Binder parameter

{{short description|Kurtosis of the order parameter in statistical physics}}

The Binder parameter or Binder cumulant{{cite journal | last=Binder | first=K.|author-link=Kurt Binder | title=Finite size scaling analysis of ising model block distribution functions | journal=Zeitschrift für Physik B: Condensed Matter| volume=43 | issue=2 | year=1981 | issn=0340-224X | doi=10.1007/bf01293604 | pages=119–140| bibcode=1981ZPhyB..43..119B| s2cid=121873477}}{{cite journal | last=Binder | first=K. |author-link=Kurt Binder| title=Critical Properties from Monte Carlo Coarse Graining and Renormalization | journal=Physical Review Letters| volume=47 | issue=9 | date=1981-08-31 | issn=0031-9007 | doi=10.1103/physrevlett.47.693 | pages=693–696| bibcode=1981PhRvL..47..693B }} in statistical physics, also known as the fourth-order cumulant $U\_L=1-\backslash frac\{\{\backslash langle\; s^4\backslash rangle\}\_L\}\{3\{\backslash langle\; s^2\backslash rangle\}^2\_L\}$ is defined as the kurtosis (more precisely, minus one third times the excess 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. K. Binder, D. W. Heermann, Monte Carlo Simulation in Statistical Physics: An Introduction (2010) Springer

The phase transition point is usually identified comparing the

behavior of $U$ 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 in the thermodynamic limit. This behavior is based on the fact that

in the critical region, $T\backslash approx\; T\_c$, the Binder parameter behaves as $U(T,L)=b(\backslash epsilon\; L^\{1/\backslash nu\})$, where $\backslash epsilon=\backslash frac\{T-T\_c\}\{T\}$.

Accordingly, the cumulant may also be used to identify the universality class of the transition by determining the value of the critical exponent $\backslash nu$ of the correlation length.

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.

{{cite journal | last1=Kamieniarz | first1=G | last2=Blote | first2=H W J | title=Universal ratio of magnetization moments in two-dimensional Ising models | journal=Journal of Physics A: Mathematical and General| volume=26 | issue=2 | date=1993-01-21 | issn=0305-4470 | doi=10.1088/0305-4470/26/2/009 | pages=201–212| bibcode=1993JPhA...26..201K }}{{cite journal | last1=Chen | first1=X. S. | last2=Dohm | first2=V. | title=Nonuniversal finite-size scaling in anisotropic systems | journal=Physical Review E| volume=70 | issue=5 | date=2004-11-30 | issn=1539-3755 | doi=10.1103/physreve.70.056136 | page=056136| pmid=15600721 | arxiv=cond-mat/0408511 | bibcode=2004PhRvE..70e6136C | s2cid=44785145 }}{{cite journal | last1=Selke | first1=W |author-link=Walter Selke| last2=Shchur | first2=L N | title=Critical Binder cumulant in two-dimensional anisotropic Ising models | journal=Journal of Physics A: Mathematical and General| volume=38 | issue=44 | date=2005-10-19 | issn=0305-4470 | doi=10.1088/0305-4470/38/44/l03 | pages=L739–L744| arxiv=cond-mat/0509369 | s2cid=14774533 }}