In this note, we study the celebrated virial theorem for dissipative systems,
both classical and quantum. The classical formulation is discussed and an
intriguing effect of the random force (noise) is made explicit in the context
of the virial theorem. Subsequently, we derive a generalized virial theorem for
a dissipative quantum oscillator, i.e. a quantum oscillator coupled with a
quantum heat bath. Such a heat bath is modeled as an infinite collection of
independent quantum oscillators with a certain distribution of initial
conditions. In this situation, the non-Markovian nature of the quantum noise
leads to novel bath-induced terms in the virial theorem. We also consider the
case of an electrical circuit with thermal noise and analyze the role of
non-Markovian noise in the context of the virial theorem.Comment: v2: Revised version to appear in Physica