Thermodynamics of Photons on Fractals

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.

Figure 1

Typical small $\tau$ behavior of a fractal heat kernel trace. There are small oscillations [see inset], as a function of $\ln \tau$, about the leading behavior $\sim 1/{\tau }^{d_s/2}$.

Figure 2

The generic pole structure of the zeta function on a symmetric fractal [20, 21]. The dominant behavior comes from the real pole at $s=d_s/2$, and oscillations from the tower of complex poles in (21). There may be other poles in the shaded region to the left of this tower, but these do not contribute to the thermodynamic limit. On the Sierpinksi gasket, there is another tower of complex poles on the imaginary axis.