Non-local origin and correlations in the Johnson noise at nonuniform temperatures

We propose an alternative scenario for the propagation of thermal noise in a conductor. In this scenario, the noise in the emf (electromotive force) between two terminals cannot be described as a sum of contributions from uncorrelated regions, each in local thermal equilibrium. We review previous studies of thermal noise in circuits with nonuniform temperature. We suggest experiments that could distinguish between different scenarios. We build a workable 1D model for a gas of particles that undergo stochastic collisions with the lattice and exert distance-dependent forces on each other. We enunciate definitions of current, voltage, and emf, appropriate to a wire with limited number of particles. For uniform temperature, within appropriate length and temperature ranges, our simulations comply with Nyquist’s result. Analytic results can be obtained in the limit of strong interparticle interaction. The simulations indicate that (1) thermal noise in a resistor at uniform temperature within an electric circuit can be larger (smaller) than predicted by Nyquist due to the presence of a resistor with higher (lower) temperature in the circuit; (2) for sufficiently long circuits, the deviation from the Nyquist prediction is inversely proportional to the distance between the centers of the resistors; (3) if the resistors differ in temperature, their emf can be correlated, even if they are detached. The long-range repulsion between charges in electrically connected resistors may have conceptual and technological impact in nanodevices.

💡 Research Summary

The authors challenge the conventional Nyquist scenario (NS) for Johnson‑type thermal noise, which assumes that the fluctuating electromotive force (EMF) in a conductor can be expressed as a sum of independent contributions from locally equilibrated regions. They propose instead a non‑local scenario in which thermal motions of charge carriers generate plasma‑electromagnetic waves that propagate throughout the entire circuit, thereby coupling distant resistive elements. In this picture, the long‑range Coulomb repulsion between electrons, which enforces overall charge neutrality, acts as a fast information carrier that lets each resistor “sense” the resistance and temperature of other resistors in the same loop. Consequently, the variance of the EMF of a resistor can depend on the temperature of a remote resistor, even when the two are electrically isolated.

To test this hypothesis, the paper develops a tractable one‑dimensional classical particle model. A wire of length L contains N charged particles (charge q, mass m). Particles interact via a linear repulsive force k_c(ℓ−r) for separations r<ℓ (ℓ is a screening length) and do not interact for r>ℓ. Between stochastic “collisions” occurring at regular intervals τ, particles move under the combined action of these inter‑particle forces and any externally applied voltage. At each collision the particle velocities are reset to a Gaussian distribution with zero mean and variance k_B T/m, thereby mimicking thermalization with a lattice at temperature T. The model reduces to a modified Drude description with a dc resistance R_D = 2 m L²/(q² N τ).

Two essential definitions are introduced for the simulation: (i) instantaneous current ˜I = (q/L) Σ_i v_i, i.e. the spatial average of particle momentum, and (ii) EMF ε = V + I·R, where V is the voltage drop measured across the resistor. These definitions remain meaningful even when the particle number is too small to guarantee a uniform current in the macroscopic sense.

Simulation results fall into two regimes. In the weak‑interaction limit (k_c ℓ² ≪ k_B T) the system behaves like an ideal gas and reproduces the standard Nyquist result S(ε)=2k_B T R/Δt for uniform temperature. In the strong‑interaction regime (k_c ℓ² ≫ k_B T) the particles form an incompressible fluid‑like medium, and the long‑range repulsion transmits fluctuations across the whole wire. The authors find three characteristic non‑local effects:

1. Temperature‑dependent cross‑influence – If a circuit contains a resistor at a higher temperature than the rest, the Johnson noise of a uniform‑temperature resistor is amplified; conversely, a colder resistor suppresses the noise of its neighbors.

2. Distance scaling – For sufficiently long circuits the deviation of the noise variance from the Nyquist prediction scales inversely with the distance between the centers of the two resistors (ΔS ∝ 1/d).

3. EMF correlation – Even when two resistors are physically detached, their EMFs become positively correlated if their temperatures differ, reflecting the shared plasma‑wave field.

The paper revisits earlier experimental work by Monnet, Ciliberto and Bellon (MCB), who measured voltage fluctuations along a metallic alloy wire with alternating hot and cold segments. MCB reported that the effective fluctuation temperature T_fluc was lower than the spatial average temperature T_avg, contrary to the extended NS prediction that T_fluc > T_avg when (T−T)(ρ−ρ) > 0. The authors argue that the observed reduction can be explained by a “smearing” effect: the non‑local mechanism replaces the local temperature T(x) by a spatially averaged value, thereby diminishing the contribution of temperature‑resistivity correlations.

They also discuss multiterminal conductors studied by Sukhorukov and Loss (SL), whose Boltzmann‑Langevin treatment includes only local scattering and predicts that noise at one terminal depends on effective temperatures at the other terminals. The present non‑local framework provides a more natural explanation for such cross‑terminal dependencies, as the long‑range Coulomb interaction inherently couples all terminals.

To discriminate experimentally between the NS and the proposed non‑local scenario, the authors suggest a set of feasible tests. One involves two resistors initially kept at different temperatures and physically separated; a fast electronic switch (e.g., MOSFET) then connects them, and the voltage variance is recorded before, during, and after the connection. If the EMF variance of the colder resistor drops sharply after connection, this would support the non‑local picture. Another variant uses periodic connection/disconnection to extract a time lag between electrical coupling and the change in noise statistics, which should be absent in a purely local model.

In conclusion, the study provides a coherent theoretical and computational argument that Johnson noise in non‑uniform temperature circuits cannot be fully described by a sum of independent local contributions. Instead, the long‑range electrostatic repulsion among charge carriers creates a non‑local, wave‑mediated channel that correlates fluctuations across the entire circuit. This insight has practical implications for the design of low‑noise nano‑electronic devices, where temperature gradients and strong carrier interactions are common. Future work should focus on high‑precision, fast‑time‑scale experiments to validate the predicted distance scaling and EMF correlations, and on extending the model to fully quantum mechanical treatments relevant at cryogenic temperatures.