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]].
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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>.
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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>. 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 22:36, 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

U_4 = 1- \frac{\langle m^4 \rangle }{3\langle m^2 \rangle^2 }

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 L \rightarrow \infty, U_4 \rightarrow 0 for T > T_c, and U_4 \rightarrow 2/3 for T < T_c. 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

  1. K. Binder "Finite size scaling analysis of ising model block distribution functions", Zeitschrift für Physik B Condensed Matter 43 pp. 119-140 (1981)