Fokker-Planck formalism in magnetic resonance 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, 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.

1H line width dependence on MAS speed in solid state NMR – Comparison of experiment and simulation

“…The routines for the Fokker-Plank calculations [57] of the Spinach library were executed from a program controlling the input of the COSMOS coordinate files, data handling, powder averaging which also allowed to set the parameters of the NMR experiment. The simulation routine was complemented by a script for exporting graphics and spectra.…”

Microscopic interpretation and generalization of the Bloch-Torrey equation for diffusion magnetic resonance

“…In future work, the microscopic model can be further generalized to non-Markovian systems which may also take into account interactions between pores [45, 46]. The model can be generalized further, for example, by including hydrodynamic flow or considering more general Louiville dynamics of spins [47]. While our simulations demonstrated the effect of inhomogeneous diffusion in space, they are equally applicable to dynamic diffusion changes over time.…”

Separating the coherence transfer from chemical shift evolution in high‐resolution pure shift COSY NMR

“…The CT-COSY spectrum (Figure 6, top) was acquired using a previously described pulse sequence. [15] Theoretical simulations were carried out for the disconnected 11-spin system on the "right hand side" of rotenone molecule (Figure 1) using the Fokker-Planck theory [24] module of Spinach library. [25] IK-2 basis set was used [26] with 100 spatial discretisation points.…”