%0 Book Section %1 statphys23_0465 %A Kastening, B. %A Dohm, V. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K casimir finite-size force mean model scaling spherical statphys23 topic-2 %T Critical Casimir force scaling function of the mean spherical model in $2<d<3$ dimensions for nonperiodic boundary conditions %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=465 %X Little is known about finite-size effects of critical systems below the bulk transition temperature $T_c$ for realistic boundary conditions, such as of Dirichlet or Neumann type. However, the exactly solvable mean spherical model can play a significant role in elucidating finite-size effects in the entire scaling regime from above to below $T_c$. We present the first investigation of finite-size effects in this model with nonperiodic boundary conditions for temperatures below $T_c$. To avoid nonscaling effects present in $d=3$ dimensions 1, we continue the model to $2<d<3$, where scaling is known to hold even for nonperiodic boundary conditions 2. Explicit results are obtained in film geometry for the universal scaling functions of the excess free energy and the Casimir force for Dirichlet and Neumann boundary conditions. For appropriate bulk correlation lengths $\xi_+$ and $\xi_-$ above and below $T_c$, respectively, we find an unexpected size dependence $\propto\xi_\pm^-1/\nuL^-2$ of the Casimir force for large $L/\xi_\pm$. The behavior above $T_c$ originates from the combined action of the spherical constraint and the presence of a surface free energy. It differs from the predictions of standard finite-size theories 3. The behavior below $T_c$ arises from a nonzero energy of the lowest mode. The results for both above and below $T_c$ differ qualitatively from the predictions of current finite-size theories for the critical Casimir force above and below the lambda point of $^4$He 4. Possible applications of our results include the weakly interacting Bose gas in the precritical regions above and below $T_c$. Support by DLR is gratefully acknowledged. 1 M.N.~Barber and M.E.~Fisher, Annals of Physics (N.Y.) 77, 1 (1973). 2 X.S.~Chen and V.~Dohm, Phys.\ Rev.\ E67, 056127 (2003). 3 V.~Privman, P.C.~Hohenberg, and A.~Aharony, in Phase Transitions and Critical Phenomena, C.~Domb and J.L.~Lebowitz, eds., Vol.\ 14 (Academic, NY, 1991); M.~Krech, The Casimir Effect in Critical Systems, (World Scientific, Singapore, 1994). 4 D.~Dantchev, M.~Krech, and S.~Dietrich, Phys.\ Rev.\ Lett.\ 95, 259701 (2005); R.~Zandi, A.~Shackell, J.~Rudnick, M.~Kardar, and L.P.~Chayes, cond-mat/0703262.

@incollection{statphys23_0465, abstract = {Little is known about finite-size effects of critical systems below the bulk transition temperature $T_c$ for realistic boundary conditions, such as of Dirichlet or Neumann type. However, the exactly solvable mean spherical model can play a significant role in elucidating finite-size effects in the entire scaling regime from above to below $T_c$. We present the first investigation of finite-size effects in this model with nonperiodic boundary conditions for temperatures below $T_c$. To avoid nonscaling effects present in $d=3$ dimensions [1], we continue the model to $2<d<3$, where scaling is known to hold even for nonperiodic boundary conditions [2]. Explicit results are obtained in film geometry for the universal scaling functions of the excess free energy and the Casimir force for Dirichlet and Neumann boundary conditions. For appropriate bulk correlation lengths $\xi_+$ and $\xi_-$ above and below $T_c$, respectively, we find an unexpected size dependence $\propto\xi_\pm^{-1/\nu}L^{-2}$ of the Casimir force for large $L/\xi_\pm$. The behavior above $T_c$ originates from the combined action of the spherical constraint and the presence of a surface free energy. It differs from the predictions of standard finite-size theories [3]. The behavior below $T_c$ arises from a nonzero energy of the lowest mode. The results for both above and below $T_c$ differ qualitatively from the predictions of current finite-size theories for the critical Casimir force above and below the lambda point of $^4$He [4]. Possible applications of our results include the weakly interacting Bose gas in the precritical regions above and below $T_c$. Support by DLR is gratefully acknowledged. [1] M.N.~Barber and M.E.~Fisher, Annals of Physics (N.Y.) {\bf 77}, 1 (1973). [2] X.S.~Chen and V.~Dohm, Phys.\ Rev.\ E{\bf 67}, 056127 (2003). [3] V.~Privman, P.C.~Hohenberg, and A.~Aharony, in Phase Transitions and Critical Phenomena, C.~Domb and J.L.~Lebowitz, eds., Vol.\ 14 (Academic, NY, 1991); M.~Krech, The Casimir Effect in Critical Systems, (World Scientific, Singapore, 1994). [4] D.~Dantchev, M.~Krech, and S.~Dietrich, Phys.\ Rev.\ Lett.\ {\bf 95}, 259701 (2005); R.~Zandi, A.~Shackell, J.~Rudnick, M.~Kardar, and L.P.~Chayes, cond-mat/0703262.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Kastening, B. and Dohm, V.}, biburl = {https://www.bibsonomy.org/bibtex/22a96d0628bb6e716f5d474113664afbe/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {ad37732d7febfae94f18741acd3c40f2}, intrahash = {2a96d0628bb6e716f5d474113664afbe}, keywords = {casimir finite-size force mean model scaling spherical statphys23 topic-2}, month = {9-13 July}, timestamp = {2007-06-20T10:16:20.000+0200}, title = {Critical Casimir force scaling function of the mean spherical model in $2<d<3$ dimensions for nonperiodic boundary conditions}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=465}, year = 2007 }