1202.1274 (Joe P. Chen)

Statistical mechanics of Bose gas in Sierpinski carpets    [PDF]

Joe P. Chen

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.

View original: http://arxiv.org/abs/1202.1274