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On the Theory of Relaxation Processes
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1957
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Spectral TheoryQuantum DynamicRelaxation ProcessEngineeringPhysicsEntropySpin SystemsEntropy ProductionApplied PhysicsInteracting Particle SystemRelaxation ProcessesProbability TheoryThermodynamicsThermal BathNuclear Spin SystemSpin-lattice RelaxationStatistical Field Theory
The paper presents a general procedure to derive the density‑matrix equation of motion for a system coupled to a thermal bath, exemplified by a weakly coupled nuclear spin system. They model the thermal bath classically and quantum mechanically, describing relaxation with a linear matrix operator whose elements can be secular or nonsecular perturbations; for slow system dynamics the operator is representation‑independent, while time‑dependent Hamiltonians lead to Boltzmann relaxation or require higher‑order corrections, and the framework is applied to spin‑lattice relaxation of coupled nuclear spins in a metal under arbitrary fixed magnetic fields.
A general procedure is given for finding the equation of motion of the density matrix of a system in contact with a thermal bath, as for example a nuclear spin system weakly coupled to a crystal lattice. The thermal bath is treated both classically and quantum mechanically, and the theory is similar to, and a generalization of, conventional theories of time proportional transition probabilities. Relaxation of the system by the thermal bath is expressed by a linear matrix operator, and it is stressed that elements of this operator can be regarded as secular or nonsecular perturbations on the equation of motion and can be treated accordingly. When the motion of the system is slow compared to that of the thermal bath, the equation of motion can be expressed in an operator form which is independent of representation. If the system has a time-dependent Hamiltonian which varies slowly compared to the motion of the thermal bath, the same equation of motion is obeyed and the system is relaxed by the bath toward a Boltzmann distribution with respect to its instantaneous Hamiltonian. If the time variation of the Hamiltonian is more rapid, higher order corrections to the equation of motion must be applied. The theory is applied to spin-lattice relaxation of a coupled nuclear spin system in a metal, for arbitrary externally applied fixed magnetic field.