On the breakdown of finite-size scaling in high dimensional systems
A. Hucht, и S. Luebeck. Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)
Аннотация
Finite-size scaling functions of continuous phase transitions exhibit a
scaling anomaly above the upper critical dimension
$d_c$. This so-called breakdown of finite-size scaling is
well-established on the basis of field theoretical and numerical
approaches for system with periodic boundary conditions (BC), both in
equilibrium (e.g. the Ising model) and non-equilibrium (e.g. directed
percolation 1). Less work was done for geometric phase transitions
and for Dirichlet BC. Therefore, we numerically investigate the bond
percolation transition in $2 d 10$ dimensions with various
boundary conditions. For $d<d_c=6$ the spatial correlation
length at criticality, $\xi_c$, is limited by the systems size
$L$ for all BCs, whereas it exceeds the systems size as
$\xi_c L^d/d_c$ in systems with periodic BC above
$d_c$, the hallmark of the breakdown of finite-size
scaling. We present, to our knowledge for the first time, a
phenomenological and descriptive interpretation of this breakdown of
finite-size scaling. Using a generalized distance definition, the correlation length $\xi$ can be directly measured in simulations, even when it exceeds the linear system size $L$.
Furthermore, we show that the high-dimensional
behavior depends strongly on the boundary conditions.
1) S. Luebeck and H.-K. Janssen, Phys. Rev. E 72, 016119 (2005)
%0 Book Section
%1 statphys23_0976
%A Hucht, A.
%A Luebeck, S.
%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 correlation critical dimension finite-size length montecarlo scaling simulations statphys23 topic-2 upper
%T On the breakdown of finite-size scaling in high dimensional systems
%U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=976
%X Finite-size scaling functions of continuous phase transitions exhibit a
scaling anomaly above the upper critical dimension
$d_c$. This so-called breakdown of finite-size scaling is
well-established on the basis of field theoretical and numerical
approaches for system with periodic boundary conditions (BC), both in
equilibrium (e.g. the Ising model) and non-equilibrium (e.g. directed
percolation 1). Less work was done for geometric phase transitions
and for Dirichlet BC. Therefore, we numerically investigate the bond
percolation transition in $2 d 10$ dimensions with various
boundary conditions. For $d<d_c=6$ the spatial correlation
length at criticality, $\xi_c$, is limited by the systems size
$L$ for all BCs, whereas it exceeds the systems size as
$\xi_c L^d/d_c$ in systems with periodic BC above
$d_c$, the hallmark of the breakdown of finite-size
scaling. We present, to our knowledge for the first time, a
phenomenological and descriptive interpretation of this breakdown of
finite-size scaling. Using a generalized distance definition, the correlation length $\xi$ can be directly measured in simulations, even when it exceeds the linear system size $L$.
Furthermore, we show that the high-dimensional
behavior depends strongly on the boundary conditions.
1) S. Luebeck and H.-K. Janssen, Phys. Rev. E 72, 016119 (2005)
@incollection{statphys23_0976,
abstract = {Finite-size scaling functions of continuous phase transitions exhibit a
scaling anomaly above the upper critical dimension
$d_\mathrm{c}$. This so-called breakdown of finite-size scaling is
well-established on the basis of field theoretical and numerical
approaches for system with periodic boundary conditions (BC), both in
equilibrium (e.g. the Ising model) and non-equilibrium (e.g. directed
percolation [1]). Less work was done for geometric phase transitions
and for Dirichlet BC. Therefore, we numerically investigate the bond
percolation transition in $2 \leq d \leq 10$ dimensions with various
boundary conditions. For $d<d_\mathrm{c}=6$ the spatial correlation
length at criticality, $\xi_\mathrm{c}$, is limited by the systems size
$L$ for all BCs, whereas it exceeds the systems size as
$\xi_\mathrm{c} \sim L^{d/d_c}$ in systems with periodic BC above
$d_\mathrm{c}$, the hallmark of the breakdown of finite-size
scaling. We present, to our knowledge for the first time, a
phenomenological and descriptive interpretation of this breakdown of
finite-size scaling. Using a generalized distance definition, the correlation length $\xi$ can be directly measured in simulations, even when it exceeds the linear system size $L$.
Furthermore, we show that the high-dimensional
behavior depends strongly on the boundary conditions.
1) S. Luebeck and H.-K. Janssen, Phys. Rev. E 72, 016119 (2005)},
added-at = {2007-06-20T10:16:09.000+0200},
address = {Genova, Italy},
author = {Hucht, A. and Luebeck, S.},
biburl = {https://www.bibsonomy.org/bibtex/267bd0da9d00a6c391e539b5228fd116e/statphys23},
booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics},
editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano},
interhash = {4f422aedc7587b16c99998f14c8c70be},
intrahash = {67bd0da9d00a6c391e539b5228fd116e},
keywords = {correlation critical dimension finite-size length montecarlo scaling simulations statphys23 topic-2 upper},
month = {9-13 July},
timestamp = {2007-06-20T10:16:36.000+0200},
title = {On the breakdown of finite-size scaling in high dimensional systems},
url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=976},
year = 2007
}