The Binder parameter or Binder cumulant in

statistical physics Statistical physics is a branch of physics that evolved from a foundation of statistical mechanics, which uses methods of probability theory and statistics, and particularly the mathematical tools for dealing with large populations and approxi ...

, also known as the fourth-order cumulant

U_L=1-\frac

is defined as the

kurtosis In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kur ...

of the order parameter, ''s'', introduced by Austrian theoretical physicist

Kurt Binder Kurt Binder (10 February 1944 – 27 September 2022) was an Austrian theoretical physicist. He received his Ph.D. in 1969 at the Technical University of Vienna, and his habilitation degree 1973 at the Technical University of Munich. He decided t ...

. It is frequently used to determine accurately

phase transition In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states ...

points in numerical simulations of various models. 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 In statistical mechanics, the thermodynamic limit or macroscopic limit, of a system is the limit for a large number of particles (e.g., atoms or molecules) where the volume is taken to grow in proportion with the number of particles.S.J. Blunde ...

. This behavior is based on the fact that in the critical region,

T\approx T_c

, the Binder parameter behaves as

U(T,L)=b(\epsilon L^)

, where

\epsilon=\frac

. Accordingly, the cumulant may also be used to identify the universality class of the transition by determining the value of the

critical exponent Critical or Critically may refer to: *Critical, or critical but stable, medical states **Critical, or intensive care medicine * Critical juncture, a discontinuous change studied in the social sciences. * Critical Software, a company specializing i ...

\nu

of the

correlation length A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables. If one considers the correlation function between random variables re ...

. In the

, at the critical point, the value of the Binder parameter depends on

boundary condition In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to ...

s, the shape of the system, and

anisotropy Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's phys ...

of correlations.

References

Statistical mechanics {{statisticalmechanics-stub