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Equipartition laws of thermal noise

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Classically, the thermal noise in electricalRCcircuits andLCRseries circuits is governed by the equipartition lawfrac{1}{2}overline{CV^{2}} = frac{1}{2}kT, whereV(t)is the noise voltage developed acrossC. When quantum effects are taken into account, the equipartition law no longer holds forRCcircuits, although an equipartition law can be deemed for the measured mean square noise voltage under certain conditions. InLCRseries circuits the equipartition lawfrac{1}{2}overline{CV^{2}} = frac{1}{2}kT, changes intofrac{1}{2}overline{CV^{2}} = frac{1}{2}bar{E}(f_{0})for high-Qtuned circuits, wherebar{E}(f_{0})is the average energy of a harmonic oscillator tuned at the tuning frequency of the tuned circuit.

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Quantum Electronics, IEEE Journal of  (Volume:17 ,  Issue: 8 )