Difference between revisions of "Binder cumulant"

From SklogWiki
Jump to: navigation, search
m
((universality class))
Line 1: Line 1:
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
+
The '''Binder cumulant''' was introduced by [[Kurt Binder]] in the context of [[Finite size effects |finite size scaling]]. It is a quantity that allows
 +
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>  
  
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]], 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. 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. 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]]
 
[[category: Computer simulation techniques]]

Revision as of 22:59, 7 June 2010

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

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

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 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. 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.

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)