# Quantum dissipation and the virial theorem

## TL;DR

This paper explores the virial theorem in dissipative systems, both classical and quantum, revealing how noise and non-Markovian effects influence the theorem's formulation and applications in quantum oscillators and electrical circuits.

## Contribution

It derives a generalized virial theorem for dissipative quantum oscillators coupled with quantum heat baths, highlighting the impact of non-Markovian quantum noise.

## Key findings

- Non-Markovian quantum noise introduces novel terms in the virial theorem.
- Classical noise affects the virial theorem explicitly.
- The quantum virial theorem is extended to systems coupled with quantum heat baths.

## Abstract

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.