Phase transitions in finite systems: Influence of geometry on approach towards bulk critical behavior.

Abstract

We predict the manner in which a physical system, of size ${\mathrm{L}}^{\mathrm{d}\mathrm{\ensuremath{-}}\mathrm{d}\mathcal{'}}$\ifmmode\times\else\texttimes\fi{}${\mathrm{\ensuremath{\infty}}}^{\mathrm{d}\mathcal{'}}$, subject to periodic boundary conditions, approaches bulk critical behavior as L\ensuremath{\rightarrow}\ensuremath{\infty}. While for Tg${\mathrm{T}}_{\mathrm{c}}$(\ensuremath{\infty}) the approach is exponential, for T${\mathrm{T}}_{\mathrm{c}}$(\ensuremath{\infty}) it is generally governed by power laws whose indices are determined by the bulk exponents for the corresponding d- and d'-dimensional systems. Specific predictions on the spherical model of ferromagnetism and the relativistic Bose gas with pair production are verified by analytical results.