Alladi Ramakrishnan Hall

Energetics of the dissipative quantum oscillator

Jasleen Kaur

Indian Institute of Technology, Bhubaneswar

In classical statistical mechanics, the average kinetic energy of a particle recieves equal contribution from all the accessible degrees of freedom at thermal equilibrium, irrespective of the
(conservative) potential, a very well known fact: Classical equipartition theorem, while the case
turns out to be different in quantum world. In this discussion, I will be presenting some results
for a quantum mechanical set-up in a situation involving ‘strong’ coupling between the system and
heat bath,as opposed to the commonly-employed weak-coupling approximation in the standard
approach to equilibrium statistical mechanics. The model system consists of a quantum harmonic
oscillator linearly coupled to a passive quantum heat bath through coordinate variables. Within
the framework of linear response theory, the Fluctuation-Dissipation Theorem of Callen-Welton
type provides a method for obtaining closed-form expressions for the average energies of the system. The mean kinetic and potential energies can be expressed as two-fold averages where the
first averaging is performed over the Gibbs canonical state of the quantum heat bath, the second
one is performed over well-defined and normalized probability distribution functions defined in the
‘frequency space’ of the heat bath. Such distribution functions are sensitive to the system parameters and the system-bath coupling strength. Finally, I will also briefly discuss the generalization of
these results for a charged oscillator in a magnetic field which is a dissipative version of the Landau
problem, relevant to diamagnetism and Hall effect.