Discrete and Weyl density of states for photons and phonons

TL;DR

This paper develops and compares discrete and Weyl-based density of states models for photons and phonons with linear dispersion, highlighting their accuracy at low energies and implications for thermodynamic calculations.

Contribution

It introduces a discrete DOS model for linearly dispersing particles and compares it with Weyl's conjecture-based DOS, extending previous quadratic dispersion work.

Findings

Discrete DOS reduces to continuum expressions in limits.

Relative errors of DOS and NOS match quadratic dispersion case.

Discrete DOS improves low-energy state calculations.

Abstract

The current density of states (DOS) calculations do not take into account the essential discreteness of the state space, since they rely on the unbounded continuum approximation. Recently, discrete DOS based on the quantum-mechanically allowable minimum energy interval has been introduced for quadratic dispersion relation. In this work, we consider systems exhibiting linear dispersion relation, particularly photons and phonons, and calculate the related density and number of states (NOS). Also, a Weyl's conjecture-based DOS function is calculated for photons and phonons by considering the bounded continuum approach. We show that discrete DOS function reduces to expressions of bounded and unbounded continua in the appropriate limits. The fluctuations in discrete DOS completely disappear under accumulation operators. It's interesting that relative errors of NOS and DOS functions with respect to discrete ones are exactly the same as the ones for quadratic dispersion relation. Furthermore, the application of discrete and Weyl DOS for the calculation of internal energy of a photon gas is presented and importance of discrete DOS is discussed. It's shown that discrete DOS function given in this work needs to be used whenever the low energy levels of a physical system are heavily occupied.