Statistical mechanics of Bose gas in Sierpinski carpets

We carry out a mathematically rigorous investigation into the equilibrium thermodynamics of massless and massive bosons confined in generalized Sierpinski carpets (GSCs), a class of infinitely ramified fractals having non-integer Hausdorff dimensions $d_h$. Due to the anomalous walk dimension $d_w>2$ associated with Brownian motion on GSCs, all extensive thermodynamic quantities are shown to scale with the spectral volume with dimension $d_s = 2(d_h/d_w)$ rather than the Hausdorff volume. We prove that for a low-temperature, high-density ideal massive Bose gas in an unbounded GSC, Bose-Einstein condensation occurs if and only if $d_s>2$, or equivalently, if the Brownian motion on the GSC is transient. We also derive explicit expressions for the energy of blackbody radiation in a GSC, as well as the Casimir pressure on the parallel plate of a fractal waveguide modelled after a GSC. Our proofs involve extensive use of the spectral zeta function, obtained via a sharp estimate of the heat kernel trace. We believe that our results can be verified through photonic and cold atomic experiments on fractal structures.

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