Weak Solutions to the Bloch Equations with Distant Dipolar Field

Abstract

The distant dipolar field (DDF) is a long-range, nonlocal contribution to liquid-state spin dynamics that arises from intermolecular dipolar couplings and can generate multiple-quantum coherences and novel MRI contrast. Its sign-changing kernel makes Bloch-DDF dynamics strongly geometry dependent, and FFT-based dipolar convolutions naturally assume periodic or padded Cartesian domains rather than bounded samples with reflective diffusion boundaries. We study the Bloch equations with the DDF on bounded domains under homogeneous Neumann diffusion conditions. We derive a finite-element weak formulation that supports spatially varying diffusion and relaxation parameters and uses a short-distance regularization of the secular DDF kernel with length a>0. For fixed a we prove boundedness of the DDF operator, establish an L2 energy balance in which precession is neutral while diffusion and transverse relaxation are dissipative, and obtain local well-posedness with continuous dependence on the data, with global existence under energy-neutral transport. For the Galerkin semi-discretization we show a discrete energy identity mirroring the continuum estimate. For computation, we evaluate the DDF in real space with a matrix-free near/far scheme and advance in time using a second-order IMEX splitting method that treats diffusion and relaxation implicitly and precession explicitly. The explicit stage applies a Rodrigues rotation at DDF quadrature points followed by an L2 projection, enabling stable multi-cycle lab-frame simulations. We validate against three closed-form benchmarks and quantify curved-boundary effects by comparing mapped finite elements with a voxel-mask finite-difference baseline on spherical Neumann eigenmode decay. These results provide an analyzable and reproducible route for Bloch-DDF dynamics on bounded domains with complex geometry.

Figures (9)

• Figure 1: Uniform-mode analytical benchmark. Numerical $S(t)=\langle M_x+iM_y\rangle$ (from the FE run), where $\langle\cdot\rangle$ denotes the spatial average over $\Omega$ of the corresponding FE field, compared against the closed-form solution \ref{['eq:uniform_S_sol']} with $\kappa_{\mathrm{eff}}$ computed directly from the regularized kernel and the stated quadrature rule. Panels show $\mathrm{Re}\,S(t)$, $\mathrm{Im}\,S(t)$, and $|S(t)|$.
• Figure 2: Periodic plane-wave analytical benchmark for a single Fourier mode. Numerical evolution of the mode amplitude $A(t)$ compared against the closed-form solution \ref{['eq:planewave_A_sol']} using the kernel symbol value $\lambda=-1.3889$ (regularization length $a=0.04$, mode $(m_x,m_y,m_z)=(1,0,0)$, and periodic box $\Omega=[0,1)^3$ discretized on a $32\times 32\times 32$ uniform grid). Panels show $\mathrm{Re}\,A(t)$ and $|A(t)|$. The plotted case corresponds to the smallest time step in \ref{['tab:planewave_dt']}.
• Figure 3: Long-time lab-frame evolution of the global transverse signal $S(t)=\langle M_x+iM_y\rangle$ with DDF enabled. Panels show (a) $\mathrm{Re}\,S(t)$, (b) $\mathrm{Im}\,S(t)$, and (c) the envelope $|S(t)|$ for the same run.
• Figure 4: Longitudinal diffusion+$T_1$ analytical benchmark. Time evolution of the modal coefficient $c(t)$ for the Neumann eigenmode \ref{['eq:neumann_phi']} compared against \ref{['eq:c_sol']}. For $(1,1,1)$ in the unit box, $\lambda=3\pi^2$ and the predicted decay rate is $D\lambda+T_1^{-1}=0.2296$.
• Figure 5: Envelope comparison with DDF off ($k=0$) versus DDF on ($k=5$) at otherwise fixed parameters.

Theorems & Definitions (30)

• Definition 2.1: Weak form of the Bloch--DDF system
• Definition 2.2: Galerkin FE formulation
• Remark 1: Non-energy-neutral advection
• Proposition A.1: Boundedness of $\mathcal{T}_a$
• proof
• Proposition A.2: Energy balance and $L^2$ estimate
• proof
• Proposition A.3: Continuous dependence
• proof
• Corollary A.4: Global existence under additional conditions