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

TL;DR

This paper draws an analogy between thermal equilibration in a gas mixture and transverse relaxation in magnetic resonance, using a stochastic collision model to analyze energy exchange and interference effects.

Contribution

It introduces a novel perspective by modeling thermal equilibration as oscillations between energy states, linking it to magnetic resonance relaxation phenomena.

Findings

Molecules oscillate between energy states due to collisions.

Destructive interference explains the decay of average kinetic energy.

The model relates to Boltzmann and Ornstein-Uhlenbeck processes.

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.