`mmf.math.special.fermi`¶

`fermi_functions`	Compute various properties of free fermions.
`fermi_integral`(mu, m, T[, pmax, singlet, ...])	Return density of a single species of free fermions of mass m

The core of this module is to evaluate the generalized Fermi-Dirac integrals:

$\begin{aligned} F_{k}(\eta, \theta) &= \int_{0}^{\infty} \d{t}\; \frac{t^{k}\sqrt{1+ \theta t/2}} {1+e^{t-\eta}},\\ F^{(a)}_{k}(\beta\mu, \beta m) &= F_{k}(\eta, \theta) - F_{k}(\eta - \beta\mu, \theta),\\ &= \int_{0}^{\infty} \d{t}\; \frac{t^{k}\sqrt{1+ \theta t/2}\sinh(\beta\mu)} {\cosh(t + \beta m) + \cosh(\beta\mu)} \end{aligned}$

This allows us to compute, for example, the density and energy-density of a relativistic Fermi gas (see [Miralles:1996rt] for some explicit formulae for the thermodynamic quantities).

$\begin{aligned} x &= \beta(E-m), & \eta &= \beta(\mu-m), & \theta &= \frac{1}{\beta m}, \end{aligned}$

System Message: WARNING/2 (\begin{aligned} n_{\beta}(\mu, m) &= \int \dbar^{3}{k}\; \frac{1}{1+e^{\beta(\sqrt{k^2 + m^2} - \mu)}} = \frac{m^3\theta^{3/2}}{\sqrt{2}\pi^2} \left[F_{\frac{1}{2}}(\eta, \theta) + \theta F_{\frac{3}{2}}(\eta, \theta)\right],\\ n^{(s)}_{\beta}(\mu, m) &= \int \dbar^{3}{k}\; \frac{m}{E}\frac{1}{1+e^{\beta(\sqrt{k^2 + m^2} - \mu)}} = \frac{m^3\theta^{3/2}}{\sqrt{2}\pi^2} F_{\frac{1}{2}}(\eta, \theta) \end{aligned})

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Using $F^{(a)}$ includes the anti-particle contributions.

For more details, see the notes in section Fermi-Dirac Integrals.

[Miralles:1996rt]

J. A. Miralles and K. A. van Riper, “Accurate Evaluation of Fermi-Dirac Integrals and Their Derivatives for Arbitrary Degeneracy and Relativity”, ApJ Supp. 105, 407-418 (1996) [arXiv:astro-ph/9509124]

Following the suggestion by [Natarajan:1993rz]

class mmf.math.special.fermi.fermi_functions¶

Bases: object

Compute various properties of free fermions.

Examples

>>> fermi_functions.P(1, 1, 1)               
array(0.23482812162868...)
>>> fermi_functions.N(1, 1, 1)               
array(0.160318814464720...)
>>> fermi_functions.N_s(1, 1, 1)              
array(0.075970412559876...)
>>> fermi_functions.dN_dmu(1, 1, 1)          
array(0.193444769294...)
>>> fermi_functions.dN_dm(1, 1, 1)           
array(-0.0425632327406...)

__init__()¶: x.__init__(...) initializes x; see help(type(x)) for signature

classmethod N(mu, m, T)¶: Return the density.

classmethod N_s(mu, m, T)¶: Return the scalar density.

classmethod P(mu, m, T)¶: Return the pressure.

classmethod dN_dm(mu, m, T)¶: Return the derivative of the density wrt. m.

classmethod dN_dmu(mu, m, T)¶: Return the derivative of the density wrt. mu.

classmethod get_imt()¶: Return the class IMT instance. This is expensive to compute and so is “memoized”.

mmf.math.special.fermi.fermi_integral(mu, m, T, pmax=None, singlet=False, abs_tol=None, rel_tol=None)¶

Return density of a single species of free fermions of mass m at temperature T and chemical potential mu in natural units ( $\hbar = c = 1$ ). Does not include any degeneracy factors:

$\int_{0}^{p_{\text{max}}}\d{p}\; \frac{p^{2}}{1+e^{(\sqrt{p^2 + m^2} - \mu)/T}} - \frac{p^{2}}{1+e^{(\sqrt{p^2 + m^2} + \mu)/T}}$

Parameters :

mu : float

Chemical potential

m : float

Mass

T : float

Temperature

pmax : float

Upper limit on integration.

singlet : bool

If True, then the singlet density will be computed:

$\int_{0}^{p_{\text{max}}}\d{p}\; \frac{m}{\sqrt{p^2 + m^2}} \left( \frac{p^{2}}{1+e^{(\sqrt{p^2 + m^2} - \mu)/T}} - \frac{p^{2}}{1+e^{(\sqrt{p^2 + m^2} + \mu)/T}} \right)$

Examples

>>> import math, numpy as np
>>> mu = 3.0
>>> m = 2.0
>>> T = 0.0
>>> pF = math.sqrt(mu*mu-m*m)
>>> N = fermi_integral(mu, m, T)
>>> np.allclose(N, pF**3/6/pi**2)
True

>>> import math, numpy as np
>>> mu = 30.0
>>> m = 22.0
>>> T = 0.00001
>>> pF = math.sqrt(mu*mu-m*m)
>>> N = fermi_integral(mu, m, T)
>>> np.allclose(N, pF**3/6/pi**2)
True

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