Thermodynamics of photons on fractals

TL;DR

This paper develops a thermodynamic framework for photons confined to fractal spaces, revealing unique relations between pressure, energy, and spectral dimension, and proposing experimental tests for these phenomena.

Contribution

It introduces a thermodynamic description of photons on fractals, defining spectral volume and energy, and explores how fractal geometry affects physical properties.

Findings

Derived an equation of state relating pressure and energy via spectral dimension

Defined spectral volume as a generalization of geometric volume for fractals

Proposed experimental methods to probe fractal thermodynamics

Abstract

A thermodynamical treatment of a massless scalar field (a "photon") confined to a fractal spatial manifold leads to an equation of state relating pressure to internal energy, $P V_s=U/d_s$, where $d_s$ is the spectral dimension and $V_s$ defines the "spectral volume". For regular manifolds, $V_s$ coincides with the usual geometric spatial volume, but on a fractal this is not necessarily the case. This is further evidence that on a fractal, momentum space can have a different dimension than position space. Our analysis also provides a natural definition of the vacuum (Casimir) energy of a fractal. We suggest ways that these unusual properties might be probed experimentally.