Thermodynamics of the harmonic oscillator: Wien’s displacement law and the Planck spectrum

Abstract

A thermodynamic analysis of the harmonic oscillator is presented. Motivation for the study is provided by the blackbody radiation spectrum; when blackbody radiation is regarded as a system of noninteracting harmonic oscillator modes, the thermodynamics follows from that of the harmonic oscillators. Using the behavior of a harmonic oscillator thermodynamic system under an adiabatic change of oscillator frequency ω, we show that the thermodynamic functions can all be derived from a single function of ω/T , analogous to Wien's displacement theorem. The high-and low-frequency energy limits allow asymptotic energy forms involving T alone or ω alone, corresponding to energy equipartition and zero-point energy. It is suggested that the "smoothest possible" function which behaves as a − bx at small values of x and vanishes at large x is ae −bx/a because it is a monotonic analytic function every derivative of which is a multiple of the function itself. In this sense, it is noted that the Planck spectrum with zero-point radiation corresponds to the function satisfying the Wien displacement result which provides the smoothest possible interpolation between energy equipartition at low frequency and zero-point energy at high frequency.

Quantum Mechanics as an Emergent Property of Ergodic Systems Embedded in the Zero-point Radiation Field

“…This is because the universal constant is introduced via the energy of the modes of the zpf, which is proportional to the frequency, contrary to classical energy equipartition among a set of oscillators in equilibrium. 13 In fact, it has been shown [30][31][32][33] that Planck's law is a necessary consequence of the presence of the zpf (without the need to resort to any explicit quantum assumption) while its fluctuations ultimately impose a minimum (non-zero) value for the dispersions of the dynamic variables, thus allowing for the existence of fluctuations at temperature T = 0-which are commonly recognized as quantum fluctuations-in contrast to the classical thermodynamic view. These observations, together with our present results, lead us to conclude that the transition from classical to quantum theory requires not just the introduction of a zpf, but of a fluctuating zpf.…”

Scaling, scattering, and blackbody radiation in classical physics

“…However, thermal radiation involves not coherent but rather random radiation over a spectrum of frequencies. As pointed out in an earlier analysis [15] of the thermodynamics of the harmonic oscillator (or of radiation normal modes), the principles of thermodynamics allow the possibility of zeropoint energy. Zero-point energy is random energy which exists even at the absolute zero of temperature and which takes the form…”

“…If we consider the thermodynamics of a harmonic oscillator and ask for the smoothest interpolation between zero-point energy (1/2) ω at low temperature and k B T at high temperature, then we derive the Planck blackbody spectrum within classical physics. [15] If we compare paramagnetic behavior with diamagnetic behavior in classical zero-point radiation while using relativistic limits consistently, then we derive the Planck spectrum within classical physics. [19] If we consider time-dilating conformal transformations of thermal radiation in a Minkowski coordinate frame and in a Rindler frame, then the Planck spectrum is derived within classical physics based upon the structure of relativistic spacetime.…”

Sub-Quantum Thermodynamics as a Basis of Emergent Quantum Mechanics

“…Note, however, that there exists a possibly related explanation of the universality of based on a thermodynamic analysis of the harmonic oscillator, i.e., T. Boyer's derivation [25], given in the context of a classical physics in the presence of a (-classical‖) zero-point energy field.…”