# Extended Nyquist formula for a resistance subject to a heat flow

## TL;DR

This paper extends the Nyquist formula to account for a resistance subjected to a heat flux between two different temperature sources, accurately describing thermal noise in non-equilibrium steady states.

## Contribution

It introduces an extended Nyquist formula applicable to non-uniform temperature fields in non-equilibrium conditions, validated through experimental measurements.

## Key findings

- The extended formula accurately predicts thermal noise in non-equilibrium steady states.
- Thermal fluctuations are equivalent to equilibrium at the mean temperature.
- Experimental results confirm the formula's precision.

## Abstract

The Nyquist formula quantifies the thermal noise driven fluctuations of voltage across a resistance in equilibrium. We deal here with the case of a resistance driven out of equilibrium by putting it in contact with two thermostats at different temperatures. We reach a non-equilibrium steady state where a heat flux is flowing through the resistance. Our measurements demonstrate anyway that a simple extension of the Nyquist formula to the non uniform temperature field describes with an excellent precision the thermal noise. For a metallic ohmic material, the fluctuations are actually equivalent to those of a resistance in equilibrium with a single thermostat at the mean temperature between the hot and cold sources.