Particles ========= :ref:`O2scl ` The classes described below calculate the thermodynamic properties of interacting and non-interacting quantum and \classical particles. Particles contents ------------------ - :ref:`Particle data classes` - :ref:`Units` - :ref:`Classes for particle thermodynamics` - :ref:`Thermodynamics with derivatives` - :ref:`Other derivatives` - :ref:`Particle example` Particle data classes --------------------- The class :cpp:class:`o2scl::part_tl` is the basic structure for a particle. The classes with a ``_tl`` suffix are templates which can handle a variety of floating point types. Typedefs without a ``_tl`` suffix specialize these templates for the floating point type ``double``. - :cpp:var:`o2scl::part_tl::m` - mass (i.e. rest mass), :math:`m` - :cpp:var:`o2scl::part_tl::g` - degeneracy factor (e.g. :math:`g=2j+1`) - :cpp:var:`o2scl::part_tl::n` - number density, :math:`n` - :cpp:var:`o2scl::part_tl::ed` - energy density, :math:`\varepsilon` - :cpp:var:`o2scl::part_tl::pr` - pressure, :math:`P` - :cpp:var:`o2scl::part_tl::en` - entropy density, :math:`s` - :cpp:var:`o2scl::part_tl::ms` - effective mass, :math:`m^{*}` - :cpp:var:`o2scl::part_tl::mu` - chemical potential, :math:`\mu` - :cpp:var:`o2scl::part_tl::nu` - effective chemical potential, :math:`\nu` - :cpp:var:`o2scl::part_tl::inc_rest_mass` - True if the rest mass is included (default true) - :cpp:var:`o2scl::part_tl::non_interacting` - False if the particle includes interactions (default true) The data members :cpp:var:`o2scl::part_tl::ms` and :cpp:var:`o2scl::part_tl::nu` allow one to specify modifications to the mass and the chemical potential due to interactions. This allows one to calculate the properties of particle due to interactions so long as the basic form of the free-particle dispersion relation is unchanged, i.e. .. math:: \sqrt{k^2+m^2} - \mu \rightarrow \sqrt{k^2+m^{* 2}} - \nu If the particle is non-interacting, then :cpp:var:`o2scl::part_tl::nu` and :cpp:var:`o2scl::part_tl::ms` are sometimes used by O₂scl_part functions for temporary storage. If :cpp:var:`o2scl::part_tl::inc_rest_mass` is \c true (this is the default in all of the classes except :cpp:class:`o2scl::nucleus`), then all functions include the rest mass (stored in :cpp:var:`o2scl::part_tl::m`) energy density in the energy density, the "mu" functions expect that the rest mass is included in :cpp:var:`o2scl::part_tl::mu` or :cpp:var:`o2scl::part_tl::nu` as input and the "density" functions output :cpp:var:`o2scl::part_tl::mu` or :cpp:var:`o2scl::part_tl::nu` including the rest mass. Note that it is assumed that :cpp:var:`o2scl::part_tl::m` contains the rest mass even if the particle is interacting and an effective mass is stored in :cpp:var:`o2scl::part_tl::ms`. When :cpp:var:`o2scl::part_tl::inc_rest_mass` is true, antiparticles are implemented by choosing the antiparticle chemical potential to be :math:`- \mu`. When :cpp:var:`o2scl::part_tl::inc_rest_mass` is false, there is an ambiguity in the relative definitions of the rest mass contribution for the antiparticles and the combination of both particles and antiparticles. Define energy density for particles including the rest mass contribution as :math:`\varepsilon_+`, and the energy density without the rest mass contribution as :math:`\tilde{\varepsilon}_{+} = \varepsilon_{+} - n_{+} m` . Similarly, for antiparticles, we have :math:`\tilde{\varepsilon}_- = \varepsilon_- - n_- m`. The total energy density including the rest mass contribution is then :math:`\varepsilon = \varepsilon_{+} + \varepsilon_-` and without the rest mass contribution :math:`\tilde{\varepsilon} \equiv \varepsilon - (n_{+}-n_-) m`. Then, .. math:: \begin{eqnarray} \tilde{\varepsilon} & = & \varepsilon_+ - n_{+} m + \varepsilon_- + n_- m \nonumber \\ & = & \varepsilon_+ - n_{+} m + \varepsilon_- - n_- m + 2 n_- m \nonumber \\ & = & \tilde{\varepsilon}_+ + \tilde{\varepsilon}_- + 2 n_- m \nonumber \end{eqnarray} Similarly, for the chemical potentials, we have .. math:: \tilde{\mu}_+ \equiv \frac{\partial \tilde{\varepsilon}_+} {\partial n_+} = \mu_+ - m \quad \mathrm{and} \quad \tilde{\mu}_- \equiv \frac{\partial \tilde{\varepsilon}_-} {\partial n_-} = \mu_- - m thus :math:`\tilde{\mu}_- = - \tilde{\mu}_+ - 2 m` . This bookkeeping is handled by :cpp:func:`o2scl::part_tl::anti()`, the :cpp:func:`o2scl::fermion_thermo_tl::pair_mu()`, and the :cpp:func:`o2scl::fermion_thermo_tl::pair_density()`, functions. The thermodynamic identity used to compute the pressure for interacting particles is .. math:: P = -\varepsilon + s T + \nu n where :cpp:var:`o2scl::part_tl::nu` is used. This way, the particle class doesn't need to know about the structure of the interactions to ensure that the thermodynamic identity is satisfied. Note that in the O₂scl_eos library, where in the equations of state the normal thermodynamic identity is used .. math:: P = -\varepsilon + s T + \mu n Frequently, the interactions which create an effective chemical potential which is different than :cpp:var:`o2scl::part_tl::mu` thus create extra terms in the pressure and the energy density for the given equation of state. The :cpp:class:`o2scl::fermion_tl` class is a child of :cpp:class:`o2scl::part_tl` which contains data members for the Fermi momentum and energy gap. The :cpp:class:`o2scl::boson` class contains an extra data member for the condensate. The :cpp:class:`o2scl::quark` class is a descendant of the :cpp:class:`o2scl::fermion_tl` class which contains extra data members for the quark condensate and the contribution to the bag constant. Nuclei are represented by the :cpp:class:`o2scl::nucleus` class and documented in :ref:`Nuclei and Nuclear Masses`. Units ----- Factors of :math:`\hbar, c` and :math:`k_B` have been removed everywhere, so that mass, energy, and temperature all have the same units. Number and entropy densities have units of mass cubed (or energy cubed). The particle classes can be used with any system of units which is based on powers of one unit, i.e. :math:`[n] = [T]^3 = [m]^3 = [P]^{3/4} = [\varepsilon]^{3/4}`, etc. For O₂scl EOS classes, powers of :math:`\mathrm{fm}^{-1}` are often chosen. Classes for particle thermodynamics ----------------------------------- At zero temperature, the thermodynamic properties of fermions can be computed using :cpp:class:`o2scl::fermion_zerot_tl`. The class :cpp:class:`o2scl::classical_thermo_tl` computes the properties of particles in the classical limit. At finite temperature, there are different classes corresponding to different approaches to computing the integrals over the distribution functions. The approximation scheme from [Johns96]_ is used in :cpp:class:`o2scl::boson_eff` and :cpp:class:`o2scl::fermion_eff`. An exact method employing direct integration of the distribution functions is used in :cpp:class:`o2scl::boson_rel` and :cpp:class:`o2scl::fermion_rel_tl`, but these are necessarily quite a bit slower. All of these classes use expansions to give ensure comparably accurate results in the degenerate and non-degenerate limits. The class :cpp:class:`o2scl::fermion_eff` usually works to within about 1 part in :math:`10^4`, but can be as bad as 1 part in :math:`10^2` in some more extreme cases. The default settings for :cpp:class:`o2scl::fermion_rel_tl` give an accuracy of at least 1 part in :math:`10^6` (and frequently better than this). For :cpp:class:`o2scl::fermion_rel_tl`, the accuracy can be improved to 1 part in :math:`10^{10}` by decreasing the integration tolerances. See :ref:`Fermion Details` for more discussion on the fermion integrals. The class :cpp:class:`o2scl::fermion_nonrel_tl` assumes a non-relativistic dispersion relation for fermions. It uses an exact method for both zero and finite temperatures. The non-relativistic integrands are much simpler and :cpp:class:`o2scl::fermion_nonrel_tl` uses the appropriate GSL functions (which are nearly exact) to compute them. Thermodynamics with derivatives ------------------------------- Sometimes it is useful to know derivatives like :math:`ds/dT` in addition to the energy and pressure. The class :cpp:class:`o2scl::part_deriv_press_tl` stores the three derivatives which correspond to second derivatives of the pressure .. math:: \left(\frac{\partial n}{\partial \mu}\right)_{T}, \quad \left(\frac{\partial n}{\partial T}\right)_{\mu}, \quad \mathrm{and} \quad \left(\frac{\partial s}{\partial T}\right)_{\mu} \quad . All other first derivatives of the thermodynamic functions can be written in terms of these three. The new data classes are :cpp:class:`o2scl::part_deriv_tl` and :cpp:class:`o2scl::fermion_deriv_tl` which store the basic particle thermodynamics described above with these additional three derivatives. There are three classes which compute these derivatives for fermions and classical particles. The class :cpp:class:`o2scl::classical_deriv_thermo_tl` handles the nondegenerate limit, :cpp:class:`o2scl::fermion_deriv_rel_tl` handles fermions and :cpp:class:`o2scl::fermion_deriv_nr_tl` handles nonrelativistic fermions. The class :cpp:class:`o2scl::fermion_deriv_thermo_tl` is a base class for :cpp:class:`o2scl::fermion_deriv_rel_tl` and uses degenerate and nondegenerate expansions to evaluate both the base thermodynamic quantities and the three derivatives from :cpp:class:`o2scl::part_deriv_press_tl` . The function :cpp:func:`o2scl::part_deriv_tl::deriv_f()` computes the derivatives which are second derivatives of the free energy from the three computed above. Other derivatives ----------------- For the derivative of the entropy with respect to the chemical potential, there is a Maxwell relation .. math:: \left(\frac{\partial s}{\partial \mu}\right)_{T,V} = \left(\frac{\partial^2 P}{\partial \mu \partial T}\right)_{V} = \left(\frac{\partial^2 P}{\partial T \partial \mu}\right)_{V} = \left(\frac{\partial n}{\partial T}\right)_{\mu,V} The first derivatives of the energy density can be computed using the thermodynamic identity: .. math:: \left(\frac{\partial \varepsilon}{\partial \mu}\right)_{T,V}= \mu \left(\frac{\partial n}{\partial \mu}\right)_{T,V}+ T \left(\frac{\partial s}{\partial \mu}\right)_{T,V} .. math:: \left(\frac{\partial \varepsilon}{\partial T}\right)_{\mu,V}= \mu \left(\frac{\partial n}{\partial T}\right)_{\mu,V}+ T \left(\frac{\partial s}{\partial T}\right)_{\mu,V} Most of the other common derivatives which are used are those which can be obtained by second derivatives of the Gibbs free energy, :math:`G = F + P V`. .. math:: \begin{eqnarray} \left(\frac{\partial^2 G}{\partial T^2}\right)_{P,\{N_i\}} &=& -\left( \frac{\partial S}{\partial T} \right)_{P,\{N_i\}} = - \frac{N c_P}{T} \nonumber \\ \left(\frac{\partial^2 G}{\partial T \partial P}\right)_{\{N_i\}} &=& \left( \frac{\partial V}{\partial T} \right)_{P,\{N_i\}} = V \alpha \nonumber \\ \left(\frac{\partial^2 G}{\partial P^2}\right)_{T,\{N_i\}} &=& \left( \frac{\partial V}{\partial P} \right)_{T,\{N_i\}} = - V \kappa_T \nonumber \end{eqnarray} Other common derivatives are the heat capacity per particle at constant volume, :math:`c_V`, and the speed of sound, :math:`( d P / d \varepsilon)_{\{N_i\},S}`. These derivatives are computed by functions in :cpp:class:`o2scl::deriv_thermo_base_tl` from the three second derivatives of the pressure stored in a :cpp:class:`o2scl::part_deriv_tl` or :cpp:class:`o2scl::fermion_deriv_tl` object. - :cpp:func:`o2scl::deriv_thermo_base_tl::heat_cap_ppart_const_vol()` - :cpp:func:`o2scl::deriv_thermo_base_tl::heat_cap_ppart_const_press()` - :cpp:func:`o2scl::deriv_thermo_base_tl::compress_adiabatic()` - :cpp:func:`o2scl::deriv_thermo_base_tl::compress_const_tptr()` - :cpp:func:`o2scl::deriv_thermo_base_tl::coeff_thermal_exp()` - :cpp:func:`o2scl::deriv_thermo_base_tl::squared_sound_speed()` .. (begin comment) I think the expression below only works for fermions? I'm taking this section out until it's better commented In the case where the particle is interacting, the derivative of the density with respect to the effective mass is \f[ \left(\frac{dn}{dm^{*}}\right)_{\mu,T} = \left(\frac{3 n}{m^{*}}\right) - \frac{T}{m^{*}} \left(\frac{dn}{dT}\right)_{m^{*},\mu} - \frac{\nu}{m^{*}} \left(\frac{dn}{d\mu}\right)_{m^{*},T} \f] This relation holds whether or not the mass is included in the chemical potential :math:`\nu`, as the rest mass is held constant even though the effective mass is varying. This relation also holds in the case where the particle is non-interacting, so long as one does not allow the rest mass in the chemical potential to vary. This derivative is useful, for example, in models of quark matter where the quark mass is dynamically generated. (end comment) Particle example ---------------- p .. literalinclude:: ../../../examples/ex_part.cpp :language: c++ :start-after: sphinx-example-start