Thermodynamic fluctuations and the fluctuation-imposed limit for temperature measurement

In experimental sciences, random processes often place a fundamental limit on the achievable resolution. A well known example is the Johnson noise voltage across a resistor. In this paper, we describe the observation of temperature fluctuations in a thermometer caused by the random transfer of heat in and out of it. This process places a fundamental limit on the resolution of a thermometer. We find that the noise of the thermometer at low frequency is given by 4RkBT2 (in units of K2/H.z), where R is the thermal resistance that 1 inks the sensing element to the object whose temperature is to be measured. This implies that fo] noise reduction purposes, R is the only available engineering parameter to adjust. In a recent thermometer design, we have minimized R to achieve a resolution of 5x 10-1' K/~}lz. In thermodynamic terms, a fhermome?eris modeled as a subsystem in contact with a heat reservoir. The reservoir, the object whose temperature is to bc mc-asured, is assumed to be isothermal and to have infinite heat capacity. In equilibrium thermodynamics, thermodynamic properties of the subsystem are known to undergo spontaneous fluctuations even in equilibrium. The aim of a thermometer is to reconstruct the temperature of the reservoir from measurements of a temperature dependent quantity of the subsystem. The accuracy of temperature measurements is therefore limited by the fluctuations of the quantity as a result of random energy transfer between the subsystcm and the reservoir. In a macroscopic systelo, equilibrium thermodynamics tells us that fractional fluctuations go approximately as llfi, where N is the number of particles. N is perhaps 1022 for a typical thermometer, therefore causing a fractional fluctuations of the order of 10-1 1. Although this value is very small, it is within the range of resolution of state-of-the-art thermomctry with reported resolution of the order of 10-9 to 10-'0 Ww'llz'. In a recent paper2, the noise spcctrurn from one such thermometer was measured and was found to agree very well with the fluctuation dissipation theorem (FDT). The FDT is an extension of equilibrium thermodynamics that includes time dependent fluctuations. In an experimental setup not much different from an ordinary incandescent light bulb, the intensity of light from a hot filament was observed to fluctuate with an amplitude consistent with the theory of thermodynamic fluctuations. In the following, we describe results of two experiments which further clarify the mechanism which limits the resolution of …