Nonlocal Singular Interaction Weight

• Nonlocal singular interaction weights are functions or operators that characterize spatial coupling by combining nonlocality with singular behavior, as seen in Coulomb-type and power-law kernels.
• They enable rigorous modeling of interacting particle systems and continuum limits in PDEs by using fine analytic, variational, and entropy methods to manage divergence at zero distance.
• Applications span fields such as opinion dynamics, fluid alignment, and quantum evolution, with numerical schemes like finite element and spectral methods facilitating efficient simulation.

A nonlocal singular interaction weight is a function or operator defining the spatial coupling (interaction) between system components, characterized both by nonlocality—dependence on separation or relative configuration rather than only local values—and by a singularity, usually a divergence at zero relative distance or near-coincidence. Such objects are central in mathematical modeling of interacting particle systems, statistical mechanics, PDEs with nonlocal terms, and quantum dynamics. Their study involves fine analytic, variational, and numerical techniques to rigorously handle singularities and nonlocal contributions at both the microscopic and mean-field (continuum) levels.

1. Formal Definitions and Prototypical Forms

A nonlocal singular interaction weight is typically instantiated as a kernel $K(x-y)$ or $k(x-y)$, often entering a nonlocal operator, interaction energy, or stochastic differential equation. The singularity commonly occurs at $x=y$, and the nonlocality allows the kernel to have long-range or directionally structured effects. The most prominent examples include:

• Coulomb-type kernels: $K(x) = \mathrm{sgn}(x)$ in 1D (Coulomb/Newtonian potential), or $W_0(x) = -\log|x|$ in 2D, $W_0(x) = |x|^{2-d}$ in $d \geq 3$ (Mora, 2023).
• Power-law kernels: $k(z) \sim |z|^{-(d+\alpha)}$ for $\alpha > 0$; fractional Laplacian weights in PDEs (Abels et al., 1 Dec 2025).
• Weighted communication/interaction functions: $\Psi(x, y) = \|x-y\|^{-\alpha}$, $0 < \alpha < d$ (Luçon et al., 2013), or $\psi(r) \sim r^{-\alpha}$ for velocity-alignment in fluid/kinetic systems (Carrillo et al., 15 Oct 2025).

The singular behavior is governed by the decay or divergence rate as $|x-y| \to 0$, and the (potentially) anisotropic or inhomogeneous structure determines the nonlocal coupling.

2. Microscopic Models and Mean-Field Limits

In agent-based or particle systems, nonlocal singular weights govern the interaction terms in ODE/SDE systems:

$$\dot{x}_i = \frac{1}{N} \sum_{j=1}^N w_j(t) K\left(x_j(t) - x_i(t)\right)$$

with possible time-dependent weights $w_j(t)$ reflecting inhomogeneity or population fractions (Porat et al., 2023).

The mean-field (large-$N$) limit yields deterministic evolution equations for the empirical measure $\mu^N$ converging to a PDE with nonlocal singular terms:

$$\partial_t\mu + \partial_x\left[ \mu (K * \mu) \right] = \mu (S * \mu)$$

where $(K * \mu)(x) = \int K(x-y) \mu(y) \,dy$ (Porat et al., 2023, Bonaschi et al., 2013).

Singular weights $K$ require the use of entropy solutions and BV theory to rigorously formulate and prove well-posedness in the continuum.

3. Key Analytical and Mathematical Properties

Table: Singular Kernels and Typical Properties

Kernel Form	Singularity	Domain(s)
$K(x) = \mathrm{sgn}(x)$	Jump at $x=0$	$\mathbb{R}$
$k(z) \sim |z|^{-(d+\alpha)}$	Blow-up as $z\to0$	$\mathbb{R}^d$
$W(x)$ Coulomb/Log	Diverge as $x\to0$	$\mathbb{R}^2, \mathbb{R}^3$
General $\Psi(x,y)$	$\|x-y\|^{-\alpha}$	Lattice/Continuous

Singular weights pose challenges for definition of interaction energies, convolution operators, and for ensuring tightness and convergence of empirical measures, particularly in regimes where critical exponents (e.g., $\alpha \geq d/2$) generate anomalous fluctuations or slow propagation of chaos (Luçon et al., 2013).

Cut-off procedures, normalization, and moment constraints are imposed to ensure the kernel is integrable away from the singularity, and to guarantee desirable limits (e.g., $\Gamma$-convergence to local forms (Abels et al., 1 Dec 2025), consistency with the Laplacian or Dirichlet energies (Hurm et al., 2024)).

4. Entropy Methods, Well-posedness, and Uniqueness

The analysis of nonlocal singular weights in deterministic and stochastic PDEs relies on entropy inequalities and variational methods:

• Kružkov entropy solutions: For scalar conservation laws with singular nonlocal source terms, solutions are sought in the space of functions of bounded variation (BV), incorporating entropy inequalities to select physically relevant solutions and handle discontinuities induced by singular weights (Porat et al., 2023, Friedrich et al., 2022, Bonaschi et al., 2013).
• Energy dissipation and convexity: In interaction energy formulations, strict convexity or positivity of the Fourier transform of the singular kernel ensures uniqueness and stability of minimizers (Mora, 2023).
• Extended subdifferential calculus: For gradient flow of energies with non-smooth (e.g., $|x|$) singularities, existence and uniqueness often require moving to the framework of extended Fréchet subdifferentials at measures with atoms or singular support (Bonaschi et al., 2013).

Propagation of chaos, law of large numbers bounds, and entropy dissipation are established for particle systems interacting via singular weights under mild Lipschitz and moment assumptions, even in degenerate diffusive cases (Luçon et al., 2013).

5. Anisotropy, Inhomogeneous and Matrix-Valued Singular Weights

Nonlocal singular interaction weights may display anisotropy or matrix structure:

• Anisotropic kernels: $K(x) = |x|^{-(d+\alpha)} a(x/|x|)$ with $a$ even and uniformly elliptic, leading to directionally dependent diffusion in the limit (Abels et al., 1 Dec 2025). Shapes of minimizers in the associated energy problem can display geometric transitions (elliptic, filamentary, collapse to submanifolds) depending on the sign structure of $a$'s Fourier modes (Mora, 2023).
• Nonlocal weights in transmission/interface problems: In elliptic PDEs with nonlocal transmission conditions, the nonlocal weight may be a compact operator $K:L^2(\Sigma) \to L^2(\Sigma)$ acting in boundary conditions, potentially matrix-valued and integral operator-valued, leading to new spectral phenomena (e.g., accumulation of negative eigenvalues) (Heriban et al., 2024).
• Quantum dynamics and Wigner kernel regularization: Singular spatial potentials, e.g., $V(x) = \delta(x)$, $|x|^{-\alpha}$, generate nonlocal interaction weights in phase-space (Wigner kernel), which often regularize or redistribute singularity from position to momentum space, enabling accurate spectral discretization schemes (Shao et al., 2023).

6. Numerical Schemes and Discretization

Efficient numerical treatment of nonlocal operators with singular weights is essential for high-fidelity simulation:

• Finite element methods: For nonlocal Laplacian/Fractional Laplacian with singular kernels, the stiffness matrix entries are computed as explicit integrals involving B-splines, with singularity handled analytically near the diagonal and Gauss quadrature used elsewhere. The resulting stiffness matrix often exhibits Toeplitz/block-Toeplitz structure, allowing fast FFT-based solvers (Sheng et al., 29 Sep 2025).
• Spectral and operator-splitting methods: For quantum evolution with singular nonlocal weights (Wigner kernels), the nonlocal convolution is efficiently handled in Fourier space, as all the singularity is transferred to a smooth or weakly singular kernel, and high-order time-splitting can be used (Shao et al., 2023).

These methods typically exploit the compact support, symmetry, or moment structure of the singular weight for efficient integration and regularization.

7. Applications and Further Developments

Nonlocal singular interaction weights arise across a broad spectrum of applied and theoretical contexts:

• Opinion dynamics and social systems: Agent-based models with time-dependent weights and singular kernels capture aggregation, clustering, or alignment under strongly nonlocal (Coulombic) influence (Porat et al., 2023).
• Disordered diffusions and neural models: Power-law interaction weights are fundamental in modeling long-range dependence and synchronization phenomena in neuroscience (Luçon et al., 2013).
• Fluid alignment and flocking: Euler-alignment systems with singular communication weights exhibit sharp qualitative differences in convergence rates (exponential vs algebraic) determined by the local structure of the interaction weight (Carrillo et al., 15 Oct 2025).
• Transmission and interface problems: Nonlocal weights in boundary conditions induce novel spectral behavior in elliptic operators, relevant to quantum waveguides or interface models (Heriban et al., 2024).
• Cahn-Hilliard and phase separation: Nonlocal singular kernels govern the interface energy and dynamics in multi-phase models and converge, under suitable scaling, to anisotropic local diffusions (Abels et al., 1 Dec 2025, Hurm et al., 2024).

Singular nonlocal weights are linked to open problems in fluctuation theory, regularity of measure-valued solutions, and dimension reduction phenomena for minimizers. Comprehensive understanding and rigorous handling of such kernels are crucial for realistic and robust modeling of strongly interacting, inhomogeneous, and multi-scale systems.