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This question is similar to the following, but I have expanded the question moderately:

Nonlinearities arising from linear equations

The Bloch equations are described by the following vector equation (ignoring relaxation): $$ \frac{d}{dt}\mathbf{M}(t) = \mathbf{M}(t) \times \gamma \mathbf{B}(t) $$

It is frequently stated that the Bloch equations are non-linear.

For example,

In Principles of Magnetic Resonance Imaging - A Signal Processing Perspective by Liang and Lauterbur (pg. 89), it is stated without elaboration that :

The linear system assumption is not valid for a nuclear spin system during excitation.

Additionally, in Principles of Magnetic Resonance by Nishimura (pg. 124), it states :

"... the nonlinear behavior of the spin system becomes appreciable."

Lastly, in Magnetic Resonance Imaging - Physical Principles and Sequence Design by Brown et al. (pg. 661), "Bloch equation nonlinearities" are listed as a reason for possible measurement error.

The equation listed above can be reformulated in the following manner: $$ \frac{d}{dt}\begin{bmatrix} M_x(t)\\M_y(t)\\M_z(t) \end{bmatrix}= \begin{bmatrix} 0 & \gamma B_z(t) & -\gamma B_y(t) \\ -\gamma B_z(t) & 0 & \gamma B_x(t) \\ \gamma B_y(t) & -\gamma B_x(t) & 0 \end{bmatrix} \begin{bmatrix} M_x(t)\\M_y(t)\\M_z(t) \end{bmatrix} $$

This seems like a linear differential equation to me. What do people mean when they refer to the Bloch equations as non-linear?

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The non linearities arise when you consider the feedback loop. The magnetic moment can generate a field of its own. $\mathbf{B}$ will no longer be the externally applied field, but will rather depend on $\mathbf{M}$ hence the nonlinearity. These nonlinearities give rise to new behaviors such as synchronization and chaotic motion.

Check out this article: Abergel D. "Chaotic solutions of the feedback driven Bloch equations." Phys Lett A 2002;302:17–22.

Hope this helps and tell me if you find some mistakes.

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  • $\begingroup$ Thank you for the article recommendation! It sent me down a really helpful rabbit hole, and I think I understand what people mean now. So, instead of saying "the Bloch equations are non-linear." People should instead say the cumbersome, but more accurate: "the Bloch equations, as classically written, represent a linear system. However, this overlooks the interactions of the average spin magnetization on induced magnetic fields which likewise contribute to the equations of motion for these systems." $\endgroup$
    – jmd
    Apr 27, 2022 at 22:00
  • $\begingroup$ Helpful resources I encountered for future readers: Vlassenbroek et al. J. Chem. Phys. 103 (14) 1995.; Bowtell et al., J. Magn. Reson. 88. 1990.; Jeener. Phys. Rev. Lett. 82 (8) 1999; Krishnan and Murali. Prog Nucl Magn Reson Spectrosc. 68. 2013. From the last article: "In the absence of collective effects, Bloch's differential equations for the components of average spin maggnetization are linear. The presence of an induced magnetic field [...] is explictly dependent on the average spin magnetization, making these equations to be non-linear." $\endgroup$
    – jmd
    Apr 27, 2022 at 22:13
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First, let me note that the equations given in the OP are not the full Bloch equations, which usually include the relaxation terms with characteristic times $T_1$ and $T_2$ for the longitudinal and the transverse (in respect to the magnetic field) spin components, see the Wikipedia article on the Bloch equations.

The linear system assumption is not valid for a nuclear spin system during excitation.

This claim does not actually imply that the equations are non-linear, but that linear response theory (e.g., in the standard Kubo-Greenwood form) cannot be used (directly) to analyze the magnetic response. Indeed, if we were to exclude two of the magnetization components, we would obtain a third-order equation with time-dependent coefficient, which is generally impossible to solve.

"... the nonlinear behavior of the spin system becomes appreciable."

In this case one means that the magnetic field includes the field generated by the magnetic moments themselves, i.e., $\mathbf{B}=\mathbf{B}(\mathbf{M})$, and the equations are thus truly non-linear. However, instead of Bloch equation it is more appropriate in this case to talk about the Landau-Lifshitz-Gilbert equation.

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  • $\begingroup$ Thank you very much for your detailed and thoughtful answer! I really appreciate the high-level summary in the last paragraph. I was previously unaware of the Landau-Lifshitz-Gilbert equation. $\endgroup$
    – jmd
    Apr 27, 2022 at 22:19

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