Thermal Spectra Without Detailed Balance

TL;DR

This paper demonstrates that thermal spectra can arise without actual thermalization of the probe, challenging the common interpretation that spectra indicate detailed balance.

Contribution

It introduces a kernel-based criterion to distinguish genuine thermal equilibrium from spectra produced by emission kernel structure.

Findings

Thermal spectra can be generated without probe thermalization.

A specific class of emission kernels can produce thermal spectra.

The study provides a criterion to identify true thermal equilibrium.

Abstract

A thermal spectrum is often taken as a signature that the emitted probe has reached detailed balance with the surrounding medium. We show that this interpretation is not generally valid by studying how the microscopic emission kernel determines the macroscopic spectrum. In $3+1$ dimensions, a simple thermal spectrum can be generated without probe thermalization when the relevant kernel belongs to a thermally degenerate class. A representative case is realized when the differential cross section depends on the scattering angle but carries no additional dependence on the Mandelstam variable $s$, as in low-energy Thomson scattering. Our results provide a kernel-based criterion for distinguishing genuine probe--medium exchange equilibrium from thermal spectra produced by the structure of the emission kernel itself.