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• W2077859093 abstract "The present paper presents a derivation of the master or Boltzmann gain-loss equation from the Schrodinger equation, i.e., a derivation of the equation for the evolution in time of the probabilities of finding a physical system in its various states from the equation for the corresponding probability amplitudes. The master equation is derived for an, in effect completely self-enclosed, supersystem, [$A+B$], consisting of a system of interest, [$A$], and a surroundings, [$B$], in relatively weak mutual interaction. A discussion is given of the range of validity of the master equation for [$A+B$] and it is shown that the random phase assumption is required for the state vector of [$A+B$] at the initial time only. The normally microcanonical character of the equilibrium statistical configuration of [$A+B$] is demonstrated and a treatment is given of exceptional, extremely quantal-coherent, initial statistical distributions of [$A+B$] which may evolve away from equilibrium. Derivations are also presented of the master equation for [$A$] and of the master equation for an individual particle or quasi-particle [$q$], within [$A$]; a discussion of the range of validity of these master equations is given and the normally canonical character of the equilibrium statistical configuration of [$A$] is deduced. General solutions of the master equations for [$A+B$], [$A$], and [$q$] are worked out and the relation between the principles of microscopic reversibility and detailed balance and the nonoscillatory character of the approach to equilibrium are exhibited. A theorem is presented regarding the time variation of the entropy of [$A$].As illustrations of the general methods developed two important processes in magnetic resonance---the time variation of the longitudinal magnetization, ${〈ensuremath{mu}〉}_{t}$, and the time variation of the transverse magnetization, ${〈{ensuremath{mu}}^{ensuremath{'}}〉}_{t}$---are discussed in some detail. It is shown that the variation of ${〈ensuremath{mu}〉}_{t}$ with $t$ and of ${〈{ensuremath{mu}}^{ensuremath{'}}〉}_{t}$ with $t$ for a nonrigid lattice can be described by means of the master equation for an individual spin [$q$] and several special cases are discussed on the basis of the evaluation of the appropriate transition probabilities; a comparison with the spin-temperature procedure is also appended. On the other hand, it is demonstrated that for a rigid lattice no description of the variation of ${〈{ensuremath{mu}}^{ensuremath{'}}〉}_{t}$ with $t$ can be given on the basis of a master equation; in this case, quantal coherence effects neglected in the derivation of the master equation from the Schrodinger equation are vital and ${〈{ensuremath{mu}}^{ensuremath{'}}〉}_{t}$ must be evaluated by a rigorous calculation of Trace {[appropriate time dependent density matrix] ${ensuremath{mu}}^{ensuremath{'}}$}." @default.
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• W2077859093 title "Approach to Equilibrium in Quantal Systems: Magnetic Resonance" @default.
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• W2077859093 doi "https://doi.org/10.1103/physrev.119.178" @default.
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