Probabilities.FermiDiracPDF Method

Overload List

#SignatureDescription
1procedure FermiDiracPDF(const x: TDenseMtxVec; Mu: Double; s: Double; const y: TDenseMtxVec);Fermi-Dirac probability density function (PDF).
2function FermiDiracPDF(x: Double; Mu: Double; s: Double): Double;Fermi-Dirac occupation function (PDF form).

Overload 1: procedure FermiDiracPDF(const x: TDenseMtxVec; Mu: Double; s: Double; const y: TDenseMtxVec);

Fermi-Dirac probability density function (PDF).

#NameDescription
1xFunction domain, real value.
2MuLocation parameter, real value.
3sShape parameter, real positive value.
4yContains result.

Result: stored in self (calling object)

Returns: Fermi-Dirac probability density function (PDF).

Remarks:

Calculates the Fermi-Dirac PDF for x using the parameters Mu and s. The distribution arises in the study of half-integer spin particles in physics. Half-integer spin particles are described by Fermi-Dirac statistics and are called FERMIONS. Fermi-Dirac PDF is defined by the following equation:

PDF(xμ,s)=xsexp(xμ)+1\text{PDF}\left(x| \mu,s\right)= \cfrac{x ^s}{\exp (x-\mu)+1}
See Also: Probabilities.BoseEinsteinPDF

Overload 2: function FermiDiracPDF(x: Double; Mu: Double; s: Double): Double;

Fermi-Dirac occupation function (PDF form).

#NameDescription
1xFunction domain, real value.
2MuLocation (chemical-potential) parameter, real value.
3sShape parameter, real value.

Returns: Double - the Fermi-Dirac occupation value at x for the given Mu and s. This is NOT a normalized probability density.

Remarks:

Computes the Fermi-Dirac statistical weight that arises in the study of half-integer-spin particles (fermions):

f(x|mu,s)=x^s/(exp(x-mu)+1)

This is the physics occupation form, not a normalized PMF; no parameter validation is performed. Unlike the Bose-Einstein form the denominator never vanishes for real x.

See Also: Probabilities.BoseEinsteinPDF, Probabilities.FermiDiracCDF