%0 Generic %1 Chen2012 %A Chen, Joe P. %D 2012 %K fractals %T Statistical mechanics of Bose gas in Sierpinski carpets %U http://arxiv.org/abs/1202.1274 %X 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.

@misc{Chen2012, abstract = { 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. }, added-at = {2012-02-08T00:36:41.000+0100}, author = {Chen, Joe P.}, biburl = {https://www.bibsonomy.org/bibtex/2f25dee92b315ce3f42b3fab9472847ff/vakaryuk}, description = {Statistical mechanics of Bose gas in Sierpinski carpets}, interhash = {726becc1c6c21f2ecc4e3a4864f64986}, intrahash = {f25dee92b315ce3f42b3fab9472847ff}, keywords = {fractals}, note = {cite arxiv:1202.1274 Comment: 26 pages, 5 figures, 1 table}, timestamp = {2012-02-08T00:36:41.000+0100}, title = {Statistical mechanics of Bose gas in Sierpinski carpets}, url = {http://arxiv.org/abs/1202.1274}, year = 2012 }