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.
Description
Statistical mechanics of Bose gas in Sierpinski carpets
%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
}