Fokker-Planck formalism in magnetic resonance simulations

TL;DR

This paper reviews the Fokker-Planck formalism for magnetic resonance simulations, highlighting its advantages over traditional methods and demonstrating its application in simplifying complex NMR, EPR, and MRI problems.

Contribution

It introduces novel applications of the Fokker-Planck approach and discusses its implementation in the Spinach library for practical magnetic resonance simulations.

Findings

Fokker-Planck Hamiltonian is time-independent for relevant spatial dynamics.

Simplifies complex NMR, EPR, and MRI simulations.

Enables routine practical use of previously complex simulations.

Abstract

This paper presents an overview of the Fokker-Planck formalism for non-biological magnetic resonance simulations, describes its existing applications and proposes some novel ones. The most attractive feature of Fokker-Planck theory compared to the commonly used Liouville - von Neumann equation is that, for all relevant types of spatial dynamics (spinning, diffusion, flow, etc.), the corresponding Fokker-Planck Hamiltonian is time-independent. Many difficult NMR, EPR and MRI simulation problems (multiple rotation NMR, ultrafast NMR, gradient-based zero-quantum filters, diffusion and flow NMR, off-resonance soft microwave pulses in EPR, spin-spin coupling effects in MRI, etc.) are simplified significantly in Fokker-Planck space. The paper also summarises the author's experiences with writing and using the corresponding modules of the Spinach library - the methods described below have enabled a large variety of simulations previously considered too complicated for routine practical use.