Distribution functions¶

Various distributions using different smearing techniques.

`get_distribution`(method[, smearing, x0])	Create a distribution function, Gaussian, Lorentzian etc.
`gaussian`(x[, sigma, x0])	Gaussian distribution function
`lorentzian`(x[, gamma, x0])	Lorentzian distribution function
`fermi_dirac`(E[, kT, mu])	Fermi-Dirac distribution function
`bose_einstein`(E[, kT, mu])	Bose-Einstein distribution function
`cold`(E[, kT, mu])	Cold smearing function, Marzari-Vanderbilt, PRL 82, 16, 1999

sisl.physics.distribution.gaussian(x, sigma=0.1, x0=0.0)[source]

Gaussian distribution function

\[G(x,\sigma,x_0) = \frac{1}{\sqrt{2\pi\sigma^2}}\exp\Big[\frac{- (x - x_0)^2}{2\sigma^2}\Big]\]

Parameters:	x: array_like points at which the Gaussian distribution is calculated sigma: float, optional spread of the Gaussian x0: float, optional maximum position of the Gaussian
Returns:	numpy.ndarray the Gaussian distribution, same length as x

sisl.physics.distribution.lorentzian(x, gamma=0.1, x0=0.0)[source]

Lorentzian distribution function

\[L(x,\gamma,x_0) = \frac{1}{\pi}\frac{\gamma}{(x-x_0)^2 + \gamma^2}\]

Parameters:	x: array_like points at which the Lorentzian distribution is calculated gamma: float, optional spread of the Lorentzian x0: float, optional maximum position of the Lorentzian
Returns:	numpy.ndarray the Lorentzian distribution, same length as x

sisl.physics.distribution.fermi_dirac(E, kT=0.1, mu=0.0)[source]

Fermi-Dirac distribution function

\[n_F(E,k_BT,\mu) = \frac{1}{\exp\Big[\frac{E - \mu}{k_BT}\Big] + 1}\]

Parameters:	E: array_like energy evaluation points kT: float, optional temperature broadening mu: float, optional chemical potential
Returns:	numpy.ndarray the Fermi-Dirac distribution, same length as E

sisl.physics.distribution.bose_einstein(E, kT=0.1, mu=0.0)[source]

Bose-Einstein distribution function

\[n_B(E,k_BT,\mu) = \frac{1}{\exp\Big[\frac{E - \mu}{k_BT}\Big] - 1}\]

Parameters:	E: array_like energy evaluation points kT: float, optional temperature broadening mu: float, optional chemical potential
Returns:	numpy.ndarray the Bose-Einstein distribution, same length as E

sisl.physics.distribution.cold(E, kT=0.1, mu=0.0)[source]

Cold smearing function, Marzari-Vanderbilt, PRL 82, 16, 1999

\[C(E,k_BT,\mu) = \frac12 + \mathrm{erf}\Big(-\frac{E-\mu}{k_BT}-\frac1{\sqrt2}\Big) + \frac1{\sqrt{2\pi}} \exp\Bigg\{-\Big[\frac{E-\mu}{k_BT}+\frac1{\sqrt2}\Big]^2\Bigg\}\]

Parameters:	E: array_like energy evaluation points kT: float, optional temperature broadening mu: float, optional chemical potential
Returns:	numpy.ndarray the Cold smearing distribution function, same length as E