Equipartition laws of thermal noise

Classically, the thermal noise in electrical
RC
circuits and
LCR
series circuits is governed by the equipartition law
\frac{1}{2}\overline{CV^{2}} = \frac{1}{2}kT
, where
V(t)
is the noise voltage developed across
C
. When quantum effects are taken into account, the equipartition law no longer holds for
RC
circuits, although an equipartition law can be deemed for the measured mean square noise voltage under certain conditions. In
LCR
series circuits the equipartition law
\frac{1}{2}\overline{CV^{2}} = \frac{1}{2}kT
, changes into
\frac{1}{2}\overline{CV^{2}} = \frac{1}{2}\bar{E}(f_{0})
for high-
Q
tuned circuits, where
\bar{E}(f_{0})
is the average energy of a harmonic oscillator tuned at the tuning frequency of the tuned circuit.