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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 $<ΔT^2> = \frac{k_B T^2}{C_v}$. Upon decreasing the size of the object, we argue, one necessarily reaches the quantum regime that we call quantum temperature fluctuations. At temperatures below $T^{*}\sim \hbar/k_{B}τ$, where $τ$ 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, $T^{*}$ can be on a scale of a few Kelvin.