# 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''' was introduced by [[Kurt Binder]] in the context of [[Finite size effects |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, it 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]]. It is therefore a fourth order cumulant, related to the kurtosis. | 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>. 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 estimate for the value of the [[Critical points |critical temperature]]. | |

− | 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 | ||

==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 10:37, 12 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, it 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 estimate for the value of the critical temperature.