Kinetic theory says that the minimal temperature is zero kelvin, at which every motion is stopped. However, we do know that quantum theory says that we have unavoidable quantum fluctuations, so: what is the minimal quantum temperature that can be reach? Is there any known closed formula (for either bosons or fermions)? Remark: we do know that quantum black holes have a temperature is inversely proportional to black hole mass, and if we assume a $10^{11}-10^{12}$ solar mass black hole, that would give a minimal mass about 10 yoctokelvin. But this mass is macroscopic, is there any microscopic known limit on quantum solids?
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Approaching absolute zero means subtracting smaller and smaller amounts of energy from the system. Using the energy-time uncertainty relation, $\Delta E\Delta t\geq \hbar/2$ we conclude that the minimum attainable temperature is greater than: $$ T_{min}> \frac{\Delta E}{k_B}> \frac{\hbar}{2k_B t_{universe}}, $$ where $t_{universe}$ is the lifetime of the universe, although for amore appropriate extimate one should probably use the time since the mankind exists, or the duration of the current civilization, or, optimistically, the expected time till the end of humanity.
Roger V.
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