Function fermion_calc_mu_ndeg (o2scl)

O2scl : Function List

template<class fermion_t, class fp_t = double>
bool o2scl::fermion_calc_mu_ndeg(fermion_t &f, fp_t temper, fp_t prec = 1.0e-17, bool inc_antip = false, int verbose = 0)

Non-degenerate expansion for fermions.

Attempts to evaluate thermodynamics of a non-degenerate fermion. If the result is accurate to within the requested precision, this function returns true, and otherwise this function returns false and the values in stored in the pr, n, en, and ed field are meaningless.

If \( \mu \) is negative and sufficiently far from zero, then the thermodynamic quantities are smaller than the smallest representable double-precision number. In this case, this function will return true and report all quantities as zero.

The following uses the notation of [Johns96].

Defining \( \psi \equiv (\mu-m)/T \), \( t \equiv T/m \), and \( d \equiv g~m^4/(2 \pi^2) \) the pressure in the non-degenerate limit ( \( \psi \rightarrow - \infty \)) is

\[ P = d \sum_{n=1}^{\infty} P_n \]

where

\[ P_n \equiv \left(-1\right)^{n+1} \left(\frac{t^2}{n^2}\right) e^{n \left(\psi+1/t\right)} K_2 \left( \frac{n}{t} \right) \]

The density is then

\[ n = d \sum_{n=1}^{\infty} \frac{n P_n}{T} \]

and the entropy density is

\[ s = \frac{d}{m} \sum_{n=1}^{\infty} \left\{ \frac{2 P_n}{t} -\frac{n P_n}{t^2}+ \frac{\left(-1\right)^{n+1}}{2 n} e^{n \left(\psi+1/t\right)} \left[ K_1 \left( \frac{n}{t} \right)+K_3 \left( \frac{n}{t} \right) \right] \right\} \]

This function is accurate over a wide range of conditions when \( \psi < -4 \).

The ratio of the nth term to the first term in the pressure series is

\[ R_n \equiv \frac{P_{n}}{P_{1}} = \frac{(-1)^{n+1} e^{(n-1)(\psi+1/t)} K_2(n/t) }{n^2 K_2(1/t)} \]

This function currently uses 20 terms in the series and immediately returns false if \( |R_{20}| \) is greater than prec

In the nondegenerate and nonrelativistic ( \( t \rightarrow 0 \)) limit, the argument to the Bessel functions and the exponential becomes too large. In this case, it’s better to use the expansions, e.g. for \( x \equiv n/t \rightarrow \infty \),

\[ \sqrt{\frac{2 x}{\pi}} e^{x} K_2(x) \approx 1 + \frac{3}{8 x} - \frac{15}{128 x^2} + ... \]