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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 | 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 | ||
to | |||
:<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>. Thus, the function is discontinuous in this limit. | 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]]. | ||
==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]] | [[category: Computer simulation techniques]] | ||