#include <potentials.h>

A class that stores parameters of potentials, calculates potentials and their gradients.

Potentials are responsible for long-range interactions and stand in the left part of Boltzmann equation. Short-range interactions are taken into account in the right part of it - in the collision term.

Definition at line 31 of file potentials.h.

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Public Member Functions
	Potentials (Configuration conf, const DensityParameters &parameters)
	Potentials constructor. More...

virtual	~Potentials ()
	Standard destructor. More...

double	skyrme_pot (const double baryon_density) const
	Evaluates skyrme potential given a baryon density. More...

double	symmetry_pot (const double baryon_isospin_density, const double baryon_density) const
	Evaluates symmetry potential given baryon isospin density. More...

double	symmetry_S (const double baryon_density) const
	Calculate the factor \(S(\rho)\) in the symmetry potential. More...

double	potential (const ThreeVector &r, const ParticleList &plist, const ParticleType &acts_on) const
	Evaluates potential at point r. More...

std::pair< ThreeVector, ThreeVector >	skyrme_force (const double density, const ThreeVector grad_rho, const ThreeVector dj_dt, const ThreeVector rot_j) const
	Evaluates the electrical and magnetic components of the skyrme force. More...

std::pair< ThreeVector, ThreeVector >	symmetry_force (const double rhoI3, const ThreeVector grad_rhoI3, const ThreeVector djI3_dt, const ThreeVector rot_jI3, const double rhoB, const ThreeVector grad_rhoB, const ThreeVector djB_dt, const ThreeVector rot_jB) const
	Evaluates the electrical and magnetic components of the symmetry force. More...

virtual std::tuple< ThreeVector, ThreeVector, ThreeVector, ThreeVector >	all_forces (const ThreeVector &r, const ParticleList &plist) const
	Evaluates the electrical and magnetic components of the forces at point r. More...

virtual bool	use_skyrme () const

virtual bool	use_symmetry () const

double	skyrme_a () const

double	skyrme_b () const

double	skyrme_tau () const

Static Public Member Functions
static std::pair< double, int >	force_scale (const ParticleType &data)
	Evaluates the scaling factor of the forces acting on the particles. More...

Private Member Functions
double	dVsym_drhoI3 (const double rhoB, const double rhoI3) const
	Calculate the derivative of the symmetry potential with respect to the isospin density in GeV * fm^3. More...

double	dVsym_drhoB (const double rhoB, const double rhoI3) const
	Calculate the derivative of the symmetry potential with respect to the net baryon density in GeV * fm^3. More...

Private Attributes
const DensityParameters	param_
	Struct that contains the gaussian smearing width \(\sigma\), the distance cutoff \(r_{\rm cut}\) and the testparticle number needed for the density calculation. More...

bool	use_skyrme_
	Skyrme potential on/off. More...

bool	use_symmetry_
	Symmetry potential on/off. More...

double	skyrme_a_
	Parameter of skyrme potentials: the coefficient in front of \(\frac{\rho}{\rho_0}\) in GeV. More...

double	skyrme_b_
	Parameters of skyrme potentials: the coefficient in front of \((\frac{\rho}{\rho_0})^\tau\) in GeV. More...

double	skyrme_tau_
	Parameters of skyrme potentials: the power index. More...

double	symmetry_S_Pot_
	Parameter S_Pot in the symmetry potential in MeV. More...

bool	symmetry_is_rhoB_dependent_ = false
	Whether the baryon density dependence of the symmetry potential is included. More...

double	symmetry_gamma_
	Power \( \gamma \) in formula for \( S(\rho) \): More...

Constructor & Destructor Documentation

◆ Potentials()

smash::Potentials::Potentials	(	Configuration	conf,
		const DensityParameters &	parameters
	)

Potentials constructor.

Parameters

[in]	conf	Configuration which contains the switches determining whether to turn on the Skyrme or the symmetry potentials, and the coefficents controlling how strong the potentials are.
[in]	parameters	Struct that contains the gaussian smearing factor \(\sigma\), the distance cutoff \(r_{\rm cut}\) and the testparticle number needed for the density calculation.

◆ ~Potentials()

smash::Potentials::~Potentials ( )

virtual

Standard destructor.

Definition at line 107 of file potentials.cc.

107 {}

Member Function Documentation

◆ skyrme_pot()

double smash::Potentials::skyrme_pot ( const double baryon_density ) const

Evaluates skyrme potential given a baryon density.

Parameters

[in] baryon_density Baryon density \(\rho\) evaluated in the local rest frame in fm \(^{-3}\).

Returns: Skyrme potential
\[U_B=10^{-3}\times\frac{\rho}{|\rho|} (A\frac{\rho}{\rho_0}+B(\frac{\rho}{\rho_0})^\tau)\]
in GeV

Definition at line 109 of file potentials.cc.

                                                                {
   const double tmp = baryon_density / nuclear_density;
   /* U = U(|rho|) * sgn , because the sign of the potential changes
    * under a charge reversal transformation. */
   const int sgn = tmp > 0 ? 1 : -1;
   // Return in GeV
   return 1.0e-3 * sgn *
          (skyrme_a_ * std::abs(tmp) +
           skyrme_b_ * std::pow(std::abs(tmp), skyrme_tau_));
 }

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◆ symmetry_pot()

double smash::Potentials::symmetry_pot	(	const double	baryon_isospin_density,
		const double	baryon_density
	)		const

Evaluates symmetry potential given baryon isospin density.

Note: The second term is neglected if \(\gamma\) is not specified in the config.

Parameters

[in]	baryon_isospin_density	The difference between the proton and the neutron density in the local rest frame in fm \(^{-3}\).
[in]	baryon_density

Returns: Symmetry potential
\[U_I=2\times 10^{-3}S_{\rm sym} \frac{\rho_{I_3}}{\rho_0} + \left[12.3\left(\frac{\rho_B}{\rho_0}\right)^{2/3} + 20\left(\frac{\rho_B}{\rho_0}\right)^\gamma\right] \left(\frac{\rho_{I_3}}{\rho_B}\right)^2\]
in GeV

Definition at line 127 of file potentials.cc.

                                                                    {
   double pot =
       1.0e-3 * 2. * symmetry_S_Pot_ * baryon_isospin_density / nuclear_density;
   if (symmetry_is_rhoB_dependent_) {
     pot += 1.0e-3 * symmetry_S(baryon_density) * baryon_isospin_density *
            baryon_isospin_density / (baryon_density * baryon_density);
   }
   return pot;
 }

◆ symmetry_S()

double smash::Potentials::symmetry_S ( const double baryon_density ) const

Calculate the factor \(S(\rho)\) in the symmetry potential.

Parameters

[in] baryon_density baryon density

Returns: factor S in symmetry potenial

Definition at line 119 of file potentials.cc.

                                                                {
   if (symmetry_is_rhoB_dependent_) {
     return 12.3 * std::pow(baryon_density / nuclear_density, 2. / 3.) +
            20.0 * std::pow(baryon_density / nuclear_density, symmetry_gamma_);
   } else {
     return 0.;
   }
 }

◆ potential()

double smash::Potentials::potential	(	const ThreeVector &	r,
		const ParticleList &	plist,
		const ParticleType &	acts_on
	)		const

Evaluates potential at point r.

Potential is always taken in the local Eckart rest frame, but point r is in the computational frame.

Parameters

[in]	r	Arbitrary space point where potential is calculated
[in]	plist	List of all particles to be used in \(j^{\mu}\) calculation. If the distance between particle and calculation point r, \( \|r-r_i\| > r_{cut} \) then particle input to density will be ignored.
[in]	acts_on	Type of particle on which potential is going to act. It gives the charges (or more precisely, the scaling factors) of the particle moving in the potential field.

Returns: Total potential energy acting on the particle:
\[U_{\rm tot} =Q_BU_B+2I_3U_I\]
in GeV, where \(Q_B\) is the baryon charge scaled by the ratio of the light (u, d) quark to the total quark number and \(I_3\) is the third compnent of the isospin.

Definition at line 138 of file potentials.cc.

                                                                 {
   double total_potential = 0.0;
   const bool compute_gradient = false;
   const bool smearing = true;
   const auto scale = force_scale(acts_on);
  
   if (!(acts_on.is_baryon() || acts_on.is_nucleus())) {
     return total_potential;
   }
   const double rho_B = std::get<0>(current_eckart(
       r, plist, param_, DensityType::Baryon, compute_gradient, smearing));
   if (use_skyrme_) {
     total_potential += scale.first * skyrme_pot(rho_B);
   }
   if (use_symmetry_) {
     const double rho_iso = std::get<0>(
         current_eckart(r, plist, param_, DensityType::BaryonicIsospin,
                        compute_gradient, smearing));
     const double sym_pot =
         symmetry_pot(rho_iso, rho_B) * acts_on.isospin3_rel();
     total_potential += scale.second * sym_pot;
   }
   return total_potential;
 }

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◆ force_scale()

std::pair< double, int > smash::Potentials::force_scale ( const ParticleType & data )

static

Evaluates the scaling factor of the forces acting on the particles.

The forces are equal to the product of the scaling factor and the gradient of the potential. We need these scaling factors to describe the motions of the hyperons as well as the anti-particles in the potentials. For Lambda and Sigma, since they carry 2 light (u or d) quarks, they are affected by 2/3 of the Skyrme force. Xi carries 1 light quark, it is affected by 1/3 of the Skyrme force. Omega carries no light quark, so it's not affected by the Skyrme force. Anti-baryons are affected by the force as large as the force acting on baryons but with an opposite direction.

Parameters

[in] data Type of particle on which potential is going to act.

Returns: ( \(Q_B(1-\frac{|Q_S|}{3}), Q_B\)) where \(Q_B\) is the baryon charge and \(Q_S\) is the strangeness.

Definition at line 164 of file potentials.cc.

                                                                      {
   const auto &pdg = data.pdgcode();
   const double skyrme_scale =
       (3 - std::abs(pdg.strangeness())) / 3. * pdg.baryon_number();
   const int symmetry_scale = pdg.baryon_number();
   return std::make_pair(skyrme_scale, symmetry_scale);
 }

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◆ skyrme_force()

std::pair< ThreeVector, ThreeVector > smash::Potentials::skyrme_force	(	const double	density,
		const ThreeVector	grad_rho,
		const ThreeVector	dj_dt,
		const ThreeVector	rot_j
	)		const

Evaluates the electrical and magnetic components of the skyrme force.

Parameters

[in]	density	Eckart baryon density [fm \(^{-3}\)].
[in]	grad_rho	Gradient of baryon density [fm \(^{-4}\)]. This density is evaluated in the computational frame.
[in]	dj_dt	Time derivative of the baryon current density [fm \(^{-4}\)
[in]	rot_j	Curl of the baryon current density [fm \(^{-4}\)

Returns

( \(E_B, B_B\)), where

\[E_B = -V_B^\prime(\rho^\ast)(\nabla\rho_B + \partial_t \vec j_B)\]

is the electro component of Skyrme force and

\[B_B = V_B^\prime(\rho^\ast) \nabla\times\vec j_B\]

is the magnetic component of the Skyrme force with \(\rho^\ast\) being the Eckart baryon density.

Definition at line 172 of file potentials.cc.

                                    {
   ThreeVector E_component(0.0, 0.0, 0.0), B_component(0.0, 0.0, 0.0);
   if (use_skyrme_) {
     const double dV_drho =
         (skyrme_a_ + skyrme_b_ * skyrme_tau_ *
                          std::pow(density / nuclear_density, skyrme_tau_ - 1)) *
         mev_to_gev / nuclear_density;
     E_component -= dV_drho * (grad_rho + dj_dt);
     B_component += dV_drho * rot_j;
   }
   return std::make_pair(E_component, B_component);
 }

◆ symmetry_force()

std::pair< ThreeVector, ThreeVector > smash::Potentials::symmetry_force	(	const double	rhoI3,
		const ThreeVector	grad_rhoI3,
		const ThreeVector	djI3_dt,
		const ThreeVector	rot_jI3,
		const double	rhoB,
		const ThreeVector	grad_rhoB,
		const ThreeVector	djB_dt,
		const ThreeVector	rot_jB
	)		const

Evaluates the electrical and magnetic components of the symmetry force.

Parameters

[in]	rhoI3	Relative isospin 3 density.
[in]	grad_rhoI3	Gradient of I3/I density [fm \(^{-4}\)]. This density is evaluated in the computational frame.
[in]	djI3_dt	Time derivative of the I3/I current density [fm \(^{-4}\)]
[in]	rot_jI3	Curl of the I3/I current density [fm \(^{-4}\)]
[in]	rhoB	Net-baryon density
[in]	grad_rhoB	Gradient of the net-baryon density
[in]	djB_dt	Time derivative of the net-baryon current density
[in]	rot_jB	Curl of the net-baryon current density

Returns

( \(E_I3, B_I3\)) [GeV/fm], where

\[\vec{E} = - \frac{\partial V^\ast}{\partial\rho_{I_3}^\ast} (\nabla\rho_{I_3} + \partial_t \vec j_{I_3}) - \frac{\partial V^\ast}{\partial\rho_B^\ast}(\nabla\rho_B + \partial_t \vec j_B)\]

is the electrical component of symmetry force and

\[\vec{B} = \frac{\partial V^\ast}{\rho_{I_3}^\ast} \nabla\times\vec j_{I_3} + \frac{\partial V^\ast}{\rho_B^\ast} \nabla\times\vec j_B \]

is the magnetic component of the symmetry force with \(\rho^\ast\) being the respective Eckart density.

Definition at line 187 of file potentials.cc.

                                                               {
   ThreeVector E_component(0.0, 0.0, 0.0), B_component(0.0, 0.0, 0.0);
   if (use_symmetry_) {
     E_component -= dVsym_drhoI3(rhoB, rhoI3) * (grad_rhoI3 + djI3_dt) +
                    dVsym_drhoB(rhoB, rhoI3) * (grad_rhoB + djB_dt);
     B_component +=
         dVsym_drhoI3(rhoB, rhoI3) * rot_jI3 + dVsym_drhoB(rhoB, rhoI3) * rot_jB;
   }
   return std::make_pair(E_component, B_component);
 }

◆ all_forces()

std::tuple< ThreeVector, ThreeVector, ThreeVector, ThreeVector > smash::Potentials::all_forces	(	const ThreeVector &	r,
		const ParticleList &	plist
	)		const

virtual

Evaluates the electrical and magnetic components of the forces at point r.

Point r is in the computational frame.

Parameters

[in]	r	Arbitrary space point where potential gradient is calculated
[in]	plist	List of all particles to be used in \(j^{\mu}\) calculation. If the distance between particle and calculation point r, \( \|r-r_i\| > r_{cut} \) then particle input to density will be ignored.

Returns: ( \(E_B, B_B, E_{I3}, B_{I3}\)) [GeV/fm], where \(E_B\): the electric component of the Skyrme force \(B_B\): the magnetic component of the Skyrme force \(E_{I3}\): the electric component of the symmetry force \(B_{I3}\): the magnetic component of the symmetry force

Definition at line 225 of file potentials.cc.

                                                                             {
   const bool compute_gradient = true;
   const bool smearing = true;
   auto F_skyrme =
       std::make_pair(ThreeVector(0., 0., 0.), ThreeVector(0., 0., 0.));
   auto F_symmetry =
       std::make_pair(ThreeVector(0., 0., 0.), ThreeVector(0., 0., 0.));
  
   const auto baryon_density_and_gradient = current_eckart(
       r, plist, param_, DensityType::Baryon, compute_gradient, smearing);
   const double rhoB = std::get<0>(baryon_density_and_gradient);
   const ThreeVector grad_rhoB = std::get<2>(baryon_density_and_gradient);
   const ThreeVector djB_dt = std::get<3>(baryon_density_and_gradient);
   const ThreeVector rot_jB = std::get<4>(baryon_density_and_gradient);
   if (use_skyrme_) {
     F_skyrme = skyrme_force(rhoB, grad_rhoB, djB_dt, rot_jB);
   }
  
   if (use_symmetry_) {
     const auto density_and_gradient =
         current_eckart(r, plist, param_, DensityType::BaryonicIsospin,
                        compute_gradient, smearing);
     const double rhoI3 = std::get<0>(density_and_gradient);
     const ThreeVector grad_rhoI3 = std::get<2>(density_and_gradient);
     const ThreeVector djI3_dt = std::get<3>(density_and_gradient);
     const ThreeVector rot_jI3 = std::get<4>(density_and_gradient);
     F_symmetry = symmetry_force(rhoI3, grad_rhoI3, djI3_dt, rot_jI3, rhoB,
                                 grad_rhoB, djB_dt, rot_jB);
   }
  
   return std::make_tuple(F_skyrme.first, F_skyrme.second, F_symmetry.first,
                          F_symmetry.second);
 }

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◆ use_skyrme()

virtual bool smash::Potentials::use_skyrme ( ) const

inlinevirtual

Returns: Is Skyrme potential on?

Definition at line 190 of file potentials.h.

190 { return use_skyrme_; }

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◆ use_symmetry()

virtual bool smash::Potentials::use_symmetry ( ) const

inlinevirtual

Returns: Is symmetry potential on?

Definition at line 192 of file potentials.h.

192 { return use_symmetry_; }

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◆ skyrme_a()

double smash::Potentials::skyrme_a ( ) const

inline

Returns: Skyrme parameter skyrme_a, in MeV

Definition at line 195 of file potentials.h.

195 { return skyrme_a_; }

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◆ skyrme_b()

double smash::Potentials::skyrme_b ( ) const

inline

Returns: Skyrme parameter skyrme_b, in MeV

Definition at line 197 of file potentials.h.

197 { return skyrme_b_; }

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◆ skyrme_tau()

double smash::Potentials::skyrme_tau ( ) const

inline

Returns: Skyrme parameter skyrme_tau

Definition at line 199 of file potentials.h.

199 { return skyrme_tau_; }

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◆ dVsym_drhoI3()

double smash::Potentials::dVsym_drhoI3	(	const double	rhoB,
		const double	rhoI3
	)		const

private

Calculate the derivative of the symmetry potential with respect to the isospin density in GeV * fm^3.

\[ \frac{\partial V_\mathrm{sym}}{\partial \rho_{I_3}} = 2\frac{S_\mathrm{Pot}}{\rho_0} + \frac{2\rho_{I_3}\left[12.3\left(\frac{\rho_B}{\rho_0}\right)^{2/3} + 20 \left(\frac{\rho_B}{\rho_0}\right)^\gamma\right]}{\rho_B^2} \]

Note: The isospin 3 density here is actually the density of I3 / I.

Parameters

[in]	rhoB	net baryon density
[in]	rhoI3	isospin density

Returns: partial derivative of the symmetry potenital with respect to the isospin density.

Definition at line 201 of file potentials.cc.

                                                                            {
   double term1 = 2. * symmetry_S_Pot_ / nuclear_density;
   if (symmetry_is_rhoB_dependent_) {
     double term2 = 2. * rhoI3 * symmetry_S(rhoB) / (rhoB * rhoB);
     return mev_to_gev * (term1 + term2);
   } else {
     return mev_to_gev * term1;
   }
 }

◆ dVsym_drhoB()

double smash::Potentials::dVsym_drhoB	(	const double	rhoB,
		const double	rhoI3
	)		const

private

Calculate the derivative of the symmetry potential with respect to the net baryon density in GeV * fm^3.

\[ \frac{\partial V_\mathrm{sym}}{\partial \rho_B} = \left(\frac{\rho_{I_3}}{\rho_B}\right)^2 \left[\frac{8.2}{\rho_0}\left(\frac{\rho_B}{\rho_0}\right)^{-1/3} + \frac{20\gamma}{\rho_B}\left(\frac{\rho_B}{\rho_0}\right)^\gamma\right] -2\frac{\rho_{I_3}^2}{\rho_B^3} \left[12.3\left(\frac{\rho_B}{\rho_0}\right)^{2/3} + 20\left(\frac{\rho_B}{\rho_0}\right)^\gamma\right]\]

Note: The isospin 3 density here is actually the density of I3 / I

Parameters

[in]	rhoB	net baryon density
[in]	rhoI3	isospin density

Returns: partial derivative of the symmetry potenital with respect to the net baryon density.

Definition at line 211 of file potentials.cc.

                                                                           {
   if (symmetry_is_rhoB_dependent_) {
     double rhoB_over_rho0 = rhoB / nuclear_density;
     double term1 = 8.2 * std::pow(rhoB_over_rho0, -1. / 3.) / nuclear_density +
                    20. * symmetry_gamma_ *
                        std::pow(rhoB_over_rho0, symmetry_gamma_) / rhoB;
     double term2 = -2. * symmetry_S(rhoB) / rhoB;
     return mev_to_gev * (term1 + term2) * rhoI3 * rhoI3 / (rhoB * rhoB);
   } else {
     return 0.;
   }
 }

Member Data Documentation

◆ param_

const DensityParameters smash::Potentials::param_

private

Struct that contains the gaussian smearing width \(\sigma\), the distance cutoff \(r_{\rm cut}\) and the testparticle number needed for the density calculation.

Definition at line 207 of file potentials.h.

◆ use_skyrme_

bool smash::Potentials::use_skyrme_

private

Skyrme potential on/off.

Definition at line 210 of file potentials.h.

◆ use_symmetry_

bool smash::Potentials::use_symmetry_

private

Symmetry potential on/off.

Definition at line 213 of file potentials.h.

◆ skyrme_a_

double smash::Potentials::skyrme_a_

private

Parameter of skyrme potentials: the coefficient in front of \(\frac{\rho}{\rho_0}\) in GeV.

Definition at line 219 of file potentials.h.

◆ skyrme_b_

double smash::Potentials::skyrme_b_

private

Parameters of skyrme potentials: the coefficient in front of \((\frac{\rho}{\rho_0})^\tau\) in GeV.

Definition at line 225 of file potentials.h.

◆ skyrme_tau_

double smash::Potentials::skyrme_tau_

private

Parameters of skyrme potentials: the power index.

Definition at line 231 of file potentials.h.

◆ symmetry_S_Pot_

double smash::Potentials::symmetry_S_Pot_

private

Parameter S_Pot in the symmetry potential in MeV.

Definition at line 234 of file potentials.h.

◆ symmetry_is_rhoB_dependent_

bool smash::Potentials::symmetry_is_rhoB_dependent_ = false

private

Whether the baryon density dependence of the symmetry potential is included.

Definition at line 240 of file potentials.h.

◆ symmetry_gamma_

double smash::Potentials::symmetry_gamma_

private

Power \( \gamma \) in formula for \( S(\rho) \):

\[ S(\rho)=12.3\,\mathrm{MeV}\times \left(\frac{\rho}{\rho_0}\right)^{2/3}+20\,\mathrm{MeV}\times \left(\frac{\rho}{\rho_0}\right)^\gamma \]

Definition at line 247 of file potentials.h.

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

src/include/smash/potentials.h
src/potentials.cc

Public Member Functions

Static Public Member Functions

Private Member Functions

Private Attributes

Constructor & Destructor Documentation

◆ Potentials()

◆ ~Potentials()

Member Function Documentation

◆ skyrme_pot()

◆ symmetry_pot()

◆ symmetry_S()

◆ potential()

◆ force_scale()

◆ skyrme_force()

◆ symmetry_force()

◆ all_forces()

◆ use_skyrme()

◆ use_symmetry()

◆ skyrme_a()

◆ skyrme_b()

◆ skyrme_tau()

◆ dVsym_drhoI3()

◆ dVsym_drhoB()

Member Data Documentation

◆ param_

◆ use_skyrme_

◆ use_symmetry_

◆ skyrme_a_

◆ skyrme_b_

◆ skyrme_tau_

◆ symmetry_S_Pot_

◆ symmetry_is_rhoB_dependent_

◆ symmetry_gamma_