Binder parameter

The Binder parameter^[1] in statistical physics, also known as the fourth-order culumant $U_L=1-\frac{{\langle s^4\rangle}_L}{3{\langle s^2\rangle}^2_L}$ 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

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