# Difference between revisions of "Binder cumulant"

Carl McBride (talk | contribs) (New page: {{Stub-general}} The '''Binder cumulant''' for an Ising model with zero field, is given by :<math>U_4 = 1- \frac{\langle m^4 \rangle }{3\langle m^2 \rangle^2 }</math> . ...) |
Carl McBride (talk | contribs) m (Reverted edits by 70.135.118.126 (talk) to last revision by 134.130.4.242) |
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− | + | The '''Binder cumulant''' was introduced by [[Kurt Binder]] in the context of [[Finite size effects |finite size scaling]]. It is a quantity that allows | |

− | The '''Binder cumulant''' | + | to locate the critical point and critical exponents. 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> |

− | 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>. | + | where ''m'' is the [[Order parameters |order parameter]], i.e. the magnetization. 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. An important observation is that the intersection points of the cumulants for different system sizes usually depend only rather weakly on those sizes, providing a convenient estimate for the value of the [[Critical points |critical temperature]]. Caution is needed in identifying the universality class from | ||

+ | the critical value of the Binder cumulant, because that value depends on boundary condition, system shape, and anisotropy of correlations. | ||

==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)] | ||

+ | [[category: Computer simulation techniques]] |

## Latest revision as of 15:31, 4 January 2012

The **Binder cumulant** was introduced by Kurt Binder in the context of finite size scaling. It is a quantity that allows
to locate the critical point and critical exponents. For an Ising model with zero field, it is given by

where *m* is the order parameter, i.e. the magnetization. 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. An important observation is that the intersection points of the cumulants for different system sizes usually depend only rather weakly on those sizes, providing a convenient estimate for the value of the critical temperature. Caution is needed in identifying the universality class from
the critical value of the Binder cumulant, because that value depends on boundary condition, system shape, and anisotropy of correlations.