A generalization and modification of a previous discussion is given. By requiring that the steady states obtained in magnetic resonance experiments correspond to the minimum rate of entropy production, a condition is obtained which must be satisfied by the equation describing the rate of change of the spin density matrix due to interactions with the surroundings. This enables one to test the adequacy of the form of equations proposed to describe relaxation phenomena in resonance by finding whether they satisfy this condition, and, conversely, one can derive directly possible forms of macroscopic terms from its solutions. As examples of this procedure, the modified Bloch equations are derived, and the equation of Codrington, Olds, and Torrey is shown to be consistent with this condition. A corrected form of the Wangsness-Bloch equation for the rate of change of the density matrix is shown to be more adequate for arbitrary values of the fields, and the justification of the modified Bloch equations is also given in this way. Other examples, as well as the general relation of this work to irreversible thermodynamics, are discussed, and a generalized reciprocal relation is derived. The steady-state solutions of the new equations of motion are given in an appendix.
It is well known that the concept of spin temperature may not be used to describe a statistical ensemble if the off-diagonal elements of the density matrix are not zero. This concept does not provide a valid description of masers, lasers, or conventional NMR or electron-spin resonance (ESR). It may be used to characterize transient behavior in a spin system "prepared" by sudden reversal of the magnetic field or a saturating pulse. Spin temperature, in such cases, is well defined after a time
T2. In nuclear-spin systems this is generally much shorter than the time
T1 required to reach thermal equilibrium with the lattice. The thermodynamics of irreversible processes provides an alternate and more comprehensive theoretical model for the description of spin systems. Examples are given where it may be used when the off-diagonal elements of the density matrix are not zero. The advantages and limitations of the spin-temperature formalism have been experimentally studied, primarily by means of the techniques of NMR. It is suggested that the predictions of irreversible thermodynamics be tested by observing the steady-state heat capacity of a maser at low temperature or the rate of evaporation of liquid helium in a transient experiment on a nuclear-spin system in which heat is being delivered irreversibly to the bath.
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] ${\ensuremath{\mu}}^{\ensuremath{'}}$}.
Thermodynamics is a phenomenological theory describing the energy conversion of work and heat. At its origins thermodynamics was developed in order to understand and improve heat engines. In conventional thermodynamics, however, only such processes are completely describable which are slow enough to keep the system of interest in an equilibrium state with its thermal environment at all times. On the contrary, all real physical processes are accompanied by non-equilibrium phenomena. These are mathematically described with the help of the irreversible entropy production, i.e. the dissipated work divided by the temperature of the bath. The second law of thermodynamics merely states that the irreversible entropy production is on average always positive. The modern trend of miniaturization, however, leads to smaller and smaller devices. On short length scales thermal noise as well as quantum fluctuations become important. Thus, usual thermodynamic quantities as work and heat acquire stochastic nature. Moreover, in the quantum regime a completely new theory had to be invented, since classical notions of work and heat are no longer valid. The present dissertation contributes to this prevailing field by the derivation of analytical expressions for the entropy production in open and closed quantum system far from thermal equilibrium. The theoretical treatment is motivated by an experimental point of view. The obtained results were derived from the reduced dynamics of the quantum mechanical system, which may be coupled to a thermal bath. To this end, it was dealt with methods and approaches of statistical physics, conventional thermodynamics, quantum information theory and the theory of open quantum systems. In this dissertation the most important physical systems are treated. First, an isolated system is considered before later on a heat bath is included. The limit of weak as well as of strong coupling are analyzed in detail. For isolated systems it turns out that a geometric approach is able to capture the thermodynamic properties, completely. To this end, the irreversible entropy production is sharply estimated from below by the angle between the real non-equilibrium state and the corresponding equilibrium one. Further, it is shown that the Heisenberg uncertainty relation for energy and time can be formulated more precisely in terms of that angle. As experimental system cold ion traps are introduced, in which the theoretical predictions can be verified. For weakly coupled systems, then, the irreversible entropy production is identified and a fluctuation theorem is derived by merely thermodynamic arguments. As experimental system we, again, propose the cold ion traps, where the heat bath can be simulated with the help of LASER light. In the limit of strong coupling a semiclassical description of the reduced quantum system is discussed. Again, an analytically exact expression for the entropy production and the corresponding fluctuation theorem is derived. A physical system, in which strong coupling to the environment can be analyzed, is given by Josephson junctions. This dissertation discusses the entropy production for isolated, weakly coupled, and overdamped quantum systems. For these physical regimes analytical relation are derived and experimental systems proposed, with which the predictions can be verified.
Von Neumann's macroscopic entropy definition is not wholly satisfactory because (
a) it rests on an unsatisfactory characterization of macroscopic variables, (
b) it fails to make satisfactory connection with the Nernst heat theorem, (
c) it is based on an unsatisfactory definition of thermal equilibrium, (
d) it is not applicable to microscopic systems. It is proved that if two systems initially in independent states are placed in thermal contact and isolated, the sum of their microscopic entropies cannot decrease, but will in general increase. This theorem is a substitute for the usual rule regarding the increase in entropy of a single isolated system. Appreciable departures from the canonical state are shown to have a vanishingly small probability in the case of a system subjected to the thermostat experimental procedure described in a preceding paper.