# Difference between revisions of "Binder cumulant"

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− | + | The '''Binder cumulant''' was introduced by Kurt Binder in the context of finite size scaling. It is a quantity that is supposed to be invariant for different system sizes at criticality. For an [[Ising Models |Ising model]] with zero field, is given by | |

− | The '''Binder cumulant''' for an [[Ising Models |Ising model]] with zero field, is given by | ||

:<math>U_4 = 1- \frac{\langle m^4 \rangle }{3\langle m^2 \rangle^2 }</math> | :<math>U_4 = 1- \frac{\langle m^4 \rangle }{3\langle m^2 \rangle^2 }</math> | ||

− | where ''m'' is the [[Order parameters |order parameter]]. | + | where ''m'' is the [[Order parameters |order parameter]]. It is therefore a fourth order cumulant, related to the kurtosis. |

− | In the [[thermodynamic limit]], where the system size <math>L \rightarrow \infty</math>, <math>U_4 \rightarrow 0</math> for <math>T > T_c</math>, and <math>U_4 \rightarrow 2/3</math> for <math>T < T_c</math>. | + | |

+ | In the [[thermodynamic limit]], where the system size <math>L \rightarrow \infty</math>, <math>U_4 \rightarrow 0</math> for <math>T > T_c</math>, and <math>U_4 \rightarrow 2/3</math> for <math>T < T_c</math>. Thus, the function is discontinuous in this limit --- the useful fact is that curves corresponding to different system sizes (which are, of course, continuous) all intersect at approximately the same temperature, which provides a convenient estimatation of the critical temperature. | ||

==References== | ==References== | ||

#[http://dx.doi.org/10.1007/BF01293604 K. Binder "Finite size scaling analysis of ising model block distribution functions", Zeitschrift für Physik B Condensed Matter '''43''' pp. 119-140 (1981)] | #[http://dx.doi.org/10.1007/BF01293604 K. Binder "Finite size scaling analysis of ising model block distribution functions", Zeitschrift für Physik B Condensed Matter '''43''' pp. 119-140 (1981)] |

## Revision as of 23:36, 8 November 2007

The **Binder cumulant** was introduced by Kurt Binder in the context of finite size scaling. It is a quantity that is supposed to be invariant for different system sizes at criticality. For an Ising model with zero field, is given by

where *m* is the order parameter. It is therefore a fourth order cumulant, related to the kurtosis.

In the thermodynamic limit, where the system size , for , and for . Thus, the function is discontinuous in this limit --- the useful fact is that curves corresponding to different system sizes (which are, of course, continuous) all intersect at approximately the same temperature, which provides a convenient estimatation of the critical temperature.