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

TL;DR

The paper questions the locality of Johnson noise in circuits with nonuniform temperatures by proposing a nonlocal propagation mechanism for thermal fluctuations and by constructing a tractable 1D particle-gas model to simulate emf and current fluctuations. It demonstrates that, under nonuniform temperature, emf variances can deviate from Nyquist predictions and can exhibit correlations across distant segments, with deviations scaling inversely with the separation and increasing with interparticle coupling. The work finds that a hotter resistor can influence fluctuations in a colder region and that correlations between detatched segments emerge, implying a long-range coupling mediated by the circuit. These results suggest potential corrections to Johnson noise thermometry and point to new routes for noise control in nanodevices, while also acknowledging the limitations of a one-dimensional, classical model and calling for further experimental and theoretical exploration.

Abstract

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.

Figures (8)

• Figure 1: A circuit with two resistors. There are no power supplies in the circuit, but, due to thermal noise, there is a voltage drop (defined as positive at the right end) and a current (positive if clockwise).
• Figure 2: Experiment intended to determine which is the pertinent scenario. Two resistors are initially separated and their temperatures are different. The variance of the voltage between A and B is measured before and after the resistors touch each other and form a closed circuit.
• Figure 3: Expected deviations from electroneutrality in a closed circuit, for several parameters. In all cases, $k_c=50m/\tau^2$, $L=10\ell$, and there was a constant emf applied by an external source, equal to $0.5m\ell^2/q\tau^2$.
• Figure 4: Variances of the normalized efm [as defined in Eq. (\ref{['hatemf']})] in a closed circuit, as functions of the temperature. As shown in the inset, $OO$ ($OB$, $OA$, $BO$) has length $L$ ($3L/4$, $L/2$, $L/4$), where $L=40\ell$. The points for which the temperature is an odd (even) multiple of 0.125 were obtained setting $k_c=50m/\tau^2$ ($k_c=25m/\tau^2$); for other parameters, see text. For visibility, the results for different segments have been shifted upwards in steps of 2 units. The straight lines show the values predicted by Eq. (\ref{['varep']}). Since all the points in the circuit are equivalent, $\langle\hat{\varepsilon}\rangle =0$ for every segment.
• Figure 5: Variances of the normalized efm [as defined in Eq. (\ref{['hatemf']})] in an open circuit, as functions of the temperature. As shown in the inset, $O_1O_2$ ($O_1B$, $O_1A$, $BO_2$) has length $L$ ($3L/4$, $L/2$, $L/4$), where $L=40\ell$. All the parameters are the same as in Fig. \ref{['varT']}. For visibility, the results for different segments have been shifted upwards in steps of 2 units. The straight lines show the values predicted by Eq. (\ref{['varep']}).