Beyond the Formulations of the Fluctuation Dissipation Theorem Given by Callen and Welton (1951) and Expanded by Kubo (1966)

Lino Reggiani,Eleonora Alfinito

doi:10.3389/fphy.2020.00238

Abstract

The quantum formula of the fluctuation dissipation theorem (FDT) was given by Callen and Welton in 1951 for the case of conductors, and then expanded by Kubo in 1966. The drawback of these quantum relations concerns with the appearance of a zero-point contribution, ${\hbar \omega}/{2}$ with $\hbar$ the reduced Planck constant and $\omega$ the angular frequency of the considered photon, which implies a divergence of the fluctuation spectrum at increasing frequencies. This divergence is responsible for a vacuum-catastrophe, to keep the analogy with the well-known ultraviolet catastrophe of the classical black-body radiation spectrum. As a consequence, the quantum formulation of the FDT as given by Callen-Welton and Kubo introduces a Field Grand Challenge associated with the existence or less of a vacuum-fluctuations catastrophe for the energy-density spectrum. Here we propose a solution to this challenge by taking into account of the Casimir energy that, in turns, is found to be responsible for a quantum correction of the Stefan-Boltzmann law.

Highlights

The quantum form of the fluctuation dissipation theorem (FDT) was given by Callen and Welton in 1951 [1] for the case of conductors, and expanded by Kubo in 1957–1966 [2, 3]
The quantum form of the FDT as formulated by Callen-Welton and Kubo (CWK) introduces a Field Grand Challenge associated with the existence or less of a vacuum-fluctuations catastrophe for the energy spectrum
We reformulate the FDT by accounting for the presence of the Casimir energy and the associated Casimir force, a genuine quantum phenomena neglected by CWK

Summary

INTRODUCTION

The quantum form of the fluctuation dissipation theorem (FDT) was given by Callen and Welton in 1951 [1] for the case of conductors, and expanded by Kubo in 1957–1966 [2, 3]. The drawback of these quantum relations concerns with the appearance of a zero-point contribution, hω/2 with hthe reduced Planck constant and ω = 2πf the angular frequency of the considered photon, which implies a divergence of the energy-spectrum radiated by a physical system at a given temperature at increasing frequencies. By adding the zero-point contribution, the quantum formulation including zero-point contribution leads to UP,CW,K → ∞

FORMULATION OF THE FLUCTUATION DISSIPATION THEOREM

CONCLUSIONS