Binder Parameter
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Binder Parameter

The Binder parameter or Binder cumulant[1][2] in statistical physics, also known as the fourth-order cumulant ${\displaystyle U_{L}=1-{\frac {{\langle s^{4}\rangle }_{L}}{3{\langle s^{2}\rangle }_{L}^{2}}}}$ is defined as the kurtosis of the order parameter, s introduced by Austrian Theoretical Physicist Kurt Binder. It is frequently used to determine accurately phase transition points in numerical simulations of various models. [3]

The phase transition point is usually identified comparing the behavior of ${\displaystyle U}$ as a function of the temperature for different values of the system size ${\displaystyle L}$. The transition temperature is the unique point where the different curves cross in the thermodynamic limit. This behavior is based on the fact that in the critical region, ${\displaystyle T\approx T_{c}}$, the Binder parameter behaves as ${\displaystyle U(T,L)=b(\epsilon L^{1/\nu })}$, where ${\displaystyle \epsilon ={\frac {T-T_{c}}{T}}}$.

Accordingly, the cumulant may also be used to identify the universality class of the transition by determining the value of the critical exponent ${\displaystyle \nu }$ of the correlation length. [1]

In the thermodynamic limit, at the critical point, the value of the Binder parameter depends on boundary conditions, the shape of the system, and anisotropy of correlations. [1][4][5][6]

## References

1. ^ a b c Binder, K. (1981). "Finite size scaling analysis of ising model block distribution functions". Zeitschrift für Physik B Condensed Matter. Springer Science and Business Media LLC. 43 (2): 119-140. doi:10.1007/bf01293604. ISSN 0340-224X.
2. ^ Binder, K. (1981-08-31). "Critical Properties from Monte Carlo Coarse Graining and Renormalization". Physical Review Letters. American Physical Society (APS). 47 (9): 693-696. doi:10.1103/physrevlett.47.693. ISSN 0031-9007.
3. ^ K. Binder, D. W. Heermann, Monte Carlo Simulation in Statistical Physics: An Introduction (2010) Springer
4. ^ Kamieniarz, G; Blote, H W J (1993-01-21). "Universal ratio of magnetization moments in two-dimensional Ising models". Journal of Physics A: Mathematical and General. IOP Publishing. 26 (2): 201-212. doi:10.1088/0305-4470/26/2/009. ISSN 0305-4470.
5. ^ Chen, X. S.; Dohm, V. (2004-11-30). "Nonuniversal finite-size scaling in anisotropic systems". Physical Review E. American Physical Society (APS). 70 (5): 056136. doi:10.1103/physreve.70.056136. ISSN 1539-3755.
6. ^ Selke, W; Shchur, L N (2005-10-19). "Critical Binder cumulant in two-dimensional anisotropic Ising models". Journal of Physics A: Mathematical and General. IOP Publishing. 38 (44): L739-L744. doi:10.1088/0305-4470/38/44/l03. ISSN 0305-4470.