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Voltage and current density in a rectangle with point in- and ejection via Green's functions and $q$ -analysis

This paper presents an exact and computationally efficient analytic formula for the potential and current density in a 2D rectangular domain with point source and sink, overcoming the slow convergence of traditional Fourier series by employing $q$ -analysis techniques related to Gaussian polynomials and Jacobi's theta-functions.

Original authors: P. Van Mieghem

Published 2026-09-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: P. Van Mieghem

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a flat, rectangular sheet of material, like a thin slice of metal or a board, where electricity flows from one specific point to another. In the real world, electricity does not travel in a straight line like a laser beam; instead, it spreads out, taking every possible path through the material to get from the starting point to the finish. This spreading behavior is governed by fundamental laws of physics that describe how electric fields and currents interact with space. Scientists have long been interested in predicting exactly how this current distributes itself across a surface, because knowing the path of least resistance helps in understanding everything from how power grids handle stress to how data moves through complex networks. When the material is a simple rectangle, the problem seems straightforward, but the mathematics required to describe the flow at every single point turns out to be surprisingly difficult, often leading to calculations that take forever to settle on a precise answer.

A researcher at the Delft University of Technology has now solved this classic problem by finding a new way to calculate the voltage and the strength of the electric current at any point inside a rectangular sheet. The study focuses on a scenario where a unit of current is injected at one specific spot and removed at another, creating a flow that spreads across the entire rectangle. While previous methods relied on adding up thousands of wave-like patterns to approximate the answer, these calculations were notoriously slow and often inaccurate, especially near the points where the current enters and leaves. The researcher discovered that by shifting the mathematical approach from standard wave patterns to a different class of functions known as elliptic functions, the problem could be solved with an exact formula that is also incredibly fast to compute.

The key to this breakthrough was realizing that the standard way of solving the equations, which involves summing up double layers of waves, converges too slowly to be practical. It is like trying to fill a bucket by pouring in water drop by drop when the bucket is huge; you would wait an eternity to see it full. The researcher instead turned to a mathematical tool called q-analysis, which uses a specific type of product sequence related to Gaussian polynomials and theta functions. These are sophisticated mathematical objects that have appeared in other famous physics problems, such as the study of how atoms arrange themselves in crystals. By expressing the solution in terms of these functions, the researcher derived a formula where the answer stabilizes almost instantly. In fact, for a square-shaped rectangle, the formula is so efficient that using just the very first term of the calculation produces a result that is visually indistinguishable from using hundreds of terms. This means that the complex behavior of the current can be captured with a simple, closed-form expression rather than a massive, slow-moving calculation.

The study also addresses a subtle but important physical limitation. When trying to model current entering a two-dimensional sheet at a single, infinitely small point, the mathematics suggests that the resistance between the entry and exit points becomes infinite, which is physically impossible. The researcher explains that this is a flaw in the idealized model of a "point" injection, as real-world currents always enter over a small, finite area. By adjusting the model to account for a small rectangular area of entry rather than a single point, the infinite resistance disappears, and the calculations become physically meaningful. The final result provides a precise map of the current density, showing exactly how strong the flow is at every location on the rectangle. This map reveals that the current is not uniform; it concentrates heavily along the path between the source and the destination, but also spreads out in a specific, predictable pattern that fills the rest of the rectangle.

This work is significant because it offers a practical tool for understanding flow in networks. The researcher suggests that in large, complex networks where the exact connections are not fully known, the path that data or resources take is likely very similar to the path of least resistance in this continuous electrical model. By knowing the exact distribution of current in a rectangle, scientists can better estimate which nodes in a network are most critical for the shortest path between two points. This has implications for securing networks against attacks or optimizing the flow of traffic in transportation systems. The paper does not just offer a theoretical curiosity; it provides a computationally efficient method to calculate these values, turning a problem that was once a computational bottleneck into one that can be solved with minimal effort. The findings confirm that while the underlying physics is complex, the mathematical description can be surprisingly elegant and fast, provided one uses the right set of tools to look at it.

Technical Summary: Voltage and Current Density in a Rectangle with Point In- and Ejection via Green's Functions and q-analysis

Problem Statement
The paper addresses the classical problem of computing the electric potential and current density within a two-dimensional rectangular domain where a unit current is injected at a source point $s$ and ejected at a destination point $d$ . While the potential in a 3D rectangular beam can be solved via standard separation of variables leading to a double Fourier series, the reduction to a 2D limit (where the beam thickness $c \to 0$ ) presents significant computational challenges. Specifically, the double Fourier series representation of the current density converges extremely slowly, rendering it impractical for numerical evaluation. Furthermore, the paper notes that the "effective resistance" between two point charges in a 2D continuum diverges, a modeling artifact arising from the zero-dimensional nature of point sources.

Methodology
The author employs a multi-stage analytical approach:

3D Formulation: The problem is first solved in a 3D rectangular parallelepiped with Neumann boundary conditions (no current flow through the sides). Using separation of variables, the potential is expressed as a double Fourier series involving cosine terms in $x$ and $y$ , and hyperbolic cosine terms in $z$ .
Dimensional Reduction: The 2D solution is derived by taking the limit as the thickness $c \to 0$ . This requires a renormalization argument ( $\rho I / c = 1$ ) to handle the singularity of point injection, effectively converting the problem into a Poisson equation with opposite delta functions at $s$ and $d$ .
Quasi-Green's Function: Recognizing that a standard Green's function does not exist for Neumann boundary conditions with a single delta source (due to the requirement of zero net charge), the author constructs a "quasi-Green's function" representing the difference between two opposite delta sources. This is initially formulated as a single Fourier series, which converges faster than the double series but still exhibits slow convergence and discontinuities in derivatives at the source/sink lines.
q-Analysis and Elliptic Functions: The core methodological innovation involves transforming the single Fourier series into a $q$ -series. By utilizing properties of Gaussian polynomials and Jacobi's theta functions, the infinite series is re-expressed as an infinite product involving a parameter $q = e^{-2\pi b/a}$ . This transformation leverages the rapid convergence of $q$ -products (where $q$ is typically very small, e.g., $e^{-2\pi} \approx 1/535$ for a square).

Key Contributions and Results

Exact Analytic Formula: The paper derives an exact, closed-form analytic formula for the potential and the magnitude of the current density in a rectangle. The potential is expressed via a quasi-Green's function $K(x, y; \xi, \eta)$ , which is decomposed into a linear term and a $q$ -function $T_q$ .
Computational Efficiency: The primary result is the demonstration that the current density can be computed with extreme efficiency using the $q$ -series. The paper shows that for a square domain, the series converges so rapidly that a single term ( $k=0$ ) provides visual and numerical accuracy indistinguishable from series with many terms. This contrasts sharply with the slow convergence of the traditional double Fourier series.
Resolution of Divergence: The paper rigorously demonstrates that the effective resistance between two point sources in a 2D continuum is infinite (divergent). It clarifies that this is a consequence of the point-source model (delta functions) rather than a physical reality, and suggests that modeling injection over a finite area resolves this divergence.
Connection to Special Functions: The work explicitly links the solution of the Laplacian in a rectangle to the theory of elliptic functions, specifically showing how the solution relates to Jacobi theta functions and Gaussian polynomials.

Significance and Claims
The paper positions itself as an exploratory journey from classical Fourier analysis to the more advanced field of $q$ -analysis. The author claims that the derived $q$ -series formula is the "best accuracy with least computational effort" for this problem. The significance lies in:

Providing a computationally tractable solution to a problem where standard methods (double Fourier series) fail due to slow convergence.
Offering a mathematical bridge between electrostatics in bounded domains and the theory of elliptic functions, echoing the work of Onsager on the 2D Ising model.
Highlighting the limitations of point-source models in 2D continuum networks (specifically regarding effective resistance) and the necessity of finite-area modeling for physical consistency.

The author does not propose new experimental setups or broad applications beyond the context of analyzing shortest paths in random geometric graphs (HRGG), where the current density is conjectured to be proportional to the probability of a node lying on a shortest path. The work remains strictly within the realm of analytical mathematics and theoretical physics.

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