Binder parameter

The Binder parameter^[1] in statistical physics, also known as the fourth-order cumulant $U_{L}=1-{\frac {{\langle s^{4}\rangle }_{L}}{3{\langle s^{2}\rangle }_{L}^{2}}}$ in Ising model,^[2] is used to identify phase transition points in numerical simulations. It is defined as the kurtosis of the order parameter. For example in spin glasses one defines the Binder as

${B={\frac {1}{2}}\left(3-{\frac {\overline {\langle q^{4}\rangle }}{{\overline {\langle q^{2}\rangle }}^{2}}}\right)}$

where $\langle\cdot\rangle$ stands for Boltzmann average, $\overline{\cdot}$ for average over the disorder and $q$ is the overlap between two identical replicas of the system. The phase transition point is usually identified comparing the behavior of $B$ as a function of the temperature for different values of the system size $L$ . The transition temperature is the unique point where the different curves cross. This is based on finite size scaling hypothesis, according to which, in the critical region $T\approx T_{c}$ the Binder behaves as $B(T,L)=b(\epsilon L^{{1/\nu }})$ , where $\epsilon ={\frac {T-T_{c}}{T}}$ .

References

↑ K. Binder, Z. Phys. B 43, 119 (1981)
↑ K. Binder & D. W. Heermann, Monte Carlo Simulation in Statistical Physics An Introduction, Ed. 4, Spring

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] K. Binder, Z. Phys. B 43, 119 (1981)

[2] K. Binder & D. W. Heermann, Monte Carlo Simulation in Statistical Physics An Introduction, Ed. 4, Spring