🔬 condensed matter

Nonlocal thermal noise in electrically coupled conductors: A microscopic two-dimensional study

Microscopic two-dimensional simulations of capacitively coupled conductors at different temperatures reveal that finite correlations between distant electromotive forces arise, challenging the validity of the standard local-noise representation of thermal noise in nonequilibrium electrically coupled systems.

Original authors: Jorge Berger

Published 2026-08-27

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CC BY 4.0

Original authors: Jorge Berger

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

In the quiet hum of a warm wire, invisible chaos reigns. Charged particles, such as electrons, jitter and collide constantly due to heat, creating a faint, random electrical signal known as thermal noise. For nearly a century, scientists have relied on a simple rule to describe this phenomenon: the noise generated in any small part of a conductor depends only on the temperature of that specific part. This idea, rooted in the work of physicists like John Johnson and Harry Nyquist, suggests that if you have two pieces of wire sitting side by side but kept at different temperatures, the electrical noise in the cold wire should care nothing about the heat in the hot wire, provided they are not directly touching. They are viewed as independent sources of static, each whispering its own story based on its local warmth. This assumption allows engineers to treat complex circuits as simple collections of independent parts, a view that has held up well in equilibrium. But what happens when these parts are electrically linked, yet held at different temperatures? Does the heat from one side truly stay on its own side, or does the electrical connection allow the noise to leak across, defying the old rule?

A researcher named Jorge Berger set out to test this boundary using a virtual laboratory built from the ground up. Instead of measuring real wires, which would be incredibly difficult to isolate from other environmental factors, he constructed a detailed computer simulation of charged particles moving inside two long, thin rectangular channels. These channels represent the wires. The particles inside were programmed to behave like electrons in a metal: they repel each other, they bounce off the walls, and they collide with an invisible lattice that acts as a heat bath, resetting their speed to match a specific temperature. To link the two wires without letting the particles themselves jump from one to the other, Berger placed ideal capacitors between them. In the real world, a capacitor allows electrical current to flow while blocking the physical passage of matter, acting like a bridge for electricity but a wall for particles. By keeping the two wires at different temperatures—one cool, one hot—and connecting them only through these capacitors, he created a scenario where the wires were electrically coupled but physically separated.

The simulation ran for a vast amount of virtual time, tracking the motion of hundreds of particles as they collided billions of times. The goal was to measure the random electrical fluctuations, or noise, generated in each wire. When the wires were kept at the same temperature, the simulation behaved exactly as the established theory predicted: the noise in each wire depended only on its own temperature, and the fluctuations in the two wires were completely uncorrelated, as if they were strangers in a crowd. However, the moment the temperatures were set to be different, the story changed. The simulation revealed that the electrical noise in the cold wire was no longer determined solely by its own chill. It was subtly influenced by the heat of the neighboring wire. The random electrical pushes, known as electromotive forces, generated in distant segments of the two wires began to show a measurable connection. They started to move in a coordinated, albeit weak, dance, suggesting that the electrical link allowed the thermal agitation of one wire to reach across the gap and affect the other.

This finding challenges the long-held belief that thermal noise is strictly a local phenomenon. The research suggests that even when particles interact only over very short distances and the wires are separated by a barrier that stops physical contact, the electrical coupling is enough to create a shared noise environment. The effect was not a dramatic takeover of one wire by the other; the change in the noise level was small, shifting the apparent temperature of the noise by a tiny fraction compared to the actual temperature difference. Yet, the correlation was real and systematic. The strength of this connection depended on the length of the wires and the difference in their temperatures. In the simulation, the wires were long enough that the particles could pass one another, ensuring that the result was not an artifact of a one-dimensional bottleneck where particles are forced to line up. The effect persisted even though the particles themselves never crossed the gap, driven only by the electrostatic forces transmitted through the capacitors.

The study also looked at how energy moved between the wires. It found that while some heat did transfer from the hot wire to the cold one, the change in the noise pattern was larger than what could be explained by this simple heat flow alone. This implies that the mechanism linking the noise is distinct from the mechanism that equalizes temperature. The researchers suspect that the repulsion between the charged particles, which keeps them spread out and maintains electrical neutrality, acts as the medium that transmits this influence. It is as if the particles in the cold wire feel the pressure of the crowded, energetic particles in the hot wire through the electric field, even without touching. The results suggest that in circuits where components are at different temperatures, the standard method of calculating noise by simply adding up local contributions may be slightly inaccurate. The noise in one part of the circuit may carry a faint echo of the temperature of another part, a reminder that in the world of electricity, nothing is ever truly isolated.

Technical Summary: Nonlocal Thermal Noise in Electrically Coupled Conductors

Problem Statement
The Johnson-Nyquist theory of thermal noise typically models a conductor as a collection of independent local noise sources, where the noise strength is determined solely by the local temperature. This "local-noise" representation, often justified by the Fluctuation-Dissipation Theorem (FDT) in equilibrium, assumes that the electromotive force (emf) generated in a conductor depends only on the agitation of charges within that specific conductor. The central question addressed in this work is whether this locality assumption remains valid for electrically coupled conductors maintained at different temperatures. Specifically, the study investigates if electrical coupling and short-range particle interactions can induce correlations between the thermal noise of distant segments, thereby invalidating the independent-local-source picture in nonequilibrium steady states.

Methodology
The author employs microscopic two-dimensional simulations of interacting classical charge carriers to model conducting wires.

Particle Model: The system consists of $N$ particles with mass $m$ and charge $q$ , interacting via a short-range, screened repulsive potential designed to mimic electron-electron repulsion and ensure near-perfect electroneutrality.
Thermalization: Particles collide with lattice deformations (phonons/defects) at regular intervals $\tau$ . Upon collision, velocities are randomized to a Maxwellian distribution corresponding to a local bath temperature.
Geometry: Wires are modeled as rectangles with length $L \gg$ width $W$ . To simulate open circuits and avoid particle leakage between coupled wires, the wires are connected via ideal capacitors. These capacitors allow current continuity while preventing particle transfer, maintaining distinct temperatures ( $T_1$ and $T_2$ ) in the two wires.
Definitions: Instantaneous current and voltage are defined via spatial averages of particle momentum and forces. The emf ( $\epsilon$ ) is derived from the measured voltage ( $V$ ) and current ( $I$ ) using the relation $\epsilon = V + RI$ .
Validation: The model is first validated against Ohm's law, Drude model impedance, and the standard Nyquist formula for single wires at uniform temperatures. The 2D nature of the simulation is specifically chosen to ensure particles can bypass one another, avoiding artifacts associated with strictly 1D "Luttinger liquid" behavior found in previous work.

Key Contributions and Results

Validation of Local Noise in Equilibrium: When both wires are at the same temperature, the simulations reproduce Ohm's law, equilibrium Johnson noise, and vanishing correlations between detached wire segments, consistent with the standard local-noise picture.
Nonlocal Noise in Nonequilibrium: When the coupled wires are held at different temperatures ( $T_1 \neq T_2$ $T_{1} \neq = T_{2}$ ), the study finds:
- Systematic Deviations: The Johnson temperature inferred from the local-noise picture deviates systematically from the actual local temperature. Specifically, the variance of the emf in a wire at temperature $T_i$ is influenced by the temperature of the coupled wire.
- Correlations: Finite correlations develop between the electromotive forces generated in distant segments of the two wires. The correlation coefficient $r(\epsilon_1, \epsilon_2)$ is non-zero and depends on the temperature difference and the relative lengths of the wires.
- Persistence: These effects persist despite the microscopic particle interactions being short-ranged and the wires interacting only through capacitive coupling. The effect extends to distances significantly larger than the particle interaction range.
Mechanism: The results suggest that the strong repulsion between charges, which enforces approximate incompressibility (Kirchhoff's law for particle flow), acts as the agent transmitting the influence between resistors. The study distinguishes this noise transfer from simple energy transfer, noting that the change in Johnson noise cannot be fully explained by the difference between particle and lattice temperatures.

Significance and Claims
The paper claims that the independent-local-source representation of thermal noise may not remain valid in electrically coupled nonequilibrium conductors.

Challenge to Locality: The findings challenge the assumption that thermal noise in a resistor is determined solely by its local temperature, even when the coupling is purely capacitive and particle exchange is absent.
Distinction from Previous Work: Unlike the author's previous 1D study, this 2D investigation eliminates the possibility that correlations are artifacts of particles being unable to pass one another. The results are presented as a "cleaner" test of the nonlocal noise hypothesis.
Scope: The study is limited to the quasistatic regime where radiation is negligible. The author notes that the propagation speed of this noise remains an open question, conjecturing it may be acoustic-like rather than diffusive.
Modesty: The author acknowledges that the magnitude of the effect in their simulations is small (shifts in Johnson temperature are orders of magnitude smaller than the temperature difference). They suggest that in real experiments, where incompressibility is better approximated, the influence between resistors might be observable at even longer distances than predicted by their specific simulation parameters. The paper concludes that while the effect is small, it indicates a fundamental limitation in applying the local-noise picture to coupled nonequilibrium systems.

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