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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
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{{Stub-general}}
to locate the critical point and critical exponents. For an [[Ising Models |Ising model]] with zero field, it is given by
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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>  
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:<math>U_4 = 1- \frac{\langle m^4 \rangle }{3\langle m^2 \rangle^2 }</math> .
  
where ''m'' is the [[Order parameters |order parameter]], i.e. the magnetization. It is therefore a fourth order cumulant, related to the kurtosis.
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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. 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]]
 

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