Ensemble average relaxation matrix

In addition to corrections tising an appropriate spectral density function, in principle one also needs to consider an ensemble of structures. Bonvin et al. U993) used an ensemble iterative relaxation matrix approach in which the NOE is measured as an ensemble property. A relaxation matrix is built from an ensemble of structures, using averaging of contributions from different structures. The needed order parameters for fast motions were obtained fi um a 50-ps molecular dynamics calculation. The relaxation matrix is then used to refine individual structures. The new structures are used again to reconstruct the relaxation matrix, and a second new set of structures is defined. One repeats the process until the ensemble of structures is converged. The caveat espressed earlier that the accuracy of the result is limited by the accuracy of the spectral density function applies to all calculations of this typ . 

In the linearized LB equation Eq. (7), the ensemble averaged effect of the particle-particle collision is now represented by a relaxation of the distribution function ft to the equilibrium function /fq, where the matrix Ly does not necessarily have to correspond to an existing set of collision rules. The question now arises if L can be simplified even further to the form Ly = aSy, so that the LB equation takes the form (with x — -St/a)... 

The symbols Rauto and Rcross within the relaxation matrix are the auto- and cross-relaxation rates, respectively. and (l2Z) are the longitudinal magnetizations of spin 1 and 2, respectively, and the brackets indicate averaging over the whole ensemble of spins. Rcross in terms of the spectral densities is given by... 

On less formal grounds a Markovian type of equation was also obtained by Redfield [167]. Redheld used perturbation theory up to second order and derived a Master equation for the evolution of the density matrix Oaa y where a, a denote quantum states of the molecule with energy defined by ha. The simple equation for the relaxation of the density matrix was obtained by neglecting the ensemble-averaged first-order terms. Thus we have [167]... 

The excitation of a quantum system under conditions of population and phase relaxation cannot be described by the probability amplitudes a (t). Instead, the time evolution of the particle is described by the combinations a (t)aj(t) averaged over the ensemble. This is a standard procedure in quantum mechanics, according to which the quantity a t)aj(t)) is an element Pji t) of a density matrix (see, for example, Sargent et al. 1974) ... 

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