# Dynamics of position-phase probability density in magnetic resonance

## TL;DR

This paper introduces a new probability density framework for describing the joint behavior of molecular position and phase in magnetic resonance, providing a more fundamental and intuitive understanding of magnetization dynamics.

## Contribution

It develops a position-phase probability density approach that encodes solutions to Bloch-Torrey equations, offering a conceptual advantage over traditional magnetization density models.

## Key findings

- Derived the evolution equation for the position-phase density
- Showed the PPD encodes solutions to Bloch-Torrey equations
- Provided analytical solutions for special cases

## Abstract

We consider the behaviour of precessional angle (phase) carried by molecules of a diffusing specimen under magnetic fields typical of magnetic resonance experiments. An evolution equation for the ensemble of particles is constructed, which treats the phase as well as the position of the molecules as random variables. This "position-phase (probability) density" (PPD) is shown to encode solutions to a family of Bloch-Torrey equations (BTE) for transverse magnetization density, which is because the PPD is a more fundamental quantity than magnetization density; the latter emerges from the former upon averaging. The present paradigm represents a conceptual advantage, since the PPD is a true probability density subject to Markovian dynamics, rather than an aggregate magnetization density whose evolution is less intuitive. We also work out the analytical solution for suitable special cases.