It is shown by statistical mechanical methods that in the steady state attained by a system of magnetic moments in a resonance experiment the rate of entropy production is a minimum. It is assumed that the external field has a circularly polarized component perpendicular to a constant component, and that the contact between the spin system and the heat bath is provided only by strong collisions. Because of the coherent relations among the phases of the probability amplitudes, the average off-diagonal elements of the density matrix do not vanish in this case, and it is shown to be necessary to extend the usual statistical mechanical definition of entropy to include terms involving these elements.
We define three general conditions that the relaxation operator in the von Neumann equation for the density matrix of an open quantum system should satisfy. The first one provides the correct relation between the macroscopic current and polarization in a quantum medium, the second is defined by thermodynamic arguments and the third one is imposed by gauge invariance of the solution. The corrected phenomenological relaxation operator is proposed for any quantum system with a discrete number of basis functions, as well as for a quantum system described by the density matrix in coordinate representation. The dissipative three-dimensional oscillator in an arbitrary oriented magnetic field is considered in detail. We have shown that the minimally modified relaxation operator in comparison with the standard relaxation operator, corresponding to tau-approximation, eventually leads to simpler and noncontradictory equations for the dynamics of averaged values of physical quantities.
The present paper presents a derivation of the "master" or Boltzmann "gain-loss" equation from the Schr\"odinger 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 Schr\"odinger equation are vital and ${〈{\ensuremath{\mu}}^{\ensuremath{'}}〉}_{t}$ must be evaluated by a rigorous calculation of Trace {[appropriate time dependent density matrix] ${\ensuremath{\mu}}^{\ensuremath{'}}$}.
The well-known magnetic or optical Bloch equations are extended to the case of unstable systems such as, e.g., spontaneously decaying particles and molecules undergoing chemical reactions or decaying from excited states back to states other than those involved in resonance. The modified equations are derived from a general theory of irreversible processes where quantum Markovian master equations and the corresponding completely positive dynamical semigroups dictate the time evolution of density operators. The results are of particular importance in those situations where decay occurs level-selectively and all relaxation and decay constants are of comparable order of magnitude. The obtained generalized Bloch equations are of direct applicability to problems of time-resolved magnetic or optical resonance spectroscopy.
In the framework of the recent quest for well-behaved nonlinear extensions of the traditional Schrödinger-von Neumann unitary dynamics that could provide fundamental explanations of recent experimental evidence of loss of quantum coherence at the microscopic level, a recent paper [Gheorghiu-Svirschevski 2001 Phys. Rev. A 63 054102] reproposes the nonlinear equation of motion proposed by the present author [see Beretta G P 1987 Found. Phys. 17 365 and references therein] for quantum (thermo)dynamics of a single isolated indivisible constituent system, such as a single particle, qubit, qudit, spin or atomic system, or a Bose-Einstein or Fermi-Dirac field. As already proved, such nonlinear dynamics entails a fundamental unifying microscopic proof and extension of Onsager's reciprocity and Callen's fluctuation-dissipation relations to all nonequilibrium states, close and far from thermodynamic equilibrium. In this paper we propose a brief but self-contained review of the main results already proved, including the explicit geometrical construction of the equation of motion from the steepest-entropy-ascent ansatz and its exact mathematical and conceptual equivalence with the maximal-entropy-generation variational-principle formulation presented in Gheorghiu-Svirschevski S 2001 Phys. Rev. A 63 022105. Moreover, we show how it can be extended to the case of a composite system to obtain the general form of the equation of motion, consistent with the demanding requirements of strong separability and of compatibility with general thermodynamics principles. The irreversible term in the equation of motion describes the spontaneous attraction of the state operator in the direction of steepest entropy ascent, thus implementing the maximum entropy production principle in quantum theory. The time rate at which the path of steepest entropy ascent is followed has so far been left unspecified. As a step towards the identification of such rate, here we propose a possible, well-behaved and intriguing, general closure of the dynamics, compatible with the nontrivial requirements of strong separability. Based on the time–energy Heisenberg uncertainty relation, we derive a lower bound to the internal-relaxation-time functionals that determine the rate of entropy generation. This bound entails an upper bound to the rate of entropy generation. By this extreme maximal-entropy-generation-rate ansatz, each indivisible subsystem follows the direction of steepest locally perceived entropy ascent at the highest rate compatible with the time– energy uncertainty principle.
The solution of the time-dependent wave equation describing the interaction of two coupled spins with a rotating magnetic field in the presence of a steady field, as in the usual magnetic resonance situation, has been reexamined. By extending the method previously developed by Salwen, an exact expression is derived for all components of the wave function describing a three-state system under magnetic-resonance conditions. This solution describes in analytical form all the single and multiple resonances of which the system is capable for fields of any intensity. It is shown that when
z is the axis of quantization, the projection of the wave function on an eigenstate of
Fx oscillates at twice the resonance frequency at exact double quantum resonance. An approximate form is derived for the transition probability near double quantum resonance, showing that our solution is consistent with Salwen's in the limit of well-separated single and double quantum resonances. The theory is applied to the uppermost three substates of the upper hyperfine multiplet of an alkali metal of arbitrary nuclear spin, a case of interest to the analysis of magnetic resonance in optically pumped vapors.