An analogy between the thermal equilibration of a gas mixture and transverse relaxation in magnetic resonance spectroscopy

Abstract

We study a gas containing two components, a small component P and a large component Q. P is selectively heated to a high temperature and then returns to equilibrium via collisions with Q. This thermal equilibration process is analysed in a new way. We divide the kinetic energy space of the molecules of P into two regions F and D, and show that the molecules of P randomly switch (`oscillate') between the two states as time proceeds due to collisions with the molecules of Q. Initially, the molecules of P are all in the state D, however because each molecule in P collides with the molecules of Q at different times, the oscillations occur out of step with each other. There is a net destructive interference between the oscillations, and so they are not observed when monitoring the average kinetic energy of the molecules of P as a function of time. We will explain the similarities and differences between this observation and transverse relaxation processes that occur in magnetic resonance spectroscopy. This study employs a stochastic model of elastic collisions between the molecules of P and Q, and for completeness we examine its relationship with the two major models of thermal equilibration in statistical physics, namely the Boltzmann equation and the Ornstein-Uhlenbeck process.