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, stationary 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.
Highlights
Good theory papers have two essential features: they are readable and computable
This impedes the first step normally taken in magnetic resonance simulations – the rotating frame transformation – and makes Liouville - von Neumann equation simulations slow
This section contains a speculative overview of other types of magnetic resonance simulations and applications that could be enabled, or made simpler, by the Fokker-Planck formalism
Summary
Good theory papers have two essential features: they are readable and computable. That is the reason why the Liouville - von Neumann equation (on the coherent side) and Bloch-RedfieldWangsness / Lipari-Szabo theories (on the relaxation side) dominate magnetic resonance – there are papers and books that describe them with the eloquence and elegance of a well written detective story [1,2,3,4]. KÃqðtÞ; ð2Þ where the overbar indicates ensemble averaging and K is the kinetics superoperator that accounts for the possible presence of chemical processes in the system [2] This equation is currently the central pillar of most magnetic resonance simulation frameworks [12,13,14,15,16,17]. The biggest source of complications here is that the ‘‘invisible hand” of spatial dynamics makes even the ensemble-averaged spin Hamiltonian timedependent in non-trivial ways This can lead to spectacularly complex analytical solutions – the excellent Eq (38) in the recent paper by Scholz, Meier and Ernst [20] provides some encouragement to explore methods where the transition from the analytical to the numerical treatment happens earlier in the simulation flowchart. It produces major generalisations and simplifications across the simulation code – recent versions of Spinach [12] owe much of their versatility to the Fokker-Planck formalism
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