1 Time - Resolved Measurements of Thermodynamic Fluctuations of the Particle Number in a Dilute Fermi Gas

We report on time-resolved measurements of thermodynamic fluctuations in the number of particles in a dilute Fermi gas. The gas is comprised of thermal quasiparticles, confined in a superconducting Al box by large-gap Ta leads. The average number of quasiparticles is about 10 5. This number fluctuates due to the quasiparticle generation and recombination processes. The number is measured from the tunneling current through a barrier that bisects the box. The recombination time is independently measured by single-photon excitation and agrees with the frequency dependence of the fluctuations. Statistical mechanics elucidates the microscopic origins of the laws of thermodynamics, connecting the thermodynamic quantities of a system to ensemble averages of microscopic quantities. Thermodynamics is successful because the fluctuations of thermodynamic quantities are relatively small in macroscopic systems. However, fluctuations of thermodynamic quantities can be significant in "small" systems. For example, a small container filled with a gas that can exchange energy and particles with a reservoir is described by the grand canonical ensemble of statistical mechanics. In equilibrium, the chemical potential and temperature of the gas and the reservoir are equal and constant. The energy and number of particles in the container are variables that fluctuate [1]. If there are N o particles on average, the rms amplitude of the fluctuation in the number is exactly (N o) 1/2. Thus, the relative fluctuations are large if N o is small. To assess the consequences of the fluctuations, we must also consider their dynamics. For example, when the gas is in equilibrium with the reservoir, the average flow of particles entering the gas from the reservoir cancels the average flow of particles leaving the gas. However, these two flows do not exactly cancel at all times and the number of particles fluctuates. If the characteristic time that a particle spends in the gas before returning to the reservoir is τ, then the spectrum of the fluctuations extends to a characteristic frequency ~ 1/τ. In this Letter we present time-resolved measurements of thermodynamic fluctuations in a small, grand canonical system. We measure the number of quasiparticles, as a function of time,