Correlation functions scalar-velocity

To simplify the discussion, a scalar even variable x will be used. In this case the most likely terminal position is x (x, x) = —Q(x)Sx, where the correlation function is <2(x) = 1 (x(t + x)x(f))0. The most likely terminal velocity is... 

Like the velocity spatial correlation function discussed in Section 2.1, the scalar spatial correlation function provides length-scale information about the underlying scalar field. For a homogeneous, isotropic scalar field, the spatial correlation function will depend only on r = r, i.e., R,p(r, t). The scalar integral scale L and the scalar Taylor microscale >-,p can then be computed based on the normalized scalar spatial correlation function fp, defined by... 

The scalar relativistic contribution is computed as the first-order Darwin and mass-velocity corrections from the ACPF/MTsmall wave function, including inner-shell correlation. 

In many respects, at a superficial level, the theory for the chemical reaction problem is much simpler than for the velocity autocorrelation function. The simplifications arise because we are now dealing with a scalar transport phenomenon, and it is the diffusive modes of the solute molecules that are coupled. In the case of the velocity autocorrelation function, the coupling of the test particle motion to the collective fluid fields (e.g., the viscous mode) must be taken into account. At a deeper level, of course, the same effects must enter into the description of the reaction problem, and one is faced with the problem of the microscopic treatment of the correlated motion of a pair of molecules that may react. In the following sections, we attempt to clarify and expand on these parallels. 

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