Quantum Nyquist Temperature Fluctuations

We consider the temperature fluctuations of a small object. Classical fluctuations of the temperature have been considered for a long time. Using the Nyquist approach, we show that the temperature of an object fluctuates when in a thermal contact with a reservoir. For large temperatures or large specific heat of the object $C_v$, we recover standard results of classical thermodynamic fluctuations $<\mathrm{\Delta }{T}^2>=\frac{k_B{T}^2}{C_v}$. Upon decreasing the size of the object, we argue, one necessarily reaches the quantum regime that we call quantum temperature fluctuations. At temperatures below $T^{\ast }\sim \mathrm{\hslash }\mathrm{/}{k}_B\tau$, where $\tau$ is the thermal relaxation time of the system, the fluctuations change the character and become quantum. For a nano-scale metallic particle in a good thermal contact with a reservoir, $T^{\ast }$ can be on a scale of a few Kelvin.