Thermodynamics of photons on fractals

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.