EquationsOfState::Enthalpy< LowDensityEoS > Class Template Reference

An equation of state given by parametrized enthalpy. More...

#include <Enthalpy.hpp>

Classes
struct	CosCoefficients
struct	MaximumDensity
struct	MinimumDensity
struct	PolynomialCoefficients
struct	ReferenceDensity
struct	SinCoefficients
struct	StitchedLowDensityEoS
struct	TransitionDeltaEpsilon
struct	TrigScaling

Public Types
using	options = implementation defined

Public Member Functions
	Enthalpy (const Enthalpy &)=default
Enthalpy &	operator= (const Enthalpy &)=default
	Enthalpy (Enthalpy &&)=default
Enthalpy &	operator= (Enthalpy &&)=default
	Enthalpy (double reference_density, double max_density, double min_density, double trig_scale, const std::vector< double > &polynomial_coefficients, const std::vector< double > &sin_coefficients, const std::vector< double > &cos_coefficients, const LowDensityEoS &low_density_eos, const double transition_delta_epsilon)
std::unique_ptr< EquationOfState< true, 1 > >	get_clone () const override
std::unique_ptr< EquationOfState< true, 3 > >	promote_to_3d_eos () const override
std::unique_ptr< EquationOfState< true, 2 > >	promote_to_2d_eos () const override
bool	is_equal (const EquationOfState< true, 1 > &rhs) const override
bool	operator== (const Enthalpy< LowDensityEoS > &rhs) const
bool	operator!= (const Enthalpy< LowDensityEoS > &rhs) const
	WRAPPED_PUPable_decl_base_template (SINGLE_ARG(EquationOfState< true, 1 >), Enthalpy)
double	rest_mass_density_lower_bound () const override
	The lower bound of the rest mass density that is valid for this EOS.
double	rest_mass_density_upper_bound () const override
	The upper bound of the rest mass density that is valid for this EOS.
double	specific_enthalpy_lower_bound () const override
	The lower bound of the specific enthalpy that is valid for this EOS.
double	specific_internal_energy_lower_bound () const override
	The lower bound of the specific internal energy that is valid for this EOS.
double	specific_internal_energy_upper_bound () const override
	The upper bound of the specific internal energy that is valid for this EOS.
double	baryon_mass () const override
	The vacuum baryon mass for this EoS.

Static Public Member Functions
static std::string	name ()

Static Public Attributes
static constexpr size_t	thermodynamic_dim = 1
static constexpr bool	is_relativistic = true
static constexpr Options::String	help

Detailed Description

template<typename LowDensityEoS>
class EquationsOfState::Enthalpy< LowDensityEoS >

An equation of state given by parametrized enthalpy.

This equation of state is determined as a function of $x=\ln \left(\rho /{\rho }_0\right)$ where $\rho$ is the rest mass density and ${\rho }_0$ is the provided reference density. The pseudo-enthalpy $h\equiv (p+rho+u)/rho$ is expanded as

$$\begin{array}{}\text{(1)}& h(x)=\sum _i{a}_i{x}^i+\sum _j{b}_j\sin \left(jkx\right)+c_j\cos \left(jkx\right)\end{array}$$

This form allows for convenient calculation of thermodynamic quantities for a cold equation of state. For example

$$\begin{array}{}\text{(2)}& h(x)=\frac{de}{d\rho }{|}_{x=\log \left(\rho /{\rho }_0\right)}\end{array}$$

where $e$ is the total energy density. At the same time $dx=d\rho /\rho$ so ${\rho }_0{e}^{x}dx=d\rho$ Therefore,

$$\begin{array}{}\text{(3)}& e(x)-e(x_0)={\int }_{x_0}^{x}h(x^{\prime })e^{x^{\prime }}d{x}^{\prime }\end{array}$$

This can be computed analytically because

$$\begin{array}{}\text{(4)}& \int a_i\frac{x^i}{i!}{e}^{x}dx=\sum _{j\le i}{a}_i(-1{)}^{i-j}\frac{(x{)}^j}{j!}+C\end{array}$$

and

$$\begin{array}{}\text{(5)}& \int b_j\sin \left(jkx\right)e^{x}dx=b_j{e}^x\frac{\sin \left(jkx\right)-jk\cos \left(jkx\right)}{j^2{k}^2+1}\end{array}$$

$$\begin{array}{}\text{(6)}& \int c_j\cos \left(jkx\right)e^{x}dx=b_j{e}^x\frac{\cos \left(jkx\right)+jk\sin \left(jkx\right)}{j^2{k}^2+1}\end{array}$$

From this most other thermodynamic quantities can be computed analytically

The internal energy density

$$\begin{array}{}\text{(7)}& ϵ(x)\rho (x)=e(x)-\rho (x)\end{array}$$

The pressure

$$\begin{array}{}\text{(8)}& p(x)=\rho (x)h(x)-e(x)\end{array}$$

The derivative of the pressure with respect to the rest mass density

$$\begin{array}{}\text{(9)}& \chi (x)=\frac{dp}{d\rho }{|}_{x=x(\rho )}=\frac{dh}{dx}\end{array}$$

Below the minimum density, a spectral parameterization is used.

Member Data Documentation

help

template<typename LowDensityEoS >

constexpr Options::String EquationsOfState::Enthalpy< LowDensityEoS >::help

staticconstexpr

Initial value:

= {
      "An EoS with a parametrized value h(log(rho/rho_0)) with h the specific "
      "enthalpy and rho the baryon rest mass density.  The enthalpy is "
      "expanded as a sum of polynomial terms and trigonometric corrections. "
      "let x = log(rho/rho_0) in"
      "h(x) = \sum_i a_ix^i + \sum_j b_jsin(k * j * x) + c_jcos(k * j * x) "
      "Note that rho(x)(1+epsilon(x)) = int_0^x e^x' h((x') dx' can be "
      "computed "
      "analytically, and therefore so can "
      "P(x) = rho(x) * (h(x) - (1 + epsilon(x))) "}

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The documentation for this class was generated from the following file:

• src/PointwiseFunctions/Hydro/EquationsOfState/Enthalpy.hpp