0) = 1. The mutual Coulomb repulsion of similar particles results in a rapid disappearance of close AA (BB) pairs separated by a distance r < L (seen in Fig. 6.47 as a decay of X (r, t) at short r with time). On the other hand, it stimulates strongly the mutual approach (aggregation) of dissimilar particles leading to the maximum for Y(r, t) at intermediate distances observed in Fig. 6.47. 

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